Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 23 Jul 2005 21:19 In article <1122161361.019072.50650(a)f14g2000cwb.googlegroups.com>, "david petry" <david_lawrence_petry(a)yahoo.com> wrote: > My argument is that the mathematics that is relevant to real > world problems necessarily has a property of "observability" - that is, > mathematical statements must have observable implications, where > we think of the computer as the mathematician's microscope which > lets us make observations. Before Cantor came along, mathematicians > would have listened to that argument. Now they don't. G.H. Hardy worked in number theory, which he regarded as so pure that it would never be 'sullied' by any application to the real world. Today's electronic commerce would be impossible without the developments of number theory. IIRC, group theory was pure math until quantum mechanics, among other "observabilities", discovered a need for it. So that the "observability" of an area of pure mathematics may not be observable until after, possibly long after, the time of its developement, and no one at that time of development is likely to be a good judge of whether it will ever become "observable".
From: Vaughan Pratt on 24 Jul 2005 01:58 While I'm really enjoying this great thread, at the same time I'm really bothered that no one has mentioned what to me is by far the biggest problem with Cantor's theory, namely its claim to universality. Cantor's paradise, as Hilbert fondly called it, is a mathematical universe every object of which is a set. The basic objection to this that I want to raise here can be summarized in the slogan "Groups are not sets." Now I have no problem with set theory as a subject in its own right, to be pursued like group theory, linear algebra, or Boolean algebra. Where I run into difficulty is the idea that groups, vector spaces, and Boolean algebras are somehow reducible to sets. On closer examination this idea is incoherent. Not because it's incoherent to mix categories, I'm fine with that as you'll see below, but for more subtle reasons. (This is the idiosyncratic part of this post; most people who raise this objection do so on the ground that it is incoherent to mix categories.) Just to put things in perspective, there's a separate issue that I'm not proposing to object to here, namely whether set theory creates unnecessarily much mathematics having little intrinsic interest beyond set theory itself. For example set theory makes exactly one of the following true: either there exists a family of nonempty sets whose cartesian product is empty (how on earth could that happen?), or any set can be arranged in some order such that every nonempty subset of it has a least member with respect to that order (how on earth could you rearrange the reals in such a way, i.e. well-order them?). This dichotomy is that of Choice, which is either false or true according to which of the above disjuncts holds, respectively. It is nice to imagine, as Goedel did, that some future insight into mathematics will decide Choice, along with the continuum hypothesis, one way or the other. A more modern view is that Choice is a choice-point for mathematics: accepting it leads to one mathematics in which there exists a well-ordering of the reals, rejecting it leads to another in which there exists a family of nonempty sets whose product is empty. Many mathematicians prefer not to go out on either limb, and some would go further by saying that neither limb was real mathematics, an interesting position in its own right that I don't propose to go into here. What I'm more concerned about can happen just as well with tiny sets as with huge. Let's take the Klein group V_4 as an example. Is V_4 a set, and if so how? Well, V_4 would seem to have 4 elements, so presumably it must be the set {0,1,2,3} or whatever we want to call those four elements, perhaps {00,01,10,11}. But it can't *just* be that since how would we know it was V_4 and not Z_4, the other group of order 4? A better answer would be that V_4 is the group (V_4,+) where + is exclusive-or of the binary representation of 0,1,2,3 (as opposed to addition mod 4, which would give Z_4). But this is a pair consisting of V_4 and the operation +. How can a pair be a set? Well, in Cantor's paradise the pair (V_4,+) is actually the set {{V_4,+},{V_4}}. The intersection and the set difference of its elements are the singletons {V_4} and {+} respectively, guaranteeing that each component of the pair can be extracted, regardless of what's in those singletons. This is not terribly satisfactory. First, saying that the group V_4 is really the set {{V_4,+},{V_4}} seems to expose structure that has nothing to do with group theory per se. It's a set with two elements, and so on. Second, calling a pair a set is essentially the same trick as calling a natural number a string of decimal digits, or hexadecimal, or