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:)

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, and a clearer way to consider randomly choosing a natural number may 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:)

Zero's signs

Maths begins with 1, 2, 3, and so forth; and a natural next step is to include fractions, and at some point we include 0 and negative numbers. In my last post I wondered if the adjunction of negative numbers should be construed as the introduction of directions. If so then when we include negative numbers we should also be exchanging our original unsigned numbers for explicitly positive numbers (the unsigned amount in the positive direction), but the question then arises of what we should do with 0. There is no numerical difference between negative zero and positive zero, both are just zero, and so we might leave 0 unsigned; but in my 2005 paper I presumed (on page 99) that 0 could be assigned both directions (and that in the case of complex numbers, 0 could have all the infinitely many directions of the complex plane). Intuitions for both views come from the use of 0 to label the origin of geometrical coordinates: To get to 0 from 0 we don't actually go in any direction; but then, to get to 0 from 0 we could travel no distance in any direction. So I'm wondering if there are any good reasons to favour one view over the other (aside from my 2005, which is a reason to favour the latter).
(PS: This post is linked to in the Carnival of Mathematics 63:)

Does Mathematics Need New Directions?

I've been wondering recently: For most people, mathematics is primarily the study of numbers. For most mathematicians it seems to be the study of standard axiomatic sets, but let's go with what most people think to begin with (after all, even if we began with sets we would have to explain why some of them are regarded as more important, as standard).
......Perhaps most people would think (with Mill) that the meaning of '2 + 2 = 4' is that two objects plus another two makes four objects. Now, it might be supposed that negative numbers present a problem for that view (e.g. Mark Balaguer thinks it nonsense to think of minus two pebbles being added to minus two pebbles to make minus four pebbles), but think of a tiled floor, with one tile missing, as that is a pretty good picture of minus one tile. And positive numbers of tiles could be represented by tiles placed on top of the tiled floor (cf. holes and electrons in semiconductors).
......Note that positive numbers of tiles are not quite the same as numbers of tiles, in that representation, as there are tiles within the tiled floor. That is not a problem, however, because positive numbers are different from unsigned numbers. The former exist within a mathematical space such as the integers, in which '2 - 4 = -2' makes sense (where +2 and +4 have, as usual, been written as '2' and '4' for convenience, because of the obvious isometry), whereas the latter do not, of course, allow you to take 4 away if you only have 2 to start with.
......Now, when we extend the negative numbers to include negative lengths (as we extend whole numbers to include fractions and other lengths), a better representation would have sea-level in place of the tiled floor, and consider extending positive heights of land above sea-level to include depths (negative heights) of the sea-floor. Or we might, of course, look at money, with negative money representing either a flow of money out of an account or, within the account, a debt. And we might, more generally, think of 'positive' and 'negative' as the names of two directions.
......That also fits nicely with the extension of the real number line to the complex number plane, in which there are not just two but infinitely many directions. Historically, the complex numbers (e.g. the square roots of minus one) were only taken seriously when mathematicians could picture them as extra directions. And now we find the complex numbers being as applicable in science (e.g. in quantum mechanics, the foundation of chemistry, electronics, lazers and so forth) as the negative numbers are in everyday situations, which indicates that we are here cutting nature at its joints, so to speak.
......Although Balaguer thinks we must use sets to get the negative (and similarly, the real) numbers, surely we use a concept of direction to make sense of, e.g., 2 - 4 (and similarly, of the square roots of minus one), and surely that also accounts for how useful we find such numbers, as we make our ways in the world. (And incidentally, the real numbers show how our number concepts might be judged empirically, as Mill thought.) In other words, signed numbers and complex numbers may best be thought of as elementary vectors (another fundamental mathematical concept with many applications).
......And so the question of what we mean by 'direction' arises. A very ordinary use of 'direction' is when one gives someone directions to get somewhere. When you get to the next crossroads, take the first left, for example. Note that the road to go down is picked out by a direction relative to where you will be when you have to go that way. And roads can be travelled in two directions (so I am not here thinking of Fregean directions, which are sets of parallel lines), and so I am wondering how deeply the concept of a direction is related to the concept of an option.
......Were there no choices, there would just be all those roads, where they are (much as sets are abstract objects that just sit there in the Platonic realm, so to speak), and little sense to logic. So we might expect choice to be a fundamental mathematical concept. Certainly the word 'choice' appears in the foundations of mathematics, there being an axiom of choice in standard set theory, and choice sequences replacing real numbers in constructive mathematics. But more deeply, our mathematical concepts are likely to be grounded in mathematical intuitions that we share with other intelligent animals (e.g. via the number sense), and intelligent animals make choices.

