Galileo's Paradox
in [Math]
|
From: Lester Zick on 2 Dec 2006 14:27 Tonico, let me revisit my comments below to present the same analysis in slightly different terms. If we take some conclusion, C, and want to know if it's true there are two different ways to proceed.Aristotelian syllogistic inference bases the truth of C on the truth of its constituent premises such as B. In other words it says "if B then C" which is a truism because C would certainly be true if B were true. However then we're just faced with exactly the same problem with B. So we further regress analysis of the truth of C to the truth of B to find the truth of B relies on the truth of some constituent premise of B such as A with the result that we wind up with "if A then B then C". This is exactly how classical syllogistic inference has always worked in the context of science and mathematics. To support the truth of some conclusion such as C there is an indefinite regression of problematic premises and this regression is what I call empiricism. In ordinary science this regression stops at what would appear a logical boundary of sensory and perceptual experience whereas in mathematics it stops with axioms and axiomatic assumptions of truth. Now this doesn't mean that truisms like "if A then B then C" cannot be true only that their truth can never be known in exhaustive terms. The most we can hope for is to stumble on some syllogistic regression that turns out to be true and employ it to ground further speculations. In effect syllogistic regressions such as "if A then B then C" become a line of reasoning or in the parlance of modern math a "model" of truth because the truth can never be known absolutely with such a method. Now I analyze the same problem from exactly the opposite perspective. Instead of asserting the truth of C relies on the truth of constituent premises I maintain the truth of any conclusion such as C relies on the falsity of alternatives to C, in other words what is "not C'. Thus we form a tautological regression of "C, not C" instead of the syllogistic regression "if A then B then C" and find that C can and must be true only if "not C" must be false and "not C" must be false only if it is self contradictory. In any event I hope this clears up what I mean by empiricism and truth in the context of mathematics and science. On Fri, 01 Dec 2006 12:04:42 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On 30 Nov 2006 23:55:39 -0800, "Tonico" <Tonicopm(a)yahoo.com> wrote: [. . .] >>Perhaps it is that we don't really understand what each other means by >>"empirical". For example, what empirical evidence (of what, where, >>when...?) does the axiom stating the existence of a unit element in >>group theory have? Or the axiom in Topology that states that the empty >>set is part of the set of open sets? > >I agree we don't understand the term "empirical" the same way. Not my >fault since I've discussed the subject at length over the past couple >years here and elsewhere. Basically any tautologically undemonstrated >judgment is empirical. Doesn't matter whether the judgment is sensory, >perceptual, cognitive, or whatever. If you assume an axiom such as "a >straight line is the shortest distance between points" the assumption >is empirical until and unless demonstrated true analytically. The same >applies to definitions. > >Most people completely misunderstand the meaning of an empirical >judgment. Most think it means getting out the tape measure, scales, >and so forth. The problem originated with Aristotle and his concept of >syllogistic inference. Aristotle was history's first empiricist in >formal terms. He found he could not establish the truth of any >conclusion syllogistically except by regression to further premises >whose truth he could not establish either except by further regression >ad infinitum. Which meant he could establish no truth syllogistically >at all without some kind of true basic premises which he set out to >find in unreducible perceptual terms. Which left us epistemologically >exactly where we are today in terms of all kinds of mathematical and >scientific methodologies. > >In point of fact however empirical judgment is nothing more than input >to a process of tautological regression whose ultimate goal is >reduction to self contradictory alternatives. That's how the mind and >brain work, tautological rather than syllogistic inference because it >can produce reductions to truth in exhaustive mechanical terms. > >~v~~ ~v~~
From: Virgil on 2 Dec 2006 19:24 In article <4571c22b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Bob Kolker wrote: > >>> Tony Orlow wrote: > >>> > >>>> Are x and y ordered? The Cartesian plane is ordered in two dimensions, > >>>> not a linear order, but a 2D ordered plane with origin. > >>> > >>> The plane is not a linearly ordered set of points. > >>> > >>> Bob Kolker > >> That's what I just said. > > > > So what kind of ordering do you contend it is? > > > > MoeBlee > > > > It's the superposition of two continuous linear orders, such that each > point obeys trichotomy with each other point, along each of the two > dimensions, or linear orderings of each element of the n-tuple > describing each point. Surely, you don't consider the Cartesian plane to > be completely unordered??? According to any standard definition of being a (totally) ordered set, the Cartesian plane is not an ordered set any more that a spherical surface is an ordered set. That some subsets of the Cartesian plane have an obvious ordering is irrelevant.
