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 P. 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 plausibilities of some 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:)


Anonymous said...

An endless sequence of radioactive particles may be unrealistic, but it is not a big problem for mathematics because the supposedly paradoxical outcome is only impossible. The probability of two real numbers being on the same list is 0, so it never happens. And the uniform probability distribution over real numbers between 0 and 1 only allows ranges of real numbers to be observed, as in the real world where observations are always within some finite limits, because the probability of any particular real number is also 0.

enigMan said...

The standard uniform probability distribution over [0. 1] is the unit square, which does assign a probability of 0 to any particular outcome from an endless sequence of radioactive particles during half-lives. But that does not mean that each particular outcome is impossible. If we do take a probability of 0 to mean impossibility, then we just have to find another distribution when using probabilities to describe such a scenario, or not use probabilities (or use them differently). The values that probabilities can take are given by mathematical theories of probability, while the meaning of ‘probability’ is given by some philosophical theory. Some philosophers use such scenarios as those particles to argue that a probability of 0 does not necessarily mean impossibility, and so that question does not really affect Levy’s paradox. If we could have aleph-null particles, in some logically possible space, then surely they could be radioactive particles, and then it seems to follow directly that we would have some random sequence of decays and non-decays, or 1s and 0s (using 0 here to stand for non-decay). While any particular outcome is certainly extremely unlikely, it is far from unlikely but certain that there will be some outcome (and that it will be the particular outcome that it is).

As for the chance of two such outcomes being on the same list, that depends on how many lists a god could know. Prima facie he could know uncountably many. If so there would then be the problem (for my paradox) that the outcomes might appear on several lists together, but given the axiom of Choice there is some ordering of the uncountably many lists, and so even then the god could simply use the first list in that ordering (the one he knows). Now, the axiom of Choice could be denied (e.g. Freiling resolves his paradox that way), but to do that would lose us a lot of standard maths anyway, and so it would make the alternative resolution (that I am suggesting we should not simply ignore, as is usually done) a relatively better alternative anyway.

And consider what might happen, given such an endless sequence of radioactive particles. The first particle might decay, during the following half-life, or it might not. So there is a chance of ½ that our sequence begins 0.1..., and a chance of ½ that it begins 0.0...; and similarly, there are chances of ¼ that it begins 0.11..., 0.10..., 0.01..., and 0.00...; and so on. The standard limits are chances of 0 (non-standard limits might be various infinitesimals), but the main thing is that the outcomes are prima facie equally likely.

And an outcome might be 0.1110, followed by some particular random sequence (prs) of infinitely many 1s and infinitely many 0s. Other possible outcomes are 0.10 followed by that prs, 0.110 followed by that prs, 0.11110 followed by it, and so on. (Indeed, there is a chance of ½ that our outcome will be of that form, for some prs, since those are the outcomes that begin with a decay, and since those with only finitely many 1s or finitely many 0s have measure 0.) And if two outcomes were of that form for the same prs, then they would certainly have been equally likely, via Finite Additivity, because they would differ in only finitely many places. And while the chance of two random outcomes having that form (with the same prs) has a standard measure of 0, that does not necessarily make it impossible, only highly unlikely (like any particular outcome), and so the mere possibility should not be contradictory (however unlikely it is). (And intuitively, there seems to be no reason why some prs should be more or less likely to occur than any other, which indicates intuitively that any two outcomes are equally likely (and how plausible is the standard axiom of infinity, intuitively?))

scott roberts said...

I fail to see what it could mean that "standard mathematics is true" (or false). I thought, since axiomatic mathematics came to be worked out, that the word "true" ceased to be meaningful with respect to mathematics -- just used loosely as a substitute for "provable from these axioms".

It could still be the case that our mathematics may not correspond to God's (on which more below). But you seem to have something different in mind when you say "Standard mathematics being the language of science, it would be quite surprising were it very likely to be wrong about the real numbers." Actually, given quantum mechanics, I would be very surprised if the real numbers were "right", that is, have physical meaning. All that it takes for numbers to be useful in science is that we can extend them out to the limit of our ability to physically measure. Past the point of the Planck limits, we can not assume it is meaningful to speak of spatiotemporal measurements. Physical reality might be discrete.

