mm-488 === >I could not find any algebra book that really defines quotient rings >in polynomials. Run some of the unreal definitions past us, so we can snicker. >For example, I can't visualize these: Why in the world would you want to visualize them? There's nothing (on the face of it) visual about them! They're just what they're defined to be! (I can visualize james dolan right now...) Maybe you can better understand what's going on by trying to find a normal form for each equivalence class in the quotient ring. >a. Z[x]/(2, x) Here, for instance, the normal form of a+bx+cx^2+... could be a mod 2. (Don't reflexively ask us, What do you mean? or What does that mean?; spend a minute or so asking yourself What might he mean? What might that mean?) That would leave you with just two distinct normal forms, the two distinct integers mod 2. What's the ring structure on this set with two elements? >b. Z[x]/(x) Here, the normal form of a+bx+cx^2+... still throws out b, c, and so on, but it keeps a intact. So...? >c. Q[x]/(y - x^2) What's y? Does it appear elsewhere in what is appearing more and more like a homework assignment? >d. Z[x]/(2, x^2-x+1). How do I show that this is a finite field? How >many elements does it have? How can I show maximality of (2, x^2-x-1)? >or finiteness of the quotient? Yep, homework. Go ask the instructor, kid; someone's paying hir on your behalf. No one is paying sci.math. Lee Rudolph === Subject: Re: Quotient rings - help! And it is not a homework assignment: I am studying for an upcoming exam. N.C. >I could not find any algebra book that really defines quotient rings >in polynomials. > Run some of the unreal definitions past us, so we can snicker. . . .. . > Yep, homework. > Go ask the instructor, kid; someone's paying hir on your behalf. > No one is paying sci.math. > Lee Rudolph === Subject: Re: Quotient rings - help! OK, so you can trade y for x^2. So any polynomial in x and y is equivalent to a polynomial in x alone... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Quotient rings - help! >I could not find any algebra book that really defines quotient rings >in polynomials. I know the elements of quotient rings are all the >left/right cosets, but how do they look like? I wish some book would >make this clear. It is so frustrating. >If R[x] is a polynomial ring, and I[x] is some ideal of R[x], how do >the elements of R[x]/I[x] look like? I would think of it this way: you're looking at the polynomial ring R[x], but considering two polynomials as equivalent if their difference is in the ideal. >For example, I can't visualize these: >a. Z[x]/(2, x) for instance; or OK, let's take this example. You're looking at polynomials in x with integer coefficients, and (1) anything with all its coefficients even is equivalent to 0. So every polynomial is equivalent to one with coefficients in {0,1}. (2) any multiple of x is equivalent to 0. So you can throw away all the non-constant terms of a polynomial. Conclusion: Z[x]/(2,x) has only two members, (the equivalence classes for) 0 and 1. >c. Q[x]/(y - x^2); or What is y? It can't be an indeterminate since you said Q[x], not Q[x,y]. So I guess it is some fixed member of Q. Now for any polynomial of degree >= 2, you can get an equivalent polynomial of lower degree by adding an appropriate multiple of y - x^2. On the other hand, two distinct polynomials of degree <= 1 can't be equivalent. So here the quotient ring corresponds to everything of the form a x + b, a and b in Q, with multiplication (a x + b)*(c x + d) = (a d + b c) x + b d + a c y Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Quotient rings - help! > I could not find any algebra book that really defines quotient rings > in polynomials. I know the elements of quotient rings are all the > left/right cosets, but how do they look like? I wish some book would > make this clear. It is so frustrating. > If R[x] is a polynomial ring, and I[x] is some ideal of R[x], how do > the elements of R[x]/I[x] look like? > For example, I can't visualize these: > a. Z[x]/(2, x) for instance; or > b. Z[x]/(x); or > c. Q[x]/(y - x^2); or > d. Z[x]/(2, x^2-x+1). How do I show that this is a finite field? How > many elements does it have? How can I show maximality of (2, x^2-x-1)? > or finiteness of the quotient? > N.C. One thing that can help is to think of them as the polynomial ring R[x] with the relationship that all polynomials in the idea are equal to zero. so for example Z[X]/(2,x) is the set of all polynomials in Z with 2=0 and x=0, so you get Z/2Z Z[x]/(x) is just Z I'm not sure what you mean by Q[x]/(y-x^2) since y is not an element of Q[x]. For (d) to show finiteness, first note that since we are modding out the element 2, we can think of it as Z_2[x]/(x^2-x+1). Then the fact that x^2-x+1=0 means that x^2=x-1, and thus one can represent every element by a+bx, with a,b elements of Z_2. Hope this helps. -Ron === Subject: help with transposition, dot product Hiya, Im really struggling to transpose this to find 1 of the vectors, my maths is very basic with now further education to support it... (BTW before anyone says this is homework its not! im just dumb, and why would anyone do homework in the summer holidays!) ok well this is what i want to transpose - u.v / |u||v| = I where I is a scaler and U and V are unit vectors, I want to get V on its own on one side, but im stuck! any help would be really appreciated! === Subject: Re: help with transposition, dot product > Hiya, > Im really struggling to transpose this to find 1 of the vectors, my > maths is very basic with now further education to support it... (BTW > before anyone says this is homework its not! im just dumb, and why > would anyone do homework in the summer holidays!) > ok well this is what i want to transpose - > u.v / |u||v| = I > where I is a scaler and U and V are unit vectors, I want to get V on > its own on one side, but im stuck! > any help would be really appreciated! The most you can do is rearrange to get |v| on its own. u.v = I|u||v| so |v| = u.v / I|u| u.v is a dot product where if u = (a,b) and v = (c,d) then u.v = ac + bd |v| = (ac + bd) / I|u| where |v| = sqrt(c^2+d^2) PH === Subject: Re: help with transposition, dot product u (dot) v = |u| |v| cos(theta), where theta is the angle between the vectors u and v. If you know u, you can find the direction, but not the magnitude, of v; it is at an angle of arccos(I) from u. === Subject: Re: help with transposition, dot product > u (dot) v = |u| |v| cos(theta), where theta is the angle between the > vectors u and v. If you know u, you can find the direction, but not > the magnitude, of v; it is at an angle of arccos(I) from u. Does it determine the direction uniquely? I think not. Does it not give the choice of two directions in 2-space and a cone of directions in 2-space? === Subject: Question: Divisors of the difference of unknown primes Hello. I've been given the product of two large primes and tasked with finding divisors of the difference between the two (unknown to me) primes. I don't even know if it's possible, though. That is, I've been given M = N(N+P), where N and N+P are large primes. And I need to find divisors of P. Clearly 2 | P. Also 3 | P iff M = 1 (mod 3). This is because N^2 = 1 (mod 3) if N is relatively prime to 3. The same is true for 4, 6, and 8. So I know whether P is divisible by 2, 3, 4, 6, and 8 (and obviously 12 and 24). But this trick doesn't work for 5, 7, or 9, and I've run out of ideas. I especially want to know about 9. Any thoughts? === Subject: Re: parabola - conic section I would be interested to see similar derivations for the ellipse and hyperbola; if you happen to have such derivations handy on-hand, could you please post these as well. Michael === Subject: Re: parabola - conic section > I would be interested to see similar derivations for the ellipse and > hyperbola; if you happen to have such derivations handy on-hand, could > you please post these as well. All you have to do differently is change the angle by which the y-axis is rotated. Instead of 45 degrees, let it be some angle f, between 0 and 90 degrees. The substitution is then: x = cu - sv z = su + cv where c = cos(f) and s = sin(f). The equation of the cone then becomes: (cu)^2 - 2scuv + (sv)^2 + y^2 - ((su)^2 + 2scuv + (cv)^2) = 0 Simplifying: y^2 = (c^2 - s^2)(u^2 - v^2) - 4scuv = C(u^2 - v^2) - 2Suv where C = c^2 - s^2 = cos(2f) and S = 2sc = sin(2f). Completing the square in u, and bringing it to the left, you get: (u - Sv/C)^2 - y^2/C = (1 - (S/C)^2)v^2 For constant v, this equation describes a hyperbola for C > 0, and an ellipse for C < 0. Recalling C = cos(2f), this occurs when f < 45 degrees and f > 45 degrees, respectively. === Subject: Re: parabola - conic section > I was just reading a book on conic sections. How can one prove that if > cone is > sliced by a plane parallel to a lateral side of the cone, we get a > parabola? > Let pi be some fixed plane. > Let D be some fixed line in pi. > Let F be some fixed point in pi, but not in D. > A parabola may be defined as the locus of a point, in pi, which moves so > that its distance from F is equal to its distance from D. > F is called the focus of the parabola. > D is called the directrix of the parabola. > We may use a Dandelin sphere to prove that if a cone, C, is intersected by > a plane, pi, which is parallel to one of the generators, G, of C, then the > curve of intersection is a parabola. > Let S be a sphere (a Dandelin sphere) inscribed in C and touching pi. [I > will not prove the existence (or uniqueness) of such a sphere.] > Let delta be the plane in which S touches C. > Let D be the line of intersection of delta and pi. > Let F be the point at which S touches pi. > Let the vertex of the cone be the point V. > Let A be the point at which G (the generator of C to which pi is parallel) > intersects delta. > Let L be the line in pi, and through F, which is parallel to G. > Let B be the point of intersection of L and D. > Clearly, angle VAB = angle ABF (since VA [the line G] is parallel to BF [the > line L]). > Let us call this angle t. > Clearly, all the generators of C meet delta at angle t. > Let P be any point on the intersection of C and pi. > Let the generator of C which passes through P intersect delta at Q. > Let the foot of the perpendicular from P to delta be the point R. > Let the foot of the perpendicular from P to D be the point C. > Clearly, the plane PCR is parallel to the plane FBA and so the angle PCR is > equal to t. > Hence the triangles PCR and PQR are congruent (they are both right-angled > with a common side PR and equal corresponding angles [since angle PQR = t = > PCR]). > So PQ=PC. > Also, PF=PQ (they are both tangents from P to S, and hence of equal length). > Hence, PF=PC. > So, the distance from P to F is the same as the distance from P to D. > Thus, P lies on a parabola with focus F and directrix D. > Q.E.D. See http://www.pisquaredoversix.force9.co.uk/Cone1.gif for an animation that may help you visualise the above proof. Imagine the chequered background to be a vertical wall. A glass sphere (of refractive index 1), diameter d, touches the wall at the point marked by the small red ball. The white ball is at a distance d from the wall. The yellow ball moves across the back of the sphere in such a way that the green line is always a tangent to the sphere. The path of the yellow ball is a semicircle. The plane of that semicircle intersects the wall in the red line. The green line intersects the wall at a point marked by the green ball. It can be shown that the distance from the green ball to the red ball is the same as the distance from the green ball to the red line. Hence, the locus of the green ball is part of a parabola. The correspondence between this animation and the proof above is as follows: The white ball: the point V The green ball: the point P The blue ball: the point B The red ball: the point F, the focus The cyan ball: the point C The magenta ball: the point R The yellow ball: the point Q The red line: the line D, the directrix -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Status of Waring-problem Am 20.08.04 23:23 schrieb Gottfried Helms: > Hi - > googling about waring's problem brought two different > results: > some stated, that 1986 the last unsolved case of G(4) > was solved, > mathworld stated, that it needs a certain inequality > to prove, that warings problem is completely solved. > Is mathworld 18 years behind or are that different facts? > Gottfried Helms Well, it seems there is no more discussion for this subject. The replies were very informative for me, summarized, what I did understand. My starting point was the connection between one critical condition for the existence of a primitive loop in the collatz-problem and one critical inequality of the waring-problem. The condition can be stated, that the following equation must hold 3^N*i - 1 2^A = ----------- (1) 2^N*i - 1 and also 3^N*i - 1 powerceil2(3^N) <= 2^(A+N) = 2^N * ---------- 2^N*i - 1 or, rewritten: powerceil2(3^N) 2^N 3^N*i - 1 ---------------- <= --- * ---------- (2) 3^N 3^N 2^N*i - 1 One translation of this condition forms the inequality, which is common to both problems: express a power 3^N in terms of 2^N as d*q + r where q is 2^N, d is floor(3^N/q) and r is the remainder then if d+r< q (3) then this is equivalent to the mathworld-formula of a waring-condition as well as to a nonexistence-condition of the primitive-loop/1-cycle in the collatz-problem. 1) This condition is still not proven, while the connected question of the g()-function of the waring-problem is answered, and this condition (*) plays only a role as a selector between two possible formulations of the g()- function. 2) The primitive loop (in the notation of 1-cycle) was already disproven by Ray Steiner 1978, by showing that (1) cannot hold. While (3) only negates, that the rhs of (1) could be integer, Steiner showed, that it cannot satisfy the stronger requirement to be also a power of 2. 3) A line of arguing in this matter is to analyze the coefficients of the continued fraction of log(3)/log(2) and see, whether the partial quotients agree with the approximation of (2) dependent on N. Empirically (2) is never satisfied for N up to 2^40, and the approximation of the rhs of (2) to 1 is far better than that of the lhs. 4) The cf-approximation-argument holds, if the coefficients of the cf do not increase too fast. While we have a statistical expectation for the height of the cf-coeffi- cients in such problems provided by Gauss, we seem not to have a rigid proof for the current case, that the lhs in (2) indeed approximates worse then the rhs to 1. ---------- I hope, I got this summary right. This answeres also my opening question concerning mathworlds actuality in that way, that initially I had a misconception of the relation between the open question of the inequality and the status of the solution of the g()- function. of David, and am now watching out for more information regarding 4) and a possible limiting to the cf-coefficients in general and similar cases of approximation of logarithms and naturally for this specific case. Gottfried Helms === have used a clean-new needle === === Subject: continued fractions: upper bounds for coefficients? and: something known about cf with coefficients other than integer? Hi , in the discussion of approximation of log(3)/log(2) with means of continued fractions the problem arises, whether there exist known upper bounds for the coefficients a_n of the cf of a real number x depending on its indexposition n. Is something known about that in regard to certain classes of x, say x is a square-root of a rational (here we have periodicity) or of higher algebraic degree, or if x is of transcendental classes like exponential of a rational or logarithm, etc? Fiddling a bit with that I remembered my solutions for nonstandard cf-s (rational coefficients) for rational powers of e, and noticed, that even for transcendent powers of e this simple scheme holds. Unfortunately nonstandard cf-s do not hold for the condition of best rational approximation as simple cf-s do. So I'm looking for information on such constructions. Is something discussed about such cf-s with rational (or even irrational) coefficients ? Gottfried Helms === Subject: Re: continued fractions: upper bounds for coefficients? and: something Gottfried Helms helms@uni-kassel.de > in the discussion of approximation of log(3)/log(2) with > means of continued fractions the problem arises, whether > there exist known upper bounds for the coefficients a_n > of the cf of a real number x depending on its index position > n. > Is something known about that in regard to certain classes > of x, say x is a square-root of a rational (here we have > periodicity) ... For quadratic irrationals, (P_0 + sqrt(D))/Q_0, [P_0^2 == D (mod Q_0), D > 0 not a square, Q_0 <> 0] with complete quotients (P_i + sqrt(D))/Q_i, in the periodic part of the continued fraction expansion, if Q_i <> 1, then a_i < sqrt(D), while if Q_i = 1, sqrt(D) < a_i < 2sqrt(D). Prior to the periodic part, anything can happen. For example, (10095 + sqrt(5))/5098 has a_2 = 50, while all other a_i = 1. John Robertson === Subject: Re: continued fractions: upper bounds for coefficients? and: something Am 07.09.04 01:47 schrieb Jpr2718: > Gottfried Helms helms@uni-kassel.de >>in the discussion of approximation of log(3)/log(2) with >>means of continued fractions the problem arises, whether >>there exist known upper bounds for the coefficients a_n >>of the cf of a real number x depending on its index position >>n. >>Is something known about that in regard to certain classes >>of x, say x is a square-root of a rational (here we have >>periodicity) ... > For quadratic irrationals, (P_0 + sqrt(D))/Q_0, [P_0^2 == D (mod Q_0), D > 0 > not a square, Q_0 <> 0] with complete quotients (P_i + sqrt(D))/Q_i, in the > periodic part of the continued fraction expansion, if Q_i <> 1, then a_i < > sqrt(D), while if Q_i = 1, sqrt(D) < a_i < 2sqrt(D). > Prior to the periodic part, anything can happen. For example, > (10095 + sqrt(5))/5098 > has a_2 = 50, while all other a_i = 1. > John Robertson Yes, surely. I should limit the question to the actual problems of approximations of a^n - b^m (all integer) for instance, although I mean it in some more generality. Examples show, that for the approximation of, say 3^m - 2^n the convergents of log(3)/log(2) can be used, and vice versa. Since the integer difference can at least be 1, the precision of the convergents is also limited to something related to 3^m, and thus the value of the coefficients. I'm not able to formulate that in a concise way currently, but for that specific case it seems obvious to me, that the convergents must be high-bounded by their index-position, as well as then in turn the coefficients. Isn't it that way? Gottfried Helms === Subject: Problem with inner product by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i86NG9j07960; Here is one little problem. H is Hilbert space and A is linear and bounded operator on H. If (Ax,x)=0 show that Ax=0. How can I show it? J. === Subject: re:Problem with inner product I posted a reply to this question on mathforum. What you are trying to show is not true. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: 2 triginometric equations with 2 unknowns (angles) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i86NGAj07989; >For spatial management, I need to figure out how to simplify (solve) a >system of 2 triginometric equations with 2 unknowns: > A cos(a) - B cos(b) = C1 (1) > A sin(a) - B sin(b) = C2 (2) >A, B, C1, C2 are constants, the only unknowns are the angles (a) and >(b). A simple system of 2 equations with 2 unknowns, but the problem >gains complexity when trasforming one of the sines to a cosine (or >vise versa), where you end up with a messy quadratic equation to the >4th power: >cos(a) = sqrt( 1 - sin(a)*sin(a) ) >Is there a way to simplify equations (1) and (2), trying to solve (a) >and (b)? A w - B x = C1 so w = (C1 + B x)/A ..eq(1) A y - B z = C2 so z = -(C2 - A y)/B ..eq(2) w^2 + y^2 = 1 and substitute from eq(1) x^2 + z+2 = 1 and substitute from eq(2) Gives 2 eqns with unknowns x and y phil === Subject: Trying to solve 2 homogenous quadratics How would one solve the following system of equations? (1) 17*u^2 - 144*u*v + 1904*v^2 = 43*U^2 - 60*U*V + 258*V^2 (2) 162*u^2 - 8568*u*v + 18144*v^2 = 10*U^2 - 172*U*V + 60*V^2 and u,v,U,V are in Z. We seek solutions so that (u,v) and (U,V) have the same value in either (1) or (2) (coincidental solutions?) This is from trying to factor an homogenous quartic space from the elliptic curve [0,6800,0,11559996,0] Randall === Subject: Re: Trying to solve 2 homogenous quadratics Content-Length: 1185 Originator: rusin@vesuvius > How would one solve the following system of equations? > (1) 17*u^2 - 144*u*v + 1904*v^2 = 43*U^2 - 60*U*V + 258*V^2 > (2) 162*u^2 - 8568*u*v + 18144*v^2 = 10*U^2 - 172*U*V + 60*V^2 > and u,v,U,V are in Z. > We seek solutions so that (u,v) and (U,V) have the same value in either (1) or > (2) (coincidental solutions?) > This is from trying to factor an homogenous quartic space from the elliptic > curve [0,6800,0,11559996,0] > Randall Provided the discriminants of each equation satisfy suitable mutual conditions (equal, or have the same square-free factor?) I think it's possible to find a linear transform of u, v, U, V, or maybe a birational transform of the dehomogenized pair, that reduces the original pair to the more canonical form: ax^2 + bxy + cy^2, dx^2 + exy + fy^2 = z^2, t^2 resp However, I'm moving house at the moment and all my notes are in boxes. [sci.math.research added - hope you don't mind, but I'd be interested in an expert answer to this myself!] John R Ramsden (jramsden@glasshouse.cam) com not cam === Subject: Definition of Axiom of Choice? Choice. I have been lead to feel that it means this (please tell me if it is wrong): For any set X with an infinite amount of elements, a set Y exists with any arbitrary elements selected from set X. Yes, no? === Subject: Re: Definition of Axiom of Choice? charset=iso-8859-1 > Choice. I have been lead to feel that it means this (please tell me if > it is wrong): > For any set X with an infinite amount of elements, a set Y exists with > any arbitrary elements selected from set X. > Yes, no? Well, apart from a rock group of the same name, MathWorld gives this statement for the AoC given any set of mutually exclusive nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets The original set need not be infinite, but it must be non-empty and it's a set of sets. The AoC then guarantees that you can construct a new set that contains one and only one element from each of the original sets in the collection, i.e. a representative collection. Norm === Subject: Re: Definition of Axiom of Choice? >> Choice. I have been lead to feel that it means this (please tell me if >> it is wrong): >> For any set X with an infinite amount of elements, a set Y exists with >> any arbitrary elements selected from set X. >> Yes, no? >Well, apart from a rock group of the same name, MathWorld gives this >statement for the AoC >given any set of mutually exclusive nonempty sets, there exists at least >one set that contains exactly one element in common with each of the >nonempty sets >The original set need not be infinite, but it must be non-empty no, it's trivial if the original set is empty. >and it's a >set of sets. The AoC then guarantees that you can construct a new set that >contains one and only one element from each of the original sets in the >collection, i.e. a representative collection. > Norm ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: Re: Definition of Axiom of Choice? > Choice. I have been lead to feel that it means this (please tell me if > it is wrong): > For any set X with an infinite amount of elements, a set Y exists with > any arbitrary elements selected from set X. > Yes, no? No. What the axiom of choice says is that for any collection of sets X_i with i ranging over any indexing set I, that the product of all the X_i is non-empty, or equivalently, that one can define a function f:I->Union (X_i) with the property that f(i) is an element of X_i. The name choice comes from the idea that if you have an arbitrary collection of sets you can choose one element from each set. -Ron === Subject: Re: Definition of Axiom of Choice? >> Choice. I have been lead to feel that it means this (please tell me if >> it is wrong): >> For any set X with an infinite amount of elements, a set Y exists with >> any arbitrary elements selected from set X. >> Yes, no? >No. >What the axiom of choice says is that for any collection of non-empty >sets X_i >with i ranging over any indexing set I, that the product of all the X_i >is non-empty, or equivalently, that one can define a function f:I->Union >(X_i) with the property that f(i) is an element of X_i. The name choice >comes from the idea that if you have an arbitrary collection of sets you >can choose one element from each set. >-Ron ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: Re: Definition of Axiom of Choice? > Choice. I have been lead to feel that it means this (please tell me if > it is wrong): > For any set X with an infinite amount of elements, a set Y exists with > any arbitrary elements selected from set X. > Yes, no? No. First the grammar nitpick: You mean an infinite number of elements, or simply any infinite set X. Then the logical bug: It doesn't make sense to talk about a set Y with any arbitrary elements selected from X, unless you specify what you mean by the phrase with any arbitrary elements. The way I've heard it phrased most concisely is: Suppose we have a non-empty set I. This is going to be called our index set. Now suppose we are given a whole bunch of other non-empty sets indexed by I --- that is, we have one set S_x for each and every x in I. (To take a trivial example, suppose I was the set {1,2,3}. Then we would have the sets S_1, S_2, and S_3, which could be whatever non-empty sets we like: for example, S_1 could be the set of natural numbers, S_2 could be the set of real numbers, and S_3 could be the set {Monday, Wednesday, Friday}. Of course, that's a boring example, and it would be more interesting to take I to be the set of natural numbers, or the set of real numbers, or something big like that.) Now, what the Axiom of Choice says is that the Cartesian product of all S_x (x in I) is non-empty. That is, we can produce at least one example of an ordered I-tuple (E_x | x in I) such that each and every E_x is a member of S_x. (For example, the 3-tuple (7, pi, Monday) is a member of the Cartesian product of our S_1, S_2, S_3. The 3-tuple (1, 1, Wednesday) is another. See, it's really boring and trivial when I has only a finite number of elements!) So that's all the Axiom of Choice says. If you have a set of non-empty sets S_{x in I}, then you can construct an I-tuple containing one element from each of those sets. That's all. A corollary to the Axiom of Choice is that if you have a single non-empty set, then it is possible to construct a new set containing only one of that set's elements. In other words, you can pick a singleton out of the original set. Again, it looks trivial and obvious, but it is really a separate axiom from the other axioms of set theory (or the real number system, or whatever application you're interested in). HTH, -Arthur === Subject: Re: Definition of Axiom of Choice? > Choice. I have been lead to feel that it means this (please tell me if > it is wrong): > For any set X with an infinite amount of elements, a set Y exists with > any arbitrary elements selected from set X. > Yes, no? > No. First the grammar nitpick: You mean an infinite number of > elements, or simply any infinite set X. Then the logical bug: > It doesn't make sense to talk about a set Y with any arbitrary elements > selected from X, unless you specify what you mean by the phrase with any > arbitrary elements. > The way I've heard it phrased most concisely is: > Suppose we have a non-empty set I. This is going to be called > our index set. Now suppose we are given a whole bunch of other > non-empty sets indexed by I --- that is, we have one set S_x for > each and every x in I. > (To take a trivial example, suppose I was the set {1,2,3}. Then > we would have the sets S_1, S_2, and S_3, which could be whatever > non-empty sets we like: for example, S_1 could be the set of natural > numbers, S_2 could be the set of real numbers, and S_3 could be the > set {Monday, Wednesday, Friday}. Of course, that's a boring example, > and it would be more interesting to take I to be the set of natural > numbers, or the set of real numbers, or something big like that.) > Now, what the Axiom of Choice says is that the Cartesian product > of all S_x (x in I) is non-empty. That is, we can produce at least > one example of an ordered I-tuple (E_x | x in I) such that each and > every E_x is a member of S_x. > (For example, the 3-tuple (7, pi, Monday) is a member of the > Cartesian product of our S_1, S_2, S_3. The 3-tuple (1, 1, Wednesday) > is another. See, it's really boring and trivial when I has only a > finite number of elements!) > So that's all the Axiom of Choice says. If you have a set of > non-empty sets S_{x in I}, then you can construct an I-tuple containing > one element from each of those sets. That's all. > A corollary to the Axiom of Choice is that if you have a single > non-empty set, then it is possible to construct a new set containing > only one of that set's elements. In other words, you can pick a > singleton out of the original set. Again, it looks trivial and obvious, > but it is really a separate axiom from the other axioms of set theory > (or the real number system, or whatever application you're interested > in). > HTH, > -Arthur of all S_x (x in I) is non-empty. (Somewhat nitpicking): Technically speaking the above version is called Russell's Multiplicative Principle (MP) (due to B. Russell). It is equivalent to Zermelo's Axiom of Choice (AC). They are all equivalent to the Well Ordering Principle (WO) (due to Zermelo??, I think), and Kuratowski-Zorn's Lemma (ZL). Calling MP the Axiom of Choice might confuse the beginning student. (AC) If A is any set of NONEMPTY sets, then there exists a choice function on A (that is, there exists a function f with domain A such that for every x in A, f(x) is in x. (MP) If (A_i | i in I) is a family of NONEMPTY sets, and if the index set I is a set, then the cartesian product set is nonempty (that is, there is an I-tuple (x_i | i in I) such that for every i in I, x_i is in A_i). (ZL) Every inductive partially-ordered set has a maximal element. [Recall: A partially-ordered set P is inductive if every chain in P has an upper bound in P. Note: The term inductive here is not related to induction. A chain in P is just a linearly ordered subset of P.] (WO) For every set A, there exists a well-ordering on A. [Def of a well-ordering: A partial order on a set P is a well-order if every NONEMPTY subset of P has a least element.] Motto: All animals are equal but some are more equal than others. (Orwell) We rank the animals as follows: 1. ZL (Sure) 2. MP,AC (OK, if you twist my arm.) 3. ... n. WO [Wow, it's awesome (or, secretly, crazy)!] === Subject: Re: Problem with inner product by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8700Vo12665; >Here is one little problem. >H is Hilbert space and A is linear and bounded operator on H. >If (Ax,x)=0 show that Ax=0. >How can I show it? Are you sure its true? For 2-space (a very simple Hilbert space), let A=matrix with 0 along main diagonal and (1,-1) for the off-diagonal. Then A(x,y)=(y,-x), so the inner product is xy-yx=0. In the more general case, let the operator do as above for the first 2 coordinates and make all the others 0 or alternatively do the same kind of switch (i.e with one sign reversed) on pairs of coordinates. === Subject: Re: Problem with inner product >>H is Hilbert space and A is linear and bounded operator on H. >>If (Ax,x)=0 show that Ax=0. >>How can I show it? > Are you sure its true? For 2-space (a very simple Hilbert space), > let A=matrix with 0 along main diagonal and (1,-1) for the off-diagonal. > Then A(x,y)=(y,-x), so the inner product is xy-yx=0. > In the more general case, let the operator do as above > for the first 2 coordinates and make all the others 0 or alternatively > do the same kind of switch (i.e with one sign reversed) > on pairs of coordinates. It is only true if you add the condition that A by symmetric. You can prove this by polarization. [Well, unless the field underlying H is of characteristic 2. -- I don't know what happens in this case. ] === Subject: Re: Probability(X is Prime) > Your explanation is not really the correct one. > The reason that the product does not given > the correct probability is that the probability > that p_i divides X is not independent from > the probability that p_j divides x once there are > sufficiently many primes in the product of (1-1/p) > that the direct product of the primes exceeds X. > You can lead a horse's ass to knowledge, but you can't make him think. Hi Bob, I think your answer corresponds to my original statement actually. You are saying that one can use the formual prod (1-1/p) providing the product of the primes used in this formula doesnt exceed X. I would agree with this because you are now considering the interval [1,X]. As soon as you use the next prime that such that the product takes you beyond X the formula cannot Prod (1-1/p) cannot be used. However we are trying to come up with a formula for Prob(X is Prime)and we seemed to have failed so far. === Subject: Re: Probability(X is Prime) > How about the following statement that using PNT pi(x)=x/log x > suggests the > Prob(x is prime) = 1/log (x). If this statement is to be accurate we > should have Sum (1/log x) over x=3,5,7,9,11.... should be 1 in order > to define a probability distribution on the integers. I dont know what > this sum actually is but my guess is that it is not 1. Does it > converge? > Only when the events are mutually exclsive the total probability is the sum of individual probabilities but a number can be divisible by many primes simultaneously. By Mertens theorem :(For N ---> inf.) Prob.