From the Incarnation to the Trinity

For monotheists, the idea of the Trinity (the one God being three individuals) can seem like polytheism. A popular model of the Trinity regards each person (Father, Son and Holy Spirit) as a relation between the other two, but not only is that quite mysterious, it hardly answers the objection because either such relations are distinct (because what is related is) or else they are not. Still, God is presumably more real than the mundane world, if He created it, and it may seem that our relationships are what make us truly real, more than mere things. And if we think of the Creator of the world as to us a bit like a dreamer is to his dreams, then again we have the sense of Him as more real than we are. And furthermore, even the Trinity may then come to seem less objectionable.
......Let us assume that there is a God who created us and the world around us ex nihilo. Such a Being (the ground of being) might be apprehended philosophically, although to many even this sparse conception of God seems paradoxical. Many have preferred to think of creation in terms of emanations from God's being, or of the forming of an energy (formless matter) that coexisted with God from eternity. But maybe God is, to the world, not too different from how we are to our daydreams. Our dreams are hardly real, but the thought is not that the world is just a dream, but that as we are to our dreams so God is, in some ways, to us (and the material world). God is, then, more real than what we ordinarily think of as real (the obvious objectivity of our world deriving from its dependence upon Him). And then we do not have to think of how mere matter could give rise, in some arrangement, to mind, because the original Being was Spirit.
......Let us add to that rather philosophical picture the idea that God became a man, Jesus. Many religions have stories of avatars of gods or goddesses, or even of God, and the divinity of Jesus did occur to the early Christians for some reason. But again, God's incarnation seems impossible (or blasphemous) to many. Still, it is quite reasonable to think of ourselves as spirits in a material world (and it is up to God what He does with His creation). That dualistic approach to psychology is unpopular amongst scientists at present, but modern science actually supports it (and there is an underlying monism for theists) because chemistry being fundamentally quantum mechanics all but solves the old problem of how spirit could interact with a human brain to give us our human minds. So, if God is also Spirit (a spirit more real than we are) then it is not unthinkable that He might similarly incarnate, giving Him a human mind (and body).
......A problem with the idea of God incarnating is how He sustains the universe while He is wandering around as Jesus. But how does the Incarnation look if God is (in some ways) to the world as we are to dreams? Well, dreams come in a great variety, but some end in something more like a daydream, in that the dreamer can deliberately alter them as she wakes up. Such dreams can be very vivid, so they seem very real to oneself, inside their world (so to speak) to begin with. The self in such a dream is at home in the dream, and can even be quite unlike the waking self. But one might surprise oneself in a nightmarish dream, for example, by falling and then finding that one can fly. In later dreams, flying might seem a realistic option, within the dream (whereas other daydreamish possibilities might not). But the more vivid one's control over one's dream, the less realistic the dream seems, and the more one wakes up. There seems to be a play-off in our dreams, between losing oneself in the dream (it seeming real) and one dreaming of whatever one wants (it being like a daydream).
......Now, God is presumably not much like us (or any created thing), but we do have something like a threefold aspect with respect to our dreaming. There is the theme of the dream, which we may have more control over as we wake up (cf. the Father), the dreamer aware of herself as lost in her dream (cf. the Son), and the person one is when awake or daydreaming (cf. the Holy Spirit). Since dreams naturally happen to us, the theme is naturally an unconscious aspect of our dreams, as is the mechanics of how we dream. And unlike God we do not create other people by dreaming. But if God incarnating as Jesus is a bit like our being in our dreams, with something like that sense of two different selves, then the difference that creation involves created people like us could plausibly be associated with what we would perceive as the glory of the Holy, the paradoxical presence in creation of the transcendent ground of being, revealing Himself to us as directly as any of our perceptions could be. It seems that it is by such theophanies that prophets become aware of the God of Abraham, and in similarly direct ways that we become aware of the reality of the Holy Spirit amongst us. We may then wonder how this more real than real personal presence could be related to the creator of the world, the ground of all mundane beings, and to the life and history of the works of Jesus.
