mm-2769 === Subject: LOOK YOU MORONS my favorite subject heading! Give me one example of something that is 1/ TRUE 2/ NOT PROVABLE 3/ IS PROVEN TO NOT BE PROVABLE G is one. Any others? NO, because its a rigged question, its impossible to know something is true without proving it, so if you told me something was ABSOLUTELY TRUE, and PROVEN IT COULD NEVER BE PROVEN I'm not going to believe you. Hence Godels proof is a fallacy, by definition there can be no examples to support it. Herc ps not provable means it absolutaly has no proof, it doesn't mean your grandma found it unprovable. === Subject: Re: LOOK YOU MORONS > my favorite subject heading! > Give me one example of something that is > 1/ TRUE > 2/ NOT PROVABLE Provable is a relative term. With Euclidean geometry without the parallel postulate, you cannot prove that the sum of the angles in a triangle is 180 degrees. With Euclidean geometry with the parallel postulate, you _can_ prove the sum of the angles in a triangle is 180 degrees. The Pythagorean Theorem works the same way: You need the parallel postulate. In spherical geometry, it's not even true. (Construct a triangle on the surface of the earth with vetices at the north pole, 0 degrees W on the equator, and 90 degrees W on the equator. Then the Pythagorean Theorem would say that R^2 = R^2 + R^2, where R is the distance between any two vertices, 10000 km give or take a few millimeters.) An even more radical example is: You can prove 2*3 = 6 in Peano arithmetic, but if you leave out the multiplication definition, you can't even _state_ the result. > 3/ IS PROVEN TO NOT BE PROVABLE The provable in 2/ is not the proven in 3/. More precisely, you should ask, for a finitely (people usually leave out this property) describable axiom system S which you can do number theory in, whether there is a statement G(S) which is 1. True 2. Not provable IN S, 3. Proven (using a system OTHER THAN S) to not be provable IN S. and the answer is YES. Godel tells how to find the statement G(S), which is a non-trivial thing to do (you have to pack the description of S into a form that the system S can understand; read Hofstadter's _Godel, Escher, Bach_, or Rudner's _Infinity And The Mind_). > G is one. > Any others? There is one G for every finitely describable axiom system where you can do number theory, and I agree, the formulation of G is a let-down. There are more, but they're harder to find. One final rant: Why couldn't G(PeanoAxioms) be the Goldbach Conjecture, for instance? Tell us something useful, Kurt! --- Christopher Heckman === Subject: Re: LOOK YOU MORONS : : > my favorite subject heading! : > Give me one example of something that is : > 1/ TRUE : > 2/ NOT PROVABLE : : Provable is a relative term. no, that's a weak notion of provable. a predicate has a proof or it doesn't. mathematics is the set of disciplines that are without error. I'm not interested which. : : With Euclidean geometry without the parallel postulate, you cannot : prove that the sum of the angles in a triangle is 180 degrees. With : Euclidean geometry with the parallel postulate, you _can_ prove the sum : of the angles in a triangle is 180 degrees. : : The Pythagorean Theorem works the same way: You need the parallel : postulate. In spherical geometry, it's not even true. (Construct a : triangle on the surface of the earth with vetices at the north pole, 0 : degrees W on the equator, and 90 degrees W on the equator. Then the : Pythagorean Theorem would say that : R^2 = R^2 + R^2, where R is the distance between any two vertices, : 10000 km give or take a few millimeters.) : : An even more radical example is: You can prove 2*3 = 6 in Peano : arithmetic, but if you leave out the multiplication definition, you : can't even _state_ the result. : : > 3/ IS PROVEN TO NOT BE PROVABLE : : The provable in 2/ is not the proven in 3/. More precisely, you : should ask, for a finitely (people usually leave out this property) : describable axiom system S which you can do number theory in, whether : there is a statement G(S) which is : : 1. True : 2. Not provable IN S, : 3. Proven (using a system OTHER THAN S) to not be provable IN S. : : and the answer is YES. Godel tells how to find the statement G(S), : which is a non-trivial thing to do (you have to pack the description of : S into a form that the system S can understand; read Hofstadter's : _Godel, Escher, Bach_, or Rudner's _Infinity And The Mind_). : : > G is one. : > Any others? : : There is one G for every finitely describable axiom system where you : can do number theory, and I agree, the formulation of G is a let-down. : There are more, but they're harder to find. : : One final rant: Why couldn't G(PeanoAxioms) be the Goldbach Conjecture, : for instance? Tell us something useful, Kurt! : : --- Christopher Heckman Because G is no different to this is false, its self deprecated and inherently contains no information. That's why you have failed to give an example. Herc === Subject: Re: Integration over 0,infinity The question in point is the second one. If there is any closed form about Integrate[Exp[-(at+b)^2]Erf[ct+d],{t,0,Infinity}]. Manuela === Subject: