Thursday, July 26, 2018

The Unbelievable Proof

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 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. And 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 Georg 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. The 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, 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 to be resolved.

The following version shows, to begin with, that certain possibilities become more and more numerous. Of course, 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 follows. 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 ‘time-or-super-time’ 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, 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.
     Let us begin with a self-evident truth: these words are distinct from each other. That is self-evident because that is how we were able to read them. There are, then, numbers of things, e.g. ‘happy’, ‘summer’ and ‘days’ are three words. Now, pairs of those three words – {‘happy’, ‘summer’}, {‘summer’, ‘days’} and {‘happy’, ‘days’} – are just as distinct from each other as those words were, because those three pairs differ in just those three words. And similarly, pairs of those pairs – e.g. {{‘happy’, ‘summer’}, {‘summer’, ‘days’}} – are just as distinct; as are pairs of those, and so on. Because of that ‘and so on’ we will have infinitely many, equally distinct things, if we can indeed count pairs as things. Is there 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. So such possibilities are such 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 ‘{‘happy’, ‘summer’}’ to be the name of the combinatorially possible selection of ‘happy’ and ‘summer’ 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 {‘happy’, ‘summer’} 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 {‘happy’, ‘summer’} is a possible selection is that ‘happy’ and ‘summer’ 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, before choosing to contemplate creating some particular 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. It makes sense that a being that could create things ex nihilo would know so much about them. Standard set theory would therefore be a very good mathematical model of the more imaginable levels. And 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. Also note that the existence of such a Creator would also resolve similar paradoxes, such as the Burali-Forti paradox, which concerns ordinal numbers. A dynamic Creator would be able to construct – and would probably enjoy constructing – ordinal arithmetic forever.
     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.

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