Valid Enough

The essence of analytical philosophy is the presentation of a valid argument. But often the result is a lot of boring nonsense, many of us find. Why? Well, the reason may be that we are encouraged to work with an absurd definition of ‘valid’. A technically valid argument is, for example, since x and y, therefore x.
......But what about, since the sky is blue, and there’s little wind, I won’t need my umbrella? Technically, that’s invalid because it’s not impossible that it suddenly clouds over and rains. Some philosophers would therefore call it an induction. But I don’t see any generalisation over lots of observations there. And while such a generalisation may well be one of the argument’s many implicit premises, surely it is all the obvious implicit premises that make the argument sufficiently valid for human communication.
......Or, for a more philosophical example, consider Moore’s argument: Since that looks like a tree, therefore that is a tree. Now, we usually make such a deduction subconsciously, but nevertheless, surely such arguments are usually valid enough. And even in more rigorous contexts, how else are we to do science except by taking our readings to be as we read them? What would make such arguments invalid is something like bad lighting, not the mere possibility that we’ve just been taken into the Matrix.
......Indeed, even if we had been, our argument might be valid enough, because we would then be using words in a new external world, and ‘tree’ would usually refer to the new object. Our argument would only be invalidated if we were aware that we were in the Matrix, and if that aspect of our situation was the most apposite.

Definite enough

Reference occurs when someone refers another person to something—or to some things (the first thing that Russell looked at, after thinking about Cantor’s paradox, was reference). Reference is fundamental to language (and mathematics). And even before creatures had language, there was being seen to be looking at something. There was pointing and then naming and thinking. But for Russell, names were definite descriptions, and vagueness was a logical problem. I think that vagueness is not so much a problem in logic, as the possibility of a logical solution to most if not all of the philosophical problems in logic. Consider, for example, the Lottery paradox:

If I believe, of each ticket, that it won’t win, then I believe that none of the tickets will win. Whereas, I know that one of them will win.

But I actually believe, of each ticket, that it very probably won’t win, and when I describe those beliefs with that amount of precision, there is no paradox. What about the Preface paradox; or more simply, the likelihood of some of my beliefs being wrong? I certainly believe all of my beliefs, so my belief that some of them are likely to be wrong might appear paradoxical. However, as soon as I come to the realisation that some of my beliefs are likely to be wrong, I find that I am holding those beliefs less strongly. For a more epistemological problem:

I can see very clearly that there is a horse trotting past a tree, outside my window (I live in a village in Bedfordshire), so I know that there is a horse there. I could conceivably be looking at a painted zebra that is about to be eaten by an alien stick-insectoid, though: that is a logical possibility. And if I don’t know how rare such creatures are, how can I know how unlikely this possibility is?

Well, I can simply assume that it is unlikely, until I get evidence to the contrary. We naturally assume that things remain more or less the same, in the absence of evidence to the contrary, when we name things. And we only have to be definite enough for the purposes of communication. We do not need a precise definition of “horse” in order to know that something is a horse, so why would we need to rule out all conceivable stick-insectoids?
......Quite generally, the power of natural language lies in its flexibility, which derives from the inherent vagueness of its terms: our words arrive sufficiently well defined for their usual uses, but they can always be defined with more precision if that becomes necessary. The ubiquity of vagueness in natural language is not a logical problem (as Russell believed), it is the possibility of a logical solution to most if not all of our philosophical problems.
......Philosophers have a strong bias towards bivalence because as we philosophize we clarify, aiming to maintain an adequate bivalence. But intuitions that logic ought to be bivalent are therefore quite compatible with logic not being perfectly bivalent. After all, questions of truth are essentially questions of how well our words describe the world; so, the logical primitive is not True (or T, in some mathematical model of logical reasoning) but True Enough. When we say “that’s true” we usually mean that it’s true enough. And it is implausible that statements are bound to be either true enough or else sufficiently false.

Do Inconsistent Objects Exist?