your favorite radix. It's just an arbitrary encoding having no mathematical significance. We can certainly define the natural number 77 to be the string 1001101 over the alphabet {0,1}. In fact we could mathematically define the instantaneous state of the whole internet, naughty pictures and all, to be a string over {0,1}, by associating to each computer on the internet its current storage expressed as a string of bits, and concatenating those strings in the order of the internet addresses of all the computers on the internet, suitably delimited (e.g. with a third symbol 2). Are set theorists willing to define the natural numbers as strings of bits? If they were it would save a lot of discussion about Peano arithmetic, which tries to capture the essence of number more abstractly, independently of its many possible encodings. It seems to me that those trying to make Cantor's paradise a foundation for their mathematics don't really know what they want. Sometimes they seem to want to express the essence of mathematical objects axiomatically, as with Peano arithmetic, sometimes they are willing to settle for arbitrarily chosen ad hoc encodings such as {{x,y},{x}} for (x,y). Whatever the group V_4 might be, it surely can't be exactly the set {{V_4,+},{V_4}}, which has properties having no bearing on group theory. If you were to ask me what I thought V_4 was, I'd say first of all that there's no way of doing this in a completely representation-free way. I'd then say that representation theory for abelian groups offers a mathematically cleaner definition than most of the alternatives. In particular I'd represent V_4 as the matrix 1111 1-1- 11-- 1--1 whose rows represent the elements and whose columns represent the group characters, where - abbreviates -1. This is obviously not the Cayley table of V_4, yet it nevertheless represents the group multiplication of V_4, namely via pointwise multiplication of rows, which as the symmetry makes obvious is isomorphic to its dual group represented in the same way by the columns. (In general any locally compact Abelian group is going to be such a matrix, always symmetric when finite, with matrix entries being complex numbers z with |z| = 1, i.e. points on the unit circle, such that both the rows and the columns form an abelian group under pointwise complex multiplication). Let me shift gears now and focus on a rather different issue that also comes up with tiny sets just as readily as with huge ones: the concept of power set. In Cantor's paradise, when A is a set, 2^A is also a set. I object. I have no problem with 2^A being a set *within set theory*. Just as in group theory we expect everything to be a group, so in set theory it is perfectly reasonable for everything to be a set. What I have difficulty with is the idea that 2^A should be a set *in the larger arena of mathematical objects in general*. Now I don't have any problem with AxB being a set when A and B are sets, either in set theory *or* in mathematics. When A and B are graphs, AxB should be a graph, and its edges can be obtained as pairs of edges from A and B. If A and B are discrete graphs, i.e. have no edges, AxB should be a discrete graph too. This extends to many other kinds of mathematical object that admit discrete structure. So if A and B are sets, it is perfectly reasonable to expect AxB to be a set as well. (Returning for the moment to the preceding issue, this is how I would propose to define a pair, namely as an element of the set AxB. AxB of course needs a definition, but defining it as a set of objects obtained by some ad hoc encoding like {{x,y},{x}} is mathematically uncouth.) This reasoning about AxB does not extend to 2^A. It would if 2 were a set, since then we would just have 2x2x...x2 A times, which should be a set if AxB is a set. (In pure set theory, 2 has no choice but to be a set, there being no other kind of object in that universe.) However mathematics does not use 2 that way; instead it is treated as a Boolean algebra. 2^A as 2x2x...x2 is then the product of A many Boolean algebras (where "A many" means the cardinality of A, which we're assuming here to be a set), and as such a Boolean algebra itself. All mathematicians know this intuitively, in particular they know that the members of 2^A can be combined with union, intersection, and complement relative to A. Ok, we all know that, Cantor included. How does this change anything about Cantor's paradise? What have we said that is inconsistent with how we reason about it? Nothing so far, until we face the question of what happens next as we ascend the ladder to Cantor's paradise. Let's take the next step up the ladder, to 2^(2^A). If 2^A is a Boolean algebra, say B, what is 2^B? We don't yet have a meaning for "B many" when B is a Boolean algebra. Is it just another Boolean algebra? Is it something beyond Boolean algebras, having even more