On the Sorites

          A drop of water falling on a hill does not wash it away.
So, if we start with a hill, then after a drop of water we still have a hill.
After another drop, we still have a hill; and many repeated applications
of the first, italicised line means that after lots of drops the hill remains.
But, after enough drops the hill will, of course, have been eroded away.

That is basically a Sorites paradox. Similarly, all real-world calculations will, if long enough, become swamped by error bounds. All measurements should come with error bounds, and while a short calculation will result in only slightly larger error bounds on the result, a very long calculation will be useless. Now, logic is supposed to be different, more like Geometry, where given certain lengths, geometrical manipulations can be arbitrarily long. But that will only be the case if the terms that the logic is applying to are definite. In the real world, there is a ubiquitous, if usually very slight, vagueness (it is there because it is so slight: nothing has acted to remove it). Consequently logical arguments that are about real things should not be too long. It is an interesting question, how long they can be; but certainly, those of the Sorites paradoxes are too long.

Why Merit Lacks Merit

Everywhere has been invaded, usually several times. As a rule, the invaders steal everything, killing some and enslaving the rest. A few of the natives help them. There is a continuum, between such collaborators and the dead, on which the majority of the original people find themselves. Most of them tend not to volunteer for anything, after invasion. But among the new ruling class there are many volunteers, eager to prove their metal. People pair up, and new generations give way to even newer ones. Now, some people do well, and often it is in large part because their ancestors inherited stolen goods. Of course, such people have still, themselves, done well. Perhaps they also had good genes. Perhaps they were lucky. They hardly thereby deserve advantages over people who suffer, while doing badly, because, in large part, their ancestors had stuff stolen from them, or because their genetic inheritance was also poor, or because they were unlucky. Note that according to standard evolutionary theory, every genetically fit individual owes that fitness to the immense sufferings of a huge number of other individuals who died without leaving any offspring behind. Of course, it is in general much more complicated than that, too complicated for simple words to do it justice; but it would certainly, therefore, be quite unjust, to say the least, to talk up meritocracy. (More details: this from last year.)

The Signature of God

What follows is a proof of the (probable) existence of God.
     Such an extraordinary claim requires extraordinary evidence, of course, and so this post is a bit long, but most of the heavy lifting has already been done by those who have been failing, for over a hundred years, to find atheistic explanations of certain basic mathematical facts.
     Evidence for the existence of God must be extraordinary, and of an appropriate kind. Suppose we saw letters of unearthly fire in the sky, spelling out a claim that there is a God; the most likely explanation would be pranksters, or, at a push, aliens. Evidence for the existence of the Creator of all things, including such things as the human mind, should therefore include something more like a logical proof. There are already several arguments that claim to be such, e.g. the ontological argument, and you might think of the following as another (we could expect there to be several logical proofs because when we find one proof of a mathematical theorem there are usually others to be found).

What follows is based on the nineteenth century mathematics of Georg Cantor, and in particular, his famous logical paradox.
     Logical paradoxes are chains of thought that seem logical but which take us from self-evident truths to contradictions. Nothing, you might think, could be further from a proof; but it is precisely because logical thoughts take truths to truths, not to contradictions, that it follows that in every such paradox there must be some false assumption(s). The harder the paradox is to resolve, the stronger – and more surprising – will be the chain of thought from the false assumption(s) to the contradiction. A very tough paradox can therefore amount to a rigorous chain of thought that takes some very plausible assumption(s) to a contradiction, thereby proving by reductio ad absurdum the assumption(s) to be – surprisingly – false. In particular, Cantor’s paradox refutes atheism (and classical theism, which I take to be the view that there is a being who is omnipotent, omniscient, immutable and so forth).
     Things that are as Cantor’s famous diagonal argument shows them to be could, just possibly, exist within the creation of a Creator of all things (were that Creator not classically immutable). You will see why below; and while that fact may not seem like much, it yields a reason why there is probably such a Creator because there is very probably no other way in which things as we know them to be could exist. That high probability comes from the fact that mathematicians and logicians have been looking for a more intuitively satisfying resolution of Cantor’s paradox for over a hundred years, working within their background assumptions – atheism, for the most part (although also classical theism, especially in Cantor’s day) – and in all that time they have found no better way of avoiding paradoxical contradictions than the formalization of mathematics and logic.

     Cantor was working on Fourier analysis, in the 1870s, when he found it necessary to extend arithmetic into the infinite, despite various paradoxes. He resolved those paradoxes by extending arithmetic in a rigorously logical way, throughout the 1880s, but sometime in the 1890s he found his own paradox. Naturally he worried that he had refuted his own work, but he had been very rigorous, and so there was little the mathematical community could do – given their background assumptions – but formalize the foundations of mathematics. The question of what numbers really are was left to philosophers; in mathematics, there is no paradox: there are formal proofs, in most axiomatic set theories, that there is no set of all the other sets: were there such a set, its subsets would outnumber the sets, via a diagonal argument (see below), whereas subsets are sets. Formalization enables the paradox to be avoided, but it does not resolve the underlying problem: whenever we have a lot of sets, we do have their collection, because a collection of things is, intuitively, just those things being referred to collectively; and since each of its sub-collections is, intuitively, just some of those sets, we also have all of those sub-collections. Intuitive versions of Cantor’s paradox remain, then, to be resolved.

The following version, in particular, works by way of showing that certain possibilities become more and more numerous (see my earlier sketch of this version). Now, if something is ever possible, then it was always possible; but, possibilities of various kinds can grow in number by becoming more finely differentiated, as you will see in the following two paragraphs. But to begin with, an initial worry might be that even if some possibilities were differentiated in the future, those differentiated possibilities would already exist in spacetime (so that their number would actually be constant). So note that while presentism – the view that only presently existing things really exist – is not popular, it is generally agreed to be logically possible. Let us therefore use ‘time-or-super-time’ to name time if presentism is true, and something isomorphic to presentist time – at a mere moment of which the whole of spacetime could exist – if the whole of spacetime really does exist. The point of that definition is that time-or-super-time might exist even if presentism is false; either way, ever more possibilities could, just possibly, be individuated (in time-or-super-time).

     For a simple example of differentiation, suppose that spacetimes come into being randomly, in time-or-super-time, with some of them happening to be exactly the same as our spacetime. Someone exactly the same as you exists in each of those spacetimes. And of course, each of those identical copies of you was always possible in time-or-super-time. As we consider any one of them, it seems as though there must always have been the individual possibility of that particular person; and certainly, that individual was always possible. But what about the copies of you in future spacetimes? How could their individual possibilities be already distinguished from the more general possibility of someone exactly the same as you? Such copies of you do not yet exist, to be directly referred to, and indeed, they may never exist. So for such random beings, in presentist time-or-super-time, it would not make sense for their particular possibilities to exist. So despite our hindsight, the possibilities of such people must originally have been undifferentiated parts of the more general possibility of someone just like you. It is only with hindsight – after differentiation – that we see the differentiated possibility in the past.