(N is prime) = (1-1/2)(1-1/3)x....x(1-1/p)x e^g = 1/log(N) p = prime prox. inferior to N e = 2.718281828.. ; g = Euler's Constant = .577215... === Subject: Re: Probability(X is Prime) > How about the following statement that using PNT pi(x)=x/log x > suggests the Prob(x is prime) = 1/log (x). No, it doesn't, not unless you take a lot of care in setting up your definitions. > If this statement is to be accurate we should have Sum (1/log x) over > x=3,5,7,9,11.... should be 1 No, it shouldn't, it should sum to the number of primes, which it does. > in order to define a probability distribution on the integers. I dont > know what this sum actually is but my guess is that it is not 1. Does > it converge? No. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Probability(X is Prime) > How about the following statement that using PNT pi(x)=x/log x > suggests the Prob(x is prime) = 1/log (x). If a randomly select a 20-digit number, what is the probability that it is prime? Answer: approximately 1/log(10^20) = 0.0217147 . The actual value is (according to Maple) (pi(10^20)-pi(10^19))/(10^20-10^19) = 662253978428191411/30000000000000000000 = 0.02207513261 Actually, a better approximation is 1/(log(0.5*10^20)) = 0.02204655 === Subject: Re: Probability(X is Prime) > How about the following statement that using PNT pi(x)=x/log x > suggests the Prob(x is prime) = 1/log (x). -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Uncountable sets in CZF? >This is essentially the same as my beliefs. If a bijection can be >constructed between N in R[V] in some model V', then given that the >size of N, and of R[V], has not been changed, that would seem to say >that R[V] is countable, only unprovably so in V. Since we know that >the real R is not countable, that would seem to say that ZFC is >incomplete, or wrong. >>But that is already well known for a long time. >>There is no way to characterize 'the real model of ZFC' (whatever that >>means) with new axioms or so. It was once known as the Skolem-Loewenheim >>paradox. > > I see, ZFC is incomplete. Is this related to Godel's theorem? > The Skolem-Loewenheim theorem/paradox is older (early 1920s). I meant, is it proved by the same type of argument? If it is, it confirms my sentiment that logic is incapable of talking about the uncountable. Andrew Usher === Subject: Re: Uncountable sets in CZF? >This is essentially the same as my beliefs. If a bijection can be >constructed between N in R[V] in some model V', then given that the >size of N, and of R[V], has not been changed, that would seem to say >that R[V] is countable, only unprovably so in V. Since we know that >the real R is not countable, that would seem to say that ZFC is >incomplete, or wrong. >>But that is already well known for a long time. >>There is no way to characterize 'the real model of ZFC' (whatever that >>means) with new axioms or so. It was once known as the Skolem-Loewenheim >>paradox. >I see, ZFC is incomplete. Is this related to Godel's theorem? >>The Skolem-Loewenheim theorem/paradox is older (early 1920s). > I meant, is it proved by the same type of argument? No, not at all. In the form any consistent theory has a countable model, it shows more resemblance with the ordinary completeness theorem of FOL. If it is, it > confirms my sentiment that logic is incapable of talking about the > uncountable. FOL is incapable of distinguishing between uncountable and countable models. That doesn't mean, however, that there exist no FOL-theories that are capable of talking 'about' the uncountable, in the way ZFC does. (FYI: your 'sentiment' was well known in the 1920s, but is currently no longer generally held. The S-L 'paradox' simply is no longer felt to be paradoxical anymore. Set-domain models of ZFC are only interesting in so far as they allow one to establish the consistency of certain theories (independence of certain assumptions from others). They are no realistic models anyway, as the 'real' universe of sets has a proper class for its domain.) -- Herman Jurjus === Subject: Re: Uncountable sets in CZF? By plain set theory I meant theory where the only primary objects >>were sets as opposed to a theory where numbers are primary objects. >ZF and ZFC are therefore what you call plain set theories. In ZF and >ZFC, numbers are defined AS SETS. I wonder where you got the erroneous >idea that numbers were primary objects in ZF and ZFC??? >A natural number is defined to be either the empty set or a successor >ordinal whose elements are all either empty or successor ordinals, a >set n is a natural number if >n is empty or [n is a successor ordinal and for all m in n (m is empty >or m is a successor ordinal)]. >Equivalently, a natural number is an element of the smallest limit >ordinal. Equivalently, a natural number is an element of the unique smallest inductive set, where a set x is called inductive if the empty set is an element of x and for all y in x exists z in x for all t (t in z iff (t in y or t = y)). Sufficient axioms for ZF are: (1) exists x (x in x) - this asserts the existence of at least one set, so it can be called the Axiom of Existence; (2) for all x for all y (for all z (z in x <=> z in y) => x = y); (3) for all x exists y for all z (exists t (z in t and t in x) => z in y); (4) for all x exists y for all z (for all t (t in z => t in x) => z in y); (5) for any formula , P(u,v), in ZF, and for arbitrary values for all free variables in P(u,v), except for u and v, for all u for all v for all w ((P(u,v) and P(u,w)) => v = w) => for all x exists y for all v (v in y <=> exists u (u in x and P(u,v))); (6) exists x (empty set in x and for all y in x exists z in x for all t (t in z <=> (t in y or t = y))) - Axioms 1 and 5 guarantee the existence of an empty set, and Axiom 2 guarantees its uniqueness; (7) for all x (exists y (y in x) => exists y in x for all z not (z in x and z in y)). For ZFC, this is augmented by another Axiom: (8) for all x ([exists y (y in x) and for all y in x [exists t (t in y) and for all z in x (y = z or for all t not (t in y and t in z))]] => exists u for all y in x exists v for all t ((t in y and t in u) <=> t = v)). David ----- === Subject: Re: Uncountable sets in CZF? Regarding IZF, [...] | But most especially, there can't be a surjection from the whole of the | naturals to the reals. |Cantor-Schroeder-Bernstein: it works both ways. | |What that means is that one of the reasons that people call the reals |uncountable is because they've figured out a bijection between the |reals and the powerset of the naturals, thus they reason that there |are no bijections between the reals and the naturals, because |Cantor-Schroeder-Bernstein says the existence of a surjection either |way between two sets is proof of the existence of a bijection between |those two sets. I thought the theorem was about sets A and B where there exist injections from A to B and from B to A. I don't know why one would use either result (either Cantor-Bernstein or the analogous result about surjections) to argue against the existence of a bijection between the natural numbers and the reals. It's so much more direct to just use one of the two proofs Cantor had to show that there is no surjection from the naturals to the reals. I see no point in complicating the issue. |That is to say, the existence of a surjection from A to B and from B |to A implies that A and B are equivalent, and as well from A to B to C |and C to B to A through composition. Moreover, in the context of IZF, you can't use Cantor-Schroeder- Bernstein; it's not a theorem of IZF. The bijection described in the proof has to be defined by cases. As often is the case, when a function has been defined by cases, the law of excluded middle is implicitly invoked to claim that the domain of the function is the whole original domain. That is, if X is a subset of A and we say f=g on X and f=h on A-X, then the domain of f is the set of elements of A that either are members of X or are not members of X. And of course IZF does not have the law of excluded middle. |That implies it is not a mathematical fact and to promote the other |view as gospel, immutable, written in stone, etcetera, would thus be |deceitful. I'm rather angered that you would suggest the acceptance |of a mathematical falsehood as mathematical fact. Wouldn't you be? Not always, no. I already have plenty of experience with seeing people who are confused like you are offering nonsense as mathematical fact. If I were angry about it, I would be angry quite a lot of the time. Of course, it's worse when the person who is confused is so confused, like you are, that he not only thinks that what he's saying is correct, but he's really *sure* of its being correct, and that hence everyone who disagrees with it is wrong. But worse still are the people like you who not only are confused and sure they're right, but angry at other people who are sure they're right too. |Reexamine the claim about there being only one or none proper classes. Nobody claims there's exactly one proper class. The ordinals are a proper class, and so is class of all sets. They may believe that there is no such thing as a proper class, but if they agree there are any, they agree there are many. I think you're giving a confused version of the claim that any two proper classes can be put into one-to-one correspondence. | Consider why that demands dual representation of the ur-element as |both zero and Ord, regardless of whether Ord is N. (Steve, infinite |sets are equivalent.) | |Unitize the analog! It's better to have a tool, even a primitive |tool, to measure sparse points of the continuum, than none, and bar |further consideration of the matter by fiat. | |Why don't you consider that the direct sum of infinitely many copies |of N is zero? I can't make any sense out of this. |union of countable sets, You should try to write more precisely (not that I expect you to), especially when you're allegedly stating what someone else said. This sounds like you're describing him as saying that each uncountable set is a countable union of countable sets, which he surely didn't something about its being consistent with ZF for there to exist an uncountable set that's a countable union of countable sets. |to that the constructivist powerset mapping |argument is shoddy or fails, How about getting rid of some of the shoddy that you keep distributing? |to these metatheoretical hand-wavings of |yes and no and ignorance fo the excluded middle, we've seen some of |what I would call progress in the state of discussion of these issues |here on sci.math, an open public forum where the participants are |often professional mathematicians and as such loathe to promote |untruth, particularly in the stark, concrete world of mathematics, |that is already a paradise with no need for the transfinite: |inconsistent and thus hellish. The only appearance of inconsistency is because you're confused! [...] |You're intelligent and understand the basic tenets of rational |discourse, and by now you're probably familiar with the rather few |arguments of transfinite set theory: Set theory is a rather huge subject. Keith Ramsay === Subject: Re: Uncountable sets in CZF? Discussion, linux) > |Reexamine the claim about there being only one or none proper classes. > Nobody claims there's exactly one proper class. Ross does. -- But remember, as long as one human being follows the rules of mathematics, then mathematics as a human discipline survives. Right now I'm that one human being, so mathematics survives. -- James S. Harris === Subject: Re: Uncountable sets in CZF? David, I want to compliment you, those are some very informative posts and that post should be recommended reading. There are notions of theories where numbers are primary objects in the theory and are not sets. Keith presented a statement that he could map a proper subset of the naturals bijectively to the reals. What's the deal with that? That reminds me that the direct sum of infinitely many copies of N is empty, but not, by exceptional definition. Then again having {} as an element of a set implies regularity by many naive constructions of that predicate. I think the ur-element is zero, and sometimes infinity, and one. We've probably all had calculus instruction, or at least the pre-calculus instruction about limits and the definition of a of the antiderivative, the integral. Familiarity with a strict and perhaps overly strict concept of limit is a given, where that is a relatively modern response to the vagaries of infinitesimals of Newton and Leibniz' functional, useful, empirical result driving infinitesimal analysis: the integral calculus. An even more modern response arose last century in the consideration of the infinitesimals and the establishment of definitions to allow the consideration of hyperreals. The hyperreals as a set contain no elements that are not elements of the reals, and they do include infinitesimals. Now before everyone hauls out umpteen proofs that 0.999... = 1, they are perhaps true in a similar manner as to how some geometrical results are true (sound, valid, correct), in the Euclidean geometry, and not true. McAnally, I don't know how long you lurked on sci.math before beginning your voluminous well-informed postings, you may well know that I have since shortly after discovering this unmediated forum argued the notion that infinite sets are equivalent, ignorantly, naively. My response to the declaration that there were not mappings from the naturals to the reals was the definition of the Natural/Unit Equivalency Function. It's defined in a way that the domain is the naturals and the range is the unit interval of reals. EF(0) = 0 lim EF(oo) = 1 EF(n+1) > EF(n) lim (EF(n+1)-EF(n)) = 0 People ask then what is EF(1) and I say iota and they say what is (EF(0)+EF(1))/2 and I say undefined and then (EF(0)+EF(2))/2 = EF(1) and then (EF(0)+EF(n))/m is defined only when m divides into n. Anyways that leads to the consideration of iota, the least positive real infinitesimal, with properties of an x-transcendental real, and the vague fugue. Claims are made that no function bijects between the naturals and reals for a) the antidiagonal argument, b) the nested intervals argument, and c) Cantor-Bernstein transitivity. The antidiagonal argument and nested intervals argument are shown to not apply, for dual representation and enforced list order, and for functions with properties similar to EF in monotonically mapping. The Cantor-Schroeder-Bernstein theorem(s), about surjection either way implying a bijection, presents an inconsistency in mapping between the reals and naturals and reals and powerset of the naturals. To that end I described a model (obviously part of the theory in disguise) of ubiquitous ordinals or naturals. Another alternate consideration even to that has the ur-element represented as zero and infinity. In the model of ubiquitous ordinals, the order type is the successor is the powerset, and a a set X and P(X) f(P(X)) = f(X)+1, and f(x)=x+1 for x E X is a bijective mapping, with S = {}, which while being zero when zero={} is also the ur-element and equal to X+1. About as looking from outside, metatheories/metamodels, extensions, etcetera, you have in them a set N, the set of all naturals integers, in one of those models mapping bijectively to a set R, the set of all real numbers. Ross F. === Subject: Make money easily truely works X-Library: Indy 9.00.10 Comments Follow the directions below and in two weeks you'll have up to $20000.00 in your PayPal account. There is a very high rate of participation in the program because of its low investment and high rate of return. 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Become a part of the donation program and help people help people. This program is about helping each other! Success is a journey - Not a destination! Start Your Journey TODAY!!!! === Subject: simple pde question.. For some reason I can't solve one of the simplest equations.. Is there a solution? The equation: B = curl(A) Or in coordinate form: B_i = e_ijk* d/dx_j ( A_k ) (3 dimensions OK for now, e_ijk is anti-symmetric unit tensor or Levi-Civita tensor) The problem is to solve for the vector field A given a vector field B. I know that if A is a solution, so is A' = A + grad(f) where f is any function of the coordinates. I also know that there is no solution if div(B) != 0. But how can I write some (any) solution A in terms of divergenceless B? Author-Supplied-Address: bds ipp mpg de === Subject: Re: simple pde question.. Mail-To-News-Contact: abuse@dizum.com shevek asked: |> For some reason I can't solve one of the simplest equations.. |> Is there a solution? |> The equation: |> B = curl(A) |> The problem is to solve for the vector field A given a vector field B. |> I know that if A is a solution, so is A' = A + grad(f) where f is any |> function of the coordinates. I also know that there is no solution if |> div(B) != 0. |> But how can I write some (any) solution A in terms of divergenceless |> B? If you're willing to buy a divergenceless A, then you can do one further curl operation, leaving - Laplacian A = J = curl B Then you solve this elliptic equation given the appropriate boundary conditions. -- cu, Bruce drift wave turbulence: http://www.rzg.mpg.de/~bds/ === Subject: Re: simple pde question.. shevek: >For some reason I can't solve one of the simplest equations.. >Is there a solution? >The equation: >B = curl(A) >Or in coordinate form: >B_i = e_ijk* d/dx_j ( A_k ) >(3 dimensions OK for now, e_ijk is anti-symmetric unit tensor or >Levi-Civita tensor) >The problem is to solve for the vector field A given a vector field B. >I know that if A is a solution, so is A' = A + grad(f) where f is any >function of the coordinates. I also know that there is no solution if >div(B) != 0. >But how can I write some (any) solution A in terms of divergenceless >B? It depends upon what you are given _explicitly_. One way, is to take the curl of B = curl A to get J, curl V = curl curl A = grad (div A) - grad^2 A = (4pi/c) J Use the gauge freedom to set div A = 0, then solve the three poisson equations for the components of A, -grad^2 A = (4pi/c) J Another way is to exploit some symmetry and solve for the line integral of A, B = curl A => integral B.dS = integral (curl A).dS Using the identity, integral (curl A).dS = integral A.dl, I get, integral A.dl = integral B.dS If you can write down A in a coordinate system that allows you to take A outside the integral sign in order to evaluate the line integral easily, then this will work. === Subject: Re: simple pde question.. >For some reason I can't solve one of the simplest equations.. >Is there a solution? >The equation: >B = curl(A) >Or in coordinate form: >B_i = e_ijk* d/dx_j ( A_k ) >(3 dimensions OK for now, e_ijk is anti-symmetric unit tensor or >Levi-Civita tensor) >The problem is to solve for the vector field A given a vector field B. >I know that if A is a solution, so is A' = A + grad(f) where f is any >function of the coordinates. I also know that there is no solution if >div(B) != 0. >But how can I write some (any) solution A in terms of divergenceless If B is defined on all of R^3, then one answer is: A(r) = - int_0^1 du u r x B(u r), i.e. A_x = int_0^1 du [u z B_y(u x,u y,u z) - u y B_z(u x,u y,u z)], A_y = int_0^1 du [u x B_z(u x,u y,u z) - u z B_x(u x,u y,u z)], A_z = int_0^1 du [u y B_x(u x,u y,u z) - u x B_y(u x,u y,u z)]. David ----- === Subject: Re: simple pde question.. >B = curl(A) >The problem is to solve for the vector field A given a vector field B. >I know that if A is a solution, so is A' = A + grad(f) where f is any >function of the coordinates. I also know that there is no solution if >div(B) != 0. >But how can I write some (any) solution A in terms of divergenceless In other words, you want to construct a vector potential for the solenoidal vector field B. In general, you can do this on a domain D such that every closed surface in D is the boundary of a domain contained in D (i.e. D has no holes). I suppose your domain is all of R^3. The trick is to choose gauge conditions to reduce the amount of arbitrariness in the solution. One possible gauge condition is A_3 = 0. Then you want dA_2/dx_3 = -B_1 dA_1/dx_3 = B_2 dA_2/dx_1 - dA_1/dx_2 = B_3 The first and second conditions determine A_1 and A_2 up to functions of x_1 and x_2, i.e. A_1 = int B_2 dx_3 + a_1(x_1,x_2) A_2 = -int B_1 dx_3 + a_2(x_1,x_2) and then the third equation says da_1/dx_1 - da_2/dx_2 = g(x_1,x_2) for a certain function g (exercise: show that if div(B) = 0, this is a function of x_1 and x_2 only). Now you could e.g. take a_2 = 0 and a_1 = int g(x_1,x_2) dx_1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: The real numbers, and general comments Perhaps I should try to explain again my argument. (Provided that you haven't just decided to run away from me.) R is normally conceptualised as an infinite geometric line. I say that what we call a number is a point on this line, and so many points as we can locate can be called numbers. But, it is not a valid inference from this to state that the line is made up of numbers! This is also why hyperbolic geometry is inconsistent; it requires an uncountable number of (rational) numbers. Andrew Usher === Subject: Re: The real numbers, and general comments > haven't just decided to run away from me.) > R is normally conceptualised as an infinite geometric line. I say that > what we call a number is a point on this line, and so many points as > we can locate can be called numbers. But, it is not a valid inference > from this to state that the line is made up of numbers! > This is also why hyperbolic geometry is inconsistent; it requires an > uncountable number of (rational) numbers. > Andrew Usher Why on earth is hyperbolic geometry inconsistent? and how does it require an uncountable number of rational numbers? It doesn't and there aren't. === Subject: Re: The real numbers, and general comments > You can specify any one Dedekind cut, but you can't give the rule to > construct them all. A finite definition is acceptable, of course, as: > > N = {0, 1, 2, ...} > > is understood by all of us. > But is it a finite definition? I mean it only gives the first 3. > Since the rationals are countable, they must be allowable, and since > sequences involve a countable number of terms they must be acceptable, so > are cauchy sequences of rationals not acceptable. It gives the rule to construct all the other terms, therefore it's a finite definition. Andrew Usher === Subject: Re: The real numbers, and general comments >> You can specify any one Dedekind cut, but you can't give the rule to >> construct them all. A finite definition is acceptable, of course, as: >> N = {0, 1, 2, ...} >> is understood by all of us. >> But is it a finite definition? I mean it only gives the first 3. >> Since the rationals are countable, they must be allowable, and since >> sequences involve a countable number of terms they must be acceptable, so >> are cauchy sequences of rationals not acceptable. > It gives the rule to construct all the other terms, therefore it's a > finite definition. > Andrew Usher And it is a way of defining the real numbers contradicting your original claim, I think. I don't have the original post anymore and can't be === Subject: Re: The real numbers, and general comments > You can specify any one Dedekind cut, but you can't give the rule to > construct them all. A finite definition is acceptable, of course, as: > > N = {0, 1, 2, ...} > > is understood by all of us. > Z = { ... -3, -2, -1, 0, 1, 2, 3, ...} > Q = { m/n | m in Z, n in Z, n <> 0} These are fine ... > R = { | A union B = Q; for all a in A, for all b in B, a is not in A} > For a in R, if GLB(B) in B, then is identified with GLB(B) > which is in Q. > How's that? but this is not. It doesn't specify a procedure for producing these sets. As you know, I believe that quantify over all subsets of an infinite set is illegitimate. > I would found analysis on any countable dense set in R, and use Cauchy > sequences where necessary. It is obvious that analysis over any such > countable dense set is equivalent to any other, and to analysis over > R, as long as all functions are piecewise continuous. > There's a simple problem with that: Let f be the following function: > f(x) = 0 when x is irrational, f(x) = 1 when x is rational. This is not piecewise continuous. (Note the restriction above.) > You are suggesting we just use the rationals, but the integral of f(x) > from 0 to 1 is actually 0. True, the above integral taken over Q gives 1. But who really cares about this function anyway? >No, you can't diagonalise on the computables, S, because S itself is >not computable. Hence you can't construct a number that isn't >computable. I see no reason to consider a thing that can't be >constructed to be a number. Also, this 'procedure' can't show that >there are uncountably many 'numbers'. >>S is enumerable, which is sufficient to demonstrate a number that is >>enumerable but not computable. > > Such a number is provably not constructible. Even if you accept > non-constructibles as 'numbers', you can't show that there are > uncountably many 'numbers'. > not constructible -> not computable. Diagonal arguments to show that > there are uncountably many reals, combined with only countably many > computables, implies that there are uncountably many non-computables. Yes, R is uncountable. But you haven't justified considering all of R to be numbers. If you use the common sense definition of a 'number' as something you can calculate with, then obviously all numbers are in S. >Euclidean geometry naturally embeds in projective geometry. I perceive >metric and projective techniques is simply different methods of >proving results in geometry, not as different fields. Use the extended parallel postulate: > > P. Every pair of lines meet in a point; this point is at infinity iff > the interior angles make two right angles. > > This includes both Euclidean and projective geometry. > You are modifying the set of postulates for Euclidean geometry when you > do that. As soon as you add a postulate, you get *different* results, > which means you are no longer in Euclidean geometry. It is a geometry > that is clearly very similar, but it is not the same. What different results? As far as I can see, adopting my postulate means that all the results of Euclidean and of projective geometry hold good. >No, math is not a science but is closer allied to philosophy. I >wouldn't doubt that modern philosophy suffers from the same defect. >>Having studied philosophy, I can't agree with this statement, either. >>It does offer insight into your approach to math, however. > > Would you mind expanding on that last? > The defect that philosophy suffers, from what I've seen, is that whoever > can string the best looking line of BS gets famous, regardless of how > clearly the philosopher doesn't believe it. > Solipsism would be a great example. If you don't believe anything exists > besides yourself, why would you care if someone attacks you, hurts you, > etc? If everything is in your mind and nothing is real, then don't > worry about anything. Well, not quite. Even if the real world has no independent existence, you still can logically want to act so as to avoid being hurt. Also, solipsism may be ridiculous, but the consideration of the possibility increasing our understanding of knowledge. > Math, in contrast, is about finding logicly consistent axioms and > finding their conclusions. As a result, math is much like the other > sciences in that it seeks to find truth. Philosophy, on the other hand, > may claim to seek truth but seems to do anything but that. I see both mathematics and philosophy as attempting to find absolute truth (outside of the physical world). To the extent either deviates from that, I consider it wrong. Andrew Usher === Subject: Speculative, but at least interesting Long Standing Math Puzzle May be Solves http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html === Subject: Re: Speculative, but at least interesting Discussion, linux) > Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html ,---- | Mathematicians could be on the verge of solving two separate million | dollar problems. If they are right - still a big if - and somebody | really has cracked the so-called Riemann hypothesis, financial | disaster might follow. Suddenly all cryptic codes could be | breakable. No internet transaction would be safe. | | [...] | | The Riemann hypothesis would explain the apparently random pattern | of prime numbers - numbers such as 3, 17 and 31, for instance, are | all prime numbers: they are divisible only by themselves and | one. Prime numbers are the atoms of arithmetic. They are also the | key to internet cryptography: in effect they keep banks safe and | credit cards secure. `---- *Would* the Riemann hypothesis explain the pattern of primes? If so, would this really threaten cryptography? Sorry, I've always been fuzzy on this Riemann stuff. But the connection to primes is just about the value of pi(x) as x goes to infinity, right? Are there any actual results showing that a proof of the Riemann hypothesis would yield a fast means of factoring large primes[1]? Footnotes: [1] If Bill Gates says that's the issue, then that's the issue. Don't argue. ,---- | ``The obvious mathematical breakthrough would be development of an | easy way to factor large prime numbers.'' -Bill Gates, The Road | Ahead, pg. 265 `---- -- Jesse F. Hughes One is not superior merely because one sees the world as odious. -- Chateaubriand (1768-1848) === Subject: Re: Speculative, but at least interesting charset=iso-8859-1; reply-type=response > Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html Louis de Branges sounds like their version of James Harris.... blimey! I wonder if that IS James Harris using a pseudonym! === Subject: Re: Speculative, but at least interesting > Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html > Louis de Branges sounds like their version of James Harris.... blimey! I > wonder if that IS James Harris using a pseudonym! No, based on his work on the Bieberbach Conjecture, de Branges is VERY legit. He has, in the past, published erroneous proofs of the Riemann hypothesis, but so have other mathematicians. Myxococcus xanthus === Subject: Re: Speculative, but at least interesting > Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html > Louis de Branges sounds like their version of James Harris.... blimey! I > wonder if that IS James Harris using a pseudonym! Not quite. It's true that he has in common with James an obsessive interest in one of the great unsolved problems. However, de Branges has actually contributed. He is on the faculty at Purdue, and is credited with solving another unsolved problem, the Bierbach conjecture. His proof has not been dismissed outright but is getting serious attention. If he's gone off the deep end late in his mathematical career, at least he had a distinguished mathematical career first. - Randy === Subject: Re: Speculative, but at least interesting >Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html >> Louis de Branges sounds like their version of James Harris.... blimey! I >>wonder if that IS James Harris using a pseudonym! > Not quite. It's true that he has in common with James an obsessive > interest in one of the great unsolved problems. However, de Branges > has actually contributed. He is on the faculty at Purdue, and is > credited with solving another unsolved problem, the Bierbach conjecture. > His proof has not been dismissed outright but is getting serious > attention. > If he's gone off the deep end late in his mathematical career, at > least he had a distinguished mathematical career first. > - Randy Reminds me of Abian. Alexander Abian apparently still made useful contributions to the mathematics literature even after he evolved into a crackpot. http://www.math.ucdavis.edu/~suh/abian/abian-list.html === Subject: Re: Speculative, but at least interesting Discussion, linux) >> Long Standing Math Puzzle May be Solves >> http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html >> Louis de Branges sounds like their version of James Harris.... blimey! I >> wonder if that