structure than Boolean algebras? Might it be what sigma-algebras in measure theory are all about? Well, what is 2^A? We viewed it as 2x2x...x2, A many copies of 2, which is fine when A is a set. It can equally well be defined as consisting of the functions from the set A to 2, which pointwise form a Boolean algebra thanks to 2 doing so. These functions can be viewed as homomorphisms from the set A to the Boolean algebra 2: A being discrete, there is no structure to preserve so every function is allowed. This is true independently of whether we view 2 as a set or a Boolean algebra; in the latter case there is structure in 2 waiting to receive structure from A, but if there is none in A this is not a problem. This situation is the same as for a monotone function from a set (qua discrete poset) to a chain (qua linearly ordered poset, qua distributive lattice if you like); even though a discrete poset has no structure and a chain has lots, this doesn't prevent having a map from the one to the other, and every function is monotone in this case. There is a tradition of not allowing maps between different types of objects, such as sets and Boolean algebras but this reflects more a lack of imagination as to how to define such maps than any substantive mathematical objection. On this basis, one reasonable meaning for 2^B where B=2^A, consistent with the definition of 2^A itself, is that it is some structure whose elements are the maps from B to 2. If B is a Boolean algebra, a map from it should respect the structure of B, i.e. a Boolean homomorphism. To that end 2 will need to be a Boolean algebra in order to receive that structure. (So the elements of whatever 2^A is are maps from A to the Boolean algebra 2 regardless of what A is, whether a set, a Boolean algebra, or something else again.) But when B is a finite Boolean algebra these homomorphisms are in 1-1 correspondence with the atoms of B. The atoms of the Boolean algebra B = 2^A are in 1-1 correspondence with A. Furthermore when we come to calculate the structure on 2^(2^A) we find that it is completely discrete, that is, 2^(2^A) is a set. So 2^(2^A) brings us back to A, or at least to a set having the same number of elements as A. When A is infinite, 2^A is not just any old Boolean algebra but a complete one, in that one can form the union or intersection of infinitely many sets at a time, not just two at a time. Homomorphisms from it should therefore be complete too, respecting not only the binary meets and joins but also the infinite ones. The complete Boolean homomorphisms from 2^A to 2 are still in 1-1 correspondence with A, even when A is infinite. (If we'd only considered ordinary Boolean homomorphisms and not the complete ones for 2^(2^A) we'd have arrived not exactly back at A but rather at a totally disconnected compact Hausdorff space, or Stone space, embedding A as a discrete subspace, so named for Marshall Stone who observed this in 1936.) So if instead of pretending that mathematics lives entirely inside set theory, with all its objects being discrete, we recognize the structure that is created by such operations as 2^A and respect that structure appropriately, Cantor's paradise collapses: double exponentiation is the identity (up to isomorphism). But this need not mean that this system must forgo Cantor's original paradise altogether. If we add a new mathematical operation !A which gives the underlying set of the object A, we can recreate however much of set theory is useful to our application by forming the underlying set of a power set, namely !(2^A). We can then get a larger Boolean algebra out of A as 2^!(2^A). This is in fact the free complete atomic Boolean algebra (CABA) generated by the set A (we have to say atomic here, Gaifman and Hales independently showed in 1964 that there was no such thing as an infinite free complete Boolean algebra). The idea of underlying set U(B) or |B| is of course as old as the notion of an algebra, only the ! notation for it is new, due to Girard and used in his system of linear logic, which informs part but not all of these foundational ideas. The only difference between U and ! is that U typically maps objects of one kind to objects of another, e.g. Boolean algebras to sets, whereas !B and B are of the same kind in Girard's setup. Since we're looking for the homogeneity implicit in the concept "mathematical object," !A is the more appropriate notation here. ! can be related to U as ! = FU (a comonad) where UA forms the underlying object of A, of a different kind from A (set say) and then FUA forms the free object, on UA, of the same kind as A. In the sort of homogeneous world envisaged here it doesn't make sense to try to factor ! as FU, other than perhaps by taking ! = U and F = I (identity). There is also the possibility of other !'s besides underlying set. For example a Boolean algebra has an underlying set, but it also has an underlying poset, a different object. If instead of the underlying set of B we take !B to mean the underlying poset, 2^!