mm-1849 Ah, yes, well, I think a concept is a kind of experience, actually. It's an experience whose content is other experiences. IIRC, tghis occurred to me whne I read Kant, because I couldn't see how thinking about an experience was necessary to having it. If this was/is misreading Kant, so be it. It makes sense to me. :-) voY/zkqp3jee/Z1L8hfEx0E= =sHRY Sure. But that could be then matched and exceeded with another recount of the evens. When we try to generalize from finite sets to infinite sets, there are often surprises. Sure. We can certainly say that there are more reals than there are integers. Where? Where are there more reals than integers? Neither exists outside our heads, and only specific ones can exist inside our heads. -- Mercifully free of the ravages of intelligence -- Time Bandits But perhaps they *do* exist outside our heads. In fact the assumption that they do leads to specific falsifiable conclusions, not yet falsified, and with no equally satisfying alternative explanation. So how do they get inside our heads? Or is that one of the falsifiable conclusions that hasn't yet been falsified? The criterion of falsifiability is applied only to quantitative assertions about what happens in the world external to our consciousness. In short it is a criterion applied only to objective assertions about external reality. Bob Kolker truth and the idea of truth in our heads?Jesus! And to think I already had the epitaph for this thread ready. I shoulda knowd you'd come to the rescue with more idiotic garbage. That, of course, is the sticky point, at least for materialists. As it happens, I'm not a materialist, but neither am I prepared to give an account of how we come to know mathematical truths that depends on my non-materialism. Nevertheless, a materialist can still observe the truth of my claim that you quoted. For that matter, much the same thing could be said about our notions of, say, electrons. How do *those* notions get inside our heads? I hope it's obvious that the mere fact that there are *actual* electrons inside our heads is quite irrelevant to the question. I don't really follow what you're asking here. Of course. But they pontificate nonetheless. Well that makes one of us. Do tell. And how can materialists observe the truth of such claims? Absolutely. Extremely pertinent question for those who maintain science and truth exist regardless of where they suppose them to exist. Good point. Quite. A facetious comment which can safely be ignored if you can frame the question of science and truth inside our heads correctly. You are being obtuse. Mathematical existence is conceptual and abstract, not physical. Conceptually one can prove it is impossible to map the integers one to one onto the reals. The assumption that one can leads to a contradiction. The method of proof is the so-called diagonal argument, which in slightly different form is used to prove that the halting problem for Turing Machines is algorithmically unsolvable. This is of some use to computer types. It means one cannot write a general program that determines whether any other program will halt for a given input or go into a loop or an uninteruptable wait state. In a slightly different form one can show there is not general program for deciding whether two programs are equivalent, i.e. give the same answers when they halt or both not halt. So you see there is some use for Cantor's form of argumentation. Bob Kolker Hence Bob's observation that Euclidean geometry is not true. As an applied discipline it does not describe the physical world corretly. We know that spacetime is not flat (Euclidean, 0 curvature) experimentally. Bob Kolker A somewhat different claim than Euclidean geometry is not true. When all else fails, by all means change the claim and claim no change has been made. I was unaware Euclidean geometry had a spacetime. No offense, but the statement as it stands is incorrect. A mathematical theory cannot be empirically falsified. Only a mapping between a mathematical theory and a set of observable phenomena can be statement that the geometry of physical space is Euclidean, is empirically false. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same up. I meant that Euclidean geometry interpreted empirically gives wrong answers at certain scales. Correct. That is the right statement. Bob Kolker Euclidean geometry is emprically false. Assuming the space time continuum is euclidean leads to results disproven by experiment. Is that not true enough for you? Bob Kolker I still want to say that this is not true, and I think I can perhaps explain it better than I did last time. In order to express the laws of physics in Euclidean geometry, very complicated formulas are needed. I get the impression that you are impressed by the fact that in some cases part of the required formula will be the same as what the whole formula will be in a suitable non-Euclidean geometry (or what physicists thought the whole formula was before they discovered relativity). At one point, you suggested that doing a Euclidean version of the laws of physics requires extra forces; I suspect what you meant is that the Euclidean laws will often have these parts which correspond to the easier formulations of the laws, and you have some sort of view that those are the real laws, so the additional components of the Euclidean laws are extra. However, there is no basis for making such a distinction. The complicated