Mark Colyvan (2009: ‘Applying Inconsistent Mathematics,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 160–172), while looking at Inconsistent Mathematics, tentatively suggested that (p. 163): ‘There are times when we ought to believe in inconsistent objects.’
......His example was the infinitesimals of the early calculus, which are widely believed to have been inconsistent. E.g. they were equated to both zero and non-zero quantities (while Newton’s fluxions varied but were inconsistently equated to constants). Nevertheless, the early calculus was widely applied throughout the eighteenth century. So if, as many philosophers assert, ‘we should be committed to the existence of all and only the entities that are indispensible to our best scientific theories’ (p. 162), then it seems that Colyvan’s suggestion makes sense.
......But can we know what our best theories are, without hindsight? Epicycles, for example, were an arbitrarily effective way of coping with real-world ellipses, given a mathematical language of circles. So the astronomy of Copernicus, with his circular orbits and no epicycles, was originally less accurate than Ptolemaic astronomy. But of course, the former was a better theory. It was a step in the right direction. If we can’t, then, know what our best theories are without the benefit of hindsight, then insofar as we now have better theories without such inconsistencies in them, perhaps we should now say that we should not then have believed in such objects. After all, we could always take inconsistencies to be good indications that we need a better theory.
......It is perhaps easier to see that asking if inconsistent objects exist is a bit like asking if impossible things can happen. And of course, if something happens then it must have been possible. Nevertheless, things that seem impossible can happen. And similarly, existing objects may well have descriptions that, while true enough individually (for our usual purposes), are collectively inconsistent (at least on the surface). An apparent inconsistency usually means that our descriptions stand in need of more precision. But it doesn’t mean that they’re too bad to use most of the time. Nor does it mean that the objects so described don’t exist.
......For a ubiquitous example, light as we sense it can be bright (or dull), but photons are dense (or sparse). Our word ‘light’ equivocates between our sensations of light and the light itself. Furthermore, light itself is lots of photons, but it’s also electromagnetic waves, and waves aren’t particles. But it isn’t that light doesn’t exist, of course, and eventually quantum physics described such behaviour consistently enough. It certainly makes sense for us to believe that photons exist (and to be even surer that light exists). And although light also behaves according to relativity physics, which may well be inconsistent with quantum physics, such inconsistency may just be a reason to pursue an even better theory (of light).
......For a more apposite example, the infinitesimals of the early calculus, more precisely described, may be the so-called irreal infinitesimals that were informally introduced in my To Continue with Continuity (pp. 105–107). Suppose there are such continua as I describe in that paper (e.g. space, perhaps). And let x be a real number (e.g. pi), in the sense of an integer (e.g. 3) plus, after the decimal point, an endless sequence of digits in all the decimal places: the first (e.g. 1), the second (4), the third (1) and so forth (59265...). While that isn’t the standard definition of a real number, it’s a definite enough concept, and highly applicable.
......And since the standard axiom of infinity—which says that the natural numbers are collectively a standard set—goes well beyond the Peano axioms, and is rather prone to paradox (e.g. Levy’s paradox), let us further suppose that the natural numbers are collectively indefinitely extensible. If that’s indeed the case, then x is an (infinitesimally) imprecise description of many (infinitesimally) different lengths.
......So if l is a non-zero irreal infinitesimal then x + l = x because, quite generally, infinitesimals are smaller than 1/n for any natural number n, so that adding them to x affects x in none of its decimal places. (And incidentally, the word ‘infinitesimal’ derives from a Latin word that originally meant the infiniteth term in a sequence.) There are, then, consistent (if informal) mathematical objects—irreal infinitesimals—that for the purposes of the early calculus could be adequately described by its descriptions of its infinitesimals.
......So if such continua as do exist are well enough described by such (informal) theories as mine, then surely we could take irreal infinitesimals to be the referents of ‘infinitesimal’ in the early calculus. (Points were not then the same as real numbers, and the natural numbers were widely regarded as indefinitely extensible, so that infinite space would contain infinite lengths and hence geometrical infinitesimals.) Furthermore, insofar as the natural numbers can be said to exist, we could then think of such infinitesimals as existing.
......Of course, if mathematical existence is equated with consistency, relative to some axioms, then inconsistent mathematical objects exist when, and only when, we have inconsistent axioms; and of course, such inconsistent objects shouldn’t exist because they would be too trivial.