IS James Harris using a pseudonym! > Not quite. It's true that he has in common with James an obsessive > interest in one of the great unsolved problems. However, de Branges > has actually contributed. He is on the faculty at Purdue, and is > credited with solving another unsolved problem, the Bierbach conjecture. > His proof has not been dismissed outright but is getting serious > attention. > If he's gone off the deep end late in his mathematical career, at > least he had a distinguished mathematical career first. But when he proved the Bierbach conjecture, it took some time for mathematicians to accept the proof. Similarly, once again, it has taken time for mathematicians to really look at his proposed proof of Riemann (partly because it's not the first time he's announced a proof of the Riemann hypothesis). paint such a rosy picture of mathematicians' reactions to de Branges and it also doesn't show him as having a distinguished mathematical career free of controversy prior to this. Note: I'm not saying whether mathematicians' reluctance to study his proposed proof is regrettable or not. I'm not really familiar with a fuller picture of the situation. -- Jesse Hughes But nothing's being Dr. Ullrich is a particular case of something's being such that nothing is it: (Ex)~(Ey)(y = x) -- John Correy on the failings of first order logic === Subject: Re: Speculative, but at least interesting charset=iso-8859-1; > Long Standing Math Puzzle May be Solves > http://www.guardian.co.uk/uk_news/story/0,3604,1298728,00.html >> Louis de Branges sounds like their version of James Harris.... blimey! I >> wonder if that IS James Harris using a pseudonym! > Not quite. It's true that he has in common with James an obsessive > interest in one of the great unsolved problems. However, de Branges > has actually contributed. He is on the faculty at Purdue, and is > credited with solving another unsolved problem, the Bierbach conjecture. > His proof has not been dismissed outright but is getting serious > attention. > If he's gone off the deep end late in his mathematical career, at > least he had a distinguished mathematical career first. Cool, just checking.. === Subject: Re: Definition of Axiom of Choice? > A corollary to the Axiom of Choice is that if you have a single > non-empty set, then it is possible to construct a new set containing > only one of that set's elements. In other words, you can pick a > singleton out of the original set. This does *not* require the axiom of choice. That is, the following statement is true in ZF: For all x and y, if y is in x, then there exists z such that for all w, w is in z if and only if w = y. -SJH === Subject: Re: Eigenfunctions of the Laplacian >Consider the Laplacian operator Delta on some bounded two-dimensional >domain G. There is a L^2 orthonormal set of eigenfunctions of Delta >on G. >If G is rectangular and if a function f on G is smooth up to the >boundary then the (generalised) Fourier series of f (in terms of the >eigenfunctions of Delta) is uniformly convergent on G. >Under which conditions does that yield for more general domains G? >Tobias Nahring You don't mention the boundary conditions -- that's a necessary part of formulating the eigenvalue problem. Let's take Dirichlet conditions (so the eigenfunctions are products of certain sine functions when G is a rectangle, which = 0 on the boundary). So the eigenfunctions all are zero on the boundary, and there is no way a series of them can converge uniformly to f unless f too is zero on the boundary. /dan === Subject: Weird Trigonometric Identity After reading another post (unfortunately, I can't remember which) I was playing around with the function f(n) = 2arctan( cot(2n)) + 4n and it seems that for n integer, f(n) is always an odd multiple of pi. Does anyone know if this is true, and if so why? Andrew === Subject: Re: Weird Trigonometric Identity > After reading another post (unfortunately, I can't remember which) I > was playing around with the function > f(n) = 2arctan( cot(2n)) + 4n > and it seems that for n integer, f(n) is always an odd multiple of > pi. > Does anyone know if this is true, and if so why? A simpler related identity is arctan(cot(t)) = pi/2 - t, for -pi/2 < t <= pi/2 The odd multiples come from the branches of arctan (just add 2pi), sort of an artifact of forcing a computation. Why is the simpler identity true? the proof by picture (using words!) is that cot and tan are reciprocals, so are reflections of each other over the line x==y. Which is exactly what pi/2 - t does. I'm sure you could prove it formally by converting to exponentials and using tedious algebra, but it's way too early in the morning for me to bother. -- Mitch Harris (remove q to reply) === Subject: Re: Weird Trigonometric Identity schrieb Andrew Duncan : > After reading another post (unfortunately, I can't remember which) I > was playing around with the function > f(n) = 2arctan( cot(2n)) + 4n > and it seems that for n integer, f(n) is always an odd multiple of > pi. > Does anyone know if this is true, and if so why? Try to express arctan in terms on arccot; then use 2arccot( cot(2n)) = 4n. === Subject: Fokker-Plank and Diffusive equations Let us compare Fokker-Plank and Diffusive equations in case of no drift (for simplicity). Fokker-Plank dn/dt = 1/2*ddQn/dxx Diffusive dn/dt = 1/2*dQdn/dxx n- density of distribution Q- diffusive coefficient Question: Why position of Q is different? ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Time, the need for the lack of it. > There are other forces,true, but all are actions that require other > forces or reactions, to me this leaves the only force with enourh > ability to jump start the BIG BANG, gravity. For one small moment, > gravity was the only game in town. Who or what caused gravity can only > be Prayed to. As the first force,it, gravity, needs no other force to > exist. If it, gravity, does not require time, speed, energy,or mass > ,to be,then if you remove, these other forces from considerion in > determing the outcome of any problem, in communication or movement by > or of ones self across vast distances(more than 2 feet) the solution > will be correct. The conventional/current view of gravity is that it is due to bends in space-time. Einstein discovered that when people are travelling at different velocities time and space get mixed up. As an example, suppose Captain Kirk put clocks all along the outside of Enterprise from front to back, set them all to the same time, then measured the distance between them. The Klingons coming towards him at half light speed would be laughing because they would see that the clocks were all progressively reading different times from front to back of the ship. They would laugh even more as they watched Kirk measuring distances, he would be putting his ruler down at a slant so that the two ends would never touch the ship when the clocks read the same instant. The Klingons would communicate with Kirk telling him that he should measure lengths with both ends of the ruler applied at the same time. If Science Officer Spock took the call he would tell the Klingons that neither space nor time exist, they are inextricably bound together as 'space-time'. One man's space is another Klingon's time. If you put synchronised clocks along a flag pole you would find that they all read very, very, very slightly different times. It turns out that mass changes the distribution of space and time in space-time. If you fell to earth from a few thousand miles up your acceleration would actually be adjusting your velocity to take account of the changing rate of clocks as you got nearer to earth. The changes in clock rate with velocity and height are very tiny but have huge effects on energy and hence the forces occurring in nature because mass changes slightly with velocity and e=mc^2 where c is an enormous number. Here is a conundrum. How could you see a movement or hear a word if you only exist for the present instant, if you are at the durationless boundary between the past and future? Best Wishes Alex Green === Subject: Re: Time, the need for the lack of it. > It has become increasingly ovious that the sceince comunity has > gotten it totaly wrong. Quite aside from your apparent delusions of grandeur, the fact that you are unwilling to even check your spelling/grammar and have not even spelt 'science' correctly resulted in me giving up reading this far through your post. If you're not going to take what you write seriously, why do you expect us to? alex === Subject: Re: Time, the need for the lack of it. > It has become increasingly ovious that the sceince comunity has > gotten it totaly wrong. > Quite aside from your apparent delusions of grandeur, the fact that you are > unwilling to even check your spelling/grammar and have not even spelt 'science' > correctly resulted in me giving up reading this far through your post. > If you're not going to take what you write seriously, why do you expect us to? > alex `Had I known I was to be dealing with perfection, not only would I have invested my life savings in a spell checker, but I would even put on clean drawers, so you could kiss my ass without care or bother of skid marks. Had I visions of granderur, I would mirical you ass to a far warmer climate. Some people, like myself, do not have the ability or learing to afford but a good effort at proper spelling, sorry about that. === Subject: theorems/problems with lots of quantifiers Most mathematical theorems and conjectures seem to have low quantifier depth, that is, the nesting of quantifiers is not very deep. For example, if you formalize the fundamental thm of arithmetic, you get something like for all integers n, there exists a unique factorization into primes. However, I would like to have some examples of not too obscure, fairly accessible problems (conjectures or thms) where it -is- nested deeply. For example, the pumping lemma for regular languages (notorious to students of automata/computability) has (at least) 4 quantifiers. I suppose I care more about natural problems rather than er...unmotivated ones. (I suppose a related notion is algorithmic problems with running time a higher degree polynomial: most of the algorithms we care about are degree 3 or less. what are some high degree ones? like the latest n^12 primality testing, or some of those n^6 group theory algorithms) -- Mitch Harris (remove q to reply) === Subject: Re: theorems/problems with lots of quantifiers > Most mathematical theorems and conjectures seem to have low quantifier > depth, that is, the nesting of quantifiers is not very deep. > For example, if you formalize the fundamental thm of arithmetic, you > get something like for all integers n, there exists a unique > factorization into primes. > However, I would like to have some examples of not too obscure, fairly > accessible problems (conjectures or thms) where it -is- nested deeply. > For example, the pumping lemma for regular languages (notorious to > students of automata/computability) has (at least) 4 quantifiers. > I suppose I care more about natural problems rather than > er...unmotivated ones. > (I suppose a related notion is algorithmic problems with running time > a higher degree polynomial: most of the algorithms we care about are > degree 3 or less. what are some high degree ones? like the latest n^12 > primality testing, or some of those n^6 group theory algorithms) If x and the a_i's are integers it is true that: (Ax) ((Ea_1)((Ea_2)((Ea_3)((Ea_4)((Ea_5)((Ea_6)((Ea_7)((Ea_8)((Ea_9) ((Ea_10)((Ea_11)((Ea_12)((Ea_13)((Ea_14)((Ea_15)((Ea_16)((Ea_ 17) ((Ea_18)((Ea_19)((Ea_20)((Ea_21)((Ea_22)((Ea_23)((Ea_24)((Ea_ 25) ((Ea_26)((Ea_27)((Ea_28)((Ea_29)((Ea_30)((Ea_31)((Ea_32)((Ea_ 33) ((Ea_34)((Ea_35)((Ea_36)((Ea_37) (x=a_1^5+a_2^5+a_3^5+a_4^5+a_5^5+a_6^5+a_7^5+a_8^5+a_9^5+ a_10^5+a_11^5+a_12^5+a_13^5+a_14^5+a_15^5+a_16^5+a_17^5+ a_18^5+a_19^5+a_20^5+a_21^5+a_22^5+a_23^5+a_24^5+a_25^5+ a_26^5+a_27^5+a_28^5+a_29^5+a_30^5+a_31^5+a_32^5+a_33^5+ a_34^5+a_35^5+a_36^5+a_37^5) ))))))))))))))))))))))))))))))))))))) -- Clive Tooth http://www.clivetooth.dk === Subject: Re: theorems/problems with lots of quantifiers >>However, I would like to have some examples of not too obscure, fairly >>accessible problems (conjectures or thms) where it -is- nested deeply. > If x and the a_i's are integers it is true that: > (Ax) > ((Ea_1)((Ea_2)((Ea_3)((Ea_4)((Ea_5)((Ea_6)((Ea_7)((Ea_8)((Ea_9) > ((Ea_10)((Ea_11)((Ea_12)((Ea_13)((Ea_14)((Ea_15)((Ea_16)((Ea_ 17) > ((Ea_18)((Ea_19)((Ea_20)((Ea_21)((Ea_22)((Ea_23)((Ea_24)((Ea_ 25) > ((Ea_26)((Ea_27)((Ea_28)((Ea_29)((Ea_30)((Ea_31)((Ea_32)((Ea_ 33) > ((Ea_34)((Ea_35)((Ea_36)((Ea_37) > (x=a_1^5+a_2^5+a_3^5+a_4^5+a_5^5+a_6^5+a_7^5+a_8^5+a_9^5+ > a_10^5+a_11^5+a_12^5+a_13^5+a_14^5+a_15^5+a_16^5+a_17^5+ > a_18^5+a_19^5+a_20^5+a_21^5+a_22^5+a_23^5+a_24^5+a_25^5+ > a_26^5+a_27^5+a_28^5+a_29^5+a_30^5+a_31^5+a_32^5+a_33^5+ > a_34^5+a_35^5+a_36^5+a_37^5) > ))))))))))))))))))))))))))))))))))))) Hm. OK, Waring's problem. so you can get arbitrarily large #'s of quantifiers. OK, how about the extra rule (which I conveniently forgot to mention) that the quantifiers need to -alternate- (because in a logical sense, a string of there exists really just collapses to one there exists with a possibly nasty encoding of the variables. -- Mitch Harris (remove q to reply) === Subject: Re: theorems/problems with lots of quantifiers |OK, how about the extra rule (which I conveniently forgot to mention) |that the quantifiers need to -alternate- (because in a logical sense, |a string of there exists really just collapses to one there exists |with a possibly nasty encoding of the variables. you can ask things like can the first player force a win in go-moku within 30 moves?. probably not of really great mathematical interest, but some such questions can be rather challenging. -- [e-mail address jdolan@math.ucr.edu] === Subject: Localization without using Wavelet One of the merits using Wavelet is a localization. But in return, the correspondence between frequency and basis becomes a bit complicated. Are there any methods which bring localization with keeping the simple relation between frequency and basis like Fourier analysis? I tried to construct such basis in the following site: http://139.134.5.123/tiddler2/fbt/fbt.htm === Subject: Re: The multiplicative groups Z*_n; One more time... === > Subject: Re: The multiplicative groups Z*_n; One more time... > Use > http://mathquest.com/discuss/sci.math > http://mathforum.org/epigone/sci.math >> 2 <= m, g generates (Z_p^m)^*, g^phi(p^m) /= 1 (mod p^(m+1)) >You assume this is true, right? If so, why do you need m => 2? >> ==> g generates (Z_p^(m+1))^*, g^phi(p^(m+1)) /= 1 (mod p^(m+2)) >> let b = g^phi(p^m); if b^p = g^phi(p^(m+1)) = 1 (mod p^(m+2)): >> (b - 1)(b^(p-1) +..+ 1) = b^p - 1 = 0 (mod p^(m+2)) >> p^m | b-1; not p^(m+1) | b-1; p^2 | b^(p-1) +..+ 1 >> b = 1 (mod p^m); b = 1 (mod p^2); b^j = 1 (mod p^2) > 2 <= m is needed for this step^^^^^^^ >> 0 = b^(p-1) +..+ 1 = p (mod p^2); p^2 | p, whoops >So g^phi(p^(m+1)) /= 1 (mod p^(m+2)), right? >This is what you are showing here--do I have that right? > Yes, the indented statements after if....: make contradiction. >> g generates Z_p^* ==> g or g + p generates (Z_p^2)^* >> if (g + p)^(p-1) = 1 = g^(p-1) (mod p^2) >(g + p)^(p-1) = [g^(p-1) - pg^(p-2)] > Huh? >> 1 = g^(p-1) + (p-1) g^(p-2) p + ... = 1 - p.g^(p-2) (mod p^2) >g^(p-1) /= 1 mod p^2 ; g^(p-1) = 1 mod p == >g^(p-1) = 1 + ap ==> (g + p)^(p-1) = 1 + p[a - g^(p-2)]. > Where does a come from? It is just an adjustable constant I introduced from g^(p-1) = 1 mod p ==> p | g^(p-1) - 1 ==> g^(p-1) = 1 + ap ==> (g + p)^(p-1) = 1 + p[a - g^(p-2)]. The following is much like your case of p = 3, g = 2 below. ----------- Consider p = 5, g = 2. 2^4 = 16 = 1 + 3.5 ==> a = 3 In Z*_p^2 = Z*_25, g^10 = -1, so g generates Z*_25. g + p = 7, (g+p)^4 = 7^4 = 1 mod 25, so order(g+p) = p-1, as one can see from (g + p)^(p-1) = 1 + p[a - g^(p-2)] with a = 3, 2^3 = 8, p[a - g^(p-2)] = 5[3 - 8] mod 25 = 0. I have lost track of what I was doing. ------------- I think you are correct in saying that either g or g + p generates Z*_p^2. When I did this part, I said that one can find some a so that o(g + ap) = p - 1, and o(p+1) = p, so x = (g + ap)(p+1) generates Z*p^2. >I think its best to use g + kp and then choose k so that >either o(g) = p-1 or o(g) = p(p-1) in Z*_p^2, i.e we can always >find a k so that g generates Z*_p^2. > That's what I said, k = 0 or k = 1 in paticular. >> p.g^(p-2) = 0 (mod p^2) >> p = p.g^(p-1) = 0 (mod p^2); p^2 | p, whoops >> g generates (Z_p^2)^* ==> g + p^2 generates (Z_p^2)^*, Do you mean g + p rather than g + p^2 ? >I don't understand. In Z*_p^2, g + p^2 = g. > 2 generates Z_3^*, so does 5 and 8. 5 = 2, 5^2 = 1 (mod 3) > In Z_3^2, 2^2 = 4, 5^2 = 7, 8^2 = 1 (mod 9) > Thus, 2 and 5 but not 8 generate (Z_3^2)^* as from above: Right. You must have meant g + p. OK. > g generates Z_p^*, g^(p-1) /= 1 (mod p^2) ==> g generates (Z_p^2)^* > g generates Z_p^* ==> g or g + p generates (Z_p^2)^* > g + p generates Z_p^*, thus g + p or g + 2p generates (Z_p^2)^* If you really mean g + p^2 below, I don't understand. >> (g + p^2)^phi(p^2) = 1 (mod p^3). > and (g + p^2)^phi(p^2) /= 1 (mod p^3). >> otherwise: let b = (g + p^2)^p >> 1 = b^(p-1) + (p-1) b^(p-2) p^2 + ... = 1 - b^(p-2) p^2 (mod p^3) >> (only time p > 2 is used); b^(p-2) p^2 = 0 (mod p^3) >> p^2 = b^(p-1) p^2 = 0 (mod p^3); p^3 | p^2 whoops >I will go over this last again. > Don't bother. Instead find for all odd prime p, > some g generates (Z_p^2)^* with g^(p-1) /= 1 (mod p^3). OK. Do you mean try g, g + p^2, and find one such that g^(p-1) /= 1 (mod p^3). Is this why you talk of g + p^2 ? >I am not clear what you are doing here. > Me neither. > ---- === Subject: Re: The multiplicative groups Z*_n; One more time... >> g generates Z_p^* ==> g or g + p generates (Z_p^2)^* >> if (g + p)^(p-1) = 1 = g^(p-1) (mod p^2) >(g + p)^(p-1) = [g^(p-1) - pg^(p-2)] > Huh? >> 1 = g^(p-1) + (p-1) g^(p-2) p + ... = 1 - p.g^(p-2) (mod p^2) >g^(p-1) /= 1 mod p^2 ; g^(p-1) = 1 mod p == >g^(p-1) = 1 + ap ==> (g + p)^(p-1) = 1 + p[a - g^(p-2)]. > Where does a come from? > It is just an adjustable constant I introduced from > g^(p-1) = 1 mod p ==> p | g^(p-1) - 1 ==> g^(p-1) = 1 + ap ==> Then say so: ==> some a with g^(p-1) = 1 + ap. > (g + p)^(p-1) = 1 + p[a - g^(p-2)]. -- > The following is much like your case of p = 3, g = 2 below. > Consider p = 5, g = 2. 2^4 = 16 = 1 + 3.5 ==> a = 3 ... > I have lost track of what I was doing. End of story, I'm not a mind reader. -- > Don't bother. Instead find for all odd prime p, > some g generates (Z_p^2)^* with g^(p-1) /= 1 (mod p^3). > OK. Do you mean try g, g + p^2, and find one such that No, I say don't bother with that approach. As you see, I've resolved all the issues of this second method of showing (Z_p^n)^* for odd prime p, except for finding generators as noted above. So put your mind to finding them, I haven't found a way. What can you come up with? === Subject: Re: Raatikainen's critique of Chaitin >>I guess that doesn't mean that it is true for *no* reason, but for >>no humanly comprehensible reason. > > BUt this kind of true-for-no-reason can't be what Chaitin means. I > think he has in mind a particular statement which he apprehends as > true for no reason. This means that in particular, he knows it's true > but that there is no reason for it being true. > You mean Chaitin is an intuitionist (or at least predicativist)? > Otherwise, all he needs to do is to derive a contradiction from assuming > that every true statement is true for a reason. His book Omega does not seem too far from an intuitionist position. He says in the book that theorem proving/compression doesn't work! But he also embraces good old empiricism, e.g. the only position he clearly states is that he supports a quasi-empirical view of mathematics. But how to reach new axioms? Through intuition? What is intuition? :) That is a burning question in my mind. I have posted a link to my review of his book on ai-philosophy at yahoogroups.com which may be relevant to these issues. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin <87k6vge4sj.fsf@phiwumbda.org> <87pt57hbsf.fsf@phiwumbda.org> Discussion, linux) >>>I guess that doesn't mean that it is true for *no* reason, but for >>>no humanly comprehensible reason. >> BUt this kind of true-for-no-reason can't be what Chaitin means. I >> think he has in mind a particular statement which he apprehends as >> true for no reason. This means that in particular, he knows it's true >> but that there is no reason for it being true. >> You mean Chaitin is an intuitionist (or at least predicativist)? >> Otherwise, all he needs to do is to derive a contradiction from assuming >> that every true statement is true for a reason. > His book Omega does not seem too far from an intuitionist position. He > says in the book that theorem proving/compression doesn't work! What the heck does that have to do with intuitionism? Do intuitionists say that theorem proving doesn't work? > But he also embraces good old empiricism, e.g. the only position he > clearly states is that he supports a quasi-empirical view of > mathematics. But how to reach new axioms? Through intuition? What is > intuition? :) That is a burning question in my mind. Can you give a single example of an application of empiricism of the sort he advocates? Of course, Chaitin quotes that great philosopher of mathematics, Albert Einstein, in support of his claim that mathematics is empirical (Two Philosophical Applications of Algorithmic Information Theory). In the evolution of philosophic thought through the centuries the following question has played a major role: What knowledge is pure thought able to supply independently of sense perception? Is there any such knowledge?. . . I am convinced that. . . the concepts which arise in our thought and in our linguistic expressions are all. . . the free creations of thought which can not inductively be gained from sense-experiences. . . Thus, for example, the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences. Whatever Chaitin means by empirical, it's rather different that the term used by philosophers. For most philosophers, if math is a creation of the human mind, then it is not an example of empiricism. But, what the hell, he's redefined every other term he's come across. Let him have at empiricism. -- They are anti-mathematicians, evil incarnate, dedicated to undermining intellectual development in this area. If you never thought such people could actually exist, outside of myths or legends, welcome to the real world. --James S Harris on evil incarnate's Usenet presence === Subject: Re: Raatikainen's critique of Chaitin > The great insight that Chaitin brings is that > there is randomness in arithmetic, that there are certain facts in > mathematics (as an example, the bits of Omega) that are analagous to a > coin toss, that some theorems are true for no reason, that they are > true only because they are true. > How do you define or explain the distinction between mathematical > statements that are true for a reason and mathematical statements that > are true for no reason? > This in fact is a great insight into > the nature of mathematics, as it destroys the Greek idea of > axiom->proof->theorem. > What does it mean to say that Chaitin's insight destroys the Greek > idea of axiom->proof->theorem? It doesn't destroy that, since that's not even a Greek idea, that's an Aristotle idea, mathemo-morons. Since the *Greeks* didn't even have the idea of *proof*. It was the *Latins*, who put the [proof] box in between the non-existent Zeno arrow category referents: axiom-> and ->theorem. === Subject: Re: Raatikainen's critique of Chaitin Originator: mtx014@linux.services.coventry.ac.uk (Robert Low) > It doesn't destroy that, since that's not even a > Greek idea, that's an Aristotle idea, mathemo-morons. You just have to love Usenet. -- Rob. http://www.mis.coventry.ac.uk/~mtx014/ === Subject: Re: Raatikainen's critique of Chaitin > It doesn't destroy that, since that's not even a > Greek idea, that's an Aristotle idea, mathemo-morons. > You just have to love Usenet. That's probably supposed to mean that Aristotle *didn't* *invent* *Logic*. A Capitol is used, since lower-case l is only used in Greece. as in lambda calculi. You have to love mathema-morons. Since they probability do not even know who Pythagorus IS. But, they are probably even MORE infinetly clueless, that Pythagorus' proof that sqrt(2) is irrational is the REASON, The Prime Motive Theorem of Integers, The Modus Operandi of FUNCTION Theory AND Relation Theory, that Set Theory was also invented. === Subject: Re: Raatikainen's critique of Chaitin X-RFC2646: Original >> It doesn't destroy that, since that's not even a >> Greek idea, that's an Aristotle idea, mathemo-morons. >> You just have to love Usenet. > That's probably supposed to mean that Aristotle > *didn't* *invent* *Logic*. A Capitol is used, > since lower-case l is only used in Greece. > as in lambda calculi. > You have to love mathema-morons. > Since they probability do not even know who Pythagorus IS. > But, they are probably even MORE infinetly > clueless, that Pythagorus' proof that > sqrt(2) is irrational is the > REASON, The Prime Motive Theorem of Integers, > The Modus Operandi of FUNCTION Theory AND Relation Theory, > that Set Theory was also invented. When insulting people, it's generally wise not to include a mistake in your post, let alone many mistakes. Michael === Subject: Re: Raatikainen's critique of Chaitin >> It doesn't destroy that, since that's not even a >> Greek idea, that's an Aristotle idea, mathemo-morons. >> You just have to love Usenet. > That's probably supposed to mean that Aristotle > *didn't* *invent* *Logic*. A Capitol is used, > since lower-case l is only used in Greece. > as in lambda calculi. > You have to love mathema-morons. > Since they probability do not even know who Pythagorus IS. > But, they are probably even MORE infinetly > clueless, that Pythagorus' proof that > sqrt(2) is irrational is the > REASON, The Prime Motive Theorem of Integers, > The Modus Operandi of FUNCTION Theory AND Relation Theory, > that Set Theory was also invented. > When insulting people, it's generally wise not to include a mistake in your > post, let alone many mistakes. There is no mistake, since Pythagorus didn't really have a proof. He had what Goedel later came to call, Recursive Nymber Theory. And what Gauss called the Queen of Mathematic when he was doing Inf figures, but not when he was doing real math. Since he once called Groups, Mathematics answer to all things simple. And Fields and their medals as merely Mathematics explanation of why two is a number, but three fell the loophole and is not a number. And Functions as merely Mathematics Heluim laughing-gas answer of to why Newton invented Calculus, but Leibnitz didn't. > Michael === Subject: Re: Raatikainen's critique of Chaitin >> Chaitin's argument is that the entropy of a proposition >> derived from the axioms must be <= the entropy of the axioms, >> since one can construct a Turing machine based on the axioms >> which will output the proposition. > Chaitin doesn't say this, and it doesn't make much sense. Suppose > the empty string is the only axiom. How do you conclude that a theorem > must have complexity smaller than or equal to the complexity of the > empty string? I said I would check on the accuracy of my formulation above of Chaitin's version of Godel's theorem. I have done so, and I find that my account was completely accurate. Let me quote the entire abstract of Chaitin's paper Godel's theorem and information (International Journal of Theoretical Physics 22 [1982], pp 941-954). Godel's theorem may be demonstrated using arguments having an information-theoretic flavour. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Godel is natural and widespread rather than pathological and unusual. Given that Chaitin uses the terms informational content and algorithmic entropy interchangeably, I think that my version is an exact statement of Chaitin's result. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin > I think that my version is an exact statement of Chaitin's result. Well, it's not a statement of any result, but you're right that Chaitin's formulation is just as silly as yours. === Subject: Re: Raatikainen's critique of Chaitin >> I think that my version is an exact statement of Chaitin's result. > Well, it's not a statement of any result, but you're right that > Chaitin's formulation is just as silly as yours. (1) You said Chaitin made no such statement. Perhaps an apology is in order? (2) Let me restate one sentence from the abstract I gave in full: If a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. I don't see how you can claim that is not a statement of any result. (3) What do you mean by saying it is silly? Do you mean you think it is wrong? Or do you not understand what it means? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin > (1) You said Chaitin made no such statement. > Perhaps an apology is in order? Yes, my profond apologies. > I don't see how you can claim that is not a statement of any result. There isn't any such result, as you will see if you try to look for it. === Subject: Re: Raatikainen's critique of Chaitin >> I don't see how you can claim that is not a statement of any result. > There isn't any such result, as you will see if you try to look for > it. I'm not sure what you mean. I've seen a purported proof of the result at several points in Chaitin's work. Are you saying you have never seen such an argument, or are you saying that it is wrong? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin Timothy Murphy says... >I've seen a purported proof of the result >at several points in Chaitin's work. Here's the point that is in dispute: I think you are interpreting Chaitin as proving (where T is a theory, and Phi is a statement in the language of T, and where T |- Phi means T proves Phi), (T |- Phi) -> (H(T) >= H(Phi)) (T can only prove statements with lower entropy than T). That interpretation is not true. Any theory with an infinite number of theorems has theorems with arbitrarily large entropy. That's easy to see by considering statements of the form n = n with n some huge numeral with billions of digits. The entropy of such a statement can be very high, but all such statements are provable from a theory with very low entropy. The correct statement of Chaitin's result is this: For any theory T, there is a constant real r such that for any Phi, T does not prove H(Phi) > r. It doesn't matter whether Phi is a theorem of T or not, T cannot prove that any particular statement has a very high entropy. The connection between entropy of a theory and strength of a theory (what it can prove) is very loose. -- Daryl McCullough Ithaca, NY === Subject: Re: Raatikainen's critique of Chaitin > The correct statement of Chaitin's result is this: > For any theory T, there is a constant real r such that > for any Phi, T does not prove H(Phi) > r. Oh my God! What real? H : {0,1}* -> Z^+ Hello? What do you mean by Chaitin's result? That is only the very first, and the easiest theorem in the incompleteness chapter of AIT! I suppose you did not read his work! Let's not talk about second hand interpretations of his work (like done by the master of logic Raatikainen who didn't read his work extensively, either)! > It doesn't matter whether Phi is a theorem of T or not, T > cannot prove that any particular statement has a very high > entropy. > The connection between entropy of a theory and strength of > a theory (what it can prove) is very loose. Please read Theorem C in incompleteness chapter of AIT! Here, page 210: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/cup.html The theorem you are referring to above is Theorem LB, page 198. That is the weakest incompleteness result that Chaitin gets! You don't even know what Chaitin is talking about, and hence all this nonsense!!!! What a shame! When you are criticizing his philosophy of mathematics, you are supposed to actually know the content of his theorems and proofs. Otherwise, it's all small talk. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin Eray Ozkural exa says... >H : {0,1}* -> Z^+ In general, entropy is a real number. The fact that Chaitin defines them so that they are always integers is not very important. >> The connection between entropy of a theory and strength of >> a theory (what it can prove) is very loose. >Please read Theorem C in incompleteness chapter of AIT! That theorem does *not* establish anything more than a very loose connection between the entropy of a theory and the strength of a theory. What it shows is that no theory with entropy less than n can determine more than (approximately) n bits of Omega. It doesn't imply that the theorems of a theory cannot have higher algorithmic complexity than the axioms. It doesn't imply that if theory A has higher complexity than theory B, then A is a stronger theory than theory B. So the connection between entropy and strength is very loose. -- Daryl McCullough Ithaca, NY === Subject: Re: Raatikainen's critique of Chaitin > Eray Ozkural exa says... >H : {0,1}* -> Z^+ > In general, entropy is a real number. The fact that Chaitin > defines them so that they are always integers is not very > important. We have been talking about Kolmogorov/Chaitin complexity all along, that's why. >> The connection between entropy of a theory and strength of >> a theory (what it can prove) is very loose. >Please read Theorem C in incompleteness chapter of AIT! > That theorem does *not* establish anything more than a > very loose connection between the entropy of a theory and > the strength of a theory. > What it shows is that no theory with entropy less than n > can determine more than (approximately) n bits of Omega. > It doesn't imply that the theorems of a theory cannot > have higher algorithmic complexity than the axioms. Yes, rules of inference have their own algorithmic complexity, but that is handled in the theory. If that is the only reason, then it's not a very good reason for the supposed weakness of the theorem. So, indeed, the theorems of a theory can have higher algorithmic complexity than that of the axioms, but by only an additive factor if we fix the inference rules. (But we can choose any inference rule as we like, as long as they are truth preserving, in the mathematical sense!) > It > doesn't imply that if theory A has higher complexity than > theory B, then A is a stronger theory than theory B. So > the connection between entropy and strength is very loose. It could be made to imply exactly that given a few more extra definitions I suppose, but this is not the place for that. As I said, H(A) > H(B), and still B could prove many bits of Omega, while A can prove exactly 0 bits, because A is written by a really really bad programmer, or because the input was *simply random*, with absolutely *no mathematical meaning*. [*] I completely agree with that. (And there are even mathematical proofs of that which are not very hard to accomplish I suppose because we just did it.) -- Eray Ozkural [*] Can be formalized as follows, there is a random axiom string (finite or infinite) that gives 0 bits of information about the halting problem. === Subject: Re: Raatikainen's critique of Chaitin > So, indeed, the theorems of a theory can have higher algorithmic > complexity than that of the axioms, but by only an additive factor if > we fix the inference rules. The complexity of n=n goes to infinity with n. === Subject: Re: Raatikainen's critique of Chaitin > So, indeed, the theorems of a theory can have higher algorithmic > complexity than that of the axioms, but by only an additive factor if > we fix the inference rules. > The complexity of n=n goes to infinity with n. It can, yes. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > It can, yes. There is no can involved, just the simple observation that it does. === Subject: Re: Raatikainen's critique of Chaitin Eray Ozkural says... >So, indeed, the theorems of a theory can have higher algorithmic >complexity than that of the axioms, but by only an additive factor if >we fix the inference rules. What do you mean by an additive factor? As has been pointed out, the axiom forall x, x=x has theorems such as 123049810123980928340192305103290928340192834091283409182304981 2 = 123049810123980928340192305103290928340192834091283409182304981 2 which have a *much* higher entropy. >> It doesn't imply that if theory A has higher complexity than >> theory B, then A is a stronger theory than theory B. So >> the connection between entropy and strength is very loose. >It could be made to imply exactly that given a few more extra >definitions I suppose, but this is not the place for that. No, I don't think it can. >As I said, H(A) > H(B), and still B could prove many bits of Omega, >while A can prove exactly 0 bits, because A is written by a really >really bad programmer, or because the input was *simply random*, with >absolutely *no mathematical meaning*. No, the example is PA vs ZFC. H(PA) is not much different from H(ZFC), but ZFC is *much* more powerful. As I said, the connection between strength of a theory and its computational complexity seems very loose. -- Daryl McCullough Ithaca, NY === Subject: Re: Raatikainen's critique of Chaitin Hi Daryl, cleared up things very well in your post. > Eray Ozkural says... >So, indeed, the theorems of a theory can have higher algorithmic >complexity than that of the axioms, but by only an additive factor if >we fix the inference rules. > What do you mean by an additive factor? As has been pointed > out, the axiom forall x, x=x has theorems such as > 123049810123980928340192305103290928340192834091283409182304981 2 > 123049810123980928340192305103290928340192834091283409182304981 2 > which have a *much* higher entropy. A particular inference rule can achieve this, I agree. In AIT, we can say that the axiom forall x, x=x is a program that generates all x=x in lexicographical order, to be later processed by any inference engine of our choice. (which can do anything it wants on the particular x=x) I think you are basically talking about an interesting property of Kolmogorov complexity, which is evident to beginners like us in Kolmogorov complexity. A very small nonhalting program with k program-size complexity can generate all strings in {0,1}*, and parts of its output can have higher program-size complexity than k. (ie. the output necessarily includes all Kolmogorov-random finite strings, whose complexity goes to infinity... the additive factor I am talking about might be infinity :( ) Maybe a better way to deal with nonhalting computations is generalized Kolmogorov complexity which uses computability in the limit: http://www.idsia.ch/~juergen/ijfcs2002/ I object to none of these. The problem that Chaitin does not explain to us in intuitive terms is that this is only one *specific* inference rule. For instance, if the inference rule which you have picked is classical logic, then the case about theorem proving power explained in Raatikainen's paper does occur. But it does not hold in general! That's why Raatikanen's argument is flawed I believe. (That is: I can invent an inference rule that simply discards forall x, x=x....) Chaitin's theory has a syntactical flavor, it completely discards semantic concerns! That might not be as drastic a shortcoming as it sounds I think, if you would be willing to discuss on a separate thread. On the other hand, I do think that a theory with more semantic sensibility might be desirable! I would welcome work on that. I also think time should be put back into the equation. We're doing all these timeless equations, but we are no Gods. >> It doesn't imply that if theory A has higher complexity than >> theory B, then A is a stronger theory than theory B. So >> the connection between entropy and strength is very loose. >It could be made to imply exactly that given a few more extra >definitions I suppose, but this is not the place for that. > No, I don't think it can. Okay. I cannot precisely see how it can be done, you may be right. And even if we did, it may be too patchy. >As I said, H(A) > H(B), and still B could prove many bits of Omega, >while A can prove exactly 0 bits, because A is written by a really >really bad programmer, or because the input was *simply random*, with >absolutely *no mathematical meaning*. > No, the example is PA vs ZFC. H(PA) is not much different from H(ZFC), > but ZFC is *much* more powerful. As I said, the connection between > strength of a theory and its computational complexity seems very loose. I don't think Chaitin's theory is a good theory to reason about PA and ZFC, elsewhere I've suggested that these are theories with too small complexity to be handled by the asymptotic analysis of Kolmogorov complexity. To be precise, their complexity might vanish in the bias of a practical programming language we might use for our decompressor. Elsewhere, I have asked for alternative measures of theorem proving power, which might be suitable for systems like PA or ZFC. Could you please give me some references? I would be most delighted. So, I'm not saying that Chaitin's results can help the mundane world of axiomatic set theory. I don't believe that is exactly the case. I think he just points to the very limits of the whole proof business, that's all. While we're trying to build aircraft, he's pointing to the limits of space travel. So, it would be like using a bazooka to shoot a fly, if we tried to apply AIT to PA, a simple axiomatic system. Indeed, I think Chaitin has been working on limits of mind, rather than limits of mathematics for the most part. I think working out the limits of mind is more interesting. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin Hi Daryl, cleared up things very well in your post. > Eray Ozkural says... >So, indeed, the theorems of a theory can have higher algorithmic >complexity than that of the axioms, but by only an additive factor if >we fix the inference rules. > What do you mean by an additive factor? As has been pointed > out, the axiom forall x, x=x has theorems such as > 123049810123980928340192305103290928340192834091283409182304981 2 > 123049810123980928340192305103290928340192834091283409182304981 2 > which have a *much* higher entropy. A particular inference rule can achieve this, I agree. In AIT, we can say that the axiom forall x, x=x is a program that generates all x=x in lexicographical order, to be later processed by any inference engine of our choice. (which can do anything it wants on the particular x=x) I think you are basically talking about an interesting property of Kolmogorov complexity, which is evident to beginners like us in Kolmogorov complexity. A very small nonhalting program with k program-size complexity can generate all strings in {0,1}*, and parts of its output can have higher program-size complexity than k. (ie. the output necessarily includes all Kolmogorov-random finite strings, whose complexity goes to infinity... the additive factor I am talking about might be infinity :( ) Maybe a better way to deal with nonhalting computations is generalized Kolmogorov complexity which uses computability in the limit: http://www.idsia.ch/~juergen/ijfcs2002/ I object to none of these. The problem that Chaitin does not explain to us in intuitive terms is that this is only one *specific* inference rule. For instance, if the inference rule which you have picked is classical logic, then the case about theorem proving power explained in Raatikainen's paper does occur. But it does not hold in general! That's why Raatikanen's argument is flawed I believe. (That is: I can invent an inference rule that simply discards forall x, x=x....) Chaitin's theory has a syntactical flavor, it completely discards semantic concerns! That might not be as drastic a shortcoming as it sounds I think, if you would be willing to discuss on a separate thread. On the other hand, I do think that a theory with more semantic sensibility might be desirable! I would welcome work on that. I also think time should be put back into the equation. We're doing all these timeless equations, but we are no Gods. >> It doesn't imply that if theory A has higher complexity than >> theory B, then A is a stronger theory than theory B. So >> the connection between entropy and strength is very loose. >It could be made to imply exactly that given a few more extra >definitions I suppose, but this is not the place for that. > No, I don't think it can. Okay. I cannot precisely see how it can be done, you may be right. And even if we did, it may be too patchy. >As I said, H(A) > H(B), and still B could prove many bits of Omega, >while A can prove exactly 0 bits, because A is written by a really >really bad programmer, or because the input was *simply random*, with >absolutely *no mathematical meaning*. > No, the example is PA vs ZFC. H(PA) is not much different from H(ZFC), > but ZFC is *much* more powerful. As I said, the connection between > strength of a theory and its computational complexity seems very loose. I don't think Chaitin's theory is a good theory to reason about PA and ZFC, elsewhere I've suggested that these are theories with too small complexity to be handled by the asymptotic analysis of Kolmogorov complexity. To be precise, their complexity might vanish in the bias of a practical programming language we might use for our decompressor. Elsewhere, I have asked for alternative measures of theorem proving power, which might be suitable for systems like PA or ZFC. Could you please give me some references? I would be most delighted. So, I'm not saying that Chaitin's results can help the mundane world of axiomatic set theory. I don't believe that is exactly the case. I think he just points to the very limits of the whole proof business, that's all. While we're trying to build aircraft, he's pointing to the limits of space travel. So, it would be like using a bazooka to shoot a fly, if we tried to apply AIT to PA, a simple axiomatic system. Indeed, I think Chaitin has been working on limits of mind, rather than limits of mathematics for the most part. I think working out the limits of mind is more interesting. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin Discussion, linux) > For instance, if the inference rule which you have picked is classical > logic, then the case about theorem proving power explained in > Raatikainen's paper does occur. But it does not hold in general! > That's why Raatikanen's argument is flawed I believe. (That is: I can > invent an inference rule that simply discards forall x, x=x....) So the new theorem is: There is a set of inference rules such that complexity of the axioms is a measure of the strength of the theory. Is that it? That hardly sounds so exciting as the other version, which applies to the logics that folks actually care about. -- Jesse Hughes She moaned, in pain and pleasure, as, in a confused whirlwind, she glimpsed an image of Saint Sebastian riddled with arrows, crucified and impaled. --Mario Vargas Llosa on category theory === Subject: Measuring the strength of a theory (Was Re: Raatikainen's critique of Chaitin) [First: I am not changing the target. We have been discussing the plausibility of using Kolmogorov complexity to measure theorem proving power a la Chaitin.] > For instance, if the inference rule which you have picked is classical > logic, then the case about theorem proving power explained in > Raatikainen's paper does occur. But it does not hold in general! > That's why Raatikanen's argument is flawed I believe. (That is: I can > invent an inference rule that simply discards forall x, x=x....) > So the new theorem is: There is a set of inference rules such that > complexity of the axioms is a measure of the strength of the theory. > Is that it? > That hardly sounds so exciting as the other version, which applies to > the logics that folks actually care about. Not exactly I think. But please do not blame me if my following ideas do not work for you, I will be pointing at new stuff. I am not saying that there is a set of inference rules such that complexity of the axioms is a measure of the strength of the theory, since we give no proof, either, that there is always such an FAS that obtains a mathematical truth from complex axioms. Consider my example in a previous post: Let's have this infinitesimal pin and pick a random number between 0 and 1. The probability that the random real we picked is Omega is almost 0! (But not 0!) That is, if we wanted to solve the halting problem, having a truly truly random, very real number would be of absolutely no use! Although the complexity of a random real is just as much as the number Omega, it normally contains no information about the halting problem. Unfortunately! It is obvious that a random string can have large complexity, but for relevance to theorem proving we would also like it to be meaningful. We can formalize this meaning as the following (like in computability in the limit). Consider H(a / r) ie. entropy of (the coding of) an answer a to a mathematical problem, with respect to a real number r. The meaningfulness of r for a is then mutual information among two strings H(x:y) = H(a) - H(a/r), how much information gained by looking at r. For instance a0 could be this: ith bit of a0 is 1 if program i halts (wrt a universal computer U), 1 if it does not. It is obvious that H( a0 / Omega) = O(1) since there is a nonhalting program which can solve the halting problem using omega. (the program's size is constant.) That is why, I am saying that Omega is maximally informative wrt halting problem since any conditional program could be shorter *only by a constant factor*. (and hence H(a0/Omega)=infinite, what bettter maximal than the infinite?!) Another example involving Omega. Let a1 be encoded as follows, it is a sequence of numbers such that ith number is the value of BB_LISP(k) [the maximum output size of any k-bit long LISP program]. Since H(a1/a0) = O(1), H(a1/Omega) = O(1) as well. Let's now come to Torkel's hypothetical number a3, a number encoding the truth of arithmetic sentences in first-order language of arithmetic as an exercise. ith bit of a3 is 1 if ith sentence is true, otherwise 0. What is H(a3/Omega)? Nevertheless, the probability that a random real r gives any information on the halting problem is pretty low because in an uncountably infinite number of cases H(a0 / r) = infinite, knowing r doesn't help us at all. We still need an infinite amount of information to solve it. :( How do we then incorporate truth into the theory? I suspect something like formalizing model theory in algorithmic terms will be necessary. Semantic strings as the above is a way, surely, why not, but these are our constructs, they are toys. The entropy in a semantic space must be calculated, but do not ask me how that is done. I do not know yet. -- Eray Ozkural PS: My gut feeling is that the results will be Wittgensteinean, we will find that empirical trials are painfully necessary to do mathematics! === Subject: Re: Measuring the strength of a theory <877jr8rb3e.fsf@phiwumbda.org> Discussion, linux) > Let's have this infinitesimal pin and pick a random number between 0 > and 1. The probability that the random real we picked is Omega is > almost 0! (But not 0!) Nonsense. Why is the probability of Omega greater than 0 but the probability of 0.5 is 0? What is the probability of 0.123456789012345678901234567890....? *Where* are your probabilistic claims coming from? -- Jesse F. Hughes History will hate you and love me. I'm the misunderstood and persecuted genius. You're the assholes. -- James Harris === Subject: Re: Measuring the strength of a theory > Let's have this infinitesimal pin and pick a random number between 0 > and 1. The probability that the random real we picked is Omega is > almost 0! (But not 0!) > Nonsense. Why is the probability of Omega greater than 0 but the > probability of 0.5 is 0? What is the probability of > 0.123456789012345678901234567890....? > *Where* are your probabilistic claims coming from? On second thought, perhaps my intuition failed me. The probability that the random real we picked is Omega is 0, because of the same argument for computable reals. Silly me. This would make my argument (if it can be called one, it's just a discussion) stronger, however, as you would notice, and it does not contradict other pieces of the argument. The probability that a random number you pick in [0,1] to be a particular real number (that you name/define!) is 0. How intuitive. :) Perhaps the better picture is the calculational one, instead of talking about exact probabilities like 0, let's just say they are 0 in the limit. That would suit my intuition better, but as I see it we are really implicitly talking about limits, anyway! (Isn't this infinitesimal calculus?) What confused me was the almost 0 phrase being used elsewhere... Sometimes you unconsciously believe whatever you heard. Sorry for not having been rigorous enough. -- Eray === Subject: Re: Measuring the strength of a theory <877jr8rb3e.fsf@phiwumbda.org> <87pt4ze5zi.fsf@phiwumbda.org> Discussion, linux) > What confused me was the almost 0 phrase being used elsewhere... > Sometimes you unconsciously believe whatever you heard. Sorry for not > having been rigorous enough. I don't recall anyone besides you using the phrase almost 0. -- Jesse F. Hughes [Lancelot] sighed, defeated. 'It is as practical to hurry an acorn toward treeness as to urge a damsel when her mind is set.' -- John Steinbeck, /The Acts of King Arthur and His Noble Knights/ === Subject: Re: Measuring the strength of a theory > Let's have this infinitesimal pin and pick a random number between 0 > and 1. The probability that the random real we picked is Omega is > almost 0! (But not 0!) > Nonsense. Why is the probability of Omega greater than 0 but the > probability of 0.5 is 0? What is the probability of > 0.123456789012345678901234567890....? If it's a computable real, 0. As explained in Turing's paper I suppose. There is also the measure theoretic argument that the set of computable of reals have a measure of 0, while the uncomputable ones (like Omega) has 1! (Of course, the same argument applies to any interval of real numbers) > *Where* are your probabilistic claims coming from? Not from the thin air, obviously! But why the exclamation mark in my above statement? Because, you are right, it makes no immediate sense to say that probability of Omega is >0, while a computable real is 0. That is, in my mind, an indication that a real number is a toy concept, ie. an idealization of some sort, but not necessarily a concept with which we can talk about the geometry of the real world precisely. Don't blame me for what's happening though, I didn't invent the real numbers. -- Eray Ozkural === Subject: Re: Measuring the strength of a theory >Let's have this infinitesimal pin and pick a random number between 0 >and 1. The probability that the random real we picked is Omega is >almost 0! (But not 0!) >>Nonsense. Why is the probability of Omega greater than 0 but the >>probability of 0.5 is 0? What is the probability of >>0.123456789012345678901234567890....? >If it's a computable real, 0. As explained in Turing's paper I >suppose. There is also the measure theoretic argument that the set of >computable of reals have a measure of 0, while the uncomputable ones >(like Omega) has 1! (Of course, the same argument applies to any >interval of real numbers) >>*Where* are your probabilistic claims coming from? >Not from the thin air, obviously! >But why the exclamation mark in my above statement? Because, you are >right, it makes no immediate sense to say that probability of Omega is >>0, while a computable real is 0. That is, in my mind, an indication >that a real number is a toy concept, ie. an idealization of some sort, >but not necessarily a concept with which we can talk about the >geometry of the real world precisely. Don't blame me for what's >happening though, I didn't invent the real numbers. >Eray Ozkural If you choose a real number in the interval (0,1) at random, the probability of choosing any particular number is 0 (assuming the uniform distribution). Omega is a real number in the interval (0,1). Therefore the probability choosing Omega at random is 0, the same as for any other real number. In fact, Chaitin states that it would have been better to call Omega irreducible rather than random --- I suppose in order to avoid arguments such as those above. Chaitin's original reason for calling Omega random was that the sequence of digits of Omega satisfies any reasonable statistical test of randomness. (Beacause any reasonable test can be implemented as a Turing machine, etc, ...) But this in no way implies that somehow the probability of choosing Omega at random is larger than 0. === Subject: Re: Measuring the strength of a theory <877jr8rb3e.fsf@phiwumbda.org> <87pt4ze5zi.fsf@phiwumbda.org> Discussion, linux) > In fact, Chaitin states that it would have been better to call Omega > irreducible rather than random --- I suppose in order to avoid arguments > such as those above. Does he? Do you have a reference? I think that would have been better, too. I wonder if the allusions to true for no reason are due to anything more than the misleading use of the word random. -- I have to break the code of how [mere humans] work, and I have made a lot of progress over years of effort, and I feel like I am close to figuring out all the inner details of human wiring. -- James S. Harris on the extra problems of conveying his research === Subject: Re: Measuring the strength of a theory >>In fact, Chaitin states that it would have been better to call Omega >>irreducible rather than random --- I suppose in order to avoid arguments >>such as those above. >Does he? Do you have a reference? >I think that would have been better, too. I wonder if the allusions >to true for no reason are due to anything more than the misleading >use of the word random. It's in his Omega book, Chapter 6: To say that Omega is random may be a little confusing. etc. http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html#ch6 === Subject: Re: Measuring the strength of a theory <877jr8rb3e.fsf@phiwumbda.org> <87pt4ze5zi.fsf@phiwumbda.org> Discussion, linux) >> Let's have this infinitesimal pin and pick a random number between 0 >> and 1. The probability that the random real we picked is Omega is >> almost 0! (But not 0!) >> Nonsense. Why is the probability of Omega greater than 0 but the >> probability of 0.5 is 0? What is the probability of >> 0.123456789012345678901234567890....? > If it's a computable real, 0. As explained in Turing's paper I > suppose. As explained in Turing's paper, you suppose? Most normal folk would say that for any real, the probability of selecting that real is 0, whether it is computable, random, rational or silly. > There is also the measure theoretic argument that the set of > computable of reals have a measure of 0, while the uncomputable ones > (like Omega) has 1! (Of course, the same argument applies to any > interval of real numbers) That doesn't mean that a particular uncomputable real has positive probability. >> *Where* are your probabilistic claims coming from? > Not from the thin air, obviously! Not obvious from where I sit. > But why the exclamation mark in my above statement? Because, you are > right, it makes no immediate sense to say that probability of Omega is >>0, while a computable real is 0. That is, in my mind, an indication > that a real number is a toy concept, ie. an idealization of some sort, > but not necessarily a concept with which we can talk about the > geometry of the real world precisely. Don't blame me for what's > happening though, I didn't invent the real numbers. There are some toy concepts floating around here, no doubt. But I think we disagree on where the blame for all this sloppy thinking lies. -- Do you know why I'm tall? Why? Because I eat apples. Do you know why I'm short? Why? Because I'm a kid. --Quincy P. Hughes (age almost 4) bests his father. === Subject: Re: Raatikainen's critique of Chaitin > I've seen a purported proof of the result > [that the complexity of a proposition derived from the axioms must > be <= the complexity of the axioms] at several points in Chaitin's work. No you haven't. Not even Chaitin would claim that the complexity of n=n must be <= the complexity of (x)(x=x) for every n. === Subject: Re: Raatikainen's critique of Chaitin Discussion, linux) > I said I would check on the accuracy of my formulation above > of Chaitin's version of Godel's theorem. > I have done so, and I find that my account was completely accurate. > Let me quote the entire abstract of Chaitin's paper > Godel's theorem and information > (International Journal of Theoretical Physics 22 [1982], pp 941-954). > Godel's theorem may be demonstrated using arguments > having an information-theoretic flavour. > In such an approach it is possible to argue that > if a theorem contains more information > than a given set of axioms > then it is impossible for the theorem to be derived from the axioms. > In contrast with the traditional proof based on the paradox of the liar, > this new viewpoint suggests that the incompleteness phenomenon > discovered by Godel is natural and widespread > rather than pathological and unusual. How does this work? What is the information content of a set of axioms? Does (A x)(x = x) have more or less information than z = z where z is a large initial segment of Omega[1]? > Given that Chaitin uses the terms informational content > and algorithmic entropy interchangeably, > I think that my version is an exact statement of Chaitin's result. Footnotes: [1] That is, where z is Sum_i=0^n 2^{k_i} and Omega = Sum_i=0^oo 2^{-k_i}. -- Sure, [my Usenet presence is] like Shaq playing against you in your backyard, but that has its perks, as I find ways to have my fun *and* I can send messages to certain people in the United States Government without concern that the rest of you understand them. -- James Harris === Subject: Re: Raatikainen's critique of Chaitin > What is the information content of a set of axioms? The informational content (or entropy) H(s) of a finite string s is the length of the shortest input to a chosen universal Turing machine (in Chaitin's model, which allows input and output) which outputs the string s. The entropy H(s_1,...,s_r) of a finite set of strings is defined similarly. (It is not quite the same as the entropy H(s_1...s_r) of the concatenation of the strings.) The entropy of a set of axioms is the entropy of the strings expressing those axioms. > Does (A x)(x = x) have more or less information than z = z where z is > a large initial segment of Omega[1]? I don't understand the question; but I prefer the term entropy because information comes with a lot of extraneous meaning (as Shannon suggested). I would have thought both statements would have very low entropy because they are so short. But Also it should be said that equations in algorithmic information theory are normally only true up to an additive constant dependent on the chosen universal machine. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin <87vfevs6d7.fsf@phiwumbda.org> Discussion, linux) >> What is the information content of a set of axioms? > The informational content (or entropy) H(s) of a finite string s > is the length of the shortest input to a chosen universal Turing machine > (in Chaitin's model, which allows input and output) > which outputs the string s. > The entropy H(s_1,...,s_r) of a finite set of strings is defined similarly. > (It is not quite the same as the entropy H(s_1...s_r) > of the concatenation of the strings.) > The entropy of a set of axioms > is the entropy of the strings expressing those axioms. >> Does (A x)(x = x) have more or less information than z = z where z is >> a large initial segment of Omega[1]? > I don't understand the question; > but I prefer the term entropy > because information comes with a lot of extraneous meaning > (as Shannon suggested). > I would have thought both statements would have very low entropy > because they are so short. > But > Also it should be said that equations in algorithmic information theory > are normally only true up to an additive constant > dependent on the chosen universal machine. As Daryl said, (A x)(x = x) is short, but z = z isn't short when z is a long natural number. The entropy of z = z can be arbitrarily high, no? But it is a consequence of (A x)(x = x). -- ...you are around so that I have something else to do when I'm not figuring something important out. I was especially intrigued on this iteration by cursing, which I think I'll continue at some later date as it's so amusing. --- James S. Harris === Subject: Re: Raatikainen's critique of Chaitin Timothy Murphy says... >> Does (A x)(x = x) have more or less information than z = z where z is >> a large initial segment of Omega[1]? >I don't understand the question; >but I prefer the term entropy >because information comes with a lot of extraneous meaning >(as Shannon suggested). Well, so does entropy. It has a thermodynamic meaning which has nothing to do with program complexity. >I would have thought both statements would have very low entropy >because they are so short. The statement forall x, x = x is short, and thus has a low entropy. But the statement 185009567860096092068092345767239875759214791927546591839183749 82983798734 9 = 185009567860096092068092345767239875759214791927546591839183749 829837987349 is not so short. So it would seem that a short statement (low entropy) can have a long statement (high entropy) as a theorem. -- Daryl McCullough Ithaca, NY === Subject: Re: Raatikainen's critique of Chaitin >>but I prefer the term entropy >>because information comes with a lot of extraneous meaning >>(as Shannon suggested). > Well, so does entropy. It has a thermodynamic meaning which has > nothing to do with program complexity. I think it is generally accepted - but I am not a physicist - that there is a close connection between entropy in thermodynamics and statistical mechanics and information theory. Eg consider the discussion about information escaping from black holes. The idea that the entropy of a gas (or other material) goes back a long time, I think. > But the statement 185009567860096092068092345767239875759214791927546591839183749 82983798734 9 185009567860096092068092345767239875759214791927546591839183749 829837987349 > is not so short. So it would seem that a short statement (low entropy) > can have a long statement (high entropy) as a theorem. Sorry, I didn't read your posting carefully enough. I'll have to think about your example. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin > I'll ask a question to the people who criticize Chaitin's work. What > in general about Chaitin's work, which I happen to think is some of > the best mathematics of the past century, bothers you so much? (1) I think it's an absurd exaggeration to say Chaitin's work contains some of the best mathematics of the past century. There are some moderately difficult bits of mathematics in his theory - I mean the sort of thing that might lead a good graduate student to think for an hour or two - but it would be absurd to put it beside the classification of the finite simple groups, for example. (2) I do however think algorithmic information theory is an important and beautiful theory. (3) Chaitin is given to rather fanciful applications of this theory, eg Toward a mathematical definition of 'life'. I'd say the same about the present discussion of theorems that are true for no reason. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin > I'll ask a question to the people who criticize Chaitin's work. What > in general about Chaitin's work, which I happen to think is some of > the best mathematics of the past century, bothers you so much? > (1) I think it's an absurd exaggeration to say Chaitin's work contains > some of the best mathematics of the past century. > There are some moderately difficult bits of mathematics in his theory - > I mean the sort of thing that might lead a good graduate student > to think for an hour or two - > but it would be absurd to put it beside the classification > of the finite simple groups, for example. I am more impressed with his results than classification of finite simple groups. Classification of finite simple groups is nice to have, but it doesn't shake up the foundations of mathematics like his result does. In my opinion, mathematics in the last 100 years is relatively boring compared to other centuries. His result and other results like his (for instance, Godel's Theorem) at least made things interesting. > (2) I do however think algorithmic information theory is an important > and beautiful theory. What part of it do you think is important and beautiful and why? > (3) Chaitin is given to rather fanciful applications of this theory, > eg Toward a mathematical definition of 'life'. > I'd say the same about the present discussion > of theorems that are true for no reason. I don't understand what he means by mathematical definition of life if he said that. Theorems are true for no reason makes perfect sense to me. I don't understand why people have so much trouble getting this point, that his Omega number implies this fact. Craig === Subject: Re: Raatikainen's critique of Chaitin simple groups. I'm more impressed about classification. > Classification of finite simple groups is nice to have, > but it doesn't shake up the foundations of mathematics like his result > does. Chaitin has not shaken the foundations of mathematics --- his results have no more philosophical implications than were present in the work of Godel. It was the work of Godel, Turing and Church which shook the foundations of mathematics in the 1930s not Chaitin many decades later. > In my opinion, mathematics in the last 100 years is relatively > boring compared to other centuries. I suppose you are entitled to your opinion, but it is hard to see what basis it has. > His result and other results like > his (for instance, Godel's Theorem) at least made things interesting. Ha! Godel's theorem is another result like Chaitin's! And I had always considered that Chaitin had proved various results like Godel's. (Is like transitive :-) ) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Raatikainen's critique of Chaitin > I am more impressed with his results than classification of finite > simple groups. > I'm more impressed about classification. > Classification of finite simple groups is nice to have, > but it doesn't shake up the foundations of mathematics like his result > does. > Chaitin has not shaken the foundations of mathematics --- his results > have no more philosophical implications than were present in the > work of Godel. Wrong, as a matter of fact. Did Godel show that randomness had something to do with incompleteness? Did he show the existence of a maximally unknowable number? These are not the only things, btw.... Many of these statements are actually philosophically very significant, because they have something to do with epistemology! > It was the work of Godel, Turing and Church which > shook the foundations of mathematics in the 1930s not Chaitin many > decades later. This thread started with some good observations about the weakness of Raatikainen's critique and then degenerated into a mudball about how bad and naughty Chaitin is. If you are saying such a thing, give a proper argument. He does not claim to shake the foundations of mathematics, but he thinks he has shown that incompleteness should be taken more seriously. Why do you object to that? Believe it or not, Chaitin's work has important implications in philosophy of mathematics. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin >> I am more impressed with his results than classification of finite >> simple groups. >> I'm more impressed about classification. >> Classification of finite simple groups is nice to have, >> but it doesn't shake up the foundations of mathematics like his result >> does. >> Chaitin has not shaken the foundations of mathematics --- his results >> have no more philosophical implications than were present in the >> work of Godel. > Wrong, as a matter of fact. Did Godel show that randomness had > something to do with incompleteness? Did he show the existence of a > maximally unknowable number? These are not the only things, btw.... > Many of these statements are actually philosophically very > significant, because they have something to do with epistemology! Bollocks the lot of it. All these discoveries of Chaitin's are simply repackagings of the well-established notion of completeness. >> It was the work of Godel, Turing and Church which >> shook the foundations of mathematics in the 1930s not Chaitin many >> decades later. > This thread started with some good observations about the weakness of > Raatikainen's critique and then degenerated into a mudball about how > bad and naughty Chaitin is. If you are saying such a thing, give a > proper argument. He does not claim to shake the foundations of > mathematics, but he thinks he has shown that incompleteness should be > taken more seriously. Perhaps you would be better served by learning to read. I was replying to Feinstein not Chaitin. shake up the foundations of mathematics is Feinstein's phrase. > Why do you object to that? Have you stopped beating your wife? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Raatikainen's critique of Chaitin >> > I am more impressed with his results than classification of finite >> simple groups. >> I'm more impressed about classification. >> Classification of finite simple groups is nice to have, >> but it doesn't shake up the foundations of mathematics like his result >> does. >> Chaitin has not shaken the foundations of mathematics --- his results >> have no more philosophical implications than were present in the >> work of Godel. Not true. Godel's results are that for any finite axiom system that has enough power to produce theorems about natural numbers, there must exist a statement that is not provable in the axiom system, but nevertheless true if we were to assume that the axiom system is consistent. Chaitin's Theorem (about Omega, which is really his big result) shows that only a finite number of bits of Omega are possible to determine; the only way to determine more bits is to call them axioms. Since the bits can be represented as exponential diophantine equations (1 if there are infinitely many solutions, 0 if only finitely many solutions), we see that there are some statements in number theory that are unknowable. Much stronger than Godel's result, as Godel's theorem gives a way to get around the incompleteness, by assuming that the axiom system is consistent. Chaitin's theorem shows that there is no way to get around it, since assuming that the axiom system is consistent only improves the information complexity of the axiom system by a finite amount, so you can still only know a finite number of bits of Omega. Therefore, mathematics is as Chaitin calls it, quasi-empirical, just like physics, in which theorems can only be proven by statistics and observations; they are never certain, always subject to being possibly refuted. Mathematicians don't like to hear this, since most of them almost religiously believe that mathematics is absolute certainty. But in reality, the absolute certainty is only for the trivial theorems that they prove, as Chaitin showed. The majority of mathematical facts are simply unattainable through deductive reasoning; only inductive/empirical methods work (and not so well). > > Wrong, as a matter of fact. Did Godel show that randomness had > something to do with incompleteness? Did he show the existence of a > maximally unknowable number? These are not the only things, btw.... > Many of these statements are actually philosophically very > significant, because they have something to do with epistemology! > Bollocks the lot of it. All these discoveries of Chaitin's > are simply repackagings of the well-established notion of completeness. Nope, only parts of Chaitin's discoveries are repackagings of Godel's results, i.e., using the Berry paradox instead of the liar paradox. The main part about Omega is not. >> It was the work of Godel, Turing and Church which >> shook the foundations of mathematics in the 1930s not Chaitin many >> decades later. All of them did. The only difference is that because Chaitin's results are so much stronger and are so shocking and go against the grain of how most mathematicians think, most mathematicians don't want to believe them; therefore, you get reactions that fit into the 4 categories that I mentioned when I started this thread. I should add that most mathematicians don't even understand Chaitin's results, mainly because they don't want to understand them. > > This thread started with some good observations about the weakness of > Raatikainen's critique and then degenerated into a mudball about how > bad and naughty Chaitin is. If you are saying such a thing, give a > proper argument. He does not claim to shake the foundations of > mathematics, but he thinks he has shown that incompleteness should be > taken more seriously. > Perhaps you would be better served by learning to read. I was replying > to Feinstein not Chaitin. shake up the foundations of mathematics > is Feinstein's phrase. > Why do you object to that? > Have you stopped beating your wife? Chaitin did shake up the foundations of mathematics. The only problem is mathematicians don't like his result, so they lose out and remain ignorant. (This is my opinion.) Craig === Subject: Re: Raatikainen's critique of Chaitin >Therefore, mathematics is as Chaitin calls it, quasi-empirical, just >like physics, in which theorems can only be proven by statistics and >observations; they are never certain, always subject to being possibly >refuted. Mathematicians don't like to hear this, since most of them >almost religiously believe that mathematics is absolute certainty. But >in reality, the absolute certainty is only for the trivial theorems >that they prove, as Chaitin showed. The majority of mathematical facts >are simply unattainable through deductive reasoning; only >inductive/empirical methods work (and not so well). Please explain what you mean by this. Are you saying that (1) mathematics already is like physics, as practiced by a typical mathematician today, or that (2) mathematics should be like physics, in contrast to the way it is practiced today? If you mean (2), how do you propose that mathematicians should change, so that they start doing mathematics like physics? In your message you state ... like physics, in which theorems can only be proven by statistics and observations. In fact, mathematicians do derive many results by statistics and observations; those results are called conjectures. Some conjectures are believed more strongly than others, depending on the strength of the evidence. The term theorem is reserved for the statements that have been proved. P === Subject: Re: Raatikainen's critique of Chaitin >>>> >> I am more impressed with his results than classification of finite >>> simple groups. >>>> I'm more impressed about classification. >>>> Classification of finite simple groups is nice to have, >>> but it doesn't shake up the foundations of mathematics like his >>> result does. >>>> Chaitin has not shaken the foundations of mathematics --- his results >>> have no more philosophical implications than were present in the >>> work of Godel. > Not true. Bollocks. > Chaitin's Theorem (about Omega, which is really his big result) shows > that only a finite number of bits of Omega are possible to determine; > the only way to determine more bits is to call them axioms. Since the > bits can be represented as exponential diophantine equations (1 if > there are infinitely many solutions, 0 if only finitely many > solutions), we see that there are some statements in number theory > that are unknowable. Already proved by Matiyasevich et al. > Much stronger than Godel's result, as Godel's > theorem gives a way to get around the incompleteness, by assuming that > the axiom system is consistent. That is ignorant. There is no way to get around that. > Chaitin's theorem shows that there is > no way to get around it, since assuming that the axiom system is > consistent only improves the information complexity of the axiom > system by a finite amount, so you can still only know a finite number > of bits of Omega. Who cares about Omega? All Omega does is package a lot of yes/no answers in a real number. Why bother doing that --- what utility does a real number has that a sequence of yes/no answers doesn't? It's a totally artificial construction ... why should anyone care about it? Anyone familiar with computability theory of the 1930s/1940s can give a sequence of incomputable problems. > Therefore, mathematics is as Chaitin calls it, quasi-empirical, just Another drivellous pseudo-philosophical conclusion. > like physics, in which theorems can only be proven by statistics and > observations; they are never certain, always subject to being possibly > refuted. Mathematicians don't like to hear this, since most of them > almost religiously believe that mathematics is absolute certainty. But > in reality, the absolute certainty is only for the trivial theorems > that they prove, as Chaitin showed. Like has nothing to do with it. And trivial --- more equivocation --- the only theorems that we can prove are renamed trivial --- of course there are more theorems (infinitely many) than we have written down (finitely many). Why call undecided/undecidable problems non-trivial save to advance a tendentious philosphical programme or as self-dramatization? > The majority of mathematical facts > are simply unattainable through deductive reasoning; only > inductive/empirical methods work (and not so well). Define majority. >> Wrong, as a matter of fact. Did Godel show that randomness had >> something to do with incompleteness? Did he show the existence of a >> maximally unknowable number? These are not the only things, btw.... >> Many of these statements are actually philosophically very >> significant, because they have something to do with epistemology! >> Bollocks the lot of it. All these discoveries of Chaitin's >> are simply repackagings of the well-established notion of completeness. > Nope, only parts of Chaitin's discoveries are repackagings of Godel's > results, i.e., using the Berry paradox instead of the liar paradox. > The main part about Omega is not. Yup. Diagnonalization yet again.... >>> It was the work of Godel, Turing and Church which >>> shook the foundations of mathematics in the 1930s not Chaitin many >>> decades later. > All of them did. The only difference is that because Chaitin's results > are so much stronger and are so shocking Not shocking. His theorems are unexceptionable and unexceptional. If you showed a mathematician one of Chatin's theorems and asked him/her whether they expect it to be true, they would say yes. His philosophical conclusions are tendentious. > and go against the grain of > how most mathematicians think, most mathematicians don't want to > believe them; therefore, you get reactions that fit into the 4 > categories that I mentioned when I started this thread. I should add > that most mathematicians don't even understand Chaitin's results, > mainly because they don't want to understand them. Most mathematicians don't care about Chaitin's results. >> This thread started with some good observations about the weakness of >> Raatikainen's critique and then degenerated into a mudball about how >> bad and naughty Chaitin is. If you are saying such a thing, give a >> proper argument. He does not claim to shake the foundations of >> mathematics, but he thinks he has shown that incompleteness should be >> taken more seriously. >> Perhaps you would be better served by learning to read. I was replying >> to Feinstein not Chaitin. shake up the foundations of mathematics >> is Feinstein's phrase. >> Why do you object to that? >> Have you stopped beating your wife? > Chaitin did shake up the foundations of mathematics. No. > The only problem > is mathematicians don't like his result, so they lose out and remain > ignorant. (This is my opinion.) One is entitled to one's opinion no matter how ignorant and worthless it be. The fact is that mathematicians read Chaitin and see no striking, deep nor relevant (to their work) conclusions. Now, if you can prise your head from your rectum, you may wish to explore the much more interesting philosophical/sociological ramifications of the proof of the classification of finite simple groups. This is much more of an interesting case than the proofs of Chaitin, which are relatively simple and easily surveyed and checked by an individual. In short, traditional mathematical practice. The Classification proof is nothing like this. This is a composite of the work of many mathamaticians over many years. Many of the essential parts, e.g., Feit-Thompson's work on odd order groups are very complex. There were many papers which required intricate argumentation and extensive computer calculations. This whole web of papers is too vast to be read let alone checked by any one person. They were published in many journals ... refereeing standards vary greatly, even in the same journal, as all published mathematicians realize. When in 1980 the theorem was announced, a major part of the work (Mason's classifcation of quasithin groups) remained unpublished. This MS is still unpublished (reports of its growing length have been consistent throughout this period). Recently Aschbacher has announced a new approach to these groups, and when he publishes the proof of classification will be complete (like it was in 1980?) or will it? In the meantime Gorenstein et al started publishing a series of revisonist volumes giving simplified and hopefully correct and definitive versions of some of the proofs. These volumes continue to appear although alas Gorenstein died shortly after the start of this project. This raises interesting questions. Can a proof of this complexity (not machine checkable as it stands) be accepted as true? Was the classification accepted as true in 1980? And is it accepted now? What would be the effect if a basic error was found? Those who prefer social contructivist to realist philosophies of mathematics argue that mathematical truth is a social convention (roughly something accepted by the community of mathematicians). Is the history of classification evidence for or against social constructivism? What about the ethical implications of contributing to a major distributed programme of mathematical research? Was there pressure to publish complex and hard-to-check papers which purported to/did make major contributions to the goal. I hope I have convinced readers that something like Classification raises a huge number of interesting philosophical questions. Unfortunately such issues can't be easily subsumed under a comic-book narrative of Great Men (they told you they were great themselves) making Shocking Discoveries which Shake Foundations, and so are unlikely to be discussed with any vigour on sci.math :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Raatikainen's critique of Chaitin |I hope I have convinced readers that something like Classification raises |a huge number of interesting philosophical questions. Unfortunately |such issues can't be easily subsumed under a comic-book narrative of |Great Men (they told you they were great themselves) making Shocking |Discoveries which Shake Foundations, and so are unlikely to be discussed |with any vigour on sci.math :-( It seems to me that on the issue of whether to think the result itself (the classification theorem) is true, the consensus opinion is that it is somewhat unlikely not to be true, in spite of there probably being mistakes in various parts of the proof. This leaves us in a somewhat awkward position. I don't know how one justifies this sense of confidence other than by appeals to experience, intuition, and so on. For various reasons, we prefer not to have to appeal to those things. But really, it would be very impractical to treat results as not being established until we were confident that an entirely mistake-free proof existed. Maybe someday formal verification methods will become practical enough. Keith Ramsay === Subject: Re: Raatikainen's critique of Chaitin > Chaitin's Theorem (about Omega, which is really his big result) shows > that only a finite number of bits of Omega are possible to determine; > the only way to determine more bits is to call them axioms. Since the > bits can be represented as exponential diophantine equations (1 if > there are infinitely many solutions, 0 if only finitely many > solutions), we see that there are some statements in number theory > that are unknowable. > Already proved by Matiyasevich et al. No, you missed the point. I'm surprised that you would post on a subject matter that you obviously don't understand. Matiyasevich's result was when we are to determine whether an equation has a solution or not. Chaitin's result is whether there are infinitely many solutions or not. There is a big difference, in fact the key which makes Chaitin's result stronger. Think about it or read Chaitin's work. > Much stronger than Godel's result, as Godel's > theorem gives a way to get around the incompleteness, by assuming that > the axiom system is consistent. > That is ignorant. There is no way to get around that. OK that may be so, but at least I read Chaitin's work. > Chaitin's theorem shows that there is > no way to get around it, since assuming that the axiom system is > consistent only improves the information complexity of the axiom > system by a finite amount, so you can still only know a finite number > of bits of Omega. > Who cares about Omega? > All Omega does is package a lot of yes/no answers in a real number. > Why bother doing that --- what utility does a real number has > that a sequence of yes/no answers doesn't? It's a totally artificial > construction ... why should anyone care about it? Anyone familiar > with computability theory of the 1930s/1940s can give a sequence > of incomputable problems. So you are on step 2) and 4) of the process in my original post? > Therefore, mathematics is as Chaitin calls it, quasi-empirical, just > Another drivellous pseudo-philosophical conclusion. > like physics, in which theorems can only be proven by statistics and > observations; they are never certain, always subject to being possibly > refuted. Mathematicians don't like to hear this, since most of them > almost religiously believe that mathematics is absolute certainty. But > in reality, the absolute certainty is only for the trivial theorems > that they prove, as Chaitin showed. > Like has nothing to do with it. And trivial --- more equivocation --- > the only theorems that we can prove are renamed trivial --- of course > there are more theorems (infinitely many) than we have written down > (finitely many). Why call undecided/undecidable problems non-trivial > save to advance a tendentious philosphical programme or as > self-dramatization? > The majority of mathematical facts > are simply unattainable through deductive reasoning; only > inductive/empirical methods work (and not so well). > Define majority. It's a heuristic idea, sort of like most numbers are composite. >> >> Wrong, as a matter of fact. Did Godel show that randomness had >> something to do with incompleteness? Did he show the existence of a >> maximally unknowable number? These are not the only things, btw.... >> Many of these statements are actually philosophically very >> significant, because they have something to do with epistemology! >> Bollocks the lot of it. All these discoveries of Chaitin's >> are simply repackagings of the well-established notion of completeness. > > Nope, only parts of Chaitin's discoveries are repackagings of Godel's > results, i.e., using the Berry paradox instead of the liar paradox. > The main part about Omega is not. > Yup. Diagnonalization yet again.... >>> It was the work of Godel, Turing and Church which >>> shook the foundations of mathematics in the 1930s not Chaitin many >>> decades later. > > All of them did. The only difference is that because Chaitin's results > are so much stronger and are so shocking > Not shocking. > His theorems are unexceptionable and unexceptional. > If you showed a mathematician one of Chatin's theorems and asked > him/her whether they expect it to be true, they would say yes. > His philosophical conclusions are tendentious. > and go against the grain of > how most mathematicians think, most mathematicians don't want to > believe them; therefore, you get reactions that fit into the 4 > categories that I mentioned when I started this thread. I should add > that most mathematicians don't even understand Chaitin's results, > mainly because they don't want to understand them. > Most mathematicians don't care about Chaitin's results. That is true, but a lot are upset by them. >> >> This thread started with some good observations about the weakness of >> Raatikainen's critique and then degenerated into a mudball about how >> bad and naughty Chaitin is. If you are saying such a thing, give a >> proper argument. He does not claim to shake the foundations of >> mathematics, but he thinks he has shown that incompleteness should be >> taken more seriously. >> Perhaps you would be better served by learning to read. I was replying >> to Feinstein not Chaitin. shake up the foundations of mathematics >> is Feinstein's phrase. >> Why do you object to that? >> Have you stopped beating your wife? > > Chaitin did shake up the foundations of mathematics. > No. > The only problem > is mathematicians don't like his result, so they lose out and remain > ignorant. (This is my opinion.) > One is entitled to one's opinion no matter how ignorant and worthless > it be. > The fact is that mathematicians read Chaitin and see no striking, deep > nor relevant (to their work) conclusions. Not necessarily because there are no striking deep nor relevant conclusions but more because they don't understand them. > Now, if you can prise your head from your rectum, you may wish to > explore the much more interesting philosophical/sociological > ramifications of the proof of the classification of finite simple > groups. This is much more of an interesting case than the proofs of Chaitin, > which are relatively simple and easily surveyed and checked by an > individual. In short, traditional mathematical practice. > The Classification proof is nothing like this. This is a composite > of the work of many mathamaticians over many years. Many of the essential > parts, e.g., Feit-Thompson's work on odd order groups are very complex. > There were many papers which required intricate argumentation and > extensive computer calculations. This whole web of papers is too vast to > be read let alone checked by any one person. They were published in many > journals ... refereeing standards vary greatly, even in the same journal, > as all published mathematicians realize. > When in 1980 the theorem was announced, a major part of the work > (Mason's classifcation of quasithin groups) remained unpublished. > This MS is still unpublished (reports of its growing length have been > consistent throughout this period). Recently Aschbacher has announced a > new approach to these groups, and when he publishes the proof of > classification will be complete (like it was in 1980?) or will it? > In the meantime Gorenstein et al started publishing a series of revisonist > volumes giving simplified and hopefully correct and definitive versions > of some of the proofs. These volumes continue to appear although alas > Gorenstein died shortly after the start of this project. > This raises interesting questions. > Can a proof of this complexity (not machine checkable as it stands) be > accepted as true? > Was the classification accepted as true in 1980? > And is it accepted now? > What would be the effect if a basic error was found? > Those who prefer social contructivist to realist philosophies of mathematics > argue that mathematical truth is a social convention (roughly something > accepted by the community of mathematicians). Is the history of > classification evidence for or against social constructivism? > What about the ethical implications of contributing to a major distributed > programme of mathematical research? Was there pressure to publish complex > and hard-to-check papers which purported to/did make major contributions to > the goal. > I hope I have convinced readers that something like Classification raises > a huge number of interesting philosophical questions. Unfortunately > such issues can't be easily subsumed under a comic-book narrative of > Great Men (they told you they were great themselves) making Shocking > Discoveries which Shake Foundations, and so are unlikely to be discussed > with any vigour on sci.math :-( These philosophical questions are about as interesting as the question of whether it was correct for the South-West Journal of Math to withdraw James Harris' papers even though they passed peer review (if you read that incident). In other words, it really doesn't make much difference in the grand scheme of things, especially given the fact that James Harris has probably had more people read his paper from reading usenet than any papers published in that journal. Craig === Subject: Re: Raatikainen's critique of Chaitin >> Define majority. > It's a heuristic idea, sort of like most numbers are composite. Like most numbers are too large to write down. :-( >> Most mathematicians don't care about Chaitin's results. > That is true, but a lot are upset by them. No. I am aware of no mathematicians that are upset by Chaitin's theorems. Many mathematicians can see through his quasiphilosophical hyperbole that he surrounds this theorems with. Summary: we aren't upset my his mathematics, but are slightly irritated by his dubious philosophizing. >> The fact is that mathematicians read Chaitin and see no striking, deep >> nor relevant (to their work) conclusions. > Not necessarily because there are no striking deep nor relevant > conclusions but more because they don't understand them. Because his theorems are exactly the results you would expect in the context; footnotes to Godel, Church and Turing. >> Now, if you can prise your head from your rectum, you may wish to >> explore the much more interesting philosophical/sociological >> ramifications of the proof of the classification of finite simple >> groups. This is much more of an interesting case than the proofs of >> Chaitin, which are relatively simple and easily surveyed and checked by >> an individual. In short, traditional mathematical practice. >> The Classification proof is nothing like this. This is a composite >> of the work of many mathamaticians over many years. Many of the essential >> parts, e.g., Feit-Thompson's work on odd order groups are very complex. >> There were many papers which required intricate argumentation and >> extensive computer calculations. This whole web of papers is too vast to >> be read let alone checked by any one person. They were published in many >> journals ... refereeing standards vary greatly, even in the same journal, >> as all published mathematicians realize. >> When in 1980 the theorem was announced, a major part of the work >> (Mason's classifcation of quasithin groups) remained unpublished. >> This MS is still unpublished (reports of its growing length have been >> consistent throughout this period). Recently Aschbacher has announced a >> new approach to these groups, and when he publishes the proof of >> classification will be complete (like it was in 1980?) or will it? >> In the meantime Gorenstein et al started publishing a series of >> revisonist volumes giving simplified and hopefully correct and >> definitive versions of some of the proofs. These volumes continue to >> appear although alas Gorenstein died shortly after the start of this >> project. >> This raises interesting questions. >> Can a proof of this complexity (not machine checkable as it stands) be >> accepted as true? >> Was the classification accepted as true in 1980? >> And is it accepted now? >> What would be the effect if a basic error was found? >> Those who prefer social contructivist to realist philosophies of >> mathematics argue that mathematical truth is a social convention (roughly >> something accepted by the community of mathematicians). Is the history of >> classification evidence for or against social constructivism? >> What about the ethical implications of contributing to a major >> distributed programme of mathematical research? Was there pressure to >> publish complex and hard-to-check papers which purported to/did make >> major contributions to the goal. >> I hope I have convinced readers that something like Classification raises >> a huge number of interesting philosophical questions. Unfortunately >> such issues can't be easily subsumed under a comic-book narrative of >> Great Men (they told you they were great themselves) making Shocking >> Discoveries which Shake Foundations, and so are unlikely to be discussed >> with any vigour on sci.math :-( > These philosophical questions are about as interesting as the question > of whether it was correct for the South-West Journal of Math to > withdraw James Harris' papers even though they passed peer review (if > you read that incident). > In other words, it really doesn't make much > difference in the grand scheme of things, It is your postings which epitomise not making much difference to the grand scheme of things. > especially given the fact > that James Harris has probably had more people read his paper from > reading usenet than any papers published in that journal. Why are you afraid of the philosophical consequences of the classification theorem? Is it because you don't understand it? Is it because you are Shocked by it? That it Shakes the Foundations of your world-view? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Raatikainen's critique of Chaitin >> The fact is that mathematicians read Chaitin and see no striking, deep >> nor relevant (to their work) conclusions. > > Not necessarily because there are no striking deep nor relevant > conclusions but more because they don't understand them. > Because his theorems are exactly the results you would > expect in the context; footnotes to Godel, Church and Turing. Footnotes. You call his AIT monograph a collection of footnotes? There are some quite serious and elegant theorems there. I wonder what you are working on that you would not call footnotes to X then. I can call all of 20th century mathematics as footnotes to Pythagoras, or all philosophy as footnotes to Plato and Aristotle... How accurate would that be except being an amusing reflection of the intricate relations to history? As I detected before, this thread has degenerated into a mudfest. Maybe you need another version of Peter Olcott to busy yourselves with! You are too accustomed to flamewars, rather than conducting sincere investigation of X's work. Besides, I think Chaitin has shown a great deal of effort in humility, he attributes the fundamental ideas to such great minds as Leibniz and Borel in his book Omega. His work indeed seems like formalization of Leibniz's statements, and has relevance also to questions asked by Godel. In some philosophy papers, much less rigorous and intuitive arguments about epistemology were given. For instance Putnam's somewhat horrid arguments about any computation being anywhere. It's always funny to watch how people resist to strong arguments, and have no difficulty with weak arguments when it suits them. Your posts made me realize that my task is much more difficult than it seems. I want to erase Platonist tendencies from mathematics, e.g. decisively eliminate the idea of God and heaven. If you can't understand that Chaitin's results elaborately show that there is randomness in the heart of mathematics and idealized reasoning in general, *then* it would be impossible for folks like you to appreciate that mathematical realism is bankrupt. Chaitin's work is not just a collection of footnotes, I think. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin >> The fact is that mathematicians read Chaitin and see no striking, deep >> nor relevant (to their work) conclusions. > > Not necessarily because there are no striking deep nor relevant > conclusions but more because they don't understand them. > > Because his theorems are exactly the results you would > expect in the context; footnotes to Godel, Church and Turing. > Footnotes. You call his AIT monograph a collection of footnotes? There > are some quite serious and elegant theorems there. I wonder what you > are working on that you would not call footnotes to X then. > I can call all of 20th century mathematics as footnotes to Pythagoras, > or all philosophy as footnotes to Plato and Aristotle... How accurate > would that be except being an amusing reflection of the intricate > relations to history? As I detected before, this thread has > degenerated into a mudfest. Maybe you need another version of Peter > Olcott to busy yourselves with! You are too accustomed to flamewars, > rather than conducting sincere investigation of X's work. > Besides, I think Chaitin has shown a great deal of effort in humility, > he attributes the fundamental ideas to such great minds as Leibniz and > Borel in his book Omega. His work indeed seems like formalization of > Leibniz's statements, and has relevance also to questions asked by > Godel. > In some philosophy papers, much less rigorous and intuitive > arguments about epistemology were given. For instance Putnam's > somewhat horrid arguments about any computation being anywhere. It's > always funny to watch how people resist to strong arguments, and have > no difficulty with weak arguments when it suits them. > Your posts made me realize that my task is much more difficult than it > seems. I want to erase Platonist tendencies from mathematics, e.g. > decisively eliminate the idea of God and heaven. If you can't > understand that Chaitin's results elaborately show that there is > randomness in the heart of mathematics and idealized reasoning in > general, *then* it would be impossible for folks like you to > appreciate that mathematical realism is bankrupt. Chaitin's work is > not just a collection of footnotes, I think. I thought we were on the same side. If you want to decisively eliminate the idea of God and heaven, then we're not on the same team. I think Chaitin's work demonstrates that there are some facts that only God can have access to, which is good for us to know, as it forces us to be humble if we are to be honest with ourselves. Craig === Subject: Re: Raatikainen's critique of Chaitin > Your posts made me realize that my task is much more difficult than it > seems. I want to erase Platonist tendencies from mathematics, e.g. > decisively eliminate the idea of God and heaven. If you can't > understand that Chaitin's results elaborately show that there is > randomness in the heart of mathematics and idealized reasoning in > general, *then* it would be impossible for folks like you to > appreciate that mathematical realism is bankrupt. Chaitin's work is > not just a collection of footnotes, I think. > I thought we were on the same side. If you want to decisively > eliminate the idea of God and heaven, then we're not on the same team. > I think Chaitin's work demonstrates that there are some facts that > only God can have access to, which is good for us to know, as it > forces us to be humble if we are to be honest with ourselves. I used to think like that. Then, we had a quarrel. >:-> Now I am all for a heretic interpretation of mathematics. I don't think a computation exists until it is constructed, which is to say, Omega is an imaginery number. God once told me I am the infinite mind, which I took with a good deal of suspicion. Well, God doesn't like suspicion, so he created mathematicians in his image. You see... God is a liar. -- Eray === Subject: Re: Raatikainen's critique of Chaitin >> Define majority. > > It's a heuristic idea, sort of like most numbers are composite. > Like most numbers are too large to write down. :-( No. >> Most mathematicians don't care about Chaitin's results. > > That is true, but a lot are upset by them. > No. I am aware of no mathematicians that are upset > by Chaitin's theorems. Many mathematicians can see through > his quasiphilosophical hyperbole that he surrounds this > theorems with. > Summary: we aren't upset my his mathematics, but are slightly > irritated by his dubious philosophizing. I personally am glad that he philosophizes. I learned a lot from read them. >> The fact is that mathematicians read Chaitin and see no striking, deep >> nor relevant (to their work) conclusions. > > Not necessarily because there are no striking deep nor relevant > conclusions but more because they don't understand them. > Because his theorems are exactly the results you would > expect in the context; footnotes to Godel, Church and Turing. I refuted this before. >> Now, if you can prise your head from your rectum, you may wish to >> explore the much more interesting philosophical/sociological >> ramifications of the proof of the classification of finite simple >> groups. This is much more of an interesting case than the proofs of >> Chaitin, which are relatively simple and easily surveyed and checked by >> an individual. In short, traditional mathematical practice. >> The Classification proof is nothing like this. This is a composite >> of the work of many mathamaticians over many years. Many of the essential >> parts, e.g., Feit-Thompson's work on odd order groups are very complex. >> There were many papers which required intricate argumentation and >> extensive computer calculations. This whole web of papers is too vast to >> be read let alone checked by any one person. They were published in many >> journals ... refereeing standards vary greatly, even in the same journal, >> as all published mathematicians realize. >> When in 1980 the theorem was announced, a major part of the work >> (Mason's classifcation of quasithin groups) remained unpublished. >> This MS is still unpublished (reports of its growing length have been >> consistent throughout this period). Recently Aschbacher has announced a >> new approach to these groups, and when he publishes the proof of >> classification will be complete (like it was in 1980?) or will it? >> In the meantime Gorenstein et al started publishing a series of >> revisonist volumes giving simplified and hopefully correct and >> definitive versions of some of the proofs. These volumes continue to >> appear although alas Gorenstein died shortly after the start of this >> project. >> This raises interesting questions. >> Can a proof of this complexity (not machine checkable as it stands) be >> accepted as true? >> Was the classification accepted as true in 1980? >> And is it accepted now? >> What would be the effect if a basic error was found? >> Those who prefer social contructivist to realist philosophies of >> mathematics argue that mathematical truth is a social convention (roughly >> something accepted by the community of mathematicians). Is the history of >> classification evidence for or against social constructivism? >> What about the ethical implications of contributing to a major >> distributed programme of mathematical research? Was there pressure to >> publish complex and hard-to-check papers which purported to/did make >> major contributions to the goal. >> I hope I have convinced readers that something like Classification raises >> a huge number of interesting philosophical questions. Unfortunately >> such issues can't be easily subsumed under a comic-book narrative of >> Great Men (they told you they were great themselves) making Shocking >> Discoveries which Shake Foundations, and so are unlikely to be discussed >> with any vigour on sci.math :-( > > These philosophical questions are about as interesting as the question > of whether it was correct for the South-West Journal of Math to > withdraw James Harris' papers even though they passed peer review (if > you read that incident). > In other words, it really doesn't make much > difference in the grand scheme of things, > It is your postings which epitomise not making much difference > to the grand scheme of things. You don't know that. > especially given the fact > that James Harris has probably had more people read his paper from > reading usenet than any papers published in that journal. > Why are you afraid of the philosophical consequences of > the classification theorem? Is it because you don't understand > it? Is it because you are Shocked by it? That it Shakes the > Foundations of your world-view? No, it's just a typical theorem of the 20th century from the University of Mediocrity. Boring as hell, useless, and having no philosophical consequences. The 20th century will be known to future generations as the century in which quantity triumpthed over quality in mathematics. (There are of course exceptions in applied mathematics which are useful like the FFT, Kalman's filter, finite element method, Simplex method, and also some results like Chaitin's which have philosophical consequences.) Craig === Subject: Re: Raatikainen's critique of Chaitin Discussion, linux) >> Most mathematicians don't care about Chaitin's results. > That is true, but a lot are upset by them. No. A number are annoyed by the vague, mushy but sweeping philosophical claims made about his results. His results upset no one. -- Quincy, would you rather do epistemology or conceptual analysis? You know what? I'd rather fight on an aircraft carrier.... And Mama and Baba (Papa) would fight on an aircraft carrier, too. -- Quincy P. Hughes, age 3 1/2 === Subject: Re: Raatikainen's critique of Chaitin >Chaitin's Theorem (about Omega, which is really his big result) shows >that only a finite number of bits of Omega are possible to determine; >the only way to determine more bits is to call them axioms. Since the >bits can be represented as exponential diophantine equations (1 if >there are infinitely many solutions, 0 if only finitely many >solutions), we see that there are some statements in number theory >that are unknowable. >Already proved by Matiyasevich et al. >No, you missed the point. I'm surprised that you would post on a >subject matter that you obviously don't understand. Matiyasevich's >result was when we are to determine whether an equation has a solution >or not. Chaitin's result is whether there are infinitely many >solutions or not. There is a big difference, in fact the key which >makes Chaitin's result stronger. Think about it or read Chaitin's >work. >[ snip ] For every Diophantine equation E1 there constructively exists a Diophantine equation E2 such that E2 has infinitely many solutions if and only if E1 has a solution. The proof is an easy exercise. Just add another unknown. obvious reductions. === Subject: Re: Raatikainen's critique of Chaitin > >Chaitin's Theorem (about Omega, which is really his big result) shows >that only a finite number of bits of Omega are possible to determine; >the only way to determine more bits is to call them axioms. Since the >bits can be represented as exponential diophantine equations (1 if >there are infinitely many solutions, 0 if only finitely many >solutions), we see that there are some statements in number theory >that are unknowable. > >>Already proved by Matiyasevich et al. >No, you missed the point. I'm surprised that you would post on a >subject matter that you obviously don't understand. Matiyasevich's >result was when we are to determine whether an equation has a solution >or not. Chaitin's result is whether there are infinitely many >solutions or not. There is a big difference, in fact the key which >makes Chaitin's result stronger. Think about it or read Chaitin's >work. >[ snip ] > For every Diophantine equation E1 there constructively exists > a Diophantine equation E2 such that E2 has infinitely many solutions > if and only if E1 has a solution. > The proof is an easy exercise. Just add another unknown. > obvious reductions. Again, this misses the point. Your statement may be true but is irrelevant in the context of this discussion. You cannot also say that for every Diophantine equation E2, there constructively exists a Diophantine equation E1 such that E1 has a solution iff E2 has infinitely many solutions. Because of this fact, Chaitin's result is much stronger. Craig === Subject: Re: Raatikainen's critique of Chaitin > Not true. Godel's results are that for any finite axiom system that A common misconception. > Much stronger than Godel's result, as Godel's > theorem gives a way to get around the incompleteness, by assuming that > the axiom system is consistent. This is a nonsensical comment. It is indeed a stronger result than that proved by G.9adel that any consistent formal system can only prove finitely many statements in a particular infinite class of true Pi-0-statements. This result is due to Emil Post, who proved the existence of simple sets, i.e. recursively enumerable sets A with infinite complement, such that the complement of A has no infinite recursively enumerable subset. > Therefore, mathematics is as Chaitin calls it, quasi-empirical, just > like physics, in which theorems can only be proven by statistics and > observations; they are never certain, always subject to being possibly > refuted. Therefore is just silliness. There is nothing in either Chaitin's or G.9adel's results that shows anything of the kind. === Subject: Re: Raatikainen's critique of Chaitin > Did Godel show that randomness had > something to do with incompleteness? Indeed not! But what in the particular example of incompleteness presented by Chaitin shakes the foundations of mathematics in a way that other examples of incompleteness do not? > Did he show the existence of a > maximally unknowable number? What, if anything, do you mean by maximally unknowable? === Subject: Re: Raatikainen's critique of Chaitin > Did Godel show that randomness had > something to do with incompleteness? > Indeed not! But what in the particular example of incompleteness > presented by Chaitin shakes the foundations of mathematics in a way > that other examples of incompleteness do not? I didn't say it shakes the foundations of mathematics. Neither did Chaitin. But he says that we must take incompleteness more seriously: -------------------------------------------------------------- -------------- - Just look at Bertrand Russell's three-volume magnum opus (with Whitehead) Principia Mathematica, which I keep in my office at IBM. One entire formula-filled volume to prove that 1 + 1 = 2! And by modern standards, the formal axiomatic system used there is inadequate; it's not formal enough! No wonder that Poincar.8e decried this as a kind of madness! But I think that it was an interesting philosophical/intellectual exercise to refute this madness as convincingly as possible; somehow I ended up spending my life on that! Of course, formalism fits the 20th century zeitgeist so well: Everything is meaningless, technical papers should never discuss ideas, only present the facts! A rule that I've done my best to ignore! As Vladimir Tasic says in his book Mathematics and the Roots of Postmodern Thought, a great deal of 20th century philosophy seems enamored with formalism and then senses that G.9adel has pulled the rug out from under it, and therefore truth is relative. --- Or, as he puts it, it could have happened this way. --- Tasic presents 20th century thought as, in effect, a dialogue between Hilbert and Poincar.8e... But truth is relative is not the correct conclusion. The correct conclusion is that Hilbert was wrong and that Poincar.8e was right: intuition cannot be eliminated from mathematics, or from human thought in general. And it's not all the same, all intuitions are not equally valid. Truth is not reinvented by each culture or each generation, it's not merely a question of fashion. Let me repeat: formal axiomatic systems are a failure! Theorem proving algorithms do not work. One can publish papers about them, but they only prove trivial theorems. And in the case-histories in this book, we've seen that the essence of math resides in its creativity, in imagining new concepts, in changing viewpoints, not in mindlessly and mechanically grinding away deducing all the possible consequences of a fixed set of rules and ideas. Similarly, proving correctness of software using formal methods is hopeless. Debugging is done experimentally, by trial and error. And cautious managers insist on running a new system in parallel with the old one until they believe that the new system works. -------------------------------------------------------------- -------------- -- I think he may have wiped away a good deal of nonsense. > Did he show the existence of a > maximally unknowable number? > What, if anything, do you mean by maximally unknowable? Numbers like Omega. We can talk about it, but we can never know it. Of course, the existence of random reals was shown before, but Omega is special, because it is not only random but also contains an infinite amount of mathematical information. Allow me to make it crystal clear for those who cannot see it. Here is the argument, in very precise philosophical terms. 1. Omega acts as an oracle for the halting problem, the program to solve any instance of the halting problem for a program p has constant program-size complexity, given Omega_|p|. 2. Omega is a random real, its bits are completely independent, ie. the outcomes of a hypothetical fair coin. (Which could not physically exist by the way!) 3. Therefore, we cannot reason about this number which packs all the information about the halting problem: it is beyond our cognitive capacity. Hence the adjective unknowable. (If you need more explanation here, I can point to you some online courses on epistemology) The extra claim is that 4. There are no numbers which can pack more mathematical information than Omega. And hence the adverb maximally, but this might actually be relative to a theory of mind, or a theory of physics. I am not very sure of it, actually. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin >> Did Godel show that randomness had >> something to do with incompleteness? >> Indeed not! But what in the particular example of incompleteness >> presented by Chaitin shakes the foundations of mathematics in a way >> that other examples of incompleteness do not? > I didn't say it shakes the foundations of mathematics. Neither did > Chaitin. But Feinstein did, and it's Feinstein that is contributing to this thread, not Chaitin. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Raatikainen's critique of Chaitin > Numbers like Omega. What, if anything, does it mean that Omega is maximally unknowable? === Subject: Re: Raatikainen's critique of Chaitin > Numbers like Omega. > What, if anything, does it mean that Omega is maximally unknowable? You will perhaps keep repeating the question until you make me confess the answer you want? Anyway, it goes like this. It is maximally unknowable and maximally informative because 1. It is a random real, which means that to know it you would need to know the results of all possible computations, a countably infinite number of them, something we deem impossible. And of course, there is no stronger form of unknowability if minds are discrete computers! [*] 2. It contains the answer to all instances of the halting problem in a non-redundant way. Every bit of it is absolutely necessary! So the precise wording should be maximally informative *of* the halting problem I am sincerely hoping that this explanation will clarify the terms I have used. A question whose answer that I do not yet know well is, are there problems in mathematics which require strictly more information than the halting problem to answer them? Can you please give some examples? -- Eray Ozkural [*] The minds could be continuous computers, or based on transcendental mechanisms of some kind, in which case Omega could be easily knowable! === Subject: Re: Raatikainen's critique of Chaitin > I am sincerely hoping that this explanation will clarify the terms I > have used. Your comments suffer from a weakness that seems endemic to enthusiastic talk about Chaitin's amazing discoveries (by himself and others). There is no definition or explanation of just what true for a reason/true for no reason or maximally unknowable etc means that allows a reader to judge for himself to what extent these concept apply. Ooh-ing and aah-ing does not actually amount to an explanation or definition of anything, either in mathematics or philosophy. Thus, in the present case, your comments don't even tell a reader for which bit i is 0 or 1 depending on whether the i-th arithmetical sentence is true or not. If we are to make any sense at all of Chaitin's claim to have discovered the amazing fact that there are mathematical statements that are true for no reason, we first need a definition or explanation of the distinction between mathematical statements that are true for a reason and mathematical statements that are true for no reason, a definition or explanation that is not inextricably joined to enthusiastic burbling about Chaitin's amazing discoveries. Similarly, if we are to be able to make any sense of the claim that Chaitin has unearthed a maximally unknowable number, we need a definition or explanation of how to compare numbers with regard to how unknowable they are. === Subject: Re: Raatikainen's critique of Chaitin > I am sincerely hoping that this explanation will clarify the terms I > have used. > Your comments suffer from a weakness that seems endemic to > enthusiastic talk about Chaitin's amazing discoveries (by himself and > others). There is no definition or explanation of just what true for > a reason/true for no reason or maximally unknowable etc means that > allows a reader to judge for himself to what extent these concept > apply. Ooh-ing and aah-ing does not actually amount to an explanation > or definition of anything, either in mathematics or philosophy. > Thus, in the present case, your comments don't even tell a reader > for which bit i is 0 or 1 depending on whether the i-th arithmetical > sentence is true or not. Nonsense! They do! Is that real number random? (I guess not! So it's not maximally unknowable!) So, you could not understand my quite elementary description. Fine. > If we are to make any sense at all of Chaitin's claim to have > discovered the amazing fact that there are mathematical statements > that are true for no reason, we first need a definition or explanation > of the distinction between mathematical statements that are true for a > reason and mathematical statements that are true for no reason, a > definition or explanation that is not inextricably joined to > enthusiastic burbling about Chaitin's amazing discoveries. Done already by Chaitin! You did not read Chaitin's Omega book did you!!!!!! You are trolling again!!!! > Similarly, > if we are to be able to make any sense of the claim that Chaitin has > unearthed a maximally unknowable number, we need a definition or > explanation of how to compare numbers with regard to how unknowable > they are. The definitions are there, but you can't understand them!!!! === Subject: Re: Raatikainen's critique of Chaitin > Done already by Chaitin! Alas, everything said by Chaitin in supposed explanation of true for a reason/true for no reason *is* inextricably joined to enthusiastic burbling about his amazing discoveries. Unfortunately, it seems impossible for admirers of his philosophical claims to set such burbling aside and attempt a sober explanation of the distinction between statements true for a reason and statements true for no reason, followed by a proof of the existence of statements true for no reason (the amazing discovery). Clearly you are not the person to make such an attempt. === Subject: Re: Raatikainen's critique of Chaitin > Done already by Chaitin! > Alas, everything said by Chaitin in supposed explanation of true > for a reason/true for no reason *is* inextricably joined to > enthusiastic burbling about his amazing discoveries. Unfortunately, it > seems impossible for admirers of his philosophical claims to set such > burbling aside and attempt a sober explanation of the distinction > between statements true for a reason and statements true for no > reason, followed by a proof of the existence of statements true for no > reason (the amazing discovery). Clearly you are not the person to > make such an attempt. What is it that you do not understand in an infinite chain of necessary reasons? Algorithmic days, -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin Discussion, linux) > Anyway, it goes like this. It is maximally unknowable and maximally > informative because > 1. It is a random real, which means that to know it you would need to > know the results of all possible computations, a countably infinite > number of them, something we deem impossible. And of course, there is > no stronger form of unknowability if minds are discrete computers! [*] So *every* random real is maximally unknowable? So just about every real is maximally unknowable. Huh. That's a striking property, that is. -- My proof has been checked very thoroughly, both by me and others. Those others apparently decided that they would not believe the proof was correct, but cannot support that position using mathematics. But hey, they're just human beings. --JSH, prover of Fermat's Last Thm === Subject: Re: Raatikainen's critique of Chaitin > Anyway, it goes like this. It is maximally unknowable and maximally > informative because > 1. It is a random real, which means that to know it you would need to > know the results of all possible computations, a countably infinite > number of them, something we deem impossible. And of course, there is > no stronger form of unknowability if minds are discrete computers! > [*] > So *every* random real is maximally unknowable? > So just about every real is maximally unknowable. Huh. That's a > striking property, that is. Absolutely! That's why a random real is so unreal. These exercises suggest that their reality is very far from explaining the physical qualities of space-time. (It's just not a good model, in my opinion.) -- Eray === Subject: Re: Raatikainen's critique of Chaitin >I'll ask a question to the people who criticize Chaitin's work. What >in general about Chaitin's work, which I happen to think is some of >the best mathematics of the past century, bothers you so much? >>(1) I think it's an absurd exaggeration to say Chaitin's work contains >>some of the best mathematics of the past century. >>There are some moderately difficult bits of mathematics in his theory - >>I mean the sort of thing that might lead a good graduate student >>to think for an hour or two - >>but it would be absurd to put it beside the classification >>of the finite simple groups, for example. >I am more impressed with his results than classification of finite >simple groups. Classification of finite simple groups is nice to have, >but it doesn't shake up the foundations of mathematics like his result >does. In my opinion, mathematics in the last 100 years is relatively >boring compared to other centuries. His result and other results like >his (for instance, Godel's Theorem) at least made things interesting. >[ snip ] Now of course what is _best_ mathematics is a matter of opinion, and you are entitled to your own. If the classification of finite simple groups is not deep enough for you, then there is little anyone can say to convince you otherwise. But you asked what bothers people about Chaitin's work. I think many mathematicians are bothered exactly by these exaggerated claims more than by anything in Chaitin's mathematics itself. You may not agree with that, but you asked what other people think and this is your answer. As for being boring, the mathematics of the last 100 years has certainly been less _ambiguous_ than prior to that. Not-quite-precise concepts have been formalized to the extent that there is now very little argument about admissible operations with them (e.g. probability and computability in the 20th century, continuity toward the end of 19th century). As a result of that, mainstream mathematics has seen less controversy about what is a legitimate proof, and more deep results. Sophisticated tools have been developed, long-standing open problems have been solved, new concepts have been defined and successfully