Tuesday, March 23, 2010

Lévy's Paradox

You probably learned about the real (or measuring) numbers at school. They are usually written as decimals—e.g. ½ is 0.5000..., and pi is 3.1415..., although most of them have completely random decimal expansions—and the set of them all is the real number line, R. Real number variables are ubiquitous in science, with curves in two dimensions having the form y = f(x); and three-dimensional space—that of our imagination, if not of reality (since it is infinite and Euclidean, rather than Einsteinian)—is usually modeled as R cubed, e.g. via Cartesian coordinates (x, y, z).

Standard mathematics being the language of science, it would be quite surprising were it very likely to be wrong about the real numbers. Unfortunately scientists usually take that to mean that they can safely assume that standard mathematics is unlikely to be wrong, rather than that they ought to assess its likelihood, and to develop (and use alongside it) the least unlikely alternatives. For the following look at that likelihood we first need some terminology, so let the selection of a number of some kind be completely arbitrary when any number of that kind might be selected, with none being more likely to be selected than any other, and let a Real be a real number between 0 and 1 whose selection was completely arbitrary.

Our first question is, are Reals plausible? Well, any radioactive particle has a half-life, a period of time such that its chance of decaying in that time is exactly 50%, and so an endless sequence of particles, each followed by such a period, could give us a Real in binary notation (with decays corresponding to 1s, and non-decays to 0s), if we ignore sequences with finitely many decays (since sequences with finitely many non-decays correspond to identical numbers), and if the particles are sufficiently independent (e.g. well spaced out). And such quantities of particles may well exist if space is infinite, or if there are other universes alongside ours (in a multiverse), or if the future is infinite. And physical possibility implies logical possibility, of course, and so Reals do at least seem to be logically possible.

The problem with that is we therefore get a paradox like that attributed to Paul Lévy by F. P. Cantelli (1935, ‘Considérations sur la Convergence dans le Calcul des Probabilités’, Annals de l’Institut Henri Poincaré 5, pp. 1–50).

A being limited by little but what is logically possible—say, a god—might know many endless lists of different real numbers, and so he might decide that if two Reals happened to be on the same list, he would use their natural numbered positions on that list as two completely arbitrary natural numbers. He could then write, for each number, a note promising the bearer that many days in paradise, put the notes into two envelopes, and ask someone to take one. That is paradoxical because whichever note she picks the other note was almost bound to have been the better choice because, given any natural number, there are only finitely many natural numbers that are smaller, and infinitely many equally likely natural numbers that are larger.

The standard resolution of Lévy’s paradox—and similarly, of Freiling’s paradox (and not too dissimilarly, Banach-Tarski’s)—is likely to involve the slightest possible violation of our intuitions about probabilities (and related measures). But what appears like a neat resolution given standard mathematics (assuming there is one) is likely to produce a mess of errors if standard mathematics is incorrect. To have any idea of where those errors are likely to show up—e.g. details of theories of probability (such as Popper's) make a difference to predictions in some high-energy physics (likely to be increasingly applied) and to the relative plausibility of theories of mind (and hence to some ethics)—we need, not only such standard resolutions, but also whatever other resolutions and associated theories are not too unlikely a priori. And so we first need to go back to basics.

The natural (or counting) numbers—1, 2, 3 and so forth—are elementary mathematical entities, defined by the endless reiteration of the addition of the unit, starting with the unit (where the unit corresponds to the elementary metaphysical concept of an individual thing, which is presumed by all logics). It is not such numbers but formal (or axiomatic) sets which give us the standard foundation of mathematics, but even so we are only interested in certain formal structures; and informally, a set is a quasi-spatial (or combinatorial) collection, in the following sense.

Consider some ordinary objects in a room. That is a set of objects because they coexist together in the same room. And of course, since we have all of them so we have any sub-collection (any subset) of them, coexisting in the same spatial way. To call a collection ‘quasi-spatial’ (‘combinatorial’) is essentially to say that all conceivable sub-collections of it are collections of the same basic kind.

Now, in our snapshot of those objects, in that room, everything was existing timelessly, and while their subsets were not coexisting quite like the objects were—but were rather overlapping (and were perhaps more abstract)—the subsets were also coexisting timelessly, and so we also have, in the same timeless way, a set of all those subsets—the powerset—of those objects. Natural numbers are certainly rather abstract; and the powerset of the natural numbers has the same cardinality as R.