formula for gravity in a Euclidean geometry doesn't consist of gravity plus some extra forces, it consists of a very complicated formula which all represents gravity. -- Aaron Boyden The main division between the so-called Continental and Analytic traditions has been disputes over whether the task of being unclear should be carried out in natural language or in a formal system. : I mean that, other than giving pleasure as a form of mental : masturbation for mathematicians, does it have any utility to : Science? There are no infinities in reality, only very large : numbers of things. Why the hostile tone? Does philosophy have any use other than mental masturbation for philsophers? Feynman, one of the better scientists of the 20th century had this to say about philosophers: Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong. What is useful about Cartesian dualism, or free will vs determinism? How does it help Science? And how do you know there are no infinities in reality? One real consequence of Cantor's ideas is that unsolvable problems exist. Stephen Sorry, didn't mean to sound hostile. This conversation about infinities, counting and cardinality just seemed to me to be banging it's head against a brick wall of My definitions are better than your definitions. Not often. Doesn't sound very scientific to me. Dunno. The *real* question is How would anybody know? Counting in real life is different than in mathematics. OK. Good for Cantor. But a philosopher could have told him that if he had just asked. But given that, what is the utility of defining infinities, counting and cardinalities the way he told you to? -- Mercifully free of the ravages of intelligence -- Time Bandits [...] Sure, we all know that unsolvable problems exist. We've all run into them, practically speaking. What we didn't know before modern mathematics (which includes a whole lot more than Cantor) is how to tell whether a given problem was solvable or not. A further consequence is that it's possible to tell whether a good approximation to a solution is possible, and not just in principle, but in practical terms - ie, how much computer time it takes to produce a good approximation to the solution (which can't be had, mind.) One even calculate the range of difference from the solution, but not just where in that range the calculated solution falls... Only mathematical problems. Outside of Applied Math, it's arguable whether it matters one way or the other. Oh, yes. 'Good approximations' have given us 3.57 uncracked eggs, partial pregnancies, 2.5 children, and modern economics. -- Mercifully free of the ravages of intelligence -- Time Bandits Those are -averages-, not approximations. Bob Kolker I fail to see the difference. -- Mercifully free of the ravages of intelligence -- Time Bandits Aproximation is convergence to a definite value by means of a repetative algorithm. Averaging over a set of numbers consists of adding them up and diving by the number of addends. Contrast finding the square root of two by the Newton-Rapheson approximation to finding the average age of persons living in Philadelphia on a given day. Different processes entirely. One knows ab initio that the averge of a set of numbers is not likely to be a number in that set. Bob Kolker By the cardinality of the set of addends? The only difference is that this approximation only requires a single iteration. Ab initio one knows nothing of the kind, Bob. Yet one more screwed up definition. -- Mercifully free of the ravages of intelligence -- Time Bandits Albert said: Albert, I am sorry, but if you don't see the difference between averaging a set of known values based on a simple and exact calculation, and approximating a value that cannot be calculated exactly, like pi, then you are being obtuse. You just seem to want to fight these days. -- Smiles, Tony Of course I see that difference. The difference I was referring to was the difference between mathematical and non-mathematical definitions. -- Mercifully free of the ravages of intelligence -- Time Bandits : Sorry, didn't mean to sound hostile. This conversation about : infinities, counting and cardinality just seemed to me to be : banging it's head against a brick wall of My definitions are : better than your definitions. Definitions are fundamental. If you do not define things, than how is anyone supposed to know what you are talking about? If someone wants to propose some new definitions, there is no problem with that. But arguing with standard definitions is like claiming that pineapple should mean something else. Afterall, what does a pineapple have to do with an apple? I am sure a simple web search will explain the etymology of the word, but that is not the point. : Not often. : Doesn't sound very scientific to me. : Dunno. : The *real* question is How would anybody know? : Counting in real life is different than in mathematics. How is it different? In real life when you count you pair up the items you are counting with the numbers. If you did not have the language to express large numbers, which was true for a long stretch of human history, you would likely pair up the items you are counting with some other set of objects, such as sticks, stones or shells. Historically this is what people did. Counting is all about pairing up one set of objects with another set of objects. : OK. Good for Cantor. But a philosopher could have told him that : if he had just asked. But