Galileo's Paradox
in [Math]
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From: Six on 29 Nov 2006 07:05 On 28 Nov 2006 07:40:04 -0800, "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote: >Six wrote: >> ... >> Either a proper subset of a set can be the same size as the set >> (for comparable sets or whatever technical qualification is needed), or it >> must be smaller than the set, or it makes no sense to compare infinite sets >> for size. (I suppose there could be some weirder alternative, such as the >> size of a set might depend on how it is ordered, or something like that. >> Haven't thought much about that.) Which is it, and why? >> ... > > >That's an interesting perspective. Consider a deck of cards, only >visible through a card-sized slot under a token-operated flash strobe. >When you look through the slot, insert a token, and see the >cellophane-wrapped pack, you know their order and contents, or have >some reasonable expectation thereof. After removing the cards and >perhaps shuffling them, only a few, or say, one at a time can be seen. >Are they thus "ordering-sensitive", in a sense? When it's a loose >deck, without some a priori knowledge of the disposition of the >elements, it takes 52 coins or thereabouts to know both the order and >contents of the poker deck. > >The notion of a set, for example the real numbers, being in a sense >ordering-sensitive is a reasonable one for quite a few considerations >of how and why they are, as they are. > Interesting. I will consider all that, so far as I am able. Six Letters
From: Six on 29 Nov 2006 07:02 On Tue, 28 Nov 2006 15:14:22 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: >> On Mon, 27 Nov 2006 02:21:33 +0000 (UTC), stephen(a)nomail.com wrote: > >>>Six wrote: >>>> On Fri, 24 Nov 2006 18:26:37 +0000 (UTC), stephen(a)nomail.com wrote: >>> >>>>>Six wrote: >>>>>> On Fri, 24 Nov 2006 16:04:12 +0000 (UTC), stephen(a)nomail.com wrote: >>>>> >>>>>>>Six wrote: >>>>>>> >>>>>>><snip> >>>>>>> >>>>>>>> I want to suggest there are only two sensible ways to resolve the >>>>>>>> paradox: >>>>>>> >>>>>>>> 1) So- called denumerable sets may be of different size. >>>>>>> >>>>>>>> 2) It makes no sense to compare infinite sets for size, neither to say one >>>>>>>> is bigger than the other, nor to say one is the same size as another. The >>>>>>>> infinite is just infinite. >>>>>>> >>>>>>>> >>>>>>>> My line of thought is that the 1:1C is a sacred cow. That there is >>>>>>>> no extension from the finite case. >>>>>>> >>>>>>>What do you mean by that? The one-to-one correspondence works >>>>>>>perfectly in the finite case. That is the entire idea behind >>>>>>>counting. Given any two finite sets, such as { q, x, z, r} and >>>>>>>{ #, %, * @ }, there exists a one-to-one correspondence between >>>>>>>them if and only if they have the same number of elements. >>>>>>>This is the idea that let humans count sheep using rocks long >>>>>>>before they had names for the numbers. >>>>> >>>>>> I love this quaint, homely picture of the origin of arithmetic. I >>>>>> am sure that evolutionary arithmetic will soon be taught in universities, >>>>>> if it is not already. Disregarding the anthropology, however, you have said >>>>>> absolutely nothing about whether !:!C is adequate for the infinite case. >>>>> >>>>>I was addressing your claim that there was "no extension from the >>>>>finite case". In the finite case, two sets have the same number >>>>>of elements if and only if there exists a one to one correspondence >>>>>between them. This very simple idea has been extended to the >>>>>infinite case. >>> >>>> OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of >>>> infinity is not. >>> >>>That depends on what 'idea of infinity' of you are talking about. >>>The mathematical definition of 'infinite' is as simple as the >>>idea of a 1:1 correspondence. > >> The mathematical definition of infinity may be simple, but is it >> unproblematic? It seems to me that infinity is a sublte and difficult >> concept. > >What concept of infinity? Note, I said 'infinite', not 'infinity'. >You have been talking about Cantor and one-to-one correspondences, >so you have been talking about set theory. The word 'infinity' >is generally not used in set theory. It has no formal definition. >'infinite' is used to describe sets, and it has a very simple >definition. I'm talking about mathematical meaning. Specifically I'm talking about "How many?", more or less etc.. >> And that we are entitled