By contrast, if some infinite collection (such as the natural numbers) is thought of as always growing, according to some given rule (e.g. an endless reiteration), so that we never have all of its elements—although the finite rule allows us to talk of any of them—then only those sub-collections that could be similarly specified, by a finite rule, would exist in the same kind of way. If the natural numbers are not collectively a set, but are rather as indefinitely extensible as they first appear to us, then most of the standard real numbers do not exist, not as definite numbers, because most of them correspond to completely random decimal expansions, which cannot be finitely described.

When we first think of the natural numbers, we think of them going on and on forever, so there should be some reason why standard mathematics regards them as a set (and since the use of R is ubiquitous in modern science, it should be a very good reason). Now, we very naturally think of numbers as existing timelessly, insofar as they exist (e.g. as abstractions), but we may also think of mountains as existing timelessly (and similarly languages, human rights and so forth). And such paradoxes as Cantor’s (for cardinal numbers) and Burali-Forti’s (for ordinal numbers) have shown us that, even if the natural numbers do form a set, whole numbers more generally cannot. So even if we find it hard to conceive of the indefinite extensibility of arithmetic (to use Mill’s phrase), that cannot be a good enough reason for us to have presumed so confidently that the natural numbers comprise a set.

At the heart of Lévy’s paradox is the oddity that every natural number is in roughly the first 0% of the set of all and only the natural numbers. So it would be natural for mathematicians to ask themselves whether the natural numbers go all the way to infinity; and if so, why none of them are anything like infinitely big, and if not then how we could have them all. And a reasonable way for us to think of how they could go all the way to infinity would be to use—following J. Benardete (1964, Infinity: An essay in metaphysics, Clarendon Press, p. 31)—the clear conceptual possibility of three-dimensional space, which could easily contain that many particles, e.g. one every light-year, stretching all the way across infinite space, with each being only a finite distance from anywhere.

So let us look a little more closely at that answer. Let the first particle (anywhere in space) be particle 1, the next (a light-year away) be particle 2, and so on. It seems plausible that, if we had those aleph-null particles, then particle 1 might move from some place P1 to some other place Q1, and then 2 might move from P2 to Q2, and so on. Such seems logically possible at least; and so we might have all the particles moving one by one (in the given order) from some region P (containing Pn for every natural number n) to some other region Q (containing Qn for all n). Indeed, it seems logically possible that they might do so in such a way that the move from Pn to Qn takes half as long as that from P(n – 1) to Q(n – 1), for all n, e.g. because the particles move faster, or because the distances involved are shorter. And if so then in twice the time it took particle 1 to move from P to Q we will have had all those particles moving, one by one, from P to Q. The number of particles in Q goes from 0 to aleph-null via 1, 2, 3 and so forth.

That hardly seems paradoxical, and yet if that is possible then it is surely no less plausible that such particles should move from P to Q in reverse order. E.g. if particle 1 had moved between the times of 0 and ½, and particle 2 between ½ and ¾, then we might instead have particle 1 moving between ½ and 1, and particle 2 between ¼ and ½. But in such a way we could go from having nothing in Q at time 0 to having, at any subsequent time, aleph-null things there (and finitely many remaining in P). So upon reflection it seems that having aleph-null particles in three-dimensional space is no more plausible than that we could, by gathering things one at a time, go from having nothing to having infinitely many things without at any time having any other numbers of things than zero or aleph-null.

Now, the standard view will be that the latter is plausible precisely because it has just been shown how it could be done. But even so, our clear conception of three-dimensional space only indicates the possibility of aleph-null particles if we presume that the natural numbers are not indefinitely extensible; and furthermore, that clear conception actually indicates that they are indefinitely extensible, as we will next see by using—following J. Benardete (1964, Infinity: An essay in metaphysics, p. 149)—the paradox of the Spaceship. But first, regarding that former begging of the question, note that space could be infinite, so that we could travel a light-year, and then another and another, and so on indefinitely, without our being able to travel aleph-null light-years, even in principle, precisely because the sequence 1, 2, 3, and so forth, is indefinitely extensible. The infinitude of such a space—which is what allows us to go any finite distance (relative to some unit of length), and also infinite distances—could not be a standard transfinite infinity, but there are such possibilities. In particular, there is a possibility that I have called ‘C-II’ (2005, To Continue with Continuity, Metaphysica 6, pp. 91–109), in which the infinitude of space could be the reciprocal of an irreal infinitesimal (as could the distance travelled by our Spaceship).