......Still, this is only a vestige of the Trinity (if the Trinity is real), not a good model of it. E.g. it hardly helps us to think of the one God being three selves simulataneously. But these Selves are not like created selves. And one can become aware, as one wakes, of the three aspects to one's identity when dreaming being all oneself and being all coexisting, even if one can only do so by switching one's agency and awareness between them. And since that is not like acting in three different ways, not like one's character going through three stages as one grows up, but like being in three different ways, in relation to the same dream, and since this is only an analogy, so one might see how the one transcendent Creator ex nihilo might be, in relation to His creation, what we would naturally perceive as three people. Of course, to see that possibility one must examine such experiences as one wakes up, and then think about them analogically (under the assumption that God has made us in His image and incarnated with us and is amongst us now).
......There can be a moment (which can be protracted into a series of moments) as one wakes when one is not aware of oneself in bed, in the world, but one is aware of the dream as a dream, when one can go back into the dream and continue with it, or change the dream, and go back into it, or think about why the dream was as it was. If dreams were not something that happens to us, but we were more in control of the dreaming process (as we are when daydreaming, or when walking around the real world), as God is presumably in control of His creation, then one would also at such moments be aware of oneself behind the unconscious aspects of dreaming, the creation of the dream landscape and the other characters. So if those other characters were as real as oneself (in the dream), and if one was not unconscious of much of the dreaming process, there would be something like a threefold structure to such moments, corresponding very roughly to the Trinity.
......As creatures we never relate to a God who is not relating to His creation, so it is highly speculative how He would be without creation. But presumably He would have begun then with His responsible ability to create other spirits (centres of awareness and action), and so perhaps with a Trinitarian structure (cf. one's orientation upon waking). Now, realistic thoughts about the revealed God (Jesus Christ and Holy Spirit) are difficult enough, and thoughts about the Trinity as it is in itself are plausibly beyond us, even if the Trinity has been revealed to us in history. But it is at least possible for us to see how we do not have to think of the Trinity as merely how the One appears to His creatures as He reveals Himself to them (to us).
......Suppose that we exist in a 2-dimensional world, Flatland, e.g. as thoughtful triangles, and that transcending our world is a 3-dimensional object, a cylinder. As it shows itself to us by passing through Flatland, it might appear to us as a circle suddenly appearing and disappearing, or as a rectangle slowly appearing from and disappearing into a line. As triangles we would naturally think of the reality as a circle or a rectangle, appearing from nowhere. But the reality is more than that, and the claim that the circle and the rectangle were the same being is not the claim that a circle can be square. And nor is it the case (fictionally) that the being is two shapes (like the plan of a cylinder), or an intrinsically shapeless thing that can take on the form of any shape as it appears to us.
......I don't think that either analogy, the cylinder or the dream, will yield a very accurate model of the Trinity, but they may help us to see that the Trinity might be realistic, by resolving some of the paradoxes we find with other analogies (e.g. the relational Trinity). I'll have to think about it some more (and I'd be glad of your thoughts). The world is not a dream; and God is not much like us, whether we are awake or dreaming. But all our thought about the world involves analogical reasoning, and we might expect that a good grasp of transcendent truth would involve even more of it. If God really incarnated as Jesus, why should that not give us a Trinitarian view of God? And why should God not incarnate in His world? Why should He not create people (if He could) to whom He could reveal Himself like that? Even the creation of a pebble ex nihilo can seem impossible to us, but to God it is plausibly as possible as a daydream of the seaside.
(PS: This post is linked to in the Christian Carnival CCCXII:)