to ask how well the simple mathematical >> defintion captures what we mean by it, not necessarily in all its wilder >> philosphical nuances, but what we mean by it mathematically, or if you >> like, proto- mathematically. > >A set is infinite if there exists a bijection between the set and >a proper subset of itself. That is what mathematicians mean when >they say a set is infinite. There are other equivalent definitions. I know already. >>> There is no point in dragging >>>philosophical baggage into a mathematical discussion. > >> In my opinion the philsosopy is already there, and it impoverishes >> mathematics to pretend otherwise. > >Do you have the same problem with prime numbers? Or even numbers? >The words 'prime' and 'even' have meanings outside of mathematics. >Do you feel obligated to drag those meanings into a discussion >of prime or even numbers? See above ><snip> > >>>>> >>>>>?? How do I know what the missing elements are? >>>>> >>>>>The one-to-one correspondence idea is nice because it works for any >>>>>two sets. The idea you are looking at only works if one set >>>>>is a subset of the other. >>> >>>> Yes, to set up the paradox we need to compare two sets for which >>>> there is a 1:1C and one is a subset of the other. It isn't a question of >>>> what works. It's a question of how the paradox is to be resolved. >>> >>>> Thanks, Six Letters >>> >>>There is no need to resolve the paradox. There exists a >>>one-to-correspondence between the natural numbers and the >>>perfect squares. The perfect squares are also a proper >>>subset of the natural numbers. This is not a contradiction. >>> > >> I accept that. The contradiction comes about if the one notion >> suggests equality of size and the other notion suggests inequality. Which >> they do, so there is a prima facie paradox. > >The problem is that you are using a word 'size' that you have >not defined. True. I took it that people knew what I meant. And I think they do. >> I sense a cavalierness about common sense intuitions amongst >> mathematicians (I don't mean you in particular, Stephen, it's just a >> general comment.) Yes there is such a thing as conventional, accepted, >> unexamined wisdom. Things are not always what they seem. But common sense >> is, quite literally, where we all start. The articulation of it is >> something else. > >"Common sense is the collection of prejudices acquired by age eighteen." >-- Albert Einstein. > >Common sense is often wrong. Just think where physics would be >if people relied on common sense. Already conceded. >The problem here is not so much common sense, as the use of >the word 'size' without first defining what you mean by 'size'. > >> Either a proper subset of a set can be the same size as the set >> (for comparable sets or whatever technical qualification is needed), or it >> must be smaller than the set, or it makes no sense to compare infinite sets >> for size. (I suppose there could be some weirder alternative, such as the >> size of a set might depend on how it is ordered, or something like that. >> Haven't thought much about that.) Which is it, and why? > >Why use the word size at all? Two sets have the same cardinality >if there exists a one-to-one correspondence between them. A set x >is a proper subset of a set y if every element of x is an element of y, >and there exist elements in y that are not in x. Those are two >simple definitions that apply to any two sets. I know that already. >Of course people often use 'size' informally to mean 'cardinality'. >In the finite case 'cardinality' corresponds exactly with the >common sense notion of 'same number of elements'. Of course 'size' >need not mean 'same number of elements' even in the finite case. >Size is a very vague word, even when talking about physical objects. >Does it mean height, weight, volume? If you use vague words, you >are going to get vague results. > >> First option because Cantor says so might in a way be true, it >> might be that that is where mathematicians are, but it I was going to join >> them I would want to know why. > >> Thanks, Six Letters > >Noone is doing anything because 'Cantor says so'. Childish comments >like that are a sure way to make this thread degenerate. Certainly I write things in the heat of the moment which I later regret. But this wasn't meant as a cheap jibe. I've already conceded that following Cantor might in some deep way be right, if it comes down to following productive branches and forsaking dead