     For an example without randomness, suppose that a Creator in time-or-super-time determines to create a ring of equally spaced, absolutely identical objects. None of those objects can be individuated until the ring has been created, because their Creator does not want to individuate them. So before then there is only the general possibility of such an object. Afterwards there is, for each object, the individual possibility of that object in particular, in addition to that general possibility. Once a particular object exists, there seems always to have been that particular possibility – because that particular object was always possible – even though we know, from the description of this scenario, that it was the general possibility that always existed.

I will be describing how certain possibilities might become more and more individuated by a dynamic (as opposed to immutable) Creator of all things ex nihilo. Creation of things ex nihilo is the creation of things out of nothing; it contrasts with the creation of things made out of some already existing substance (like a sentient computer making a phenomenal world out of computers and human brains). Creation ex nihilo is, at the very least, logically possible. After all, the Big Bang was clearly possible, and for all we know it could have followed nothing physical; for all we know, it could have followed some sort of creativity, such as a person. What we know for sure is that in the world there are physical objects and people. It is not easy to see how real people could be made of nothing but chemicals, but physicalism is of course a prima facie logical possibility; and it is similarly possible that spacetime and everything in it was created by a transcendent person.

     Given that such a Creator is logically possible, the following paradox then shows that the possibilities in question probably do become ever more numerous, because that is probably the only way of avoiding the contradiction derived below (other than simply ignoring it, or in other ways rejecting logic). Furthermore, it is very hard to imagine how those possibilities could possibly become more numerous if there is no such Creator. That is why this resolution of the paradox has for so long been overlooked. And that is how this paradox will show that there is probably such a Creator. So, to my intuitive but rigorous version of Cantor’s paradox.

We should begin with a self-evident truth; and clearly, these words are distinct from each other. That fact is self-evident because that is how we were able to read those words. There are, then, numbers of things; for example, ‘I’, ‘am’ and ‘lying’ are three words.
     Note that pairs of those three words – {‘I’, ‘am’}, {‘am’, ‘lying’} and {‘I’, ‘lying’} – are just as distinct from each other as those words were, because those three pairs differ in just those three words. Similarly, pairs of those pairs – e.g. {{‘I’, ‘am’}, {‘am’, ‘lying’}} – are just as distinct; as are pairs of those, and so on.
     Now, because of that ‘and so on’ we will have infinitely many, equally distinct things, if we can indeed count pairs as things. But is there really something that, for any two things, sticks them together to make a third thing? Put that way, it must seem unlikely. But, for you to pick out any two of our original three words, those two words must have already been a possible selection. Such possibilities can be our third things. In general, a combinatorially possible selection from some things corresponds to giving each of those things one of a pair of labels, e.g. the label ‘in’ if that thing is in that selection, or else the label ‘out’. If two of the labels are ‘in’, for example, we have a combinatorially possible pair. Every combination of as many such labels as there are things in some collection corresponds to some combinatorially possible selection from that collection, and vice versa.

So, let us take ‘{‘I’, ‘am’}’ to be the name of the combinatorially possible selection of ‘I’ and ‘am’ from our original three words, and similarly for the other increasingly nested pairs described above, which we may call, collectively, ‘N’. The following intuitive but rigorous version of Cantor’s diagonal argument proves that for any collection of distinct things, say T, the collection of all the combinatorially possible selections from it, say C(T), is larger than T.
     Informally, two collections are equinumerous – they have the same cardinal number of things in them – when all the things in one collection can be paired up with all of those in the other. So suppose, for the sake of the following reductio ad absurdum, that C(T) has the same cardinality as T. Each of the things in T could then be paired up with a combinatorially possible selection from T in such a way that every one of those possible selections was paired up with one of the things in T. Let P be any such pairing. We can use P to specify a possible selection, say D, as follows. For each thing in T, if the possible selection that P pairs that thing with includes that thing, then that thing is not in D, but otherwise it is, and there is nothing else in D. Since the only things in D are things in T, D is a possible selection, and so it should be in C(T). But according to its specification, D would differ from every possible selection that P pairs the things in T with, which by our hypothesis is every possible selection in C(T). That contradiction proves our hypothesis to be false: C(T) does not have the same cardinality as T. Furthermore, C(T) is not smaller than T, because for each of T’s things there is, in C(T), the possible selection of just that thing; so, C(T) is larger than T.

     As well as N, there is therefore the even larger collection C(N), and similarly C(C(N)) – which is just C(T) when T is C(N) – and so forth. All the things in all those collections are as distinct from each other as our original three words were, because they differ only in things that are just as distinct. Let the collection of all those things be called ‘U’: U is the union of N, C(N), C(C(N)) and so forth. U is larger than any of those collections because for each of them there is another of them that is larger and whose things are all in U. And since there are all of those things, there are also all of the combinatorially possible selections from them, which are just as distinct from each other, and which are collectively C(U). And so on: there is always a larger collection to be found; if not another collection of all the combinatorially possible selections from the previous collection, then another union of every collection that we have, in this way, found to be there. Those steps always take us to distinct possibilities that are fully defined by things that are already there. So, there must already be all the things that such steps could possibly get to.
     The problem is that from all of those things existing, it follows that all of the combinatorially possible selections from them also exist – since they are equally distinct possibilities, fully defined by things that are already there – and there are even more of those possible selections, as could be shown by a diagonal argument, which contradicts our having already been considering all the things that such steps could possibly get to.

Since there are no true contradictions – outside formal logic – something that seemed self-evident in the above must have been false. But the above chain of reasoning was a relatively short argument, from a self-evident premise. It is very easy to survey the whole of the argument and see how rigorous it was. The only lacuna is the one highlighted above: the obscure possibility of those combinatorially possible selections being the end results of more general possibilities becoming individuated. The following proof relies on that being the only lacuna, which you can only determine for yourself by trying – and failing – to find another. Perhaps, for example, there are no such things as possibilities? But were there no logical possibilities, logical thought would become impossible (except in some formal sense), and so we must presume that there are such things. It can be argued that there are not; but similarly, there are those who argue that there is only mind, while others argue that there is only matter. It seems to me to be self-evident that there are phenomena – our experiences – as well as physical things (e.g. those that we experience), and, similarly, that a huge range of non-formal logical thought is possible. And in particular it seems to me to be self-evident that {‘I’, ‘am’} is one of three combinatorially possible ways of making a pair of words (from our original three). Consequently the question is where a principled line should be drawn: where are the joints of nature? The reason why {‘I’, ‘am’} is a possible selection is that ‘I’ and ‘am’ are two of our original three words, and that reason generalises in an obvious way: for any things, in any given collection of things, those things are a possible selection. Note that a logically possible being could select those things from that collection.