given that, what is the utility of : defining infinities, counting and cardinalities the way he told : you to? Another definition problem here. I have a very specific definition of unsolveable problem in mind, which is likely not what you have mind. These are problems that have a solution, there is just no way to actually calculate it. However that is a whole-nother crank magnet and this thread as become far to unwieldly. Stephen [. . .] The problem everyone seems to have with infinities is that there seems to be no mathematical definition for infinity except a lack of definition. And just as obviously when mathematics deals with infinities it has to be dealing with something other than a lack of definition. Here is a lack of a definition. A set is infinite if and only if it can be mapped one to one onto a proper subset of itself. Is that undefined enough for you? Bob Kolker It's certainly undefined enough for you, Bob. Of course the question remains as to whether there are such defined subsets that are subsets of undefined sets or defined sublines of lines and whether Bob is a subset of Bob when he uses such terms without specifying what he is talking about. I really have to find a coffee-proof keyboard. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same That's hard work:-( Still, it was worth it, you proof was most ingenious. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same Sorry about that. Use a hair dryer to clean up the keyboard. Bob Kolker You are an asshole. That puts you into one to one correspondence with a proper subset of yourself. I therefore conclude you are infinite. Bob Kolker Yes, I agree. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same : You are an asshole. That puts you into one to one correspondence with a : proper subset of yourself. I therefore conclude you are infinite. : Bob Kolker That reminds me of something Einstein said: Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. I think you have just proven Einstein was correct about the latter. :) Stephen iD8DBQFCHgn2vmGe70vHPUMRApKlAJ90Y6I/Z9vDbm73IfZyPwNr+/h4CgCg0jnr BFctzwtZjheBwO93jUUkFUI= =Utrx AFAIK, in mathematics there isn't anything that is called infinity, so there is nothing to define. Mathematicians sometime may sometimes use infinity in an informal discussion, but there the context usually indicates what is intended. The symbol for infinity is used as a limit in definite integrals, sums, etc. But that usage is as part of a notational convention and there is no implication that anything exists that is properly called infinity. Not even for the length of a line or points in a line or elements in infinite sets? iD8DBQFCHjFsvmGe70vHPUMRAm0NAJ9+KS1zc9ub8otm3afFAH4At0zlWQCeLUP5 QhRgx/2CHTfSVud0ed2SQd0= =W3VL We will normally say that the line is infinite. We avoid saying that the length of the line is infinity. We perhaps might say that the length of the line is undefined. Likewise, will will say that the set is infinite, but would not say that the set contains infinity elements. We might be a little more sloppy during informal conversations. But by saying the set is infinite, I think the clear intent is that it contains infinite elements and is of infinite cardinality. iD8DBQFCHmV/vmGe70vHPUMRAg8MAJ4ujUdZmVaVl4Nat1hnmkJzpc2/wgCeI/4q vaerbYK/7SwMo7fd4WLvUbM= =4QB8 To say that a set contains infinite elements is to say that at least one of the elements of the set is infinite (whatever that might mean). Unless there is some addition context to account for infinite, then that would not make sense. To say that it is of infinite cardinality is to say that its cardinality is one of the many infinite cardinals. There isn't a particular cardinality known as infinite or as infinity. The set of intengers has no infinite elements and has infinite cardinality. Why don't you learn some mathematics. Bob Kolker : AFAIK, in mathematics there isn't anything that is called infinity, : so there is nothing to define. : Mathematicians sometime may sometimes use infinity in an informal : discussion, but there the context usually indicates what is : intended. : The symbol for infinity is used as a limit in definite integrals, : sums, etc. But that usage is as part of a notational convention and : there is no implication that anything exists that is properly called : infinity. The folks who use the extended reals have defined something called infinity. Of course the extended reals have little if anything to do with the question about the cardinality of sets. Stephen That is correct. Adding the infinities to the reals is just a topological compactification of the real numbers starting with the usual interval topology. It has nothing to do with cardinality, just as you say. Bob Kolker : That is correct. Adding the infinities to the reals is just a : topological compactification of the real numbers starting with the usual : interval topology. It has nothing to do with cardinality, just as you say. : Bob Kolker Finding a bijection between the reals and the extended reals is a neat little puzzler however. Stephen What about the cardinality of the set of integers? Bob Kolker iD8DBQFCHhgmvmGe70vHPUMRAqClAKCXZ8qcBPfHsfKgyx4j29l5QXEyiQCfUkzx eSRwCbiXGA2vWK/S9EFLDtA= =/plS It is usually called aleph zero (or aleph_0). It isn't called infinity. Indeed. And in that new definitions are so easy to invent, then why not invent some new words to go with them, rather than having one word with