ends. Look at what you've written. It consists of repeating things I already know (definitions etc.) coupled with the suggestion that I'm mixing up different notions of size. Saying that people are confusing two different notions of X is a classic manoeuvre of 20th century philosophy in the moribund analytic movement, and in every case, I'd venture to say, it sells the argument short. As if anybody that disagreed with your point of view was a complete idiot. There is an intuition that there are less squares (even numbers, primes, whatever) than naturals. We are talking here precisely of intuitions about infinite sets. It is not good enough to say: You're getting mixed up with finite sets, or: You can't rely on common sense intuitions in maths. So if there are less squares than naturals, then since they have the same cardinality, how can cardinality have anything to do with size (how many)? Why not just say there's a bijection and forget about cardinality. You suggested I conduct my argument without using the term 'infinity'. I am quite happy to do that. I suggest you conduct the rest of your argument without using the term 'cardinality'. Thanks, Six Letters
From: Jesse F. Hughes on 29 Nov 2006 08:30 Six Letters writes: > On Tue, 28 Nov 2006 15:14:22 +0000 (UTC), stephen(a)nomail.com wrote: > >>Six wrote: >>> I accept that. The contradiction comes about if the one notion >>> suggests equality of size and the other notion suggests inequality. Which >>> they do, so there is a prima facie paradox. >> >>The problem is that you are using a word 'size' that you have >>not defined. > > True. I took it that people knew what I meant. And I think > they do. Pretty standard fare for someone complaining about cardinality as set size. They assume that "size of a set" has a natural meaning and that this natural meaning should satisfy that A c B -> A is smaller than B (I'm using c for "subset"). But, no, people do not know what you mean. We are not born with flawless intuitions about the sizes of infinite sets. And so, in trying to make precise our notions about set size, we have extrapolated from the finite case by characterizing what counting means. And the analysis that is most successful to date is: two sets have the same size if there is a one-to-one correspondence between them. It seems like a well-motivated definition. Counting a finite set does amount to assigning one number to each element of the set. But it doesn't satisfy the subset condition and that seems to cause some people great anguish. Unfortunately, I have never seen any measure of set size such that the following hold: (1) If A c B then A is smaller than B. (2) Every set A and B is comparable. Cardinality satisfies (2) but not (1). There's an obvious definition that satisfies (1) but not (2): Say that A is smaller than B iff A c B. But that's not very interesting. Off-hand, I do not know of *any* interesting definition of set size that satisfies (1). Of course, (1) and (2) are mutually satisfiable by cardinality just so long as there are no infinite sets. -- "I deal with reality. It's a brutal reality. But it's the only one we've got. And people like me, do what it takes. I'm part of a long line of discoverers. So I do what it takes." -- James S. Harris channels George W. Bush
From: Six on 29 Nov 2006 09:26 On Wed, 29 Nov 2006 08:30:28 -0500, "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: >Six Letters writes: > >> On Tue, 28 Nov 2006 15:14:22 +0000 (UTC), stephen(a)nomail.com wrote: >> >>>Six wrote: >>>> I accept that. The contradiction comes about if the one notion >>>> suggests equality of size and the other notion suggests inequality. Which >>>> they do, so there is a prima facie paradox. >>> >>>The problem is that you are using a word 'size' that you have >>>not defined. >> >> True. I took it that people knew what I meant. And I think >> they do. > >Pretty standard fare for someone complaining about cardinality as set >size. They assume that "size of a set" has a natural meaning and that >this natural meaning should satisfy that A c B -> A is smaller than B >(I'm using c for "subset"). I am not claiming any originality, and if everything I have said has been said before and properly dealt with, I would be gratified to know that. It is not so much a case of assumption, though, as presenting a paradox, or dilemma. For me, anyway. I am just trying to work things out in my own mind. >But, no, people do not know what you mean. At least some of