     Regarding the possibility of the combinatorially possible selections being the end results of more general possibilities becoming individuated, it is conceivable that the Creator of all things ex nihilo would be able to individuate them because of the unique authority of such a being. Much as the individual possibilities of particular people, in the example above, could not be distinguished from the more general possibility of just such people, not until those people were there to be directly referred to, so it might be that the most unimaginably nested of the combinatorial possibilities are not individuated until such a being individuates them (by thinking of them). They need not be individually possible selections until then because who could possibly make such a selection? There is only the Creator, thinking of them in the absolutely definitive way of such a being. Naturally, such possibilities seem as immutable as the laws of physics, to us; but of course, to a God the laws of physics are mutable.
     There is not much more to be said, about such divine differentiation, though. Creation ex nihilo is totally alien to our experience, so it is essentially obscure. But, it is a relatively clear logical possibility for all that. Analogously, it is quite obscure how atoms of lifeless matter could be arranged so as to make conscious life, but that does not stop materialism being a logical possibility (for all that it might make it seem less plausible). Note that such a Creator could have existed prior to any things at all, because such a being could be, in itself, more like a Trinity than a thing. Such a being could have always known of the most general possibility of things as we know them, before choosing to contemplate creating some such things; and could then have known an awful lot about combinatorially possible selections, nested around those possible things, up to unimaginably high levels of an increasingly nested hierarchy (such levels as standard mathematicians would never contemplate). It makes sense that a being that could create things ex nihilo would know so much about them (and might even enjoy finding out more). Standard set theory would therefore be a very good mathematical model of the more imaginable levels (and of how there are unimaginably high levels, not all of which can be assumed to exist already). (Note that none of the properties of the underlying things would be made variable by the higher levels being variable; on the contrary, each level would be completely determined by those things being distinct things.)

So, since a dynamic Creator is, at the very least, a logical possibility, hence our combinatorially possible selections could, just possibly, be growing ever more numerous. And since there seems to be no other way of avoiding the contradiction, hence those possible selections are probably growing in number. Furthermore, outside the context of the absolute dependency upon their Creator of things created ex nihilo, there is no conceivable way in which those possible selections could grow in number. That is why this resolution has, for so long, gone unnoticed. And that is why it follows that there is – at least probably (in view of that long period of modern thought) – such a Creator.
     The big problem with that conclusion is, of course, that the majority of scientists are atheists. You might therefore be quite sure that there must be a flaw somewhere in the above. The most surprising thing about the above, however, is how scientific it could seem to simply ignore it, even if there is no such flaw. Many logicians take the logical paradoxes to be good reasons for not trusting pre-formal logic (and similarly, pre-formal arithmetic), however rigorously it is applied. After all, we would hardly expect primates – even highly evolved primates – to be perfectly logical. Whereas you might expect that a more formal treatment would find there to be no problem; and indeed, there is no formal paradox. Formal logic does not just look scientific, it reliably delivers desired results.

     Nevertheless, logic – our natural, pre-formal logic – is not so much an option as a necessity. Would highly evolved primates reject their own logic just because it gave them something that had seemed too good to be true? Probably not; but more importantly, it is not really an option. It is only because we believe science to be logical – in the pre-formal sense – that we believe science when it tells us that we are highly evolved primates. It is not because scientific results could be written up in a formal logic. After all, there are formal logics in which true contradictions have been formalised. And while most formal logics do not allow true contradictions, the question is: how could we determine which formal logic to use, except by applying our natural logic, as rigorously as we can? Even letting formal criteria decide the matter would be to have decided pre-formally to do so. Note that we should not do that; such formal criteria as simplicity, for example, might tell us to allow true contradictions. Indeed, the logical paradoxes could all be regarded as straightforward proofs that there really are true contradictions, unless we had already ruled that out. And we should of course rule that out, because things cannot be a certain way while not being at all that way. Being that way is precisely what ‘not being at all that way’ rules out, pre-formally.

     It was one thing to reluctantly replace logic with formal logic, and numbers with axiomatic sets, in order to avoid paradoxical contradictions; it would be quite another to jump at the chance to make such replacements just to avoid the refutation of a strongly held belief. The latter would clearly be unscientific. Of course, you may think that there is no such refutation, that God has been invoked to explain something that may well be explained by science one day. And such God-of-the-gaps arguments are indeed unsound. Before it was discovered that we are on the surface of a massive spheroid orbiting a star, for example, a sunrise might have been explained by invoking God, on the grounds that only a God could cause such an awesome event. My argument, however, is more like the Newtonian connection of the motion of planets with the motion of projectiles. That is because there is, in mathematics, a practice of defining mathematical objects in terms of human constructions; such constructivism is not popular, but it is a valid practice. I am explaining the Cantorian property of things by invoking divine constructivism, not a simplistic miracle. Note that there is no perception in modern mathematics – as there was in the early years of the twentieth century – that Cantor’s paradox might be resolved by future research within the mainstream. Rather, our axiomatic set theories and formal logics are beginning to look more and more like epicycles.
     It might be thought that I do have a God-of-the-gaps argument because I do use God to explain something scientific. So note that there were similar objections to Newton’s invocation of action-at-a-distance, in his explanation of astronomical observations, on the grounds that action at a distance is magical action. Physical action was thought to be action by physical contact (even though the physicality of such contact is primarily phenomenal). Of course, any actual action in the external world will fall under physics. And my finding of a scientific use for the hypothesis of a Creator shows that God can be a scientific hypothesis.
     Euclidean geometry was axiomatised, but that did not make it true; space is what is it. Ptolemaic astronomy could have been axiomatised, but the earth still turns. Standard mathematics is axiomatised; nevertheless, there are numbers of things.

The Liar Proof

This assertion is not true.
Let that assertion – if it is an assertion – be called ‘L’.
          If L is an assertion – the assertion that L is not true – then L is an assertion that it is not true that L is not true, and so L is also an assertion that L is true. That is unusual, to say the least; but it is clear enough what is being asserted – how else could we know that it had that unusual property? – and so L is fairly clearly an (unusual) assertion. And if L is as true as not (see below), then it is as true to say that L is true as it is to say that it is not, so there is that consistency. Note that L is not a simple conjunction of those two assertions; it is wholly the assertion that L is not true (if it is an assertion), and it thereby asserts that L is true. And note that no assertions are perfectly straightforward; all are to some extent vague, for example.

          Nevertheless, logic seems to take L to a contradiction. (By ‘logic’ I mean that which formal logics model mathematically. Formal axioms are abstracted from informal but rigorous arguments, arguments so rigorous that we regard them as proofs. Were such a proof to include a step that did not correspond to any axiom, we should have a reason to revise our formal logic; we should have no reason to reject the proof.) If L is true – if it is true that L is not true (and that L is true) – then L is not true (and true). But L cannot be true and not true, of course; the ‘not true’ rules out its being true. And so if L must be either true or else not true, then it follows that L is not true. But if L is not true – if it is not true that L is not true (and that L is true) – then L is true (and not true); and L cannot be true and not true.