A Stab at a Dogma

The Trinity, one God being three who relate to each other, seems paradoxical, prima facie, but perhaps it is only realistic (cf. how chemistry, as it became more realistic than alchemy, turned into a far stranger quantum mechanics). Let us presume that God created us, and the world around us, ex nihilo. Could He have incarnated as one of us? Well, He is presumably omnipotent; and perhaps we are essentially spirits, currently limited by our incarnation in human brains (with which we interact quantum mechanically), and perhaps God is also Spirit (who made us in His image).
......And an obvious way for us to think of creation ex nihilo is by analogy with the way we dream. When one dreams there is the creator of the dream (i.e. oneself), and the character in the dream whose point of view one has (i.e. oneself), and all the other stuff, which is not real but which corresponds (in God's creation) to us and the world around us. This is not supposed to be a very accurate model of divine incarnation; but if the other characters in one's dream were aware (as they obviously could not be) then they would naturally perceive one as a character with special powers and centrality, identified with and yet different to the creator of everything in the dream (including that special character's appearance), most strangely including themselves (which is where the analogy most obviously breaks down, but which may be where the Holy Ghost comes in).
......And quite generally perception (e.g. of a tree) seems to involve phenomena (e.g. green leaves) that are objectified (as what 'green leaves' refers to) in a rather paradoxical way (i.e. the problem of perception). Even when it is a perception of other people, so that we have a relatively direct knowledge of the kind of object, there is still an obvious distinction between how they seem to us and who they really are. And note that if there is a God then Idealism is not especially unrealistic. The connection between how something looks and what it is really like is made on the basis of wide experience and wise conjecture; and if the creator of everything else ex nihilo did incarnate as one of us, He might well be perceived by us as something more like a Trinity than not.

Not-So-Free Thinking

Enlightened critics of religion like to point to a history of intolerance of criticism in religion. So it is nice to see Terry Eagleton and Karen Armstrong in fifth place in the New Humanist's 2009 Bad Faith awards, with about six-and-a-half percent of the votes. The New Humanist's article points out "that both have written books this year criticising the New Atheists and mounting what some might call a more sophisticated defence of religion." Quite generally it seems that whenever people talk about important things, there will be those who find good criticism most irritating; and the more popular the philosophy, the more politics there will be.

Duck-poo Soup

On a green pond smooth as glass,
mallards float and pass
the time. I stare
and look away. Splashes
make me turn to see water plumes
collapsing. A pause, a floating
feather: I think of water's plumage
and up spring some ducklings, as
busy as bees in the water-lilies.

A lardy male waddles by
on lobstrous feet, evoking
with his draconian head
a Viking long boat.
A duckling ducks
submerges and emerges
like a broadside
at the adult's broad side
and nippily snatches a fly
from the bottom of the sky.
Flustered, the adult flaps
his wings, flinging off
oddly fluttering splodges. Weary of malarkey,
he fans out his butt like a pack of cards
and onto a flagstone flops. Wary of malady,
he gingerly stretches out
his white-collared neck
for soggy croutons floating
on this duck-poo soup. He wants
that sumptuous bread but he fears
being presumptuous. He does not want
to end up dead. But he cannot die.
He is a picture. I have immortalized
and immobilized him.

Unhinging winds fringe maroon-fingered moon,
like a waiter with a supernatural soup-spoon;
a crater of rubble like a burst bubble serving
as a seat of tranquillity for a duck quacking up
a soporific melody of sounds pacific:

Talk about a duck
floating on a lake,
looking like a wooden decoy does;
talk about a drake
ducking wooden ducks,
making all the ducklings he can make.

Two "proofs" that 2 + 2 = 5

The rationalist proof:

Starting with the concept of a bean, adding two beans and another two beans makes five beans because the concept of a bean is a bean (an ideal platonic bean).

The empiricist proof:

Add together two lines of length 2.36 (of our units of length) to make a line of length 4.72, and then round those lengths to the nearest whole number (of units).

Deep Thought

First things first.
......And yet I find 'zeroth' in my dictionary, to refer to the one before the first one. In Wikipedia I find: The zeroth item is the initial item of a zero-based sequence (that is, a sequence which is numbered beginning from zero rather than one), and surely 'first' would have done just as well as 'initial' there. In such a sequence, the first element is also the second; that is, a certain equivocation has been introduced, a new sense of "first" added to our initial, informal sense.
......The idea of an ordinal number zero comes from the ordering of the integers on a number line, from negative to positive infinity (exclusive). But when we use ordinal numbers to include negatives, as with years, we naturally make the direction more explicit (e.g. with BC and AD) and exclude year zero (whence all the fuss about when the millenium began).
......Modern maths does not like directions. It finds it best to begin with a collection of natural numbers {0, 1, 2, 3, ...} that are most fundamentally ordinal numbers, and are usually reduced to pure sets. Mathematics does not like directions. Following Euclid's reduction of geometry to logic, geometry was naturally reduced to analysis following Descartes, and then arithmetized, with an arithmetic reduced to set theory.
......And yet mathematicians do like directions. Imaginary numbers were only taken seriously as numbers (like the negative numbers) when it was realised that the positive and negative imaginary numbers i and -i could be regarded as unit distances (like 1 and -1), specifically in the direstions perpendicular to the positive and negative directions of the "real" number line in a "complex" plain (which is how complex numbers are introduced to students).
......Given the foundation of the concept of direction in our experience of space, I would guess that if there really are extra dimensions in physics, they are more likely to correspond to the complex numbers of quantum mechanics (e.g. by representing dimensions of actual physical possibility) than the six dimensions of phase space (nor the ten or twelve of string theory, etc.).