Thus, I would say that "standard mathematics" is simply the most interesting of axiomatic systems we have come across -- constructivist mathematics being somewhat less interesting. As to God's mathematics, I suspect that our mathematics is a curious sideshow, one that you get when fallen intelligence assumes that reality consists of a multitude of individual things. Beyond that, here's a hint of what there might be (from the mystic Franklin Merrell-Wolff): "At the deepest level of discernible thought there is a thinking that flows of itself. In its purity it employs none of the concepts that could be captured in definable words. It is fluidic rather than granular. It never isolates a definitive divided part, but everlastingly interblends them all. Every thought includes the whole of Eternity, and yet there are distinguishable thoughts." At least I find this more intriguing than thinking of divine intelligence as the ability to count to infinity.

enigMan said...

Many thanks for your comments scott, all of which I like addressing. Regarding your last statement, while I don't think of divine intelligence as merely the ability to count to infinity, that has been one of the ways in which the mainstream of the last century thought of Platonism in mathematics (e.g. Russell's talk of it being not logically but medically impossible). I think of divine intelligence as quite transcendental, incomprehensible, but of having to include some of what we know (as when we say "God knows," or "As God is my witness").

One of the facts about the world that I am personally pretty sure of is that you and I are two individuals. We seem to be quite distinct on a rather real level (at least if there is a God), e.g. I could hardly be blamed for any sins you commit. Of course, even that is not without its complications (e.g. traditional Christianity has a notion of collective sin, and modern physics might indicate something like Dean Radin's Entangled Minds), but still, I think that God probably knows 1, 2, 3, and so forth. Reality does not have to consist of atoms for there to be the possibility of individual things, and that possibility does seem to be indicated by our actuality as really distinct people (or even more so, perhaps, by one's distinction to God, although there is mystical talk even there; or as the end of your Franklin Merrell-Wolff quote says, there are distinguishable thoughts:)

By "standard mathematics" I mean the mathematics that is taught at school, which goes from natural to real and complex numbers, from arithmetic to real and complex analysis. There we assume quite a definite concept of natural number, and an informal view of the real number line. There are axiomatic systems in academic mathematics, but I am pretty sure that most mathematicians regard them as no less suspect than constructivism. After all, in such systems there are numbers of symbols, and distinguishable axioms, and all such stuff requires a definite meta-language of informal logic and informal arithmetic. Standard mathematics is so-called (by me) because it is quite well axiomatised by standard set theory, which is ZFC for most mathematicians and ZF for most logicians. It is true (false) insofar as it correctly describes how possible individuals and continua could behave, I think. But you are right that a lot of professionals in this area think that "true" in mathematics means coherence with axioms rather than correspondance with reality. Still, I have yet to see such professionals explain convincingly our implicit grasp of meta-languages (or proper classes:)

It may be that physical reality is discrete, but for all we know it might be continuous, and the correct mathematics should tell us what it means for something to be continuous. If real space is not like the standard real number line cubed, maybe it is continuous, because maybe continua are not like the standard real number line. And similiarly with the notion of reality being discrete. We need some concept of individual things to have "discrete" mean anything. I think that thoughts are very fluid and fuzzy, much as the mystics say; and yet we aim to say something sufficiently definite, about a world that is as it is and is not otherwise (although it is fairly mysterious how we can know about it). Regarding quantum mechanics, our observations are discrete (if fuzzy) observations that seem to lie (fuzzily) on rather smooth curves, indicating that the underlying wavefunctions are in reality as continuous as they are in the theory, so far as I know (maybe the weirdness enters into quantum mechanics with the background metaphysics (perhaps because of the difficulty of doing metaphysics correctly))...

Xamuel said...

Cool article, found it while searching for paradoxes, which I got interested in after writing about another paradox involving real numbers: http://www.xamuel.com/provability-paradox/

By the way, the link to Freiling leads to a 404 at Wikipedia. Maybe you meant to link to Freiling's axiom of symmetry...?