It seems reasonable to presume that an infinite space—a flat, not an Einsteinian space (and a uniformly smooth space)—is conceptually possible. We standardly think of it as not having parts at infinity—as being isomorphic to R cubed—because, given that the natural numbers are collectively a set, such parts would break that space up into such parts, with gaps (a bit like the gaps in the rational number line) between them, whereas our conception of space is that it is uniformly smooth. Such gaps follow from the gap between the finite and the parts at infinity (which might be reciprocals of hyperreal infinitesimals), and look like 1, 2, 3, …, ..., (such-an-infinity – 3), (that-infinity – 2), (that-infinity – 1), … .

Even so, there is a conceptual problem with R cubed, because a Spaceship travelling in a straight line, and covering the first light-year in one minute, and then each light-year in half the time it took to go the previous light-year would—were it capable of superluminal speeds (which is conceptually possible)—have vanished or teleported after two minutes. So, insofar as it is plausible that it should not have to vanish or teleport, it is plausible that infinite space should contain parts of space that are infinitely far away from other parts, so that our plausible Spaceship can have somewhere to have gone to. So we have a reason to favor theories that allow such spaces. The conceptual possibility of infinite space (and of our Spaceship) implies most intuitively, not that the natural numbers are a set, but that they are indefinitely extensible (as in the uniformly smooth C-II).

So, at least one argument for standard mathematics—our intuitively coherent conception of infinite space—has turned into a couple of arguments against it, and if that turns out to be the general rule then the correct resolution of Lévy’s paradox may well be the falsity of standard mathematics. Unfortunately there are surprisingly few arguments for standard mathematics. The main one appears to be that the standard mathematicians cannot all be wrong, but surely the few non-standard mathematicians that there are cannot be wrong about the elements of their profession either; and the problems with using popularity as a measure of metaphysical truth are obvious (given our history; cf. how we could have said a few years ago that bankers could not all be wrong). Those who do not like standard mathematics are far more likely to pursue careers other than pure mathematics, than they are to challenge it from within, unless they are geniuses at pure mathematics (and the numbers of such geniuses may well be evenly divided between standard and non-standard mathematics).

The final argument that I will consider here is that the main alternative to standard mathematics—constructivism (or Intuitionism)—is obviously unrealistic. So note that there are other ways of looking at the alternatives. E.g. consider how either there is a God, or else there is not. If it is the latter then our evolved concepts of number are unlikely to give us a very accurate picture of how numbers really behave at infinity. But even so we might use—following P. Kitcher (1983, The Nature of Mathematical Knowledge, OUP)—the idea of an ideal mathematician to help us to understand standard mathematics. Which brings us to the former possibility; and the commonest view of God nowadays sees Him as, whilst omnipotent, capable of change.

Such a God might be endlessly constructing arithmetic, much as He creates, in His omnipotence, all that exists (and arguably commands what is right), doing so forever whether or not standard mathematics is correct, in view of Cantor’s paradox (and Burali-Forti’s). On such a view there is at present some biggest number (finite or transfinite), but by the time we had thought of it existing (although it would be unimaginably huge) God would already have gone far beyond it, in His absolutely objective arithmetic (whence this view satisfies most of the common Platonic intuitions). Anyway, that possibility at least shows that it may well be that most of the problems that people have with constructivism do not actually apply to the most plausible way of thinking of the natural numbers as indefinitely extensible, whatever that happens to be (cf. how long it is taking standard mathematics to find a very plausible proper class theory). (PS: This post is linked to in the Carnival of Math: Mindmap Edition; and in the 106th Philosophers' Carnival: Philosophical Gourmet; and in the May issue of The Reasoner:)

