Galileo's Paradox
in [Math]
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From: Virgil on 9 Dec 2006 20:14 In article <457b568a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > >>> The absurdity is to suppose that sets have only one measurable quality. > >>> > >> Nobody says that. > > > > Then why is TO trying to exile cardinality? > > > > > > I'm not. I'm just trying to dethrone the court jester. Which one, Zick, WM, EB or TO? And what makes you think that any jester is enthroned, other than Dubya temporarily. > > >> Sets may have many measures. The question is how best > >> to formulate and compare sizes for infinite sets, the size being the > >> count of elements. > > > > For finite sets, "count" is unambiguous, but for infinite sets, it can > > have more than one meaning, and therefore more than one measure, > > depending on which "counting" process one is using. > > Yeah. some with better powers of distinction than others. Cardinality is > woefully blind. It sees what it is looking for, which is all one can require of any measure.
From: Virgil on 9 Dec 2006 20:19 In article <457b5750(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457b2f54(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> There is nothing wrong with saying E and N have the same cardinality. > >>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse > >>>> Function Rule, which indeed can parametrically compare sets mapped onto > >>>> the real line, using real valued functions, as a generalization to set > >>>> density. It works for finite and infinite sets. So, what is wrong with > >>>> trying to form a more cohesive theory of infinite set size, which > >>>> distinguishes set sizes that cardinality cannot? There certainly seems > >>>> to be intuitive impetus for such a theory. > >>> Fine. Please state your rule. Let's take a look. > >>> > >> I've been over this a lot, but hey, what's one more time. Practice makes > >> perfect. > >> > >> We start with the notion of infinite-case induction, such that a > >> equation proven true for all n greater than some finite k holds also for > >> any positive infinite n. > > > > Whatever for? Math already has both finite and transfinite induction. > > > > For the purpose of establishing an intuitively satisfying theory of > infinite sets. Duh. We already have several intuitively satisfying theories of infinite sets, at least satisfying to those who know enough about set theories to be competent to judge. Would TO require the design of bridge to be decided on only by those who know no engineering? Or surgery to be done by those with no medical training or experience? > > >> Inequalities can also be proven true for > >> infinite n, but only provided that the difference between expressions > >> which forms the inequality does not have a limit of 0 as n grows without > >> bound. If it does, the inequality holds only for finite n. Now, given > >> this extension of classical inductive proof, we can easily prove such > >> facts as, say, 2<n <-> 2 < n < 2n < n^2 < 2^n < n^n, and this ordering > >> will be true for any infinite n. Thus we have a full spectrum of > >> infinite expressions which can be ordered, provided we have some common > >> infinite n with which to express them. Okay so far? > > > > Nope. Until one has unambiguous definitions of what is meant by the > > various arithmetical operations with infinite "numbers" as arguments, > > the whole thing is nonsense. > > Not so. If we can express infinite values in a language with rules that > enable us to order the various possible expressions, then we have > ordered the values expressed in that language. You can call it nonsense > if you like, but all that means is that you can't make sense of even the > simplest concepts that you haven't already been taught before your > neurons got calcified. Tsk tsk.
From: David Marcus on 9 Dec 2006 20:32 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Fine. Please state your rule. Let's take a look. > >> I've been over this a lot, but hey, what's one more time. Practice makes > >> perfect. > >> > >> We start with the notion of infinite-case induction, such that a > >> equation proven true for all n greater than some finite k holds also for > >> any positive infinite n. > > > > What is a "positive infinite n"? > > A value greater than any finite value. > > >> Inequalities can also be proven true for > >> infinite n, but only provided that the difference between expressions > >> which forms the inequality does not have a limit of 0 as n grows without > >> bound. If it does, the inequality holds only for finite n. Now, given > >> this extension of classical inductive proof, we can easily prove such > >> facts as, say, 2<n <-> 2 < n < 2n < n^2 < 2^n < n^n, and this ordering > >> will be true for any infinite n. Thus we have a full spectrum of > >> infinite expressions which can be ordered, provided we have some common > >> infinite n with which to express them. Okay so far? > > > > No. See above. > > Why do you have a problem with the mere suggestion of an infinite value? I don't have a problem with infinite values. However, you have to do more than merely say "positive infinite n". Assuming your positive infinite things aren't the same as something that we already know about (in which case, you should just say so), you either need to give a construction of these positive infinite things or you need to specify their properties. You haven't done either. For example, how many of these things are there? How do they relate to each other? How do they interact with the natural numbers? Are the operations of addition and multiplication defined for them? -- David Marcus
From: cbrown on 9 Dec 2006 20:51 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > >> What,in mathematics, has a solution which is neither a real measure, or > >> the measure of truth of a statement, 0, 1, or somewhere in between? > > > > "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The > > solution to which is the set {(2,4), (4,2)}, which doesn't appear to be > > a "real measure", nor a "measure of the truth of a statement" (as far > > as I can understand your meaning of the terms). > > Those are specifications for two points in a 2D array of naturals, the > values within each pair denoting the distance of each point in each of > the two directions, from the origin. That's one way of imagining it. Another interpretation would be that each pair (x,y) is a specification for the point on the real line whose distance from the origin is denoted by 2^x/3^y (this is a dense set). Yet another interpretation is that the pair (x,y) is a specification of the set of all x by y chessboard positions containing 4 non-attacking rooks. Are you saying that the "solution" of {(2,4), (4,2)} in these interpretations would be somehow different from the solution in your interpretation? Why is an interpretation required at all? > Are you saying x and y and 2 and 4 > are not taken to be quantities? That's a rather strange position to take. > Well, your phrase "real measure" is a little vague. I really don't understand what you mean by it. Sure, I understand that 2 and 4 are quantities in the usual sense of the word: "2 apples", "4 apples". And I understand that when I add 1/2 cup of apple juice to 1/2 cup of apple juice, I get 1 cup of apple juice; "1/2" is a "real measure". But the "solution" to the problem given is not 2 or 4 or 1/2; it is a set of ordered pairs of naturals. Why is the set {(2,4), (4,2)} a "quantity" or a "real measure"? Can I have "{(2,4), (4,2)} apples", or "{(2,4), (4,2)} cups of apple juice"? > > > > I assume that you offer more than the trivial observation that all > > mathematical statements P, including the statement "2^4=4^2", are > > examples of the (boolean) truth valued statement "it can be proved that > > P". If so, I would claim that the mathematical question is "Find a > > proof of P, or proof of not P". And the solution is not "0" or "1"; the > > solution is either a proof of P, or a proof of not P. > > First of all, that trivial observation is plenty. All logic is subsumed > under math as a calculation of truth values between 0 and 1. I think you are confusing boolean algebra with mathematical proof. > Further, a > proof is precisely this calculation of truth value for a given > statement. To say "find a proof" is to say "define a sequence of > operations on the given statements to produce another such that its > value is 1". A proof is a series of calculations such as: from the strings "(p->q)" and "p" use the rule modus ponens to produce the string "q". It ends with a step which produces the string corresponding to the assertion of the theorem. These are not calculations producing the value 1, except in the trivial sense that the statement "It is the case that George W Bush is the current president of the US" can also be called "a calculation producing the value 1". In which case, the "solution" to just about any question you like is "a measure of truth of a statement, 0, 1, or somewhere in between"; so I don't see how it's a particular property of /mathematical/ questions as opposed to other types of questions. > Truth is a form of quantity. In different contexts there are different axioms and different rules of inference; so "truth" is certainly not a "quantity" that is fixed for some statement such as "there exists x such that 2*x = 3". > > > > > "Find all finite groups G having a maximal subgroup S, and having a > > subgroup T which is isomorphic to S but not maximal". This question was > > asked in sci.math a few days ago. The groups in question are not > > "numbers" at all; and a set of them possesses no particularly natural > > total orderings. They can be partially ordered by "size" (number of > > elements); but there are, in general, multiple distinct groups on a set > > of any given size. > > Assign each element a bit, and every group corresponds to a binary > string, which corresponds to a value. #1: There is 1 group of order 1. #2: There is 1 group of order 2. #3: There is 1 group of order 3. There are two groups of order 4, both are commutative. I will arbitrarily number them as #4 : 0 + x = x; x + x = 0; 1 + 2 = 3; 1+ 3 = 2; 2 + 3 = 1 #5 : 0 + x = x; 1 + 1 = 2; 1 + 2 = 3; 1+ 3 = 0. But how does this numbering help me figure out what the groups of order 5 are? I'm not arguing that it is impossible to count these groups (i.e., assign a unique natural to each one). I'm arguing that /before/ you can count them, you must produce them. /Producing/ them is then the "solution" to the problem, not counting them /after/ they have been produced. I think you haven't been exposed to much mathematics where this is the case; and I'm guessing that that's why you seem to assume that mathematics is about "measure". For example, "find all squares of naturals", is a case where we /can/ produce them by /counting/: "the first is 1*1 = 1, the second is 2*2 = 4, ..., the nth one is n*n = n^2, ..." A more complicated example: we can produce the Fibonacci numbers (1, 1, 2, 3, 5, 8, ...) by counting them via the function F(n) = (phi^n - (1 - phi^n))/sqrt(5). But there is no similar "closed form function" that maps naturals onto groups in this way. To find the groups with 5 elements up to isomorphism, one can examine one by one each element of the set of 5^25 (approximately 10^17) binary functions on 5 elements, which can certainly be enumerated. But it's far faster (and of more mathematical interest) to use a proof to show that there is exactly 1 such group (up to isomorphism), because 5 is a prime number. > These letters on your screen are > numbers. > And so every statement communicated through written language is a number, and therefore everything is mathematics? This again becomes true of just about any statement. And yet, I feel that there is a difference between the branches of human endeavor "history" and "mathematics". <snip> > > That's fine. I am still of the opinion that math boils down to measure, > the language of measure, and the operations allowed on that language. > Your statements would be easier to understand if you explained what you mean by "measure", "the language of measure", and the operations you allow on that language. Cheers - Chas
From: cbrown on 9 Dec 2006 21:08
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Countably infinite means potentially infinite. > >>> If you wanted to say "potentially infinite" means countably infinite, > >>> why didn't you just say so? I don't think anyone else who has been using > >>> the phrase "potentially infinite" recently shares your meaning. > >> No? Why don't you ask them. It seems pretty clear to me that that's a > >> pretty common conception. > > > > Are you using the standard meaning for "countably infinite"? > > Yes, an unbounded set (no last element), with all elements finitely > close to the beginning of the set, in whatever order is chosen. Huh? > In other > words, a set bijectible with the naturals. > Ok, that I understand. > > > > As for asking WM, EB, etc., they rarely give coherent answers when they > > are asked questions. I'm sure I've asked both WM and EB several times to > > explain what they mean by "potential infinity". I'm afraid I still have > > only the vaguest idea what they mean. > > > > I think "countable infinity" is what they mean. I think that what they mean is a set which is endless and therefore not a set. From a set which is not a set, they can proceed to prove anything they like. Cheers - Chas |