multiple contradictory definitions. Only a madman, or a mathematician, would do so, knowing the chaos it would generate in conversations with non-mathematicians. Yes. Mathematicians do that all the time. As is obvious in this thread. Or what does the word 'countable' have to do with 'infinity'? There is no etymology that can explain a mathematician's use of words. Old well used words with well known common meanings are appropriated helter-skelter and assigned new meanings that have only random and coincidental relationships to the old meanings, if that much, as anyone reading this thread will note. In real life you actually *do it*. In mathematics you simply define an algorithm for doing it, but never actually *do it*. Except that mathematician's never actually do that. See above. Which is why, I suppose, that you left your sentence above about Cantor purposely ambiguous. Like counting infinities? That is never actually done. Instead an algorithm is offered explaining how it could be done if anyone had infinite time to do it. As well it should. -- Mercifully free of the ravages of intelligence -- Time Bandits : Indeed. And in that new definitions are so easy to invent, then : why not invent some new words to go with them, rather than having : one word with multiple contradictory definitions. Only a madman, : or a mathematician, would do so, knowing the chaos it would : generate in conversations with non-mathematicians. Inventing new words is not all that easy. And anyway, in this case there are no contradictory definitions. Cardinality is well defined and does not contradict any standard definitions. Again, why the hostility towards mathematicians? Are you upset at physicists for using color or string or spin in non-standard ways? : Yes. Mathematicians do that all the time. As is obvious in this : thread. I do not see that in this thread at all. I see people upset that a well defined concept is not immediately intuitive to them. I have not see any mathematicians claiming something should be renamed. : Or what does the word 'countable' have to do with 'infinity'? What does semantic have to do with burden? Is it really beyond your imagination to combine the concepts 'countable' and 'infinity'? : There is no etymology that can explain a mathematician's use of : words. Old well used words with well known common meanings are : appropriated helter-skelter and assigned new meanings that have : only random and coincidental relationships to the old meanings, : if that much, as anyone reading this thread will note. Care to give an example? You appear to be degenerating into some sort of anti-mathematics rant. : In real life you actually *do it*. In mathematics you simply : define an algorithm for doing it, but never actually *do it*. So counting is only counting when you actually do it? I suppose than one cannot talk about 'counting', because it does not exist unless one is doing it? I really fail to see your point. : Except that mathematician's never actually do that. See above. You are ranting. Mathematicians do pair up objects from one set to objects in another set. The set { 0, 1 , 2} has the same cardinality as the set { Bob, Joe, Mary }. : Which is why, I suppose, that you left your sentence above about : Cantor purposely ambiguous. No. In my area unsolveable problem has a very clear definition and I forgot that this is likely not all that familiar to the general public. I am reading all this in sci.math, and the discussion has been largely about math, so I tend to assume people will define all these terms for you. : Like counting infinities? That is never actually done. Instead : an algorithm is offered explaining how it could be done if anyone : had infinite time to do it. No, like determining if an algorithm halts when given an input. But you seem to be in a particularly close minded mood today so I will just drop it. Stephen [. . .] Color, charm, and quark, yes, because they're frivolous and unrelated. String and spin, no, because they bear some relation even if tenuous. I assume you are equally upset with the blue state/red state designation. These are just arbitrary lables. Bob Kolker These are just arbitrary lables. Once is a typo; twice is a misspelling: labels. -- Mercifully free of the ravages of intelligence -- Time Bandits It's the easiest thing we do. My children did it all the time. What case is that? Only for finite sets. To the extent they do that, then they introduce unnecessary confusion. Then you are totally disjunct with the world outside of mathematics. If you don't see that this whole subthread has been fueled by that very practice of mathematicians, then you simply have not been paying attention. No. But it is certainly counter-intuitive and unnecessarily so. Go back and reread this thread. You will find dozens of examples. But you will require the talent of placing yourself in another's shoes. No. Not anti-mathematics. I object to unnecessary obfuscation by mathematicians. Yes. That is the original meaning of counting, before mathematicians gave it a new meaning. Which is my point. similar phrases also have well defined meanings to the general public. An invalid assumption that is obvious by noticing all of the crossposted NGs. A kludge that denies responsiblily for creating a problem. -- Mercifully free of the ravages of intelligence -- Time Bandits What really pisses me off is charm! -- Smiles, Tony It is just a lable. Bob Kolker It's just