Re: indefinite integral and area > Now my problem comes a few sections later when they relate the indefinite > integral to the definite integral. > Integral(f(x), wrt x, from a to b) = [Integral(f(x), wrt x)] evaluated at b > minus [Integral(f(x), wrt x)] evaluated at a > My qustion is, what does [Integral(f(x), wrt x)] evaluated at b mean in > terms of area? If f(x) is constant, say equal to 3, then the area represented would be that of a rectangle. What is the area of a rectangle of height 3? If you do no know both the height and the length of that rectangle, asking about the area is meaningless. If you know both a and b then the length is b-a, but if you only know one of them, you do not know the length, so asking about the area is meaningless. === Subject: Re: OK, WHAT theory can't you prove? Name one that isn't G. <4344883b$0$70955$892e7fe2@authen.white.readfreenews.net> <434493d3$0$3029$892e7fe2@authen.white.readfreenews.net> <94g9k1p5326bom2qvg8lkvtttg8fqeo9bi@4ax.com> <4344a005$0$14107$892e7fe2@authen.white.readfreenews.net> <4344ccc4$0$45681$892e7fe2@authen.white.readfreenews.net : >: > : >: >: > : >: >: >: > : >: >: >: >I'm being asked to produce the formalism of mathematics to support my argument > : >: >: >: >against your claims that there are all these formula you can't prove that are true. > : >: >: >: : >: >: >: >YOU make the claim that you prove the UNPROVABLE realm, > : >: >: >: >support it with one example. > : >: >: >: : >: >: >: >Prove 1 fact we all reasonably know is true has no proof. > : >: >: > : >: >: Fact: > : >: >: > : >: >: In the euclidean plane, given any line L1 and any point P not on the > : >: >: line L, there is one and only one line L2 through P such that L2 is > : >: >: parallel to L1. > : >: >: > : >: >: We all know this fact, and it is actually true. > : >: >: > : >: >: But you can't prove it from Euclid's axioms. > : >: >: > : >: >: Try it -- see if you can prove it. Of course, that means you have to > : >: >: look up the axioms to see what you are allowed to assume. Also, don't > : >: >: try to cheat and use the parallel postulate since that's not one of > : >: >: the axioms. In fact, the statement you are trying to prove _is_ the > : >: >: parallel postulate. > : >: >: > : >: >: For over 2000 years people tried to prove the parallel postulate and > : >: >: failed. I'm sure Euclid also tried. It wasn't until the 1800s that the > : >: >: reason for all the failures became clear. You can't prove it, not > : >: >: because it isn't true (in the euclidean plane, it is true), but > : >: >: because it's unprovable from the standard axioms. > : >: >: > : >: >: And that it's unprovable _can_ be proved. > : >: >: > : >: >: quasi > : >: : >: : >: >No it can't. > : >: > : >: Yes it can. > : >: > : >: >You just said it has a proof. > : >: > : >: I never said the parallel postulate had a proof -- in fact, I said the > : >: opposite, > : >: > : >: What I said was that it was _true_. I didn't say it was provable. > : >: > : >: Euclid knew it was true, but had no proof. For thousands of years it > : >: was an unsolved problem to find a proof of the parallel postulate. > : >: Nobody doubted that it was true -- the question was how to prove it > : >: from the axioms. > : >: > : >: But the problem was finally resolved in the 1800s, not by proving it, > : >: but rather by proving that it's unprovable. > : >: > : >: quasi > : : >Well I hate to burst your bubble but any line and point can be rotated and shifted and scaled > : >onto y=0 and p=(0,1), for which I just proved there is only one line that doesn't intersect. > : You didn't prove it based on Euclid's axioms, that's for sure. The > : cartesian plane has the parallel postulate built in, but in Euclid's > : geometry you have no coordinate system, no real number line, so you > : axioms, even though for a long time they were thought to be complete, > : in actuality, they are not strong enough. > : >You must be thinking of a specific set of axioms, > : Yes! Euclid's axioms. > : >this is not a restriction. > : It surely is a restriction. > : In any given system there has to be a declared set of axioms otherwise > : saying a statement is provable has no meaning. > : >Are you saying there is no way for you to KNOW it is true? > : What I'm saying is this: > : Knowing something is true is not the same as a formal proof. > : Euclid knew that the parallel postulate was true, but he was unable to > : prove the truth from his fundamental axioms. > : He could have resolved the issue simply by declaring the parallel > : postulate to be an additional axiom, but he didn't, presumably because > : he wasn't sure whether or not it was provable. All he knew is that he > : couldn't prove it. > : >Just line up your sights > : >and bit of trial and error... seems true? Its bloody trivial. I just proved it myself in 2 minutues. > : But your proof doesn't qualify as a proof in Euclid's system. Go look > : up Euclid's axioms so you know what the rules of the game are. If you > : play by the rules, you won't prove it in 2 minutes. You won't prove it > : in 2 years. > this is quite annoying, you just said it has no proof outright. > its not a question of axioms, its a proposition, its either true or false. Yes, but you cannot determine which it is! It's like me saying a particular equation has one solution, which is 0 or 1, but not both. What is the solution to that equation? Is the solution 0? (The answer to Is the solution 0? is, after all, yes or no.) > does the PP have a proof? Not assuming only the axioms: 1. You can draw a straight line from any point to any other. 