          So, logic takes L to a contradiction if – and as shown below, only if – we assume that assertions must be either true or else not true. The negation of that assumption is not logically impossible – see below – and so it is that assumption that logic is taking to a contradiction. That assumption is certainly very plausible, of course. To want the truth of a matter is to want things to be made clear. It is to want the vagueness to be eliminated. Nevertheless, there are a variety of abnormal situations where it would be highly implausible for the assumption in question to be true (and L is not a normal assertion). Suppose, for example, that @ is originally an apple, but that it has its molecules replaced, one by one, with molecules of beetroot. The question ‘what is @?’ is asked after each replacement, and the reply ‘it is an apple’ is always given. Originally that answer is correct: originally it is true that @ is an apple. But eventually it is incorrect. And so if the proposition that @ is an apple must be either true or else not, then an apple could (in theory) be turned into a non-apple – some mixture of apple and beetroot – by replacing just one of its original molecules with a molecule of beetroot. And that, of course, is highly implausible.

          What is surely possible, since far more plausible, is that @ is, at such a stage, no less an apple than apple/beetroot mix, that it is as much an apple as not, so that the assertion that @ is an apple is as true as not. That assertion could not be true without @ being an apple, nor not true without @ not being an apple (and we can rule out neither true nor not true, because that is just not true and true). More precisely, @ is likely to move from being an apple to being as much an apple as not in some obscure way that is, to some extent, a matter of opinion. In between true and not true we may therefore expect to find states best described as ‘about as true as not, but a bit on the true side’, ‘about as true as not’ (a description that would naturally overlap with the other descriptions) and ‘about as true as not, but a bit on the untrue side’. For such abnormal situations, formalistic precision would be quite inappropriate, because the truth predicate is indeed suited to the elimination of vagueness. It is much better to say ‘it is as much an apple as not’ instead of ‘it is an apple’ when the former is true, the latter only as true as not.

          But we cannot express L better, we have to understand it as it is. Fortunately, if we do not assume that assertions must be either true or else not true, then from the definition of L it follows only that (if L is an assertion then) L is true insofar as L is not true, and hence that L is as true as not. There is no contradiction, and so either L is not even an assertion (which seems implausible) or else the Liar paradox is a disguised proof by reductio ad absurdum that it is not the case that assertions must be either true or else not true. Note that there is no ‘revenge’ problem with this resolution. E.g. consider the strengthened assertion R, that R is not even as true as not (which is thereby also an assertion that R is at least as true as not). If R is true then R is false (and true), if R is as true as not then R is false (and true) and if R is false then R is true (and false); but, if R is about as true as not, a bit on the untrue side, then it would be about as false as not to say that R was not even as true as not (and about as true as not to say that R was at least as true as not). Greater precision than that would be inappropriate for an assertion as unnatural as R.

Definitive Selections?

Are definitive selections too odd?
      When we think of some things, and various combinations of them, it seems clear that all those combinatorial possibilities are there already, awaiting our consideration. And yet I am asking you to imagine that when a Creator, some such brilliant mind, considers some things, all those possibilities are blurred together (although none so blurry that it cannot be picked out); or am I?
      I am suggesting that for selections of selections of ... of selections, from some original collections, each possible selection from those will be a particular possibility only as it is actually selected by our Creator, independently of whom no collections of things would exist, were there such a Creator (as there provably is). The possible selections that make S(N) bigger than N (to use the terminology in my Cantorian diagonal argument) are those endless sequences of ‘I’s and ‘O’s that are pseudorandom; to make them, infinitely many selections have to be made, each one of which involves some arbitrarily large finite number of selections. They might be made instantaneously by our Creator, of course; and if so, then typical selections from S(N) could be made arbitrarily quickly.
      What about S(S(N)), which contains more things than infinite space contains points? Well, a Creator might be able to do all of that instantaneously. And similarly for selections from U, and UU, and maybe UUU; but still, you see how our Creator would have to do much more, and much, much more, and so on and so forth, without end. It is therefore quite plausible that for selection-collections that it would take me far more than mere trillions of pages to describe, our Creator would be unable or unwilling (and thence unable) to make all such selections instantaneously. After all, it is logically impossible for all possible selections to be made instantaneously. To will an incremental development of such abstract mathematics, as a necessary aspect of the creation of any things, might be regarded as a price worth paying for some such creations. And it is also quite plausible that were the Creator unable to do something (even as a consequence of such a choice) then that thing really would be impossible, given that the very possibility of it derives from that Creator.
      Solid things are solid; but mathematical properties related rather abstractly to their individuality can be works in progress; why not? Modern mathematics has a weirder story to tell of such matters! It is relatively straightforward to think of Creation as dependent upon a Creator who transcends even its mathematics. So, it may not be too odd to think of a Creator creating number by definitively adding units: 1, 2, 3 and so forth; is that any weirder than a Creator creating something ex nihilo? Number is paradoxical, so that the ultimate totalities of numbers are indefinitely extensible, and so numbers just do pop into existence, somehow; and what more reasonable way than by their being constructed by a Creator? What would be very weird indeed would be their popping into existence all by themselves, what with them being essentially structural possibilities rather than concrete things. It makes some sense to think of us creating them, as we think about the world around us, but there is something very objective about numbers of things. And again, if it makes sense for us to do it, then how can it be too odd to think of a Creator doing it, in a Platonistic way?
      There will be better ways to think of definitive selection, I am sure, but are they needed? Consider the question of how simple brute matter (just atoms, in molecules of atoms, each just some electrons around a nucleus) could possibly have feelings; a common enough answer is: Well, it must be possible, because we have such feelings, in this physical universe.

The Spectre of Logic

A famous example of a logical paradox is: “This is a lie.”
If that is a lie, then it is a lie that it is a lie, so it is not a lie.

But if it is not a lie, then what it says is false, so it is a lie.
Whereas, if it is not a lie, then it is not the case that it is a lie.

Contradiction! So, logic gives us paradoxes. Still, so what?
Would we expect highly evolved apes to be perfectly logical?

Many scientists, having taken logical looks at the evidence, while giving very low prior probabilities to the existence of a transcendent Creator (having read books by Richard Dawkins and Bertrand Russell), do think of themselves as highly evolved apes, in a purely material world that just happens to exist, so they might not be too surprised to see their natural logic being shown to be contradictory. Modern logic is, after all, much more "rigorous" than natural logic ("rigorous" in the sense of mathematical). But if their looks at the evidence were not very logical, then their lack of surprise might not be very logical. And what if there is a logical proof that there is a transcendent Creator? Of course, they would think that there is unlikely to be such a proof, because of those very low priors. But at the end of the day there is such a proof.