From: Tony Orlow on 2 Dec 2006 23:00 MoeBlee wrote: > MoeBlee wrote: >> Tony Orlow wrote: >>> Bob Kolker wrote: >>>> Tony Orlow wrote: >>>> >>>>> Are x and y ordered? The Cartesian plane is ordered in two dimensions, >>>>> not a linear order, but a 2D ordered plane with origin. >>>> >>>> The plane is not a linearly ordered set of points. >>>> >>>> Bob Kolker >>> That's what I just said. >> So what kind of ordering do you contend it is? >> >> MoeBlee > > P.S. And WHAT ordering are you referring to? The lexicographic ordering > that Kolker mentioned or some other ordering? > Lexicographic ordering corresponds to some multidimesnional ordering, such as is obvious here. :) ToeKnee
From: Tony Orlow on 2 Dec 2006 23:08 MoeBlee wrote: > Six wrote: >> On Thu, 30 Nov 2006 22:11:37 +0000 (UTC), stephen(a)nomail.com wrote: >> >>> Six wrote: >>>> On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: >>> <snip> >>> >>>>> Six wrote: >>>>> >>>>>> I suspect though that there may be no proof of the matter either >>>>>> way, as has been hinted at in other parts of this thread. It may come down >>>>>> to this: that someone who wants to take a different, but still very >>>>>> reasonable (maybe more reasonable) appoach to this paradox, would need to >>>>>> demonstrate that some interesting and viable mathematics can result from >>>>>> it. This would certainly involve having precise defintions and so forth; it >>>>>> is just that they would be more or less different ones. I certainly do not >>>>>> have the wherewithall to even begin such a task. But it's interesting to >>>>>> speculate. And it's good to keep an open mind. >>>>> You are more than welcome to come up with a definition of 'how many' >>>>> that you like. If other people like it, they may use it. It is not >>>>> going to change the fact that the naturals and the squares have the >>>>> same cardinality, nor the fact that the squares are a proper subset >>>>> of the naturals. >>>>> >>>> That is exactly what it is going to change. >>> How can that possibly change? There exists a bijection between the >>> naturals and the squares, therefore the two sets have the same cardinality. >>> Every square is a natural, and some naturals are not squares, therefore >>> the squares are a proper subset of the naturals. Those two simple facts >>> cannot change no matter what definition of 'how many' you come up with. >>> >>>> Your insistence that I define my terms in advance is back to front. >>>> There is a primitive intuition that naturals exceed squares and that they >>>> are equinumerous. There is no simple confusion here. Though undoubtedly >>>> there is something to be learned. >>> If you are not going to define your terms, then what you say is meaningless. >>> You say 'how many' but you refuse to say what you mean by that. How is anyone >>> supposed to understand you? >>> >>>> It seems to me that your understanding of the unclarity about how >>>> many is a phantom product, a reflection of the set theory you accept so >>>> uncritically. You want to pretend the ambiguity about how many pre-exists, >>>> whereas in fact it is a creation of the mathematics. Conventional set >>>> theory has no unique. proprietary rights over this paradox. and it seemed >>>> to me that, whatever else it does and no doubt does very well, it deals >>>> with this paradox rather poorly. >>>> Thanks, Six Letters >>> I am not claiming any unique proprietary rights over this paradox. >>> I am simply claiming that it is only a paradox because you are relying >>> on vague notions of 'how many' and 'size'. There is absolutely nothing >>> paradoxical about the statement >>> Every element of S is an element of N, but N contains elements not in S, and >>> There exists a bijection between S and N. >>> That is all we are saying when we say that >>> S is a proper subset of N, and >>> S has the same cardinality as N. >>> That is all set theory says on the subject. The only problem is that you have >>> some vague notion of 'size' or 'how many' that you are applying. >>> But you apparently refuse to even look at your notion of 'size' or 'how many', >>> and instead would rather complain about others being close minded. >>> >>> Stephen >> Unfortunately we don't seem to be getting anywhere. If I can think >> of a more persuasive way of getting my point across, I will reply. > > Stephen's comments are right on the mark. His latest rejoinder nails > it, even though it and his other followups should not have been needed. > There's no sensible concern that YOU persuade HIM. > > MoeBlee > Dribble bubble boil down.... okay. Here's the issue. You add elements, you get a "bigger" set, with a "greater count". You take them away, and the set becomes a smaller set, with a lesser count. Subset's a guide. So, the paradox is not whether one argument or the other is true, about infinite sets, but about how we "should" look at the problem ('though "should" is a word you should never use, much like "never", or even "use". ;)) In any case, the more faithfully we can generalize the finite case to the infinite, the more properties we can preserve, the "better". Tony
From: Tony Orlow on 2 Dec 2006 23:22
Virgil wrote: > In article <45704f2b(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Eckard Blumschein wrote: >>> On 11/29/2006 6:37 PM, Bob Kolker wrote: >>>> Tony Orlow wrote: >>>>> It has the same cardinality perhaps, but where one set contains all the >>>>> elements of another, plus more, it can rightfully be considered a larger >>>>> set. >>>> Not necessarily so, if it is an infinite set. >>>> >>>> Bob Kolker >> What, is it not necessarily so that it CAN rightfully be considered a >> larger set, with some justification? It doesn't have a right to be >> considered that way? Why? Because it contradicts theoretical >> transfinitology? > > Depends on one's standard of "size". > > Two solids of the same surface area can have differing volumes because > different qualities of the sets of points that form them are being > measured. > Absolutely true. I agree. A formula relating the surface area s to volume v of a given scalable 3D figure would boil down to a s=y^(2/3), given any linear unit of measure. > Sets can have the same cardinality but different 'subsettedness' because > different qualities are being measured. > True. Cardinality doesn't account for the subset relation, much less anything more subtle. We all do what we can, as best we can. It's not cardinality's fault. :) Sorry, that was obnoxious (but kinda fun to say). >> You do? Do you mean that the addition of elements not already in a set >> doesn't add to the size of the set in any sense? > > In the subsettedness sense yes, in the cardinality sense, not > necessarily. In the sense of well-ordered subsettedness, not necessarily. > > TO seems to want all measures to give the same results, regardless of > what is being measured. Well, Virgilium, what I want, and I don't really think this is unreasonable or even unrealistic, is to have mathematics become a single cohesive system of knowledge, with respect to facts and rules, with respect to measure and the real world, such that there exists no contradiction within this "entire" "system". I guess that makes it sort of a science, and I apologize for how distasteful that may seem to you. :) Antonius Erskineus |