2. You can produce a finite straight line continuously in a straight line. 3. You can describe a circle with any centre and distance. 4. All right angles are equal to each other. with the non-geometric axioms: 1. Things which equal the same thing also equal one another. 2. If equals are added to equals, then the wholes are equal. 3. If equals are subtracted from equals, then the remainders are equal. 4. Things which coincide with one another equal one another. 5. The whole is greater than the part. You cannot prove the Parallel Postulate using these axioms. The Parallel Postulate is neither true or false; it is possible to interpret this Geometry without PP system one way where the PP is false, and to interpret it another way where the PP is true. --- Christopher Heckman === Subject: Re: OK, WHAT theory can't you prove? Name one that isn't G. : : > : : > : >: : > : >: >: : > : >: >: >: : > : >: >: >: >I'm being asked to produce the formalism of mathematics to support my argument : > : >: >: >: >against your claims that there are all these formula you can't prove that are true. : > : >: >: >: > : > : >: >: >: >YOU make the claim that you prove the UNPROVABLE realm, : > : >: >: >: >support it with one example. : > : >: >: >: > : > : >: >: >: >Prove 1 fact we all reasonably know is true has no proof. : > : >: >: : > : >: >: Fact: : > : >: >: : > : >: >: In the euclidean plane, given any line L1 and any point P not on the : > : >: >: line L, there is one and only one line L2 through P such that L2 is : > : >: >: parallel to L1. : > : >: >: : > : >: >: We all know this fact, and it is actually true. : > : >: >: : > : >: >: But you can't prove it from Euclid's axioms. : > : >: >: : > : >: >: Try it -- see if you can prove it. Of course, that means you have to : > : >: >: look up the axioms to see what you are allowed to assume. Also, don't : > : >: >: try to cheat and use the parallel postulate since that's not one of : > : >: >: the axioms. In fact, the statement you are trying to prove _is_ the : > : >: >: parallel postulate. : > : >: >: : > : >: >: For over 2000 years people tried to prove the parallel postulate and : > : >: >: failed. I'm sure Euclid also tried. It wasn't until the 1800s that the : > : >: >: reason for all the failures became clear. You can't prove it, not : > : >: >: because it isn't true (in the euclidean plane, it is true), but : > : >: >: because it's unprovable from the standard axioms. : > : >: >: : > : >: >: And that it's unprovable _can_ be proved. : > : >: >: : > : >: >: quasi : > : >: > : > : >: > : > : >: >No it can't. : > : >: : > : >: Yes it can. : > : >: : > : >: >You just said it has a proof. : > : >: : > : >: I never said the parallel postulate had a proof -- in fact, I said the : > : >: opposite, : > : >: : > : >: What I said was that it was _true_. I didn't say it was provable. : > : >: : > : >: Euclid knew it was true, but had no proof. For thousands of years it : > : >: was an unsolved problem to find a proof of the parallel postulate. : > : >: Nobody doubted that it was true -- the question was how to prove it : > : >: from the axioms. : > : >: : > : >: But the problem was finally resolved in the 1800s, not by proving it, : > : >: but rather by proving that it's unprovable. : > : >: : > : >: quasi : > : > : > : >Well I hate to burst your bubble but any line and point can be rotated and shifted and scaled : > : >onto y=0 and p=(0,1), for which I just proved there is only one line that doesn't intersect. : > : : > : You didn't prove it based on Euclid's axioms, that's for sure. The : > : cartesian plane has the parallel postulate built in, but in Euclid's : > : geometry you have no coordinate system, no real number line, so you : > : axioms, even though for a long time they were thought to be complete, : > : in actuality, they are not strong enough. : > : : > : >You must be thinking of a specific set of axioms, : > : : > : Yes! Euclid's axioms. : > : : > : >this is not a restriction. : > : : > : It surely is a restriction. : > : : > : In any given system there has to be a declared set of axioms otherwise : > : saying a statement is provable has no meaning. : > : : > : >Are you saying there is no way for you to KNOW it is true? : > : : > : What I'm saying is this: : > : : > : Knowing something is true is not the same as a formal proof. : > : : > : Euclid knew that the parallel postulate was true, but he was unable to : > : prove the truth from his fundamental axioms. : > : : > : He could have resolved the issue simply by declaring the parallel : > : postulate to be an additional axiom, but he didn't, presumably because : > : he wasn't sure whether or not it was provable. All