Perhaps you think that it would still not be irrational of them to refuse to countenance the possibility of a transcendent Creator, much as this would not be irrational:

I see a tree, so I know it is a tree. That is certainly rational. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind empirically, of course (that is what “very convincing” means); but so what? I naturally assume that it is no such thing, and even now, after thinking of this particular possibility, I still have no idea how unlikely it really is, so I cannot do better than continue to make that assumption. Making that assumption may well make my knowledge that I am looking at a tree a sort of gamble, but such is knowledge.

What if you had evidence that that tree was an alien quasi-stick-insect, though? What if you showed me that evidence? Should I not take a logical look at it? Your discovery would be very important, if your evidence was good enough (extraordinary claims do stand in need of extraordinary evidence). What if I looked at your evidence and saw nothing wrong with it but also saw that given my assumption, it must be defective; would that be logical enough? I might think so, but then, the alien might bite my head off and you would have tried to warn me (in a very scientific way).

Suppose that you had a very good argument for something that I did not like, not at all, but suppose that I gave your argument the time of day. If I let the strength of my dislike trump the logic of your argument, would I seem logical? Suppose that I deduced from my dislike that you must have made a mistake; suppose I say that there is probably something wrong with your argument (to err is human), and that I have better things to think about. Is the latter logical enough, but not the former? Surely we owe it to ourselves to be properly logical.

Even if our logic was flawed, it would still be our logic. Assuming that it was not flawed would, even then, just be a very human error. And maybe our logic is flawless:

    “The assertion that you are now reading is not true.”
    Let us call that assertion “L”, so that     L is true if, and only if, L is not true
    or, if “is true” is a vague predicate,        L is true insofar as L is not true.
    From the latter, it follows that L is as true as not,
    which it can be, if “is true” is a vague predicate.

There is only a contradiction if “is true” is not a vague predicate. And while that might not be a proof by reductio ad absurdum that “is true” is a vague predicate, it is close to being such a proof and very far from being a reason to doubt logic.

Apparently Timeless Possibilities

Apparently timeless possibilities could, possibly,
become more numerous over time, e.g. as follows:

You were always possible,
but had you never existed,
then that possibility would have been
the possibility of someone just like you.
      It could not have been
      the possibility of you in particular 
      were you not there to refer to.
Looking back now,
we can see that there was always
that possibility, of you in particular 
as well as the more general possibility,
even before you came into being.

Now, Presentism is logically possible,
and if Presentism is true then there may
originally have been no such distinction,
even though you were always possible.
      Under Presentism it could have been
      that you might not have existed.
The distinction could therefore have
arisen when you came into being.

It is therefore logically possible
for apparently timeless possibilities
to emerge as distinct possibilities
from more general possibilities.

The Signature of God

I think belief in God reasonable only if it is based on considerations available to all humans: not if it is claimed on the basis of a special message to oneself or to the group that one belongs.

Anthony Kenny ("Knowledge, Belief, and Faith," Philosophy 82, 381-97)
      So what better signature of the creator of homo sapiens than an elementary logical proof that there is a God? In my last post, I described the argument that given some things, cardinally more selections from them are possible.
      That post ended with a brief description of how that means that paradox arises: we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection; it would follow that they were all there already, that they are collectively some impossible collection of all those possible selections.
      Logic dictates that we have made some mistake; and this version of Cantor's paradox arises because we are considering combinatorially possible selections: that is why the sub-collections that define those selections were able to become so paradoxically numerous, why the paradoxical contradiction did not just show that there are not, after all, so many extra things, over and above the original things.
      My resolution begins by observing that apparently timeless possibilities could, possibly, become more numerous over time; it begins that way because if possible selections are always becoming more numerous, then we would never have all of them. A Constructive Creator could, possibly, make the definitive selections; and if that is the only logical possibility, then that is what has been shown.
      Note that serious mathematicians have taken Constructive mathematics seriously, and when constructed by a transcendent Creator the mathematics would be much more Platonic, and much more Millian. Consider, for an analogy, how God's commands could, just possibly, define ethics. And note that such creative possibilities are not that different to the Creating of mere things ex nihilo, if you think about it: how is such Creation even possible? For us, the laws of physics present immutable limits to what can be done; for a God, such laws are, metaphorically, a brushstroke.
      We live in a world of things, and numbers of things; and for us, numbers appear timeless. But logic does seem to say that such numbers are impossible. When we first think of the origin of things, we might think of things that could have been there forever, like numbers. But logic seems to say that there was originally stuff, not things; perhaps mental stuff, perhaps a God that is not exactly one thing. There would have been some possibility of things, and more arithmetic the more that God thought about that possibility.
      I should add a note about what sort of God is being shown to exist. The proof does not show that God could not have created a four-dimensional world in a Creative act above and beyond that temporal dimension. So this God might be what we call "timeless," and might know all about the future; or not. And either way, this God could always have known all of our textbook mathematics, if only because that is essentially axiomatic.

Cantorian Diagonal Argument

Cantor’s diagonal argument that there are more real numbers than natural numbers gets its name from its picture proof (as below), and it generalizes to show that powersets are always bigger than their original sets.

Cantor originally used collections instead of sets, but they gave him a paradox. Now, a collection of things is just those things being referred to collectively, so it is hard to see how that could have been the problem. But Cantor introduced the notion of a set, or consistent collection, and most modern mathematicians use axiomatic sets, for added confidence, and define numbers from them.
      Nevertheless, there really are numbers of things, so I take the logical essentials of Cantor’s paradox, and find a lacuna. My version of the diagonal argument shows that given some things, cardinally more selections from them are possible. As is the case with Cantor’s diagonal argument, it is best to begin with small collections, and build up from there, so that the general case can be more easily understood, by comparison with those simpler cases.

There are clearly three things:
      clearly there are
Given those three words, we can select a couple, e.g. ‘clearly’ and ‘there.’ There are three ways of making a pair – three different pairs that could be made – from those three words: {‘clearly,’ ‘there’}, {‘there,’ ‘are’} and {‘clearly,’ ‘are’}.
      Each of those ways of making a pair derives from, and is therefore defined by, the presence of two particular things in the original collection. Given those two things, there is that way of making a pair, whether or not anyone would or could make it. Because the original things were distinct, those pairs are distinct, and so each can count as one thing, as when we think of those three ways of making a pair. Possible selections are, in that sense, things.
      In a similar sense, collections are things, e.g. those are three collections of words. But given some things, thinking of just some of them as yet another thing can easily seem like the weak link in a chain of reasoning that leads to a contradiction. (After all, the collection of one thing is just that thing.) It is clearer that, given some things, a way of making a pair from them is indeed another thing. (A way of selecting just one thing is a way of selecting.)