Monarchy And Democracy

Yesterday's Dispatches on Channel 4 was about "the illegal phone hacking carried out at the New of World during the six years Andy Coulson was either Deputy Editor and Editor," of current interest because "by hiring Andy Coulson David Cameron has sanctioned the News of the World culture of impunity." Rupert Murdoch certainly seems to have a worrying amount of influence over our police and politics, centre-left as well as centre-right. The origin of these revelations, in a successful prosecution for hacking a conversation between our two young princes, is of course less interesting. But the consequent importance of the fact that Murdoch could neither buy off nor intimidate the Palace does make me wonder whether the left, and other democrats, should question their traditional Republicanism, in these transitional days.

Ordinary Objects

Further to the question of whether or not Chairs Exist, I notice that Amie L. Thomasson’s 2007 book Ordinary Objects is out in paperback next month. Basically, it shows how “the claim that there are ordinary objects can form part of a coherent, reflective metaphysical view built up out of our common sense way of looking at the world, in a way that avoids the philosophical problems that have long been feared to plague a common sense ontology” (pp. 7–8).
......The first problem addressed (pp. 9–24) was that if all such things are made of atoms then there’s causal redundancy, e.g. if it’s really atoms arranged stone-wise that break a window (atoms arranged window-wise) then we shouldn’t also have a stone breaking it. But of course, the former just is the stone breaking it, if all such things are made of atoms. Thomasson’s analysis was more detailed, of course, but still readable, and seems to generalize nicely (e.g. the next section addressed the related issue of epiphenomenalism in the theory of mind).

Euler’s ‘2’

Incidentally, I’ve rewritten two of last month’s posts, and the rewrite—Did Euler’s ‘2’ refer to ZFC’s {Ø, {Ø}}?—will be appearing in next month’s issue of The Reasoner.

True Enough

What is truth? Well, snow can be brown when it is a bit muddy, and in shadows snow can be blue, but insofar as snow is white, ‘snow is white’ is true. Words are true when they are true to reality. And the obvious contrast is with falsity, with saying, of what is, that it isn’t (or of what isn’t that it is).
......But words exist in a public language and so, given how we come to acquire our linguistic skills, vague meanings are inevitably ubiquitous. Still, we invariably speak within some context, wherein we need only say enough to make our meaning clear enough. Our words can describe the world well enough—they usually do—and then what we say is true enough. And when it isn’t, we can always be more precise.
......We can even introduce new terms into our language, if we have to (as scientists and philosophers). Indeed, there seems to be no logical limit to our ability to be ever more precise. And so to say that something is true is, more precisely, to say that it’s true enough. Bivalent propositional logics—in which each sentence is either true or else false—are usefully rough approximations to the truth.
......Consider some commonplace examples: The table at which I’m typing this is flat—it isn’t warped or lopsided—but in another sense it isn’t flat, not being perfectly smooth and horizontal. To say that it’s flat is to say something that’s true enough.
......And similarly, to return to the themes of previous posts, ‘grass is green’ is true because ordinary grass (such as fills lawns and pastures) reflects the green bits of daylight (fuzzily delineated bits) ordinarily (e.g. when there’s no drought).
......And it’s insofar as chairs exist that it’s true to say of them that they do.
......And do rainbows exist? Well, in a sense they do (e.g. we can refer each other to them), but there is clearly a sense in which they don’t (much as mirages are not oases).
......What do we mean by ‘grass’ or ‘chair’? Such things form obvious kinds, which is how we come to learn such words. What most of our words have, then, are meanings that are definite enough. Indeed, such vagueness may well be logically necessary, in any possible medium of communication. But even if a more definite language was possible, it’s the vagueness we have which means that our words can be given more definite meanings as required. So a less vague language would in any case be a less useful tool in general (although it might have its uses).