Eternity

Following on from a paper I've been writing for the past year (see previous posts under 'Theology'), I've been writing an MTh essay on the paucity of the reasons given for why Christians should believe that God is timeless. The essay glances at reasons given by Boethius, Helm and Mawson, and I'm wondering what (if anything) I've overlooked...

Unretiring enigMan

Hi; tired of being retiring, I stretch my fingers with intermittent posts (on-going) and, as our droughtless summer (of many Painted Ladies) draws slowly to a close, I think its cultural highlights were: a sci-fi film of the good old 'new wave' kind, Precinct 9; and, more spook story than sci-fi, Iain Bank's Transition; and 60's nostalgiac Inherent Vice; and 90's nostalgiac Menage.

Enigman at the Crossroads

Tiring of blogging (not so much reading others, as posting my own), it's time for Enigman to retire. There's a local conference in a couple of months, but even if that inspires me with any interesting thoughts, they'd be hopeless for blogging purposes. It seems to take me months (if not years) to make my thoughts coherent. Incidentally, a lot of my links will stop working soon, as my web-pages are on Geocities, which is closing this summer. I may move those web-pages to this site, probably after re-writing them (most were written a few years ago). The other thing I'll be doing this summer is starting to study theology properly (and catch up on my reading). My philosophical interests (which were originally in physics and freedom) were re-ignited ten years ago, via the metaphysics of mathematics, but were nearly quenched by professional analytical philosophy (with few exceptions, e.g. Iris Murdoch). Anyway, thanks for the comments, which made trying my hand at blogging worthwhile.

Knowing what knowing is

When people say "I do not think it, I know it," they are saying that they are sure of it. Knowledge is important because we want bodies of knowledge that can be relied upon, and also because we need to distinguish between what we know and what we only think is the case, in all sorts of situations. When people say "I do not think it, I know it," they seem to be assuring us that what they think is indeed the case. Knowledge claims are a bit like promises. And promises can be a bit of a gamble:

Consider a boy sitting an exam, a boy who is not sure of an answer, but puts it down anyway. Suppose that his answer turns out to be correct. Would you say that he did not know the answer?

I wonder if the following has some intuitive plausibility:

Proposition p is known by subject S when S's justification for believing p guarantees that p is true.

How much of a guarantee S would need to provide would presumably depend upon the kind of uses that such knowledge might be put to, so it is conceivable that philosophers could disagree about what knowledge is without any of them being wrong!

A Linguistic Puzzle

Where, in ‘my tabletop is flat,’ is the meaning of that string of letters? Any meaning that it has must be read into it (it is not a string of magical sigils). You read those words, which are about the flatness of the table at which I am sitting, and perhaps you think of something like the tables that you know. You might think of such a thing being flat, and it being my table (whoever I really am). You get that string of letters, and you put your meanings into it. But the meaning of that phrase is, I think, the thought that I expressed with it, my thought that my tabletop is flat. And that is not the only mysterious thing about reading, of course. When we read something truly meaningful—e.g. a great novel—we do not seem to be getting out of it some rearrangement of what we already know. We seem to learn something about the world around us. Are we fooling ourselves?

Our Lewisian paradise

Modern philosophy (which began with Bacon and Descartes) looks antiquated until 1973, the year it acquired the now-standard way of analyzing counterfactuals: David Lewis’s possible worlds analysis (PW). If a one-eyed man is king in the land of the blind, Lewis seems to me to be like a blind man in the land of the one-eyed, successfully selling them a white stick:

The blind man (Lewis) tells them that they can use the white stick to find out whether things are near enough to them to be a problem. I think that they should have told him that it is just a stick, and the wrong colour for their outfits.

Lowe’s description of Lewis’s analysis is better than mine:

A counterfactual of the form ‘If it were the case that p, then it would be the case that q’ is said to be true if and only if, in the closest possible world in which p is the case, q is also the case – where the ‘closest’ possible world in question is the one in which p is the case but otherwise differs minimally from the actual world.