Making a pair is just one way of making a selection. There are, in total, 2 to the power of 3, or 2^3= 8 possible selections from a collection of 3 things (cf. its powerset). It is an elementary result in combinatorics that there are 2^3 ways of assigning 2 labels, say ‘In’ and ‘Out,’ to 3 things, e.g.
      for the pair that is ‘clearly’ and ‘there,’
      ‘clearly’ has the label ‘In,’ as does ‘there,’
      while ‘are’ has ‘Out’ because it is not in that sub-collection.
In general, for any collection of things, T, there is a selection-collection, S(T), of all the combinatorially possible selections from T, each corresponding to some combination of as many ‘In’s and ‘Out’s as there are things in T.
      There is the selection-collection of 2^8 = 256 possible selections from the aforementioned 8, for example; and there are, similarly, a further 2^256 possible selections from that collection, and so on.
      Each of those possibilities is distinguished and defined by the presence of pre-existing things in the original collection, and so they are all implicitly there already, with our original 3 things.
      There is therefore an infinitely big collection of combinatorial possibilities, say N (from which further selections might possibly be taken, giving us S(N); and so on).

Two collections of things have the same cardinal number of things when there are one-to-one mappings from each collection onto all of the other. Cardinality therefore captures some of the intuitive sense of there being as many things in one collection as there are in another. Whether cardinal numbers are rightly called ‘numbers’ or not does not matter here; for the purposes of this proof, the main thing about cardinality is that it is an equivalence relation: it is reflexive, symmetric and transitive. Cardinality therefore partitions collections into equivalence classes. In particular, S(N) is not in the same class as N, for the following reason.

For simplicity, the things in N will be given the names
‘1’ (e.g. naming {‘clearly,’ ‘there,’ ‘are’}),
‘2’ (e.g. naming {‘clearly,’ ‘there’}),
‘3’ and so forth. To get the things in S(N) we associate the things in N with either ‘In’ or else ‘Out,’ and so each thing in S(N) can be named by an infinite sequence of ‘I’s and ‘O’s.
      If S(N) had the same cardinality as N, then we could associate each combinatorially possible sequence of ‘I’s and ‘O’s with one of the names of the things in N, and thereby list all the combinatorially possible sequences of ‘I’s and ‘O’s.
      E.g. the element of S(N) whose name is the sequence I, I, O, O, … might be associated with the element of N named by ‘1,’ and so on:

 N         S(N)

 1           I           I         O         O         …

 2          O         O         I          O         …

 3           I          O        O         O         …

 4           I          O         I           I         ...

…         …         …        …         …         ...

If we can name an element of S(N) that is not in that list (for any such list), then that will show that N and S(N) are not the same size; and we can specify one that is different from each of those by specifying that:
its 1st label differs from the 1st label of the element associated with 1,
its 2nd label differs from the 2nd label of the element associated with 2,
and so on (e.g. O, I, I, O, ...). And that diagonal argument generalizes to show that every selection-collection, S(T), is cardinally bigger than its original collection, T, as follows.

We begin by supposing, counterfactually, that S(T) has the same cardinality as T, i.e. that there are one-to-one mappings from T onto all of S(T). Let M be one such mapping.
      We use M to specify a collection D as follows: for each thing in T, if the possible selection that M maps that thing to includes that thing (in other words, if that thing has the label ‘In’ in that possible selection) then D does not include it (it has the label ‘Out’ in D), but otherwise D does; and there is nothing else in D.
      Since the only things in D are things in T, D should be in S(T); but according to its specification, D would differ from every possible selection that M maps the things in T to. Consequently there is no such M; S(T) does not have the same cardinality as T. And since S(T) has at least one element for each thing in T – e.g. the selection of that thing – hence S(T) is bigger than T.

So, given any 3 things, there is an infinitely big collection, N, and an even bigger collection, S(N), and the even bigger S(S(N)) = S^2(N), and similarly S^3(N) and so on.
      All the things in all those collections are collectively the union, U, of those collections; so, there is also U. U is bigger than each of those S^n(N), for natural numbers n, because it contains all the things in each S(S^n(N)). Furthermore S(U) is even bigger, and so on. So there is also the union, say V, of all the S^n(U) for natural numbers n. S(V) is even bigger, and so on; and so forth, past W, say.
      There will be a union, UU, of U, V, W, …, and thence a union, UV, of all the S^n(UU), and similarly UW, and so on, through VU, VV, VW, ..., and so forth. There will be a union, UUU, of UU, VV, WW, …, and a union of U, UU, UUU, …, and so on, and so forth.

Paradox arises because we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection. It follows that they were all there already, that they are collectively some collection, C, of all those possible selections.
      But, there cannot be any such collection, because its elements, simply by existing, would define those of S(C); and by the diagonal argument, S(C) would contain even more things of that very kind. And we cannot – by the meaning of ‘not’ – have both that C does contain all such possible selections and that C does not contain them all.

      [Lacuna]

an inconvenient Proof ?

Do I have a proof that there is a transcendent Creator?

Well, the essence of Cantor's paradox is a logical argument for a contradiction, with an obscure lacuna: there need be no contradiction if arithmetic (that is Millian, or ordinary, common or garden arithmetic) is constructed forever. Were it not for that lacuna, we might have to throw logic away and replace it with some formal logic (symbolic calculi called "logics") or other, whilst being unable to choose logically between them. But, there is that lacuna, and so we do have a proof of the existence of a transcendent constructor of arithmetic; and of course, Millian arithmetic could only be constructed by the Creator of all other things.
      Note that a purely logical existence proof would be the appropriate signature of the Creator of homo sapiens. And consider some other kinds of proof, by way of comparison; here are a couple of examples:

Suppose there was a serious crime. Fortunately you have a suspect, and a good case against him. The defense says that your case means nothing, because you are only human, that to err is human. She goes on to detail how nice your suspect is. You point out that she should therefore doubt that opinion of him, since she is human, whilst you need not entertain any such doubts because it is not your argument; indeed, you have convicted lots of criminals on less evidence, so you ask her if they should all go free? Of course not, she says, it is only your case against this man that is thrown into doubt by our common humanity, because he is so very nice indeed! She has simply ignored your observation about her own humanity; maybe she erred in doing so! But what should we conclude from an assumption that we cannot trust our conclusions?!
      Let us suppose that your case is exceptionally water-tight: there is lots of physical evidence, and everyone else has cast-iron alibis, while your suspect has no alibi at all; and this crime is just the sort of thing that he would do. There really is no reasonable way that your suspect is innocent. He even bragged about his guilt to you in private. Since your case is so water-tight, hence all her talk about your humanity is just that: talk. It is, if anything, further evidence of his guilt, that she feels that she has to resort to such meaningless talk.

Or, suppose I say that 2 + 2 = 4. Someone says that he would prefer it to be 5, and tries to show that it really can be 5 by saying that: "If we measure two lengths and put them together, then we could find that two point three five units plus two point three five units equals four point seven units; and if we round all those measurements to the nearest integer, so that it is arithmetic, then we get two plus two equals five." Even so, there are such proofs as this: 2 + 2 = (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 = 4. And note that he only wants 5 because it really is bigger than 4 = 2 + 2. My interlocutor retorts that we have to have axiomatic arithmetic, on pain of paradox (e.g. Cantor's paradox), and that he likes those axioms that let him have 5. Could some paraconsistent logic not give arithmetical axioms the power to give him his 5 as well as us our 4, he wonders; but no, that is not really logic, and axioms that give him 5 are not arithmetical. My interlocutor will not give up though, and he has lots of friends. Even so.