Putting the green back in the greenery

The green of the greenery seems to be out there with that greenery. But what is out there is the surfaces of leaves, which send electromagnetic waves toward our eyes. The green is in the pictures of the world that our brains construct from data obtained via our sensory organs. What about the shapes of the leaves? Could the world be made of 10-dimensional strings, with our brains imposing upon the numerous sensations that come from our 10-dimensional sensory organs the shapes that we see in the world around us? Maybe; and for such reasons, some philosophers think that ordinary objects don’t exist in reality.
......Does the world (what we usually mean by ‘the world’) only exist in our heads? Well, there are certainly some ordinary objects in reality: what we mean by ‘reality’ is whatever space it is that includes the people whose language includes such expressions, so you and I are presumably two people (whatever we are made of). And there is a sense in which the greenness really is out there on the surfaces of leaves, because we learn the meaning of ‘green’ by being shown various green objects (or pictures of green objects) and being told that they are all green. It hardly matters whether you and I have the same sensations when we see them. Green is something that ordinary objects can be. Still, green is also a sensation. And a very mysterious one: do you have the same sensation as I do, when we both look at the same leaf? We have similar eyes and brains, but we know nothing about how sensations arise in our brains. Still, something is green if its surface is such that under normal lighting conditions, it would give rise to the same sort of sensations in those looking at it as they had when they learnt the meaning of ‘green’ whatever those sensations are.
......If your sensation when you see green was exactly the same as mine when I see blue, would I be wrong if I thought that you were seeing blue? Well, it seems to me that the meaning of ‘blue’ is the sensation that I have when I see blue things. But it also seems to me that the meaning of words like ‘blue’ is a public, not a private matter. So it seems to me that there is something like an equivocation here. And it seems to me that it is probably unavoidable, because our references to things in our external world are only possible via our sides of our interactions with those things. Would it be any different in the land of the blind? Suppose that they use sticks there, to feel their way around. And suppose that one of their sticks hits an unexpected obstacle. The person holding that stick might be able to tell that the other end of it had hit a bouncy object. And their word for such bounciness in an object might be the same as their word for the way that that stick had felt in that hand. But they would of course not think of the world as being full of such feelings. They would think of it as being full of objects, mysterious objects, some of them bouncy (in some non-visual sense of ‘bouncy’).
......When a tree falls in a forest at night, and there is no one around to hear it, does it make a sound? I suppose not, but what about its colour? Those trees do not look green: it is dark, and there is no one there to see them anyway. But they are green in the daytime, so I suppose that they are green trees. Suppose that you have a colour photograph of those trees on your wall. You do not think that those trees are there, in your room, but the green of those trees is there. Is it still there when you turn the light off and leave the room? Suppose that you return with a strange little light bulb: you change the bulb, turn the light on, and in that light the photograph looks blue. Is it a green photograph that looks blue? Is it still green, even though it looks blue in the strange light? And did other people learn words like ‘green’ in such a way that they would give similar answers to such questions?
......It is not implausible that philosophers could disagree about the "existence" of ordinary objects without any of them being wrong! What is clear is that a green alarm clock that goes off in a vacuum makes no sound. And if the clock is painted black, then it is no longer green. And such rooms exist in houses that are quite distinct from each other. Your house and my house are two houses, in a very precise way. When we reason logically about the world, our thoughts are as complicated as our relationships with the world. But the simplest thing to reason logically about ought to be arithmetic.

Chairs Exist

The basic contrast is with imaginary objects: Pixies don’t exist, electrons do; epicycles don’t exist, bicycles do. We learn the meaning of ‘exist’ in a world of tables and chairs, trees and cars, and so when we say that electrons exist we mean that they exist like chairs do. We can spray them onto surfaces, for example, much as we might throw chairs into a van. We can catch chairs and electrons, but not pixies.
......If we doubted that chairs exist, what could we mean by ‘exist’ if we said that electrons exist? That they are in our best theory of reality? But the thing about epicycles is not only that they aren’t fundamental objects, in our best theory. It is that they don’t exist, to be further analysed, and therefore shouldn’t have been in our best theory. Of course, pixies exist within fictions, so they exist fictionally, but that is also to say that they don’t really exist.
......Some philosophers think that chairs are imaginary, that only the atoms that make them up exist, but how could that be right? A chair made of Lego bricks would still be a chair. It would still exist, wholly composed of Lego bricks. Had it been made one brick at a time, with one brick not being a chair, and with no addition of one brick making a chair out of a non-chair, it would exist. Consider how, even though orange fades smoothly into yellow and red, with no sharp boundary, that doesn’t mean that carrots are not orange.

Do Chairs Exist?