he knew is that he : > : couldn't prove it. : > : : > : >Just line up your sights : > : >and bit of trial and error... seems true? Its bloody trivial. I just proved it myself in 2 minutues. : > : : > : But your proof doesn't qualify as a proof in Euclid's system. Go look : > : up Euclid's axioms so you know what the rules of the game are. If you : > : play by the rules, you won't prove it in 2 minutes. You won't prove it : > : in 2 years. : > : : > this is quite annoying, you just said it has no proof outright. : > its not a question of axioms, its a proposition, its either true or false. : : Yes, but you cannot determine which it is! : : It's like me saying a particular equation has one solution, which is 0 : or 1, but not both. What is the solution to that equation? Is the : solution 0? (The answer to Is the solution 0? is, after all, yes or : no.) : : > does the PP have a proof? : : Not assuming only the axioms: : : 1. You can draw a straight line from any point to any other. : 2. You can produce a finite straight line continuously in a straight : line. : 3. You can describe a circle with any centre and distance. : 4. All right angles are equal to each other. : : with the non-geometric axioms: : : 1. Things which equal the same thing also equal one another. : 2. If equals are added to equals, then the wholes are equal. : 3. If equals are subtracted from equals, then the remainders are equal. : 4. Things which coincide with one another equal one another. : 5. The whole is greater than the part. : : You cannot prove the Parallel Postulate using these axioms. : : The Parallel Postulate is neither true or false; it is possible to : interpret this Geometry without PP system one way where the PP is : false, and to interpret it another way where the PP is true. : : --- Christopher Heckman that's because you worded it vaguely. use on a cartesian plane, can ... and intersect the lines and solve the variables, must be parallel. QED Herc === Subject: The meaning of 0/0 At last I think I've reached to the explanation of what is 0/0 and x/0 in general. See: http://zaljohar.tripod.com/consistent.txt Have a nice time! === Subject: Re: Missouri State University Problem Corner >> http://math.missouristate.edu/~les/POTW.html > > Anyone tried the Challenge puzzle? I've found it to be very difficult. > Find a finite set S of at least two points in the plane, such that > the perpendicular bisector of the segment joining any pair of points > in S contains exactly two points of S > I found it rather easy. The first figure I tried, worked. > - Tim Dead easy! The boundary of any convex set with interior points. Circles and ellipses, for example. And it works for perpendiculars at any interior points (non-end-points) of the segments, they need not be bisectors. === Subject: Re: Missouri State University Problem Corner >> Find a finite set S of at least two points in the plane, such that >> the perpendicular bisector of the segment joining any pair of points >> in S contains exactly two points of S > The boundary of any convex set with interior points. > Circles and ellipses, for example. Note the third word ;^) - Tim === Subject: Re: Missouri State University Problem Corner >http://math.missouristate.edu/~les/POTW.html >>Anyone tried the Challenge puzzle? I've found it to be very difficult. > Find a finite set S of at least two points in the plane, such that > the perpendicular bisector of the segment joining any pair of points > in S contains exactly two points of S > I found it rather easy. The first figure I tried, worked. > - Tim You must have great intuition-- I've tried all the simple figures I could think of to no avail. Care to offer any hints? (I realize this is an informal competition so feel free to refuse or to contact me by email. I won't submit a solution if I receive any outside help but the solutions page hasn't been updated in a while and I'm really curious to know the solution.) === Subject: Re: Missouri State University Problem Corner > You must have great intuition-- I've tried all the simple figures I > could think of to no avail. Care to offer any hints? I started from a figure that partly satisfied the conditions, and put it inside another one. It seemed a happy coincidence that all the dangling bisectors lined up with each other's points. Since then I've seen an alternative construction that makes it more obvious how it satisfies the conditions. - Tim === Subject: Re: Anti-Cantor Cranks Your link is not readable. Besides I had discussions on the subject of a new way of representing the infinite and I had posed arguments against cantor. see http://zaljohar.tripod.com/cantor.txt and http://zaljohar.tripod.com/discussion.txt Enjoy! Zuhair === Subject: Re: Quaternions to Euler > My question is how do I mathematically convert the quaternion (Q) > back to Euler Angle format *while* still avoiding the singularities > present at theta=90 degrees? You don't. The singularities are inherent in the Euler angle representation itself. Your question is similar to asking how to find longitudes of points near the South Pole without nearby points having very different values. - Tim