the force of my claim was the rhetorical point that I am not making the simple confusion about size that some people seem to be suggesting. > We are not born with >flawless intuitions about the sizes of infinite sets. Agreed > And so, in >trying to make precise our notions about set size, we have >extrapolated from the finite case by characterizing what counting >means. And the analysis that is most successful to date is: two sets >have the same size if there is a one-to-one correspondence between >them. If it's a matter of comparative success or fruitfulness, I have no complaint. I do not have the mathematics to judge. >It seems like a well-motivated definition. Of course it does. Otherwise there would be no paradox. > Counting a finite set does >amount to assigning one number to each element of the set. But it >doesn't satisfy the subset condition and that seems to cause some >people great anguish. > >Unfortunately, I have never seen any measure of set size such that the >following hold: > >(1) If A c B then A is smaller than B. >(2) Every set A and B is comparable. > >Cardinality satisfies (2) but not (1). There's an obvious definition >that satisfies (1) but not (2): Say that A is smaller than B iff >A c B. But that's not very interesting. Off-hand, I do not know of >*any* interesting definition of set size that satisfies (1). > >Of course, (1) and (2) are mutually satisfiable by cardinality just >so long as there are no infinite sets. Thanks. That is informative. It's clear to me that I need to learn a lot more mathematics before I could usefully respond to this. Much appreciated, Six Letters
From: Tony Orlow on 29 Nov 2006 09:52
Mike Kelly wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <456C5361.40706(a)et.uni-magdeburg.de>, >>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>> >>>> On 11/28/2006 3:48 AM, Virgil wrote: >>>>> In article <456AF6F8.5020307(a)et.uni-magdeburg.de>, >>>>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>>>> >>>>>> On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: >>>>>> >>>>>>> There is no need to resolve the paradox. There exists a >>>>>>> one-to-correspondence between the natural numbers and the >>>>>>> perfect squares. The perfect squares are also a proper >>>>>>> subset of the natural numbers. This is not a contradiction. >>>>>> What is better? Being simply correct as was Galilei or being more than >>>>>> wrong? (Ueberfalsch) >>>>> Galileo was both right and wrong. He applied two standards to one >>>>> question and was confused when they gave different answers. >>>> Initially he was confused, yes. However, he found the correct answer: >>>> The relations smaller, equally large, and larger are invalid for >>>> infinite quantities. >>> For the lengths of line segments, longer, equally long, and shorter, are >>> essential to Euclidean geometry. To deny that is to "throw out the baby >>> with the bath water". And I doubt that Galileo did so. >>> >>> For the intersections of lines determining points, any two line segments >>> can be shown to have a one to one correspondence of points. >>> >>> All one needs do is divorce the "length" from the "number of points", >>> which is probably what Galileo did, as being different sorts of measures >>> (like weight versus volume), and the problem disappears. >> Does one "need" to do any such thing, or rather, does one need to >> integrate the two concepts into a coherent theory including both? > > No, one doesn't need to integrate count and measure. Cardinality is a > notion of count(bijectability) that applies to all sets. It is quite a > useful idea in some mathematical proofs. Various types of measure can > also be applied to sets, these can also be useful in mathematical > proofs. > > Nobody has provided an integrated notion of count and measure that > applies to all sets, least of all you. Nobody ever will; one does not > exist. Furthermore, it is not even clear why you consider it so > important that one exist. What new and useful mathematics could be > derived from such a thing? > >> To tie measure with count in the infinite is the task here, regarding such sets. > > Why? What is the utility of performing this "task"? What new > mathematics does it allow for? > It allows for a rich ordering of infinite sets which satisfies intuitive notions of set density in a completely generalized way. Granted, IFR only works for sets of elements with inherent measure, but where the elements have such measure, why ignore it? |