Yes they do:

A physicist will tell me that this armchair is made of vibrations and that
it’s not really here at all. But when Samuel Johnson was asked to prove
the material existence of reality, he just went up to a big stone and kicked
it. I’m with him.

......David Attenborough

No they don't:

Like many philosophers, I don't believe that tables and chairs are fundamental
objects. Like a much smaller number of philosophers, I like to say that I don't
think tables and chairs exist. I have good reasons for my denial. For instance,
it does not appear that there is an exact moment at which a table comes into
existence.

......Alexander Pruss

Euler’s ‘2’ postscript

Since our reductionists (see previous post) assume that we can refer to abstract objects, let us see if any of the ways in which such reference might occur support their view of the referent of Euler’s ‘2’. It seems not; for suppose, for example, that reference to abstract objects is correctly described by Fictionalism. Then the view in question is like taking most twentieth century utterances of ‘Sherlock Holmes’ to refer to the character recently played by Benedict Cumberbatch on the BBC. Suppose instead that Gödelian platonism is true, so that we have something like a perception of abstract objects. Then the only choice we should make in our reference to them is the choice of their names. Between those two possibilities lies a Full-Blooded platonism, according to which all possible abstract objects exist. But that position is hardly available to those who don’t want—but don’t consider impossible—non-set-theoretic numbers. So in conclusion, it seems that our reductionists are quite eccentric after all.

Euler’s ‘2’ continued

For a more eccentric analogy (see previous post for previous analogy), let a family of cooks in some dull country be introduced to the meaning of ‘orange’ by means of some imported carrots. Since our cooks want to refer only to ordinary objects, not to such things as properties, which seem to them hardly things at all, they reduce talk of orange things to talk of their carrots. But they would clearly be wrong to take us to be referring to their carrots with our uses of ‘orange’. Indeed, we are not even referring to the colour of their carrots, which might turn yellow.
......In view of the way things are—e.g. the cells of the human retina—a more realistic reduction would reduce orange to the two primary colours red and yellow. And to do something similar for the referent of ‘2’ would take us, not to standard set theory but to psychology. The letters ‘M’ and ‘N’ are angular and black and are, collectively, 2 letters, and it is by means of such examples that we came to know what ‘2’ means. Much as shapes and colours are predicated of ordinary objects, the natural numbers are predicated of finite sets.
......Could such properties be collections? Well, there is a philosophical tradition of reducing properties (e.g. orange) to extensions of properties (the class of all orange things), but there is a well known problem with reducing 2 to the class of all pairs. Set-theoretic paradoxes show that such classes are, if absolutely general (not just the class of pairs of ordinary objects), indefinitely extensible. There is no pre-existing class of all pairs to reduce 2 to. Our reductionists therefore reduce 2 to a particular pair-set. Not being eccentric, they don’t reduce it to something concrete, like a pair of carrots (although that has obvious reductionist benefits), but to something as abstract as numbers are thought to be.
......However, it is not so much a discovery as a technicality to use {Ø, {Ø}} rather than {{Ø}}, or at least, such is the choice not to use some other set, class or category theory, or indeed, constructive mathematics. The set-theoretic axiom of infinity, in particular, is true by definition of all standard sets, but is not so much a discovery as a guess about the natural numbers. Now, that assertion only makes sense insofar as numbers are not sets, but that just means that our reductionists risk losing the ability to assert that the axiom of infinity is only a guess; about what would it be a guess? Such reductionists therefore put themselves in the position of those nineteenth century scientists who, for good reasons, took ‘space’ to mean Euclidean space. Those reasons were just not good enough; and note that various supertasks and other paradoxes currently give us cause to question the truth of the axiom of infinity, construed as a property of the non-set-theoretic natural numbers.

Did Euler’s ‘2’ refer to {Ø, {Ø}}?