Wednesday, March 03, 2010

Two Envelopes

Consider, to begin with, a simple sort of Two Envelopes scenario: Mr. E. writes out a cheque for £18, another for £12, and puts each inside an envelope. Showing the two envelopes to his friend, Miss Take, he tells her that they contain money, and asks her to take one as a gift. Miss Take does so, and finds that she has £12. Then Mr. E. tells her that one of the cheques was for 50% more than the other, and asks her if she wants to swap her cheque for the other one. Now, Miss Take knows that either the amount in the other envelope is £12 + £6 = £18, or else it is £8 (since £8 + £4 = £12). So she knows that by swapping she would either gain £6 or lose £4. And she also knows that by swapping she is as likely to gain as to lose, since she has no idea which envelope contained the larger amount. So it seems to her that if she swaps she is 50% likely to gain £6 and 50% likely to lose £4. In other words, the mathematical expectation from swapping is 50% of £6 minus 50% of £4, which is £1. Since that is a positive amount, Miss Take decides to swap.
......She feels justified when she ends up with £18 but of course, she had no real reason to swap because her original choice was blind, and by swapping she is in effect just making that choice again. Still, when she swapped she did have more information about what was in the envelopes, and she seems to many to have had some mathematical reason to swap. Such is the Two Envelopes problem, which has been much discussed following M. Kraitchik’s two wallets (in 1953; for a translation of the relevant passage, see previous link). The discussion usually revolves around the problem of randomly choosing the amounts in the envelopes (with one being twice the other, usually), but it clearly does not matter how Mr. E. came by the amounts of £18 and £12. Maybe it was the 18th of December, and Miss Take’s birthday; or maybe Mr. E. was a fan of Tchaikovsky’s 1812. The main thing was that Miss Take did not know both amounts, and that the envelopes did not reveal which contained the larger amount.
......To give us some perspective on Miss Take’s mistake, Mr. E. then offers an identical pair of envelopes to his colleague, Miss Tree, who also picks the one containing £12. But this time, when he offers her the chance to swap envelopes, he tells her that the square of one third of the larger amount is twelve plus twice the smaller amount (in £s). Miss Tree calculates that either the amount in the other envelope is £18 (since 6 is one third of 18, and 6 squared is 12 plus twice 12), or else it is £2 (since 4 is one third of 12, and 4 squared is 12 plus twice 2). So swapping would either gain her £6, or else it would lose her £10. And for all she knows, a gain is as likely as a loss. So now the mathematical expectation appears to be of a loss, of £2. Even so Miss Tree swaps envelopes, precisely because a gain is as likely as a loss. Having read P. N. Johnson-Laird, Miss Tree knows how easily we can be misled by disjunctions; and she suspects that, since only one disjunct of any true disjunction—such as either a gain of £6 or else a loss of £10—needs to be true, so Mr. E. could, by telling her something true about how the two amounts are related, have led her to calculate practically any mathematical expectation.
......The problem of how we should apply mathematical expectations is neither unimportant nor uninteresting, a clearer way to consider randomly choosing a natural number may therefore be via the following variant of a paradox attributed to P. Levy (by F. P. Cantelli, in 1935), which also resembles Kraitchik’s puzzle: Suppose that a god selects two natural numbers at random—i.e. any of them might be chosen, with each being neither more nor less likely to be chosen than any other—and then writes two notes promising the bearer upon demand that many days in paradise, puts them into two identical envelopes, and asks you to pick an envelope. The puzzle is that, whichever you pick, the other is almost certain to contain a far greater gift. That is because, given any natural number, there are only finitely many natural numbers that are smaller than it, and infinitely many that are bigger. But of course, each envelope is as likely as the other to contain the larger amount. Such a random choice therefore seems to be impossible.
......Paradoxically there are, quite plausibly, random real numbers, if the standard set-theoretic axiom of infinity is not too unrealistic; and that is paradoxical because if a god could choose real numbers at random then, since any countable list of real numbers amounts to a correlation between some real numbers and the natural numbers, so a god who knew several such lists might decide that if two randomly chosen real numbers were on the same list then he would present you with the corresponding two envelopes. Regarding the aforementioned plausibility, note that any particle that could decay is, if considered one half-life into the future, mathematically akin to a perfectly fair coin-toss, whence an endless sequence of such particular half-lives could instantiate a random real number in binary notation (with decays corresponding to 1s and non-decays to 0s); and such sequences plausibly exist if space is infinite, or if it lasts forever, or if there are other universes alongside ours (in a multiverse), and so forth...
(PS: This post is linked to in the Philosophers' Carnival 105:)