Introduction

This is the second of 17 posts, which are collectively Eternity, etc.
......In this debate, “God” refers to the perfect person—or Trinity (but nothing impersonal)—who created everything else ex nihilo [i]. If such a God exists, He (they, she) created us, so such Theists naturally think of Him as the greatest conceivable being; and Open Theists are no exception [ii]. Open Theists are so-called because they believe that God keeps some of His options open, as He relates to us [iii]. Such divine openness follows from the sort of freedom (responsibility, creativity) God gave to those made in His image. Open Theists take a libertarian view of free will. And atemporalists—who think that God transcends even the possibility of change—can also take that view [iv], according to Mawson:

On the libertarian view of free will (as it is standardly construed), all that has to be true for you to be free in the future in your choice to do X is that you have the power at that time to do something other than X and on atemporalism you can have this power—have the power to make God either have the atemporal belief that you do X or the atemporal belief that you don’t do X—without having the power to make any belief He actually has false. [v]

The future being what Mawson calls “real” (“that some statements concerning what is now the future are true”) is relatively uncontroversial. Even Presentist Open Theists might think it true that, for example, the earth will still be here tomorrow. “Presentism is the view that only what exists now has any reality”[vi], but what exists now—e.g. an egg falling—could determine how something will be—e.g. the egg breaking—via the laws of nature, and God’s constancy (see section IV), and so make it true now that such will be the case.
......I argue in section IV that Presentism is not implausible under Perfect Being Theism. Mawson’s primary argument—that because of future contingents (see section III), the Open God lacks some infallible knowledge of the future—therefore lacks the significance attached to it by Mawson. Under Presentism, the Open God can be infallibly omniscient (see section III). Presentism can also explain why the Open God is essentially constant, and hence why Mawson should not have thought of the Open God as having no infallible knowledge of the future. And Mawson’s main argument—that by being incompletely omniscient, the Open God is liable to make mistakes—was based on a misunderstanding of Open divine action, according to section V.
......Section VI initiates a more direct comparison of Presentist Open Theism with libertarian atemporalism. Section VII is based on Cantor’s Paradox and is an informal mathematical argument that the numbers of things within possible creations are indefinitely extensible. God’s omnipotence therefore indicates that His knowledge of whole numbers is forever increasing. And I also suggest in section VIII that under Presentism He could, even so, be necessarily omniscient.
......Notes:
......[i] Mawson, Belief in God, pp. 9–27.
......[ii] Richard Swinburne, Is There a God? (Oxford University Press, 1996), pp. 3–19; Alan R. Rhoda, “The Philosophical Case for Open Theism,” Philosophia 35 (2007): 301–11; Alan R. Rhoda, “Generic open theism and some varieties thereof,” Religious Studies 44 (2008): 225–34.
......[iii] For more details, see Charles H. Pinnock, Most Moved Mover: A theology of God’s openness (Baker Academic, 2001).
......[iv] Not all atemporalists do, e.g. see Paul Helm, Eternal God: A study of God without time (Clarendon Press, 1988).
......[v] Mawson, “Divine eternity,” p. 40. For more details, see Timothy O’Connor, “Dualist and Agent-Causal Theories,” in Robert Kane (ed.), The Oxford Handbook of Free Will (Oxford Univ. Press, 2002), pp. 337–55.
......[vi] Rhoda, “Generic open theism,” p. 234 n. 22. For more details, see Thomas M. Crisp, “Presentism,” in Michael J. Loux & Dean W. Zimmerman (eds.), The Oxford Handbook of Metaphysics (Oxford Univ. Press, 2003), pp. 211–245; Craig Bourne, A Future for Presentism (Oxford Univ. Press, 2007).

Eternity

This is the first of 17 posts that, collectively, defend Open Theism against Mawson’s Divine eternity (2008). As I write them, I shall be polishing up what I was writing over the summer, i.e. Eternity, Mawson’s belief and Cantor’s paradox, which was itself the result of a very slow revision of the first half of my 2008 reply to Mawson, Omniscience and the Odyssey Theodicy. The Abstract for the current paper is:

T. J. Mawson believes that the God of Open Theism is liable to bodge things up, by not infallibly knowing all about the future. But I argue that Mawson misconstrued the Open Theist view of divine action. And since the God of Presentist Open Theism ould infallibly know all there is to know, I also argue that Presentism, whose falsity Mawson presupposed, is not implausible if God is the Perfect Being. Furthermore, if God can create arbitrarily many things, then because of Cantor’s Paradox, His knowledge of whole numbers is plausibly growing forever. Yet even such a powerful and hence changing God could, under Presentism, know all the truths of arithmetic. I conclude that the Presentist Open God could be the Perfect Being.