The Death of Logic

A hundred years ago, logic died: it was either logic
and a transcendent creator or neither, and not only
was atheism on the rise, the creator whose existence
was logically implied was less political than the God
of the embattled religions of those war-faring days.

Prima facie, logic took off at that time, thanks to Hilbert
and other mathematicians. There are now lots of formal logics,
so logic looks very scientific: formal logics are very mathematical.
So, they are as rigorously logical as mathematics. But is it logical
to redefine truth because of some linguistic puzzles? Is it logical
to have each number but not every number?

Suppose we get a really good string theory, S.
Should we use S to redefine all of our physical entities?
If the description of electrons in S was E then we could
say that "electron" named E. But would we?
Or do we know that electrons are electrons?

Arithmetic is a subset of the properties of possible objects:
one object and another object is one-plus-one objects, etc.
But 1 has become the name of {{}} in mathematics.
    Did Frege refute Mill's description of 1?
    Not only did he not, how could he have?
    Surely Mill (and Euler) knew what one is.
Set theory mimics mathematics well enough,
but does anyone believe that 1 is really {{}}?
Surely we all know (as Euler knew) what 1 is.
Mathematicians wanted to replace 1 with {{}}
because the logic of numbers is paradoxical.
    And because 1 was replaced, logic was too.

Logically, there was that paradoxical proof;
and although the Liar paradox now looks like
just another reason to replace truth and its logic
with something formal, that is not really scientific:
science pursues truth, and logic takes truths to truths
(where to say, of what is, that it is, is to speak the truth).
Logic takes truths to truths because logic just gives us part
of what we already had. How, then, could there be a logical
proof that there is a God? Because we already had a logical
paradox: we had to lose part of the contradiction
to get consistency. Did we have to lose the logical
possibility of there not being a God? Was there an
alternative? Yes: a hundred years ago, we lost logic
instead (because of Frege and Russell and Hilbert).

Is Logic Necessary?

I've been looking at Skepticism, following Maddy's 2017, because it connects with the topic of this month's posts: highly evolved apes are unlikely to have a perfect logic, so why should we care if logic gives us paradoxes? We see a tree, we know that it is a tree; that much is ordinary. We cannot rule out its being an alien quasi-stick-insect, of course; but then, we never thought that we had to do that, did we? And now that you come to think about it, don't you think that you could if you examined the tree more closely? Or that some scientists could? Now, we cannot ever rule out the possibility of an error of some kind or other, perhaps an error of a kind that we have never thought about: how could we rule that out? Is that is why the experts on logic think of logic as being formal logics (mathematical models of logic) nowadays? I suppose that experts could be certain about mathematical calculations. However, such logics come with modern definitions of truth, scientific theories of truth, mathematical models of truth, raising the question: What is the truth about truth? The first "truth" in that question is clearly intended to be correspondence truth, but if it does turn out to be the case that the second one cannot be correspondence, then how could the first one be? And if the first "truth" is not correspondence truth, then how satisfying could any answer to any such question be? What if we have, for example, an attractive story about how truth is an attractive story? At the end of the day, we naturally assume that truth is correspondence truth, and that logic is logic, not a mathematical model of logic. Even when it is the experts thinking about the formalities, their metalogic is simply logic; and when they give us theories of their metalogic, they do expect us to think about their presentations of those theories in logical ways. And similarly, we simply assume that trees are not alien quasi-stick-insects of some very convincing kind. We can say that they are very probably not aliens, and then try to justify that "very probably" and maybe wonder why we are doing all of that. But at the end of the day, we are simply such that for us, our logic is necessary. I see a tree, and know that it is a tree. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind, and so my "know" is a sort of gamble: I assume that it isn't an alien. I don't know that it's unlikely to be one (how could I?) but I do know that it's not mad to assume that it isn't one. Since the topic is raised, I admit that it might be an alien, that I don't know that it isn't, that I don't know, in that sense, that it is a tree; but I still claim that I do know that it is a tree, in the ordinary sense. In short, there seem to be at least two senses of "know" in play.

Doppelgangers

It seems to be logically possible for there to be an exact copy of you, say d-you, because it seems that such a thing might exist in a parallel space-time. D-you would be physically and mentally identical to you; but it would not, of course, be you. Now, we naturally assume that none of us have been instantaneously swapped with such doppelgangers. We can never have any reason to think that any of us might have been swapped; but, that is because such swapping would be undetectable, and that is why we cannot rule out the logical possibility of such swapping.

Indeed, you cannot completely rule out the possibility that you are such a doppelganger, because you would have exactly the same memories, exactly the same sense of being yourself. There would be absolutely no empirical difference; the only difference would be semantic: reference intended to be reference to you would fail to be such reference, were it to d-you, for example (and given the falsity of Functionalism, and so forth). And of course, knowledge would be lost, e.g. if I saw d-you at a bus-stop then I would not know that you were waiting for a bus. But of course, I would know that you were waiting for a bus if I saw you at a bus-stop (and you were waiting for a bus). There is no loss of knowledge caused by not ruling out the logical possibility of d-you. We simply assume that such swapping does not happen.

Note that we do not just think it unlikely (and similarly, we do not just think it unlikely that we are brains in vats, or being fooled by demons, and so on and so forth). We do not know for sure that there are no such doppelgangers, and we do not even know for sure that there are unlikely to be any (we can have no evidence for such unlikeliness). But clearly, we are assuming that there are no such things (and nothing else of that rather wide-ranging kind). That is just an obvious empirical fact about our beliefs. (We might not notice it, because being fooled by a demon would be like being a brain in an evil scientist’s vat, and a brain in a vat is like someone having a very long vivid dream; and maybe it is only highly unlikely that you are in a coma right now.)