The foundation of mainstream mathematics is standard set theory, within which ‘2’ usually refers to {Ø, {Ø}} (when we are considering the natural numbers, see comments below).
......Alexander Paseau (2009: ‘Reducing Arithmetic to Set Theory,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 35–55) thinks that those reducing arithmetic to set theory in such a way—perhaps they want their ontology to include only sets, not also non-set-theoretic numbers—may also take most mathematicians, past and present, to have been referring to the standard set {Ø, {Ø}} with their ‘2’s. He (2009: p. 42) made the following analogy: ‘When the ancient Greeks spoke about the sun, they spoke, unknowingly, about a hydrogen-helium star that generates its energy by nuclear fusion.’
......By contrast, anyone taking ‘say, a carrot to be the referent of “2” in Euler’s mouth’ should, he (2009: p. 43) thinks, ‘be an error theorist about Euler’s claims involving “2”,’ and so risk taking too many—according to Hartry Field (2001: Truth and the Absence of Fact, Clarendon Press, p. 214)—of Euler’s words to be untrue. According to Paseau (2009: p. 43), our ‘less eccentric reductionists need not interpret Euler’s arithmetical claims error-theoretically and may respect his intended truth-values.’
......They could, he thinks, take that view even if ‘2’ referring to the standard set {Ø, {Ø}} was not so much a discovery about 2 as a technical convention. As Paul Benacerraf (1965: ‘What Numbers Could Not Be,’ Philosophical Review, 74, pp. 47–73) famously observed, another possible referent is {{Ø}}. So for another analogy, suppose some chromatographers took ‘orange’ to refer to wavelengths of light within some definite range, in order to avoid vagueness and because such a stipulation was sufficient for their scientific needs. They would surely need further reasons to take us to be referring to such wavelengths with our uses of ‘orange’. And similarly, if our less eccentric reductionists have not so much discovered the referent of ‘2’ as accepted a useful technicality, then it seems to me that they should not be taking our ‘2’s—nor Euler’s—to be referring to the standard set {Ø, {Ø}}.

What Rainbows Cannot Be

Rainbows are clearly not ordinary objects, being more like mirages than oases. But they are just as clearly not fictional objects. One’s conception of a rainbow may well contain false presuppositions, but in that way rainbows do resemble ordinary objects. Consider a red apple. Perhaps what is really there is a 10-dimensional collection of particle-strings within a 4-dimensional block universe, with the red and the spheroid existing only in the minds of certain kinds of potential perceivers of that collection. (That modern scientific hypothesis is not a million miles away from theistic idealism, of course.) Anyway, suppose some fictional meteorologists defined ‘rainbow’ to be a specific sort of event. They might do so because such a technical convention suited their scientific needs better than the rather vague ordinary meaning. And of course, an event is a kind of object (especially in a 4-Dimensionalist world). Now, some relatively arbitrary choices may well have been made as they specified their referent of ‘rainbow’. But that does not mean that rainbows cannot be objects. They are, after all, intentional objects, over which we might quantify. And of course, our intuitions that rainbows are not objects all derive from the fact that they are not ordinary objects. I mention this because it strikes me as similar to Benacerraf’s famous argument about what numbers cannot be.

Modern Physical Probability

What does 'probability' mean in quantum mechanics? I address that rather philosophical question in Modern Physical Probability, a Google doc that replaces the Geocities pages onto which I had put my MLitt dissertation (all my Geocities links need replacing, I've done this one now because a rewrite of my Two Envelopes post is in this month's issue of The Reasoner:)
The doc has 9 sections:
1. Laplace’s Urn
2. Hájek’s Arguments
3. Von Mises’ Limits
4. Popper’s Propensities
5. Reichenbach’s Limits
6. Mellor’s Personalism
7. Humphreys’ Paradox
8. Eagle’s Arguments
9. Lewis’s Humeanism

Eternity, Mawson's belief and Cantor's paradox

My answer to Tim Mawson's argument for God being timeless, which I've been writing for two years, is currently the following Google Doc (updated in July): Eternity, Mawson's belief and Cantor's paradox. I defend Presentism under Anselmian theism, and use Cantor's paradox to argue that God is, if He exists, able to increase His knowledge, and hence is able to change, rather than being timeless (unless logic is even less standard than it is under Presentism); and since I also argue that, even so, He could be omniscient, in the usual sense of knowing all truths, I am also answering one of Patrick Grim's arguments against an omniscient being. And since Mawson's argument was directly against Open theism, given that we have Libertarian (agent-causal) freedom, I am thereby defending Open theism (specifically the second of Alan Rhoda's three varieties).

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.