applied to areas outside of pure mathematics, such that physics, statistics and informatics. In other words, we have stopped arguing about the rules, and now we concentrate on getting better at playing the game. So if you are looking for ambiguity and controversy then yes, modern mathematics must seem boring. But if you appreciate the insights required to solve difficult problems, the elegance of tools and concepts that have been refined through many iterations, and the unexpected connections between seemingly unrelated disciplines, then the 20th century mathematics has been very exciting. TB === Subject: Re: Raatikainen's critique of Chaitin >> (2) I do however think algorithmic information theory is an important >> and beautiful theory. > What part of it do you think is important and beautiful and why? The basic idea that to a finite string s (of 0s and 1s) one can attach a number H(s), the informational content or entropy of s, and that this function H(s) has the properties one would hope for (as detailed eg in my notes ). -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Raatikainen's critique of Chaitin <6T4_c.26503$Z14.8506@news.indigo.ie> Discussion, linux) > (2) I do however think algorithmic information theory is an important > and beautiful theory. >> What part of it do you think is important and beautiful and why? > The basic idea that to a finite string s (of 0s and 1s) > one can attach a number H(s), the informational content or entropy of s, > and that this function H(s) has the properties one would hope for > (as detailed eg in my notes > ) . While I don't share all of your admiration of AIT, I do find it an interesting development. I found Raatikainen's criticisms of AIT as a so-called Information Theory to be apt and I find many of Chaitin's claims sloppy and unfortunate, but the basic theory and the basic results still interest me (and I'd like them more, perhaps, divorced from terminology that I find misleading). The fact that H(s) shares so many properties with Shannon's H is pretty keen. use of them in preparing for my course on philosophy of information (in which I present Shannon, Chaitin and Raatikainen, among other topics). -- And the logical extension of free and open-source software in the realm of sex would certainly include publicly shared sex at a sex party,... queer sexuality and... non-proprietary sexual affection. Annalee Newitz writing in Salon.com === Subject: Re: Raatikainen's critique of Chaitin > I don't understand why people have so much trouble getting this > point, that his Omega number implies this fact. What fact is that? So far you have just mumbled vaguely about being impressed by a metaphor. === Subject: Re: Raatikainen's critique of Chaitin >>Take any acceptable indexing of Turing machines and any G.9adel numbering >>of theories. These can be as natural as you wish. The following >>observations due to Raatikainen can still be made: >> 1) There is an infinite sequence of stronger and stronger theories >> the characteristic constant of which are equal >> 2) There exists an infinite sequence of more and more complex theories >> which decide exactly the same bits of Omega > There are many problems with Raatikainen's views which I explain in > Raatikainen's Complexity Complex thread. You did not describe any problems with Raatikainen's views. All you offered was vague and confused mumblings. For example, you claimed that we shouldn't use the obsolete rules of inferences which have been shown to capture 1st order logical consequence. I noted to you that by choosing suitable rules of inference, anything can be proved from any axioms. You did not elaborate what sorts of restrictions should be placed on the new rules of inference so that the notion is not completely trivial. I strongly suspect this is because you have no idea about what such restrictions would be. If you want to establish things like the algorithmic complexity of a theory determines how complex things can be proved from it or that the strength of a theory is determined by its algorithmic complexity then you need to specify exactly what you mean, since the most obvious interpretations of these claims are easily shown to be false by Raatikainen's results. And I don't mean inane babble about mathematically intelligent computer programs. > If I restrict my attention to the quoted text in the original post of > this thread, then these do not follow. > I don't think 1) follows in any case, 2) is not a problem because that > is not our concern, those are bad theories, right? Do you have some specific notion of badness in mind? If not, how do you know that the theories relevant to mathematics are not bad or that Chaitin's claims hold for all non-bad theories? You're right in that 1) does not follow directly from Raatikainen's argument in the quoted passage. It's nevertheless true. I thought these two facts (1 and 2) would be easier for you to grasp, since you seem unable to understand the relevance of arguments based on acceptable indexings. > To make it precise, can you tell me in algorithmic terms what 1) > means? It means that there is a sequence of theories T_1, ..., T_n, ..., s.t. a. T_i+1 is stronger than T_i in the sense that it proves more arithmetical truths and b. the characteristic constants of T_i are equal. > There is no such thing as indexing in AIT. There are only programs in > bitstrings. The programming language chosen is the indexing system as you would know if you knew anything about recursion theory. > The unreal indexings are simply concocted to show that it's a real > possibility that such stuff can get published. Yes. I'm sure the referees of JPL are all complete idiots with no knowledge of the relevant mathematics and conceptual issues as you so graciously imply. I'm also sure that reviewers of Raatikainen's papers who failed to notice his grave incompetence have the mathematical maturity of a pine apple. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Raatikainen's critique of Chaitin > If you want to establish things like the algorithmic complexity of a > theory determines how complex things can be proved from it or that the > strength of a theory is determined by its algorithmic complexity then > you need to specify exactly what you mean, since the most obvious > interpretations of these claims are easily shown to be false by > Raatikainen's results. And I don't mean inane babble about > mathematically intelligent computer programs. Raatikainen does not have any results, to be precise. Nor does he establish the falsity of any interpretations.... All he has is his own interpretations, which are in my opinion, not very good ones. > If I restrict my attention to the quoted text in the original post of > this thread, then these do not follow. > > I don't think 1) follows in any case, 2) is not a problem because that > is not our concern, those are bad theories, right? > Do you have some specific notion of badness in mind? If not, how do you > know that the theories relevant to mathematics are not bad or that > Chaitin's claims hold for all non-bad theories? Sure. That they cannot prove much. Say, like a program that finds really nice theorems, but forgets to output them. Chaitin's claims... What are they really? He is talking about some upper bounds he proved. But as far as I can tell, you guys think he is talking about his own proof of Godel's first incompleteness theorem understand which theorems he is referring to. Might that indicate some lack of rigor on your part? > You're right in that 1) does not follow directly from Raatikainen's > argument in the quoted passage. It's nevertheless true. I thought these > two facts (1 and 2) would be easier for you to grasp, since you seem > unable to understand the relevance of arguments based on acceptable > indexings. It is nevertheless true. Okay. Now we are changing subject. What is an acceptable indexing, again? > To make it precise, can you tell me in algorithmic terms what 1) > means? > It means that there is a sequence of theories T_1, ..., T_n, ..., s.t. > a. T_i+1 is stronger than T_i in the sense that it proves more > arithmetical truths and b. the characteristic constants of T_i are equal. What is the characteristic constant of a theory? > There is no such thing as indexing in AIT. There are only programs in > bitstrings. > The programming language chosen is the indexing system as you would > know if you knew anything about recursion theory. We are not concerned with the concept of index to talk about Kolmogorov complexity. We are concerned with programs, and computers. (Or more generally with codings and decompressors...) I know more than sufficient about recursion theory, at least I've studied these subjects properly as a computer scientist. Do you understand anything about the invariance theorem? http://planetmath.org/encyclopedia/InvarianceTheorem.html Incidentally, do you understand what I mean by physically plausible? > The unreal indexings are simply concocted to show that it's a real > possibility that such stuff can get published. > Yes. I'm sure the referees of JPL are all complete idiots with no > knowledge of the relevant mathematics and conceptual issues as you so > graciously imply. I'm also sure that reviewers of Raatikainen's papers > who failed to notice his grave incompetence have the mathematical > maturity of a pine apple. If you are so sure, what can I say? I am not so sure. I basically think they have not thought very deep about these issues, that's all. Why do you think, for instance, do some people who read Chaitin's work and understood it say that Raatikainen's argument about Chaitin's true for no reason is not informative? Or why do some people like me say that we can put an enormous amount of mathematical knowledge in n = n (for certain n) while this information is not contained in forall x (x=x), therefore Raatikainen was wrong, those are bit strings in the theory?... Can you see how trivial that is? These are only two of several flaws in that paper, which I have discussed on three separate threads. I would be grateful if you could answer my questions about acceptable indexing and the characteristic constant of a theory at least. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > Raatikainen does not have any results, to be precise. But he does. For example: The characteristic constants of arbitrary theories T_1,...,T_n can be made to equal any number. >>Do you have some specific notion of badness in mind? If not, how do you >>know that the theories relevant to mathematics are not bad or that >>Chaitin's claims hold for all non-bad theories? > Sure. That they cannot prove much. Say, like a program that finds > really nice theorems, but forgets to output them. How informative. Could you provide a more detailed definition? > What is an acceptable indexing, again? An acceptable indexing of Turing machines is one from which can effectively go to the canonical indexing and vice versa. Alternatively, an acceptable indexing is an indexing for which the parametrization and enumeration theorems hold. > What is the characteristic constant of a theory? The characteristic constant of a theory T is the least m, s.t. T can't prove any sentences of form K(n) > m. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Raatikainen's critique of Chaitin > Raatikainen does not have any results, to be precise. > But he does. For example: > The characteristic constants of arbitrary theories T_1,...,T_n can be > made to equal any number. By means of choosing an indexing... >>Do you have some specific notion of badness in mind? If not, how do you >>know that the theories relevant to mathematics are not bad or that >>Chaitin's claims hold for all non-bad theories? > > Sure. That they cannot prove much. Say, like a program that finds > really nice theorems, but forgets to output them. > How informative. Could you provide a more detailed definition? I believe I explained the shortcomings of Chaitin's philosophical suggestions in Measuring the strength of a theory post and the other answer to your same post on this thread, alluding to case 1 and case 2 which you make. This is case 2 that Aatu was talking of, and I wholeheartedly agree with him on this issue. I think this philosophical demonstration was sufficient to show that Chaitin's theory does not give the ultimate characterization of theorem-proving strength. The part we don't agree is case 1: 1) There is an infinite sequence of stronger and stronger theories the characteristic constant of which are equal An infinite sequence of stronger theories, all of which have the same complexity? But I will stick with your definitions once again, and analyze rigorously. > What is an acceptable indexing, again? > An acceptable indexing of Turing machines is one from which can > effectively go to the canonical indexing and vice versa. Alternatively, > an acceptable indexing is an indexing for which the parametrization and > enumeration theorems hold. > What is the characteristic constant of a theory? > The characteristic constant of a theory T is the least m, s.t. T can't > prove any sentences of form K(n) > m. your remarks. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > The part we don't agree is case 1: > 1) There is an infinite sequence of stronger and stronger theories > the characteristic constant of which are equal The characteristic constant of a theory T is determined by the smallest index e, s.t. for no m does T prove Ad<=e ( {d}(0) is undefined / {d}(0)=/= m ) where {k} is the machine with index k. Given any axiomatisable theory T, say PA, a sequence of stronger and stronger theories T_0,...,T_n,... which have exactly the same characteristic constant can be defined as follows. T_0 = T T_(i+1) is obtained from T_i by adding a T_i-unprovable true Pi_1 statement expressing that a machine {e} does not halt. The machine {e} is chosen so that e > the characteristic constant of T_i (C_T_i) and {e} has no T-provable relations to the non-halting or potential output of machines <= C_T_i and T_i does not prove that {e} does not halt on empty input ({e}(0) is undefined). T_i+1 proves more Pi_1 truths about natural numbers than T_i, so it's stronger than T_i in an obvious sense. Yet T_i+1 and T_i determine exactly the same numbers to be random and have the same characteristic constant. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Raatikainen's critique of Chaitin > Why do you think, for instance, do some people who read Chaitin's work > and understood it say that Raatikainen's argument about Chaitin's > true for no reason is not informative? Or why do some people like me > say that we can put an enormous amount of mathematical knowledge in n > = n (for certain n) while this information is not contained in > forall x (x=x), therefore Raatikainen was wrong, those are bit > strings in the theory?... Can you see how trivial that is? These are > only two of several flaws in that paper, which I have discussed on > three separate threads. The above is just a summary, I've just explained the situation in some more detail to Daryl. It boils down to this. Raatikainen's criticism about strength of formal systems is not completely wrong, but his formal argument is wrong. If he had not tried to give formal arguments, his paper would have been much better. You see once you are formal, you are not allowed any mistakes, everything could crash with a single mistake. A better approach would be to outline the shortcomings of Chaitin's formalization, to show how loose the relation between the complexity of a theorem and the complexity of the axiom schema of the theory can get. (And not to try to discredit the mathematical theory) Here, I briefly indicate where Raatikainen has gone wrong with respect to his criticism of Chaitin's views on theorem proving power: 1. He is examining Theorem LB, which is the weakest incompleteness theorem in Chaitin's monograph. 2. He does not seem to be aware of the invariance theorem, or what asymptotic means. 3. He does not seem to understand that computers are supposed to be physical machines. 4. He does not seem to understand that the inference rule in formalization of axiomatic theory in Chaitin's work can be any truth-preserving system of rules, e.g. it does not have to be predicate calculus. To one set of rules quantifiers, connectives and logical symbols have a meaning, and individual numbers have *no* meaning. To another set of rules, logical symbols have absolutely *no* meaning, while numbers have a meaning. (The hardest to understand is 4.) Almost all of these mistakes are *conceptual*, hence why I think Raatikainen has not shown the proper effort to work through Chaitin's proofs in the first place. In fact 1. and 2. suggest us that he is on the wrong side of the planet, looking for New York City. Even more depressing are my recent observations on some of the loopholes in Chaitin's statements. It was sufficient that Raatikainen had demonstrated the following trivial fact philosophically (without diving into indexings and all that!): Mere complexity does not necessarily entail being informative about the world. It's really so simple that I'm cracking as I write this. Let's have this infinitesimal pin and pick a random number between 0 and 1. The probability that the random real we picked is Omega is almost 0! (But not 0!) That is, if we wanted to solve the halting problem, having a truly truly random, very real number would be of absolutely no use! Although the complexity of a random real is just as much as the number Omega, it normally contains no information about the halting problem. Unfortunately! This is case 2 that Aatu was talking of, and I wholeheartedly agree with him on this issue. I think this philosophical demonstration was sufficient to show that Chaitin's theory does not give the ultimate characterization of theorem-proving strength. (However, without giving a better mechanical alternative, it is useless to show the limitations, I believe). This can be said, because Chaitin does not talk about lower bounds in his theory, while his informal statements imply the existence of lower bounds. Discussion of case 1 is philosophically more interesting, e.g. what are the hardest kinds of mathematical problems we can imagine, what is a golden standard of theorem strength or interestingness, of axiom interestingness, of mathematical truth? But this post is not the place for it, I've tried to explain that some of the relativity described in 4. may have something to do with the solution, in other posts. I do think, however, that we might need to go above *syntax* to fully capture how and when mathematical truth manifests itself as irreducible axioms. IOW, we need to explain how humans can accomplish it, we need to explain how this second order data (as Godel suggested) is *ever* processed by mathematicians, to come up with, say, new axioms! -- Eray Ozkural PS: Humans are not computers, humans do it with divine methods. They download the data directly from God. is no explanation in my opinion! Please! === Subject: Re: Raatikainen's critique of Chaitin > The randomness definition we have for Omega is lim H(r_n) - n = > infinity. This limit neeed not in general exist. Your further comments fail to clarify Chaitin's talk of statements true for no reason. === Subject: Re: Raatikainen's critique of Chaitin > The randomness definition we have for Omega is lim H(r_n) - n = > infinity. > This limit neeed not in general exist. You mean lim_{n -> infinity} H(r_n) - n? How so? > Your further comments fail > to clarify Chaitin's talk of statements true for no reason. My comments do not fail anything. I am referring to Chaitin's own statements in his e-book. If you do not listen to what he says, you don't have the right to criticize him. Did not you see what I posted in reply to Daryl? -------------------------------------------------------------- ---- Chaitin seems to base his idea of reason on Leibniz's principle of sufficient reason in his latest e-book Omega: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html See section Coin Tosses, Randomness vs. Reason, True for no Reason, Unconnected Facts in the text. matter. -------------------------------------------------------------- --- Let's quote the section from Chaitin as well. However, bear in mind, you should read the entire chapter or even better the entire book, to understand and criticize Chaitin's point of view. -------------------------------------------------------------- ------------ Coin Tosses, Randomness vs. Reason, True for no Reason, Unconnected Facts And now, violently opposed to the Greek ideal of pure reason: Independent tosses of a fair coin, an idea from physics! A fair coin means that it is as likely for heads to turn up as for tails. Independent means that the outcome of one coin toss doesn't influence the next outcome. So each outcome of a coin toss is a unique, atomic fact that has no connection with any other fact: not with any previous outcome, not with any future outcome. And knowing the outcome of the first million coin tosses, if we're dealing with independent tosses of a fair coin, gives us absolutely no help in predicting the very next outcome. Similarly, if we could know all the even outcomes (2nd coin toss, 4th toss, 6th toss), that would be no help in predicting any of the odd outcomes (1st toss, 3rd toss, 5th toss). This idea of an infinite series of independent tosses of a fair coin may sound like a simple idea, a toy physical model, but it is a serious challenge, indeed a horrible nightmare, for any attempt to formulate a rational world view! Because each outcome is a fact that is true for no reason, that's true only by accident! Rationalist World View In the physical world, everything happens for a reason. In the world of math, everything is true for a reason. The universe is comprehensible, logical! Kurt G.9adel subscribed to this philosophical position. So rationalists like Leibniz and Wolfram have always rejected physical randomness, or contingent events, as Leibniz called them, because they cannot be understood using reason, they utterly refute the power of reason. Leibniz's solution to the problem is to claim that contingent events are also true for a reason, but in such cases there is in fact an infinite series of reasons, an infinite chain of cause and effect, that while utterly beyond the power of human comprehension, is not at all beyond the power of comprehension of the divine mind. Wolfram's solution to the problem is to say that all of the apparent randomness that we see in the world is actually only pseudo-randomness. It looks like randomness, but it is actually the result of simple laws, in the same way that the digits of ? = 3.1415926... seem to be random. Nevertheless, at this point in time quantum mechanics demands real intrinsic randomness in the physical world, real unpredictability, and chaos theory even shows that a somewhat milder form of randomness is actually present in classical, deterministic physics, if you believe in infinite precision real numbers and in the power of acute sensitivity to initial conditions to quickly amplify random bits in initial conditions into the macroscopic domain... The physicist Karl Svozil has the following interesting position on these questions. Svozil has classical, deterministic leanings and sympathizes with Einstein's assertion that God doesn't play dice. Svozil admits that in its current state quantum theory contains randomness. But he thinks that this is only temporary, and that some new, deeper, hidden-variable theory will eventually restore determinacy and law to physics. On the other hand, Svozil believes that, as he puts it, ? shows that there is real randomness in the (unreal) mental mindscape fantasy world of pure math! -------------------------------------------------------------- -------- -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > You mean lim_{n -> infinity} H(r_n) - n? How so? Why do you think it exists? > My comments do not fail anything. I am referring to Chaitin's own > statements in his e-book. If you do not listen to what he says, you > don't have the right to criticize him. I've read Chaitin's comments. His remarks about statements being true for no reason are either vacuous or unjustified. === Subject: Re: Raatikainen's critique of Chaitin > You mean lim_{n -> infinity} H(r_n) - n? How so? > Why do you think it exists? If you ask me if I think it always exists, I can't see any obvious way to make it fluctuate at the present, that's why. (Maybe you can show me) When there is a limit and it's infinite, however, it means the string is random. That was our concern, I think, but it's not relevant to our discussion at the moment. > My comments do not fail anything. I am referring to Chaitin's own > statements in his e-book. If you do not listen to what he says, you > don't have the right to criticize him. > I've read Chaitin's comments. His remarks about statements being true for > no reason are either vacuous or unjustified. It's a great opportunity for you, then! You have provided absolutely no argument till today. Neither has Raatikainen. Maybe, you believe that we must take it on faith? Here are Chaitin's arguments: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html Please show that they are either vacuous or unjustified. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > It's a great opportunity for you, then! You have provided absolutely > no argument till today. Neither has Raatikainen. This is clearly so, as far as you are concerned, and I find your comments a salutary reminder that it doesn't always make sense to argue with individuals. === Subject: Re: Raatikainen's critique of Chaitin > It's a great opportunity for you, then! You have provided absolutely > no argument till today. Neither has Raatikainen. > This is clearly so, as far as you are concerned, and I find your > comments a salutary reminder that it doesn't always make sense to > argue with individuals. Why don't you back up your statements that Chaitin's philosophy is vacuous or unjustified? You're angry at him because he deals in metaphysics? Godel did the same thing in his philosophical writings. Why approve of Godel and waste Chaitin? :) You see, the matter is not an argument among two of us. The matter is that you (Raatikainen and yourself) are accusing Chaitin of something, but do not even bother making an explanation. These are called ad hominem arguments! (I believe I merely gave the proper responses to you!) As a matter of fact, I am unable to understand your own philosophical position. One might suspect you are something of a reductionist / conventionalist, of the logical positivist variety, but then you are immersed in the works of Godel, a realist. I would appreciate if you could explain your own philosophical position and tell us if it opposes Chaitin's statements. Mine almost certainly does, but I see no need to attack Chaitin! He has proved several interesting theorems in this subject, and you are wrong: nobody thought of them before he did, *and* they are not so obvious at all! (For instance, he surely has contributed much more to the theory of incompleteness, then you have! Why are you investing in philosophical mumbo-jumbo to devalue him?) -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > Mine almost certainly does, but I see no > need to attack Chaitin! Neither do I - my concern is only to see if anybody has any kind of justification of his various utterances. Clearly you don't. === Subject: Re: Raatikainen's critique of Chaitin > Mine almost certainly does, but I see no > need to attack Chaitin! > Neither do I - my concern is only to see if anybody has any kind of > justification of his various utterances. Clearly you don't. I gave my two arguments here, and he has his arguments based on Leibniz in the Omega book. You did nothing to criticize his philosophical arguments. Now is a good time, please show us why Leibniz's principle of sufficient reason and Chaitin's interpretation of Leibniz's theory is wrong. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > I gave my two arguments here, I'm sure you did. === Subject: Re: Raatikainen's critique of Chaitin > I gave my two arguments here, > I'm sure you did. Yes, but you did not show any effort to understand them. There, please read the second argument about mental irreducibility. That is Chaitin's argument in a more rigorous way. I am saying that it needs to assume that minds are computers to achieve that conclusion, however. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin Discussion, linux) >> The randomness definition we have for Omega is lim H(r_n) - n = >> infinity. >> This limit neeed not in general exist. > You mean lim_{n -> infinity} H(r_n) - n? How so? >> Your further comments fail >> to clarify Chaitin's talk of statements true for no reason. > My comments do not fail anything. I am referring to Chaitin's own > statements in his e-book. If you do not listen to what he says, you > don't have the right to criticize him. > Did not you see what I posted in reply to Daryl? > -------------------------------------------------------------- ---- > Chaitin seems to base his idea of reason on Leibniz's principle of > sufficient reason in his latest e-book Omega: > http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html > See section Coin Tosses, Randomness vs. Reason, True for no Reason, > Unconnected Facts in the text. > matter. > -------------------------------------------------------------- --- > Let's quote the section from Chaitin as well. However, bear in mind, > you should read the entire chapter or even better the entire book, to > understand and criticize Chaitin's point of view. > -------------------------------------------------------------- ------------ > Coin Tosses, Randomness vs. Reason, True for no Reason, Unconnected > Facts > And now, violently opposed to the Greek ideal of pure reason: > Independent tosses of a fair coin, an idea from physics! > A fair coin means that it is as likely for heads to turn up as for > tails. Independent means that the outcome of one coin toss doesn't > influence the next outcome. > So each outcome of a coin toss is a unique, atomic fact that has no > connection with any other fact: not with any previous outcome, not > with any future outcome. > And knowing the outcome of the first million coin tosses, if we're > dealing with independent tosses of a fair coin, gives us absolutely no > help in predicting the very next outcome. Similarly, if we could know > all the even outcomes (2nd coin toss, 4th toss, 6th toss), that would > be no help in predicting any of the odd outcomes (1st toss, 3rd toss, > 5th toss). > This idea of an infinite series of independent tosses of a fair coin > may sound like a simple idea, a toy physical model, but it is a > serious challenge, indeed a horrible nightmare, for any attempt to > formulate a rational world view! Because each outcome is a fact that > is true for no reason, that's true only by accident! Ignoring, for a moment, the true for no reason stuff, it's funny he should mention sequences of coin flips. After all, they serve to show that his notion of random is, er, unusual. Flip a coin any finite number of times and the resulting sequence has positive probability that it is not Chaitin-random. Flip a coin an infinite number of times and the probability that the resulting sequence is Chaitin-random goes to one, but it is still perfectly possible that the resulting sequence is not Chaitin-random. (That is an event of probability 0, but it is not an impossible event.) Someone might want to fix my discussion of the infinite case, since I'm pretty naive when it comes to probabilities. In any case, sequences of coin flips seem to suggest that his notion of random is curious. -- We want a single platform. We're trying to get there using the carrot, or blackmail, or rewards, or whatever you call it. -- Madison, WI, superintendent Rainwater grasps subtlety in the operating system wars. === Subject: Re: Raatikainen's critique of Chaitin > Ignoring, for a moment, the true for no reason stuff, it's funny he > should mention sequences of coin flips. After all, they serve to show > that his notion of random is, er, unusual. > Flip a coin any finite number of times and the resulting sequence has > positive probability that it is not Chaitin-random. Your intuition fails you here, I think. See what Chaitin means by a fair coin. A fair coin generates random strings, it cannot generate any string that is not Chaitin random. That's what the theorems are all about! > Flip a coin an infinite number of times and the probability that the > resulting sequence is Chaitin-random goes to one, but it is still > perfectly possible that the resulting sequence is not Chaitin-random. > (That is an event of probability 0, but it is not an impossible > event.) All three statements of yours seem to be false. > Someone might want to fix my discussion of the infinite case, since > I'm pretty naive when it comes to probabilities. No, you just rely too much on your common sense. > In any case, sequences of coin flips seem to suggest that his notion > of random is curious. Not really. See above. And also read Chaitin's proofs that his randomness definition is the same thing as other definitions, etc. (There are 4 equivalent defs. of randomness IIRC...) Here, you should study Part III, if you are interested: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ait/index.html -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin <87d613tuhj.fsf@phiwumbda.org> Discussion, linux) > Your intuition fails you here, I think. > See what Chaitin means by a fair coin. A fair coin generates random > strings, it cannot generate any string that is not Chaitin random. > That's what the theorems are all about! Ain't no such thing as a Chaitin-fair coin then, as Denis explained. -- So, at this time, I'd like to assure you that I am not interested in I'll have prosecutors knocking on your doors. I have no problem with === Subject: Re: Raatikainen's critique of Chaitin > Your intuition fails you here, I think. > See what Chaitin means by a fair coin. A fair coin generates random > strings, it cannot generate any string that is not Chaitin random. > That's what the theorems are all about! > Ain't no such thing as a Chaitin-fair coin then, as Denis explained. I retract my statements about randomness of finite strings and fair up. My intuition failed me, I assumed that a fair coin must never be able to generate regular sequences, which was wrong. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin >> Ignoring, for a moment, the true for no reason stuff, it's funny he >> should mention sequences of coin flips. After all, they serve to >> show that his notion of random is, er, unusual. >> Flip a coin any finite number of times and the resulting sequence has >> positive probability that it is not Chaitin-random. > Your intuition fails you here, I think. > See what Chaitin means by a fair coin. A fair coin generates random > strings, it cannot generate any string that is not Chaitin random. > That's what the theorems are all about! Any fair coin, with any decent definition of fair , will produce the sequence 1111111111 once in 1024 trials , say, and that sequence has (relatively) low Kolmogorov complexity... For *finite* strings, you cannot be right..., as what kind of sequences could be generated by such a coin if it cannot ever generate a sequence *beginning* by 111111111111111 ? (and, btw, I would very much like to see an exact calculation saying *which* of those strings (of 0 and 1) is the first to be not random in PA, say : 1,11,111,1111,11111,... , or at least giving a reasonable estimation of the length of that one). In other words there is indeed (I think) a way to define random finite sequences, and to define a coin as random if it cannot generate a non-random sequence, *but* it cannot be right for length > Flip a coin an infinite number of times and the probability that the >> resulting sequence is Chaitin-random goes to one, but it is still >> perfectly possible that the resulting sequence is not Chaitin-random. >> (That is an event of probability 0, but it is not an impossible >> event.) > All three statements of yours seem to be false. Why on earth should they? When you flip a coin (an infinite number of times), the probability to get any specified number (or to get any number in a negligeable set) is 0, this doesn't mean the event is impossible... And I see only 1 statement, by the way, or at most 2 with the previous sentence... >> Someone might want to fix my discussion of the infinite case, since >> I'm pretty naive when it comes to probabilities. > No, you just rely too much on your common sense. And on basic knowledge on probability theory. Anyway, all those appeals to coins are a little off-topic, but on that one, I think you are wrong, at least with any definitions of the words I know >> In any case, sequences of coin flips seem to suggest that his notion >> of random is curious. > Not really. See above. And also read Chaitin's proofs that his > randomness definition is the same thing as other definitions, etc. > (There are 4 equivalent defs. of randomness IIRC...) > Here, you should study Part III, if you are interested: > http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ait/index.html === Subject: Re: Raatikainen's critique of Chaitin <87d613tuhj.fsf@phiwumbda.org> Discussion, linux) >> See what Chaitin means by a fair coin. A fair coin generates random >> strings, it cannot generate any string that is not Chaitin random. >> That's what the theorems are all about! > Any fair coin, with any decent definition of fair , will produce the > sequence 1111111111 once in 1024 trials , say, and that sequence has > (relatively) low Kolmogorov complexity... Eray has shown his devotion to Chaitin. He has proposed tossing out first order logic and replacing it with a logic that makes Chaitin's claims about complexity and provability correct. If he's willing to revise our basic logic so that Chaitin's claims are correct, then surely he'd be happy to scrap existing probability theory and roll a new theory so that Chaitin's use of random is sensible too. > Someone might want to fix my discussion of the infinite case, since > I'm pretty naive when it comes to probabilities. >> No, you just rely too much on your common sense. > And on basic knowledge on probability theory. Anyway, all those appeals to > coins are a little off-topic, but on that one, I think you are wrong, at > least with any definitions of the words I know It is a bit off-topic with the main thread, but I still think it's worth pointing out. Chaitin's use of the term random has little to do with the intuitive use or with basic probabilities. -- Jesse F. Hughes I thought it relevant to inform that I notified the FBI a couple of months ago about some of the math issues I've brought up here. -- James S. Harris gives Special Agent Fox a new assignment. === Subject: Re: Raatikainen's critique of Chaitin Jesse F. Hughes says... >It is a bit off-topic with the main thread, but I still think it's >worth pointing out. Chaitin's use of the term random has little to >do with the intuitive use or with basic probabilities. I don't think that's completely true. The set of reals in [0,1] that are random in the sense of Chaitin has measure 1, which means that an infinite sequence of coin tosses is almost certain to produce one. -- Daryl McCullough Ithaca, NY === Subject: Re: Raatikainen's critique of Chaitin <87vfeuq655.fsf@phiwumbda.org> Discussion, linux) > Jesse F. Hughes says... >>It is a bit off-topic with the main thread, but I still think it's >>worth pointing out. Chaitin's use of the term random has little to >>do with the intuitive use or with basic probabilities. > I don't think that's completely true. The set of reals in [0,1] that > are random in the sense of Chaitin has measure 1, which means that > an infinite sequence of coin tosses is almost certain to produce one. Yes, I guess that I exaggerated when I said little to do. Maybe a better expression would be is not the same as. -- Jesse F. Hughes Well, I don't claim to be an expert, in fact I am a fry cook with a national burger chain, but I have solved many differential and partial differential equations numerically. --C. Bond === Subject: Unit interval and random numbers > Jesse F. Hughes says... >It is a bit off-topic with the main thread, but I still think it's >worth pointing out. Chaitin's use of the term random has little to >do with the intuitive use or with basic probabilities. > I don't think that's completely true. The set of reals in [0,1] that > are random in the sense of Chaitin has measure 1, which means that > an infinite sequence of coin tosses is almost certain to produce one. Chaitin says pretty much the same thing in Omega, but I am not sure. The probability that a real number picked from the unit interval is random, is 1, which means to my simple mind it is not just *almost* certain but entirely and positively certain that it will be a random number. It might not make immediate sense, but it goes like this. Could the real we picked turn out to be 0.5? No, because then in 0.5000000000000000000000000000000000000000000.... the digits do not have equal limiting frequency, that simple. So, although it might fail our intuition, it is absolutely impossible for computable reals to be the outcome of a random choice in the unit interval. My interpretation: we cannot choose random numbers in the unit interval, in the real world, it is not a good idea! This property that Turing and others demonstrated might imply, in my opinion, that continuum does not exist in any physical sense - but it might and still exists in an imaginery sense. Of course, these are all very tangential to the main point we are discussing so I changed subject.... -- Eray Ozkural There is no perfect (continuous) circle === Subject: Re: Unit interval and random numbers <87vfeuq655.fsf@phiwumbda.org> Discussion, linux) > Could the real we picked turn out to be 0.5? > No, because then in > 0.5000000000000000000000000000000000000000000.... > the digits do not have equal limiting frequency, that simple. Huh? > So, although it might fail our intuition, it is absolutely impossible > for computable reals to be the outcome of a random choice in the unit > interval. Huh? Probability 0 doesn't mean impossible. And the selection of any particular real should have probability 0, regardless of the limiting frequency of the digites. > My interpretation: we cannot choose random numbers in the > unit interval, in the real world, it is not a good idea! This property > that Turing and others demonstrated might imply, in my opinion, that > continuum does not exist in any physical sense - but it might and > still exists in an imaginery sense. Of course, these are all very > tangential to the main point we are discussing so I changed > subject.... Er, um, yeah. -- 'Every man who has ever lived holds tight to the belief that for him alone the laws of probability are canceled out by love[...] Therefore, you will marry Guinevere. You do not want advice --- only agreement.' Merlin sighed... -- John Steinbeck === Subject: Re: Raatikainen's critique of Chaitin > I think he is not merely re-proving incompleteness, he is proving a > stronger form of incompleteness based on algorithmic randomness. What stronger form of incompleteness? Stronger how? === Subject: Re: Raatikainen's critique of Chaitin > I think he is not merely re-proving incompleteness, he is proving a > stronger form of incompleteness based on algorithmic randomness. > What stronger form of incompleteness? Stronger how? See http://www.cs.auckland.ac.nz/CDMTCS/chaitin/omega.html#ch6 Find the paragraph that explains why it is an extremely strong incompleteness result. -------------------------------------------------------------- -- ? as an Oracle for the Halting Problem Why are the bits of ? irreducible mathematical facts? Well, it's because we can use the first N bits of ? to settle the halting problem for all programs up to N bits in size. That's N bits of information, and the first N bits of ? are therefore an irredundant representation of this information. How much information is there in the first N bits of ?? Given the first N bits of ?, get better and better approximations for ? as indicated in the previous section, until the first N bits of the approximate value are correct. At that point you've seen every program up to N bits long that ever halts. Output something not included in any of the output produced by all these programs that halt. It cannot have been produced using any program having less than or equal to N bits. Therefore the first N bits of ? cannot be produced with any program having substantially less than N bits, and ? satisfies the definition of a random or irreducible real number given in Chapter V: H(the first N bits of ?) > N - c This process is written in LISP in my book The Limits of Mathematics. The LISP function that produces something with complexity greater than N bits if it is given any program that calculates the first N bits of ?, is about a page of code using the special LISP dialect that I present in that book. The size in bits of this one-page LISP function is precisely the value of that constant c with the property that H(the first N bits of ?) is greater than N - c for all N. So, the program-size complexity of the first N bits of ? never drops too far below N. Now that we know that ? is an algorithmically random or irreducible real number, the argument that a FAS can determine only finitely many bits of such a number given in Chapter V immediately applies to ?. The basic idea is that if K bits of ? could be compressed into a substantially less than K-bit FAS, then ? wouldn't really be irreducible. In fact, using the argument given in Chapter V, we can say exactly how many bits of ? a given FAS can determine. Here's the final result... An FAS can only determine as many bits of ? as its complexity As we showed in Chapter V, there is (another) constant c such that a formal axiomatic system FAS with program-size complexity H(FAS) can never determine more than H(FAS) + c bits of the value for ?. These are theorems of the form The 39th bit of ? is 0 or The 64th bit of ? is 1. (This assumes that the FAS only enables you to prove such theorems if they are true.) This is an extremely strong incompleteness result, it's the very best I can do, because it says that essentially the only way to determine bits of ? is to put that information directly into the axioms of our FAS, without using any reasoning at all, only, so to speak, table look-up to determine these finite sets of bits. In other words, the bits of ? are logically irreducible, they cannot be obtained from axioms simpler than they are. Finally! We've found a way to simulate independent tosses of a fair coin, we've found atomic mathematical facts, an infinite series of math facts that have no connection with each other and that are, so to speak, true for no reason (no reason simpler than they are). Also, this result can be interpreted informally as saying that math is random, or more precisely, contains randomness, namely the bits of ?. What a dramatic conclusion! But a number of serious caveats are in order! Math isn't random in the sense of being arbitrary, not at all---it most definitely is not the case that 2 + 2 is occasionally equal to 5 instead of 4! But math does contain irreducible information, of which ? is the prime example. To say that ? is random may be a little confusing. It's a specific well-determined real number, and technically it satisfies the definition of what I've been calling a random real. But math often uses familiar words in unfamiliar ways. Perhaps a better way to put it is to say that ? is algorithmically incompressible. Actually, I much prefer the word irreducible; I'm coming to use it more and more, although for historical reasons the word random is unavoidable. So perhaps it's best to avoid misunderstandings and to say that ? is irreducible, which is true both algorithmically or computationally, and logically, by means of proofs. And which happens to imply that ? has many of the characteristics of the typical outcome of a random process, in the physical sense of an unpredictable process amenable to statistical treatment. For example, as we'll discuss in the next section, in base two, in ?'s infinite binary expansion, each of the 2k possible k-bit blocks will appear with limiting relative frequency exactly 1/2k, and this is provably the case for the specific real number ?, even though it's only true with probability one, but not with certainty, for the outcome of an infinite series of independent tosses of a fair coin. So perhaps, in retrospect, the choice of the word random wasn't so bad after all! Also, a random real may be meaningless, or it may be extremely meaningful; my theory cannot distinguish between these two possibilities, it cannot say anything about that. If the real was produced using an independent toss of a fair coin for each bit, it'll be irreducible and it'll be meaningless. On the other hand, ? is a random real with lots of meaning, since it contains a lot of information about the halting problem, and this information is stored in ? in an irreducible fashion, with no redundancy. You see, once you compress out all the redundancy from anything meaningful, the result necessarily looks meaningless, even though it isn't, not at all, it's just jam-packed with meaning! === Subject: Re: Raatikainen's critique of Chaitin > Find the paragraph that explains why it is an extremely strong > incompleteness result. That there are true statements that cannot be logically deduced from true statements of lower complexity is a triviality. We don't need Omega for this conclusion. The dramatic conclusion that math is random is mere babbling. === Subject: Re: Raatikainen's critique of Chaitin > Find the paragraph that explains why it is an extremely strong > incompleteness result. > That there are true statements that cannot be logically deduced from > true statements of lower complexity is a triviality. We don't need > Omega for this conclusion. The dramatic conclusion that math is > random is mere babbling. Oh, yes, triviality, just like Craig said in the original post: ------------------------------------------------------ When new great ideas are presented which challenge the status quo, usually four things happen in more-or-less the following order: 1) Everyone says its wrong and its crazy. 2) Once people can't find flaws in the idea, they decide that it is correct but irrelevent. 3) When people start to realize that it is relevent, they decide that it is relevent, but it's also obvious. 4) When people start to realize that it is not obvious, they claim that the person who proposed the idea was not the first to think of it (or that they themselves thought of it first). ------------------------------------------------------- Number 3 here, it seems. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > Oh, yes, triviality, Yes, just take a statement of minimal complexity logically implying 0=0. === Subject: Re: Raatikainen's critique of Chaitin > Oh, yes, triviality, > Yes, just take a statement of minimal complexity logically implying > 0=0. This is no proof that there are true statements that cannot be logically deduced from true statements of lower complexity! And the proof is *not* trivial! It takes quite a bit of effort to construct even the *first* theorem in the incompleteness chapter of AIT. (This first theorem is Chaitin's version of Godel's first incompleteness theorem which he explains in his book Omega) The incompleteness theorems of Chaitin are not trivial at all! Some of them took me a long time to reproduce from scratch. Maybe you did not read the proofs! Neither you nor Raatikainen shows any indication of even a basic understanding of the proofs involved! Hence, all the misunderstanding, I am afraid! -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > This is no proof that there are true statements that cannot be > logically deduced from true statements of lower complexity! You claim that a true statement of minimal complexity logically implying 0=0 can be logically deduced from a true statement of lower complexity? === Subject: Re: Raatikainen's critique of Chaitin > This is no proof that there are true statements that cannot be > logically deduced from true statements of lower complexity! > You claim that a true statement of minimal complexity logically > implying 0=0 can be logically deduced from a true statement of lower > complexity? I do not, but I don't think such examples are general enough to be useful. For one thing, Chaitin's interpretation is not equivalent to saying that there are true statements that cannot be logically deduced from true statements of lower complexity, he says much stronger things instead, for instance that theorem proving does not work at all for certain mathematical facts (like showing what bits of Omega are). Read Omega, and then let's discuss. If you will not read it, then... you do not have the right to criticize him because you don't know his work well enough. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > I do not, but I don't think such examples are general enough to be > useful. OK, so you now agree that it is an utterly trivial observation that there are true statements that cannot be logically deduced from true statements of lower complexity. === Subject: Re: Raatikainen's critique of Chaitin > I do not, but I don't think such examples are general enough to be > useful. > OK, so you now agree that it is an utterly trivial observation that > there are true statements that cannot be logically deduced from > true statements of lower complexity. I am also saying that is not the only philosophical consequence of Chaitin's theorems. That's not the subject of any of the incompleteness theorems in Chaitin's work. For instance, Chaitin shows that there is a model for Leibniz's infinite sequences of reasons. Omega, a number which is meaningful, yet random. And his work in randomness (his greatest contribution!) shows that a random real has INFINITE COMPLEXITY. And then, we show that no theorem with H(Axioms) complexity can prove more than H(Axioms) + k bits of Omega, a particular maximally informative (every bit of it contains valuable information about the halting problem) random real number. Moreover, this troubling number occurs in number theory: the very cradle of mathematics. Chaitin says many more things, but rest assured his philosophical claims cannot be compressed to your utterly trivial observation! -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > I am also saying that is not the only philosophical consequence of > Chaitin's theorems. OK, so the amazing fact that some mathematical statements are true for no reason must be distinguished from the trivial fact that some true mathematical statements cannot be derived from true statements of lower complexity. But then what is the content of the amazing fact? Apparently it is impossible to say. === Subject: Re: Raatikainen's critique of Chaitin > I am also saying that is not the only philosophical consequence of > Chaitin's theorems. > OK, so the amazing fact that some mathematical statements are true > for no reason must be distinguished from the trivial fact that > some true mathematical statements cannot be derived from true > statements of lower complexity. But then what is the content of the > amazing fact? Apparently it is impossible to say. The idea that you do not like is this. Assume that minds are computers. Then, halting problem is a golden standard of reasoning. Yet, it turns out that to solve the halting problem mathematical reasoning doesn't work: an infinite amount of information is required to solve the halting problem. A mind can reason about only a finite number of halting problem instances, as determined by *its* algorithmic complexity. You accept none of the two premises therefore you will think that my mental irreducibility argument is wrong. Having been formulated in this way, it sounds plausible, but let me once again remind you that I myself am suspicious about the conclusion. For instance I do not believe the second implicit premise in Chaitin's work: halting problem is a golden standard of reasoning. But if it turns out to be such, Chaitin's work is a step towards one of the most important problems in 21st century: LIMITS OF MIND, which is a much more important subject, in my opinion, than the majority of problems mathematicians deal with. -- Eray Ozkural === Subject: Re: Raatikainen's critique of Chaitin > You accept none of the two premises therefore you will think that my > mental irreducibility argument is wrong. I'm afraid it doesn't come close to being wrong. === Subject: Re: Raatikainen's critique of Chaitin > You accept none of the two premises therefore you will think that my > mental irreducibility argument is wrong. > I'm afraid it doesn't come close to being wrong. I'm afraid your one-liners suck. === Subject: Re: Raatikainen's critique of Chaitin > You accept none of the two premises therefore you will think that my > mental irreducibility argument is wrong. > I'm afraid it doesn't come close to being wrong. I can't help you if you don't understand Leibniz's principle of sufficient reason. -- Eray Ozkural === Subject: Do you know the counter example? I have been doing battle with this problem trying to prove it but I cannot make the proof work so I am wondering if its even true. Let X be a topological space and A is a Sub set of X. Show that, The interior of the boundary of A is equal to null set. . ( B(A)^o = O/ ) It feels as if it should be true but i just cannot show it so any ideas would be welcome stephen === Subject: Re: Do you know the counter example? >I have been doing battle with this problem trying to prove it but I cannot >make the proof work so I am wondering if its even true. >Let X be a topological space and A is a Sub set of X. Show that, >The interior of the boundary of A is equal to null set. . ( >B(A)^o = O/ ) >It feels as if it should be true but i just cannot show it so any ideas >would be welcome It is false actually: the boundary of Q in R is the whole set R KP -- E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI PHONE: +31-15-2784572 TU Delft FAX: +31-15-2786178 Postbus 5031 URL: http://fa.its.tudelft.nl/~hart 2600 GA Delft the Netherlands === Subject: Co-re set with no infinite r.e. subset Is there an easy example of an infinite set of naturals that has no infinite r.e. subset? It would be better if it were co-r.e., but any example would -- Daryl McCullough Ithaca, NY === Subject: Re: Co-re set with no infinite r.e. subset >Is there an easy example of an infinite set of naturals that has no infinite >r.e. subset? not sure how easy it has to be. here's a proof that there is such a thing that can't be -too- hard, since i didn't know the answer a minute ago: let s[1], s[2], ... be an enumeration of the infinite re sets. choose n[1] in s[1]. choose m[1] > n[1]. choose n[2] > m[1] with n[2] in s[2]. choose m[2] > n[2]. etc. let s be the set of all the m[k]'s. then s is infinite, and s[k] is not a subset of s since n[k] is not in s. >It would be better if it were co-r.e., but any example would ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: Re: On the partial sums of reciprocals of primes by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i87CBIw09302; >Could we say that if >f(t) = o(g(t)), then >? int_{a}^{x} f(t) dt = o (int_{a}^{x} g(t) dt) ? >We suppose that f(t) could be negative or positive (we don't know), >while g(t) is positive for all sufficiently large positive t. >For example, if f(t) = t and g(t) = e^t, then >t = o(e^t). >And >lim x to infty int_{a}^{x} t dt / int_{a}^{x} e^t dt >= lim x ((x^2 - a^2)/2)/(e^x - e^a) = 0. >Using the hypothesis, seems to me that we could prove the question >in the positive easily. >Let me outline the proof: >Set >int_{a}^{x} f(t) dt / int_{a}^{x} g(t) dt. >By hypothesis, for any epsilon > 0, there exists M such that >for all x > M, >0 < |f(t)/g(t)| < epsilon < 1. (1) >Thus we could write >int_{a}^{x} f(t) dt / int_{a}^{x} g(t) dt >= [ int_{a}^{M} f(t) dt + int_{M}^{x}f(t) dt ] / int_{a}^{x} g(t) dt >= [ constant(M) + int_{M}^{x}f(t) dt ]/int_{a}^{x} g(t) dt >= o(1) + int_{M}^{x} f(t) dt/int_{a}^{x} g(t) dt. (2) >By (1), >|f(t)| = epsilon g(t). (3) >Integrating both sides of (3) over [a, x], we get No; instead, over [M, x]. Below are corrected; int_{M}^{x} |f(t)| dt = epsilon int_{M}^{x} g(t) dt or int_{M}^{x} |f(t)| dt / int_{M}^{x} g(t) dt = epsilon. (4) Using (4) in (2), we obtain o(1) + int_{M}^{x} f(t) dt/int_{a}^{x} g(t) dt < o(1) + int_{M}^{x} |f(t)| dt / int_{M}^{x} g(t) dt = o(1) + epsilon. (a < M) erdos fan === Subject: Re: A Non Linear Trigonometric Expression for P i . by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i87CBHk09257; >For any positive real number X: >Pi= 4*ArcTanX + 2*ArcTan[(1-X^2)/2X] , Working in Radians, >[ or , >180= 4*ArcTanX + 2*ArcTan[(1-X^2)/2X] , Working in Degrees ]. ADDENDUM -------- It is worth adding to the above formula ,my previously presented (similar in past discussions)formula : Pi/4 =ArcTAN[X] - ArcTan[(X-1)/(X+1)] , working in Radians , For X positive real number Copyright P. Stefanides >Panagiotis Stefanides >http://www.stefanides.gr P.stefanides http://www.stefanides.gr === Subject: Re: Are logarithms still useful? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i87CBJf09333; >We learn that logarithms were invented to facilitate arithmetic with big >numbers. Electronic calculators have made them unecessary for this >purpose. In what ways are logarithms still useful in mathematics? Of course computers , still, cannot give this answer. Nevertheless my theory and work on the ETIMOLOGY of the word LOGARITHM gives a meaning that it is the ratio of two numbers,and particularly the ratio of two angles, e.g: Ln 5=1.609437912 , and this is the ratio of 144.8494121..Deg /90 Deg. Ln 2=0.693147181 , this is the ratio of 62.38324625..Deg / 90 Deg,etc. Similarly ,the same applies to other ,than e, base. [Ref: http://www.stefanides.gr/why_logarithm.htm http://www.stefanides.gr/logarithm.htm http://www.stefanides.gr/nautilus.htm The above ,is still under examination and hope some care should be taken for acceptance . Concluding ,logarithms are needed and are as good as geometry trigonometry ,algebra etc., and computer cannot replace their use. Panagiotis Stefanides http://www.stefanides.gr === Subject: Infinite cardinal number Let m be an infinite cardinal number. Many proofs of m+m=m that I saw are colloraries of m x m=m. Does anyone know a direct proof ? === Subject: Math Cobweb Game I combine elements of the following games of mine, plus a little Poker, into one game. Tangle: The Game: Game involving an n-gon & lines (revised): Integer-Alterations On a Grid (fun): First, draw a regular r-gon on a piece of paper, where r is a multiple of the number of players playing this game. Each player then makes up math-functions, the same number of functions per player for a total of r functions, which transform the integers m and n (n is a positive integer) into an integer, new m. For example, m <= m + n, m <= floor(|m|/n), m <= d(|m| + n^2), d is number-of-divisors function, m <= ceiling(sqrt(m*n)), m <= numerator of reduced |m|/n, m <= Fibonacci(|m|) - n, etc. (The functions can be appropriate for any level of math-ability.) The functions are written around the perimeter of the r-gon, one function per side. it down without showing it to anybody else. (This rule is subject to change.) m begins with the value of 1. Player 1 starts at any vertex of the r-gon and draws a straight line-segment, using a straight-edge if necessary, to any side. m is modified by the function at that side and by n (which equals 1 on first move; see below on how n is calculated). Players alternate drawing line-segments, starting each segment where last player finished their segment, drawing the segment in any direction (with restrictions), and finishing the segment at the first previously drawn segment or edge of the r-gon encountered. Segments coming off of a segment hitting another player-drawn segment must pass the earlier segment, continuing on the opposite side from the most-previously drawn segment. Segments coming off of a segment drawn to the edge of the r-gon must 'bounce' (in either direction) back into the r-gon's interior. Do not draw any segments to any vertexes (of the r-gon or where previously drawn segments intersect). (See the link above about the game Tangle for a similar set of segment-drawing rules, plus for some ascii diagrams.) Now n is equal to the number of segments in the previous stretch. (A stretch is the the collection of segments between any two sides of the r-gon which are immediately connected by a path of segments {without connecting in the mean-time to any other sides of the r-gon}.) Every time a side of the r-gon is reached, m is modified by the previous m and by n and by the function at that side. Also: A stretch may intersect any particular segment at most once. Each side of the r-gon can be visited any number of times, but the game is complete when the last-to-be-visited side is finally reached. The winner is the player(s) whose secret integer is closest to the final value of m. Alternative scoring: Maybe players can make up a different situation for scoring/winning, such as: (for 2 player game) player 1 wins if the number of distinct primes dividing the final m is odd, player 2 wins if this number of primes is even. Leroy Quet === Subject: Re: Probability(X is Prime) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i87EGdl21188; come up with a formula for Prob(X is Prime)and we seemed to have >failed so far. You will not succeed at all until you define your terminology. A formula in the sense you ask is impossible; The probability, for any X is either 0 or 1. You need to define a pdf before you can begin talking about probability. One such is the following: Let S = {x | (1-eps)X < x < (1+eps) X} x in Z+ for some X. and (say) 1/2 < eps < 1 Thus, S is the set of all integers x, near X. If we select x uniformly at random from the set S, then the probability that x is prime is approximately 1/log(X). You can take x uniformly from all of Z+; no such density function exists. You have to take some bounded subset of Z+.