Truth in Dreams

In one Cartesian argument for skepticism about the reality of the world, we are to assume that if we were dreaming, then were we to see hands in that dream, those would not be hands. Still, they would be dream-hands, in a dream-world, so dream-reference to them would hardly fail, or would it? If we think of someone dreaming about hands, then clearly those are not real hands; but, were this a dream (not a dream-within-a-dream, which is what our "dreaming" would then refer to), then what is meant by "real hands" within that dream would be dream-hands. You may well wonder if that would be the case, had we fallen asleep having already learnt the meaning of "real hands" in the real world. But presumably we learnt the meaning of "real hands" in this world, and if this was a dream then this world would be a dream-world. You could counter that if this was a dream, then we would still have learnt the meanings of our words in some higher realm, but as soon as we clarify what exactly we are talking about, by describing what we mean by "an external thing," we tie the meanings of our words to this world (the photo is from last year btw :-)

Lots of Misprints

I've seen quite a few misprints recently, e.g. in TV text; also top of page 159, and again on page 169, in Maddy 2017 (" 'Proof on ..." instead of " 'Proof of ..."), just before she got to Moore's reason why pointing to each of his hands was a proof that there are two hands (and hence that there are external objects, and hence an external world), which was that he could similarly prove that there were three misprints on a certain page by:

taking the book, turning to the page, and pointing to three separate places on it, saying 'There's one misprint here, another here, and another here'

Maddy 2017: 164 (Moore 1939: 147) Although of course, while that proves that there are three misprints, it does not prove that there are three misprints. And while you might agree with Moore that those were misprints, that would not amount to a proof that they were. Moore, you will recall, does not have to show that there are two hands, nor even that there are two hands, he has to show the externality (so to speak) of such things as hands, given skeptical doubts, which is more like having to prove not just assume, that it is indeed a bad thing to have lots of misprints. And of course, why would we have to prove such a thing! Ask yourself what is meant by "external world" to see for yourself how it exists by definition (and note how one gestures as one does so). And yet, it is precisely that "proof" that is challenged by skeptical doubts (as the above-linked-to review of Maddy 2017 observes).

What do Philosophers do?

I'm half-way through Maddy's 2017 (a walk through the modern history of Skepticism), where she describes a weakness of the Argument from Dreaming:

    Although we would not be knowing the world were we now dreaming in the ordinary way, we can rule that out in quite ordinary ways; and whereas we cannot rule out that we are dreaming in some extraordinary way (e.g. a life-long coma), why should we rule it out? Maybe this is a dream-world, and my hands dream-hands within it. But should the fact that I don't know much about the fundamental substance of my hands get in the way of my knowing that I'm typing this with them because they exist (whether that is in a way that is to some unknown world much as dreams are to this world, or in some other way)?

Here's a thought though:

    If some higher power (maybe a UFO) replaced you with a pod-person who was exactly the same as you, physically and mentally, then the people of the world would of course not know, were they to see that person before them, that you were standing there. So, if the underlying substance of the world was such that things were frequently replaced with identical copies, in such ways (and note that we cannot even know that that is unlikely), then our references would frequently fail, and we would end up knowing a lot less about the world than we assume we do.

    We do assume that such does not happen, but that just means that, for example, it is at best epistemic luck that people know that you are there, when they see you. At worst it is knowledge by assumption, because we do assume as much; which reminds me of Wittgenstein's hinge propositions (which Maddy will be getting to shortly). Perhaps we assume that things generally continue to be the same things. Or perhaps we assume that things that look the same are the same.

    I would not say that we know such a proposition, but maybe we do thereby know propositions that depend logically upon it, such as that I have hands. Why not? Knowledge seems not to be some minimal amount of epistemic luck, but rather the sufficient reduction of certain kinds of epistemic luck, as required by one's context; and philosophy is a context with high standards. In philosophy we tend to accept the force of epistemic closure, because the standard is logic.

The Essence of Cantor's Paradox

(1)    There are at least three things

Clearly there are.

(2)    Given some things, there are possible selections from them

E.g. ‘clearly’ and ‘there’ are a pair of words.

(3)    There are all the things given by reiterating (2), given (1)

Note that each possible selection is a thing.

(4)    Given some things, cardinally more selections from them are possible

That is shown by a Cantorian diagonal argument.

(5)    There are cardinally more things of kind (3) than there are things of kind (3)

That follows from (4), given (3), but is contradictory, and hence false.

My resolution begins by observing that apparently timeless possibilities could possibly become more numerous over time. It begins that way because if possible selections are becoming more numerous, then that could easily change the meaning of (3) enough to avoid (5). There is no other way of avoiding the contradiction (the main resolutions were constructivism, with its potential infinities, and going axiomatic, which means not addressing numbers of things directly), which is why this is a paradox.

Consequently this is essentially a proof by reductio ad absurdum that possible selections do become more numerous over time. And how could possible selections become more numerous, if not by a transcendent creator making definitive selections, constructing arithmetic as part of the creation of all things? There is no other way that I can think of; whereas this way is a serious possibility, because
A) mathematicians have taken constructivism surprisingly seriously, and constructivism would only be more Platonistic, and Millian, were the definitive constructions made by a transcendent creator, and
B) theologians have taken the idea of God being beyond our conception of number very seriously, e.g. the Trinity.

Logic Needs That Hypothesis

In the Germany of the eighteen-nineties, Georg Cantor discovered the mathematical paradox that bears his name.
He put it down to the ineffability of God, even though he was only studying numbers; they were very big numbers.
But, the mathematical mainstream has since then replaced our natural conception of a collection with formal (or fictional) sets that are better behaved.
Whereas, the natural conceptions are fundamental to our actual thinking; in particular, if we cannot rely on our best thinking about formal sets, then why should formal sets be any better?
Consequently logical thinkers need to hypothesize God: only that allows those conceptions without paradox (as previously posted, and as sketched in my next post).
Over the next few posts I aim to scrutinize the elements of this, e.g. the essence of Cantor's paradox, and why we do still need logic in this democratic and scientific age.

Is there a Problem with Prefaces?

Suppose that in the preface to some non-fiction book, the author apologizes for whatever false statements there may be in the book, observing that there are bound to be some, even though each statement in the book is believed to be true by the author. Is this a Preface Paradox? Philosophers often "solve" this problem by taking belief to be sufficiently high credence, so that we would not believe large conjunctions of our beliefs; but of course, our believing each of our beliefs means that we believe them all. Belief is simply not sufficiently high credence (as recently posted). In fact, there is no problem here to solve. And of course, not even a philosopher would ask: Why not say that sets are simply such that sets of beliefs are well-behaved? Could we not have all that we want and nothing that we do not want by putting precisely that much into the axioms of our favourite set theory? No: that is simply not what conjunctions of beliefs are. Nevertheless, that is essentially the mainstream response to Cantor's Paradox (which I shall be posting on in my next few posts). The Preface "Paradox" may not be very paradoxical, but it does show how absurd the mainstream foundations of mathematics are.

Mrs Fox's feelings

Basking in the warmth of Heaven, she floats weightless and naked, far far above the factory chimneys and church spires of the world, in the upper reaches of a sultry sky. It's an intoxicatingly fragrant atmosphere, surging and eddying with huge gentle waves of wind and pillowy clouds – nothing like the motionless, transparent oblivion she'd always imagined Paradise would be. It's more like a breathable ocean, and she treads the heavy air, narrowing the distance between her body and that of her man who's flying beside her. When she's close enough, she spreads her thighs, wraps her arms and legs around him, and opens her lips to receive the incarnation of his love.

Michel Faber, The Crimson Petal and the White, 671