(2^A) turns out to be the free distributive lattice on A for finite A, with the counterpart of "complete atomic" being "profinite" (so in general 2^!(2^A) is the free profinite distributive lattice on A regardless of the cardinality of A). The important point to take away here is that the indispensable construct 2^A does not necessarily *commit* you to Cantor's big paradise. The real troublemaker here is not 2^A, which has a single well-defined meaning and which harmlessly puts you in a cycle of length 2, but !, which could have many meanings according to just how much structure it strips off, and which can result in really big sets, posets, topological spaces, or whatever with just a few deployments. So in general ! should be used more sparingly than 2^A. Perhaps referees and MR reviewers could enforce this point of view, if it ever takes hold, by rewarding the proper use of 2^A and punishing spurious uses of !. :) Vaughan Pratt
From: Vaughan Pratt on 24 Jul 2005 03:12 Daryl McCullough wrote: > Han de Bruijn says... > > >>No contradiction. That's perhaps the only good thing about it [set theory]. >>If that is the only thing you care about, let me tell you that most of >>us care about other things, such as a physics that has a reliable >>mathematical machinery at its disposal. > > > Give an example of a calculation in physics in which using modern > ("Cantorian") mathematics gives you the wrong answer, and using > some other kind of mathematics gives you the right answer. > > -- > Daryl McCullough > Ithaca, NY > That's pretty easy, isn't it? The Dirac delta function pervades physics, but if you used "Cantorian mathematics", which defines a function as a set of ordered pairs, you'd get either the wrong answer, an overflow error of some kind, or a type check error. Vaughan Pratt
From: David Kastrup on 24 Jul 2005 03:32 Vaughan Pratt <pratt(a)cs.stanford.edu> writes: > Daryl McCullough wrote: >> Han de Bruijn says... >> >>>No contradiction. That's perhaps the only good thing about it [set theory]. >>>If that is the only thing you care about, let me tell you that most of >>>us care about other things, such as a physics that has a reliable >>>mathematical machinery at its disposal. >> Give an example of a calculation in physics in which using modern >> ("Cantorian") mathematics gives you the wrong answer, and using >> some other kind of mathematics gives you the right answer. > > That's pretty easy, isn't it? The Dirac delta function pervades > physics, but if you used "Cantorian mathematics", which defines a > function as a set of ordered pairs, you'd get either the wrong > answer, an overflow error of some kind, or a type check error. So what? "Function" _is_ defined as a point-value mapping; that has nothing to with Cantor. If you want to work with things like Dirac delta, the way to do this is not to use functions (which can't represent them), but distributions. Those are well-defined, too. If you want to talk about infinite cardinalities, you can't use the naturals, and if you want to talk about integral sieves, you can't use functions, even though you can get arbitrarily far on the way to there. Just that you can't call a "function" which you use in contexts usually reserved to functions is not reason enough to change the concept. Rather, introduce a new one. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: klaus.schmid on 24 Jul 2005 09:05
david petry wrote: > My claim is that all the results of analysis which have observable > implications, an hence have the potential to be applied as models > of the real world, necessarily do not need Cantor's Theory. [...] > Have I every said that Cantor's Theory is "wrong"? [...] > I certainly have not claimed that it is not a formally sound theory. Nice to hear. Yet I do not see your point, if you suggest on another place to create scientific mathematics without Cantors theory. Does Cantors theory makes things so much more complicated, that a separation between scientific and pure mathematics would be justified? How should persons of each party communicate if each has a different mathematical language? Considering mathematics as a big software library, would it be worth to double the effort for maintaining, bug fixing, updating? [...] > My argument is that the mathematics that is relevant to real > world problems necessarily has a property of "observability" - that is, > mathematical statements must have observable implications, where > we think of the computer as the mathematician's microscope which > lets us make observations. If a physician gets an inspiration by remembering the sound theory of infinitiy he has learned some time ago, to develop his own physical theory, wouldn't that be an observable implication? -- Klaus |