Suppose that I’m trying to tell you something, and I know that you’ll find it hard to understand what I’m saying; I might say ‘If you knew what I was trying to tell you, you’d know how difficult this is.’ Now, if you did know, it would be trivially easy to tell you about it, so presumably I meant that after I’d told you, and you’ve understood me, you’d agree that it was difficult to tell you. But suppose it’s so hard to tell you that you never do get it. Does my meaning really depend upon which of the possible worlds in which I’ve told you is most like the actual world, in which you didn’t get it?
......What if the difference is just a few neurons that you were born with, for example, but that those neurons also make it hard for you understand why it was difficult to tell you (since you then find such things so obvious)? What if lots of things; and so basically, how could all that really affect the meaning—and hence the truth—of what I’m saying? After all, we do seem to have got bogged down in an awful lot of fallacious arguments and counter-arguments since 1973; which may be ideal for professional philosophers in a stupid economy, but less so for those applying logic to the real world.

Cartesian dualism, ii

I’ve yet to find a good philosophical argument against such substantial dualisms as (for the commonest) that our psychology results from the interaction of spiritual souls with the physical brains in which (so to speak) they’re incarnated. The two commonest arguments are (i) asserting the closure of the physical and (ii) failing to see how the spiritual could interact with the physical. Both are clearly fallacious as I’ve stated them, but I’ve yet to find a substantially fuller, non-fallacious expression of either. Now, I’ve blogged on (i) already, and have little to say about either anyway, but I’ve just been reading Lowe (Erkenntnis 65, 5–23), who put (ii) as follows (2006: 7, 11):

[...] according to Descartes, whereas the mind has beliefs, desires, and volitions, but no shape, size, or velocity, the body has shape, size, and velocity, but no beliefs, desires, or volitions. [...] it is often complained that it is completely mysterious how an unextended, non-physical substance could have any causal impact upon the body – the presumption being, perhaps, that any cause of a physical event must either be located where that event is, or at least be related to it by a chain of events connecting the location of the cause to the location of the effect.

As put, the problem seems to be one of mere conceptual possibility, which is easily answered. By typing into your keyboard you can make virtual beings move about in cyberspace. Clearly you don’t have to be where they are, in cyberspace, to be able to move them about. So it isn’t so very mysterious how such things are possible. And even if it were, why presume that would be a problem for dualism, rather than a personal failing?
......As Lowe notes, people said that Newtonian action-at-a-distance was completely mysterious, and maybe it was, and is, but there was hardly any argument there against Newtonian physics (except in the minds of some philosophers). The truth turned out to be far weirder again, and it was to be had by working through Newtonian physics. There is that other problem, of how exactly the interaction works, but the way towards answering that is the relatively hard way of science, and why should it not go through Cartesian dualism?
......My analogy only worked because of the causal link between your fingers moving on the keyboard and the consequent virtual motion (as expressed in actual space on the screen), which goes via continuous paths in space (if we include force-fields in our ontology), but still, it did work. It suggests that a possible Cartesian response is to give the body, not only a spatial location, but also another, non-spatial location, at which the soul acts. How plausible is that? In the natural theistic context of Cartesian dualism, it’s very plausible, since God created space, and is himself located elsewhere.
......And suppose that Cartesian dualism is false. Then there’s some other true theory of mind. Somehow the physical brain, which changes its form and its atomic constituents continually, is associated with a subjective unit (the mind, which we know directly), which is continuously the same person. So if there could be a non-Cartesian theory, then there’s some way of associating with the physical brain a unique continuant of some sort. It is only to that that the Cartesian theory has to associate a soul. And a very simple and natural (in the Cartesian context) way to do that would be by divine stipulation, God associating each such brain-correlate with a unique soul.
......In many ways that’s far simpler and more natural than the sort of Humean regularity approach to scientific laws that philosophers are often led to by considering how mysterious are nomological necessities (a consideration that most scientists rightly ignore). If souls are possible, then they would have individual existences, in some logical space (say heaven), and would interact in some way (say via spiritual bodies). And if so then matter would’ve been created to be such as could be used in such ways (for some reason). The details are for scientific discovery, but the mere possibility is not really so mysterious.

What's a mathematician?