a frivolous label. So is red state/blue state but it serves a purpose. Bob Kolker robert j. kolker said: Goddamn It! Have a sense of humor, Bob. -- Smiles, Tony And you know this how? I confess to being sympathetic to the view that reality consists of gunk, to use the technical term that has become standard among metaphysicians. Certainly I know of no argument that the world is not gunky. -- Aaron Boyden The main division between the so-called Continental and Analytic traditions has been disputes over whether the task of being unclear should be carried out in natural language or in a formal system. Personally I'm inclined to the somewhat heretical view that space is infinite in dimensional terms if there is nothing in space to finite it. The only other infinity I know of are the infinitessimals. Lester Zick said: I think space is ultimately infinite though I suspect our spacetime bubble is probably finite in age and size based on what we've observed. I also lean toward the infintiessimals when trying to understand infinities, since they are relative inverses; if A is relatively infinite compared to B, B is relatively infinitessimal compared to A. Zero is the flip side of infinity, the other side of the number circle. They are almost interchangeable. -- Smiles, Tony All we've observed so far, Tony, is a cosmic red shift. The rest is pure, unadulterated philosophical speculation and prejudice. I find infinitessimals considerably more useful than infinities because infinitessimals can be delimited even if there are an infinite number of them whereas infinities cannot. Aleph-0 is perfectly well defined and delimited. Bob Kolker Lester Zick said: Well, I think the Big Bang theory is probably the likeliest explanation for the observed redshift, and that it seems to indicate a definite beginning to the expansion, but I suppose it could be disproved somehow in time. Somewhat like a zero number of infinities. How about a zero number of infinitessimals? :) -- Smiles, Tony When we prove the cosmic red shift we'll disprove the big bang and cosmic expansion. The problem is that we've observed the cosmic red shift but haven't proven it true because empiricism can only falsify and doesn't prove the truth of anything. All we know for sure about the cosmic red shift is that it doesn't come from local sources and it isn't indicative of cosmic expansion unless earth is at the center of the universe. Talk about geocentric anthropocentrism! Next we'll have scientists tell us the big bang was just a wink in the eye of god. More like a finite number of infinitessimals. iD8DBQFCHmRGvmGe70vHPUMRAsyCAKDnJdRAYkp5mSTbNMQrlTquhs06awCg+z1c xVlcxo6HU7UFRODScxDq2a8= =OMpV I'm having trouble with that. Can you explain your point? I thought you had already said that cosmic red shift was observed. How does that refute BB? What does proving it true establish that is not already established by the observing? Would you care to explain why it isn't indicative of expansion? And what does earth is at the center of the universe have to do with anything? Are there large numbers of gobs of gunk? Or only one great big gob? Does gunk have a shape? Are there appendages of gunk that can be counted? Tell me more of the Theory of Gunk. -- Mercifully free of the ravages of intelligence -- Time Bandits Gunk does not have smallest parts. Any amount of gunk can always be divided into smaller amounts of gunk. So if the universe is all gunk, then far from there being no infinities in the universe, it turns out that any given thing you would care to name contains infinitely many parts, as it can be divided endlessly. -- Aaron Boyden The main division between the so-called Continental and Analytic traditions has been disputes over whether the task of being unclear should be carried out in natural language or in a formal system. The Sophist said: I like the gunk theory - very cute. Personally I would imagine the but unlike some others here, I have no problem imagining that there are an infinity of points in finite space and an infinite of moments in finite time. Even if these aren't tangible objects, they are identifiable places in the universe. Still, gunk has a good sound to it..... -- Smiles, Tony Well, the idea certainly has the advantage of continuing in the same direction as the last several thousand years of investigation. -- Mercifully free of the ravages of intelligence -- Time Bandits Archive: no Hello everybody, y} we have the so called generalized left-continuous inverse of the distribution function. Could somebody help me to prove that G(y) is effectively left-continuous, or point me to a proof of this fact? Show this without using the fact that continuous functions are integrable. How would you do this? Upper and lower sums. The Lipschitz property will help a lot. Exactly, but I don't know exactly what lower function and upper function to choose. Take a partition into equal parts. The max and min values on each part are estimated using the Lipschitz property and the length of the part. Use this information to choose how many parts are needed to get upper & lower within epsilon of each other. Say I choose Sum(k=1.....n) Sup f [x_k, x_k-1](x_k - x_(k-1) as an upper sum and Inf f[x_k, x_(k-1)](x_k - x_(k-1)) as a lower sum. Say I call l_k each Inf and L_k each Sup. Then: Sum(k=1.....n) L_k - l_k(x_k-x_(k-1))< m for any m I pick. But the lipschitz property says that for every interval [x_k, x_k-1], L_k - l_k