According to T. J. Mawson [i], even those Perfect Being Theists who take a libertarian view of free will should reject Open Theism, so long as they accept what he calls “the reality of the future,” i.e. “that some statements concerning what is now the future are true.” [ii] Mawson argued that because of future contingents, the Open God lacks “infallible knowledge of at least some aspects of the future,” [iii] and that such incomplete omniscience makes “whatever goodness (in the sense of beneficence, not just benevolence) God has a matter of luck.” [iv] I will be uncovering several lacunae in Mawson’s reasoning (in sections III to V), and then arguing more directly (in sections VI to VIII) for my conclusion, that such Theists should not reject Open Theism. The underlying question concerns how God is eternal (sections I and II). The 16 posts will be:

............I Introduction
............II Divine Attributes
............II Divine Attributes cont.
............III Future Contingents
............III Future Contingents cont.
............IV Theistic Presentism
............IV Theistic Presentism cont.
............IV Theistic Presentism again
............V Bodging Up
............V Bodging Up cont.
............VI Possible Worlds
............VII Cantor’s Paradox
............VII Cantor’s Paradox cont.
............VII Cantor’s Paradox again
............VIII Omniscience Again
............VIII Omniscience Again cont....

Notes:
......[i] T. J. Mawson, “Divine eternity,” International Journal for Philosophy of Religion 64 (2008): 35–50.
......[ii] Ibid, p. 37. Statements are basically possible assertions. For more details, see T. J. Mawson, Belief in God: An introduction to the philosophy of religion (Oxford Univ. Press, 2005), p. 239 n. 6. Questions of truth are essentially questions of how well our words describe the world, of course.
......[iii] Mawson, “Divine eternity,” p. 37.......[iv] Ibid, p. 49.

P.S.: Two years later, that unpublishable paper became Divine Eternity Revisited

The Alternative to Naturalism

...is Philosophy, which revolves around debates between Naturalists and Theists. It is a few centuries since Philosophy was the alternative to Theism, but the mainstream of analytical philosophy remains Naturalism, which is essentially atheistic. Science, which Naturalists emphasise, is not atheistic, however, but agnostic. Indeed, the particular sciences are not just agnostic about God, but about most other things too, so that scientists make the most interesting connexions. But Naturalists, no less than Theists, aim not so much to join up the sciences, but to find holes in each others’ arguments.
......As a mathematician, I am quite interested in seeing how mathematics would differ from standard set theory were we created by a Perfect Being. Indeed, I suspect there is an argument from 2 + 2 = 4 to the existence of a Perfect Being who created us. Mainstream philosophy of mathematics takes ‘2 + 2 = 4’ to be what mainstream mathematics—whose foundation is axiomatic set theory—says it is, and aims to develop an atheistic epistemology of set theory. As they do so, their concepts become implausible (whence the possibility of such an argument).
......Ironically, it is because I am essentially a mathematician that my approach to philosophical problems would strike most analytical philosophers as insufficiently mathematical (formal). And incidentally, there is a famous story that:

Czarina Catherine the Great of Russia was concerned about the deleterious effect the philosopher Diderot was having on the religious faith of the nobility who were listening to him hold forth on atheism in her court. She encouraged famous mathematician Leonard Euler to confront him, and he did, with the following challenge: “Sir, (a + bn)/z = x, hence God exists—reply!” Diderot, who, according to the story, was completely mystified by all things mathematical, fled the court and Russia in deep humiliation. Diderot and Euler actually were in Russia at the same time, both at the invitation of the Czarina, but this is a joke at Diderot’s expense that neither Euler the man nor Euler the mathematician would have made. Even if it had been, Diderot—who was actually a fairly capable mathematician himself—would not have been stumped.

Two Contraries of Naturalism

Naturalists are analytical philosophers who take the natural sciences to be giving us the best foundation for our thoughts. As philosophers, they don’t usually trust scientists to interpret their own discoveries, however, so I wonder what Naturalists contribute to the scientific enterprise. Is it logical rigour and coherence, or just the opposite?
......Naturalists emphasise the very careful observations that scientists make. And indeed, it is important that we be sure of our facts. But when it comes to the most certain thing in the world—that we are conscious entities—they favour materialistic theories that reduce such things away. They may justify their preference on the grounds of ontological parsimony, or overall theoretical elegance, or the fact that natural scientists don’t usually include subjective stuff in their scientific observations. But Naturalists rightly regard as unscientific those who would explain away such less obvious facts as the fossil record on similar grounds (e.g. not needing all those aeons of inhumanity, and there being no mention of such things in a favoured account of reality).
......Naturalists criticise religion for giving too-easy answers. And it is indeed important to get the right answers to the questions the world raises, not just the most convenient ones. But Naturalists like theories that are based on standard logic, which is simple and popular. And standard logic takes a timeless view from nowhere, which is a bit unrealistic. E.g. consider how the Copenhagen interpretation of quantum mechanics, by having an observer-shaped hole in its description of reality, is less tidy but clearly more realistic than those no-collapse interpretations that would have us doing everything it was physically possible for us to do by splitting into lots of different people all the time.

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).