A mathematician is a device for turning coffee into theorems, according to Renyi's famous joke. But since most mathematicians don't prove theorems nowadays, I tend to think of the natural mathematicians as those finding More or Less as gripping as an Agatha Christie mystery (whether or not they like set theory).
......On the news this morning, there was a story about researchers at Durham have discovered that coffee makes us hear voices. They conceded that maybe those who hear voices tend to be more stressed, and so drink more coffee, but that seems like an odd concession to me, as people who're stressed usually turn to, if not alcoholic beverages, then chocolate or cigarettes, or cannabis.
......Scientists have also claimed that cannabis can trigger psychosis. That seems more plausible, as cannabis is the famous hippy drug, but I wonder even about that. Many of the most obvious direct tests of that hypothesis would be rather unethical, and the indirect tests (e.g. statistical correlations) would be vulnerable to selection biases. It's not just that schizophrenics might be more likely to disobey the law. There is apparently a part of the brain that is involved in religious experiences, e.g. Richard Dawkins had his stimulated and experienced nothing, apparently.
......If some people are more disposed to such things (again, whether they're born that way, or whether the brain adapts to their chosen way of life, is less obvious) then that would be a relatively obscure but effective source of such a bias. Incidentally, such studies need worry religious people surprisingly little. The traditional view of God has him timelessly creating us, and so faces similar problems, e.g. from our choices to turn to him etc. And Open-theistic views must face the facts of life anyway, e.g. that some of us are born richer, or better looking, or with better brains in other ways. If they can do that, they'll be able to live with similar facts.
......Anyway (oh how tangents attract the active mind), it occurs to me that those with such a religiously inclined brain might be more inclined (statistically, not each of them of course) to hear voices and also more inclined (similarly) to drink coffee, whether because they don't drink alcohol for religious reasons, or because they like to be awake to creation, or whatever. They may also be a little more inclined (statistically) to smoke cannabis, insofar as that's associated with the mystical side of hippies, or the religious side of Rastafarians, etc. If anything, we'd expect a greater corrolation with coffee, of course (and it does seem more plausible that coffee is not actually causing us to hear voices).
......Anyway, that's one possible explanation: a common cause leading to interlinked tendencies. Another explanation is that coffee in large quantities makes some of us irritable and tense, and perhaps over-sensitive, and so maybe more likely to hear voices insofar as we're already slightly inclined towards that (although I'm not ruling out the possibility that people take coffee because they're feeling stressed), but even in that possibility there's room for biases of the former kind—a common partial cause—e.g. a weak willed person might be more likely to over indulge in coffee, and less able to resist hallucinating. Similarly, they might drink more and get into fights, for such a reason. Or they might (also) react to the drink by getting more aggressive themselves. Note that that would be no reason for those who drink to relax and socialise to drink less (you may have guessed that I drink a lot of coffee:)

Faith, a definition

Personal faith is not assent to evidence which is so strong as to be beyond reasonable doubt. It is assent to a discernment of God which is personally overwhelming but not objectively testable. This is not discernment of a historical God, timeless and unchanging. It is discernment of an active, loving God, making himself known in personal lives at specific points which become the matrix of a communal response to his will.

Keith Ward, Divine Action (London: Flame, 1990), 238.

How mysterious is Platonism?

Arithmetical Platonism is supposed to be prima facie suspect because how, it’s asked, could we have arithmetical knowledge if the objects of that knowledge are in a world apart from us, a timeless world of Platonic objects, with which we cannot interact causally? I reply by wondering, how strange are abstract objects? You have just been reading this, for the obvious example. What have you been reading? You have been reading sentences. You look at the physical instances of these words, but you see the words, you read the words, and as you do so you are (hopefully) thinking about the thoughts expressed by means of them. So, there are, in the physical world around you, those physical instances of the shapes of (written modern English) words, and there are in your mind those thoughts; so, where are the sentences? What are the sentences? Sentences are made of words, and words are parts of a language (i.e. modern British English). They can be spoken or written, and can sometimes be spelt in different ways. Furthermore they have a meaning, a sense, and they have it essentially. The mere shape of a word is not a word, no more than meaningless strings of letters are words. Words, it seems, are abstract objects (I’m not entirely sure about that, or about what abstract objects are, so I’d welcome corrections) and you’ve just been reading some words of mine (and I’ll add that words can be true, insofar as they describe the world sufficiently accurately, or not, in case anyone wants to argue that thoughts and not words are truth-bearers:)