Tuesday, August 14, 2018

Force and Foreseeability

Some thinkers think that if there is a God then God will know all about the future, because otherwise bad things might happen. About ten years ago I spent a few years trying to refute one such view as neatly as possible (see my result here), during which attempt I found a new theodicy (which I called "The Odyssey Theodicy" for no good reason) and discovered the mathematical proof that there is a God (who is not immutable) that I have recently been tidying up. Today I thought of this title to go with my original refutation; basically, my original thought was that God's power over God's creation gives God plenty of ability to know that good will definitely happen, without God needing to know all about the future. However, despite now having the sort of title that I like, for my thought, I find that I now have little interest in expressing as neatly as possible such academic thoughts. That is because my thought is so obvious that the view that I was refuting must have existed for some other reason than simply not knowing that thought. Could that view not have been clearer about its reasons, I wonder, but that is just the academic way, it seems. (I imagine that my current proof will end up the same way; but, the more things stay the same, the more they change.)

Thursday, August 09, 2018

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.)

Tuesday, August 07, 2018

A Storm in a Teacup

In last Sunday's Telegraph article Boris Johnson said, of the burka, that "it is absolutely ridiculous that people should choose to go around looking like letter boxes," and today I find that he has been jumped on by several Labour MPs (and others) and basically called a racist (see for yourself). I find this strange. If you called a nun "a penguin" you would not be called a racist, despite our history of racist jokes about stupid men called "an Irishman." And Labour MPs want Catholic votes just as much as Muslim votes. Ironically, Boris was saying that although burkas are ridiculous and sexist we should not ban the full burka. (The burka has been banned in several European countries on the grounds that it is too sexist. Ironically, it is those who wanted us to remain in the EU who have been the most eager to jump on Boris about this. Incidentally, I voted to remain.) I agree that Boris is, in general, a bit of a Johnson, but even in a competitive environment like politics, truth is not unimportant. We need to learn to give the devil his due, before it all goes to hell (as a result of the actual dynamics of competitive teamwork). And has no one noticed that the burka (like the habit) signifies that its wearer does not believe in, for example, gay marriage? Would anyone be surprised to find that most wearers of burkas are a bit racist?

Tuesday, July 31, 2018

A Hypothetical Question

What if Hitler had won the war in Europe?
Suppose that during the ensuing Cold War
Mein Kampf was regarded as a Holy Book;
and then there was a peaceful, tolerant time,
when one man used that book as toilet paper.
Would it be alright not to call him a racist?
(I agree that Richard is a bit of a Dawkins)

An Interesting Question


          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.

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.

Saturday, March 17, 2018

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.

Sunday, March 11, 2018

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, it is the case that such weaselly words are the norm nowadays. For example, how can simple brute matter (just atoms, in molecules of atoms, each just some electrons around a nucleus) have feelings, such sensitive feelings as we have? How is that possible? Am I asking for a description of a possible mechanism? Perhaps; but a common enough answer is: Well, it must be possible, because we have such feelings, in this physical universe; although I don't know how sensitive we humans really are, looking at our world! Such answers are accepted by many scientific people, as they "work" on possible mechanisms!

Friday, March 09, 2018

Logic

There are many logical paradoxes.
A famous example is the Liar paradox: “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, our logic gives us paradoxes. But, so what?
Even highly evolved apes would hardly have a perfect logic.

Most modern thinkers think of themselves as highly evolved apes, in a purely material world that just happens to exist. They/we think so because they/we have taken logical looks at the evidence; but, what happens to our image of ourselves as scientific if we can play fast and loose with logic? We want to be very careful in any choice to embrace illogicality in our thinking; we want, ironically, to make a very logical choice about any such thing.

In taking logical looks at the world, we may well have given low prior probabilities to the existence of a Creator, maybe following Richard Dawkins; but, what if there is a logical proof that there is a transcendent Creator? That would change everything! Thoughts that such a proof could not be possible are naturally based on those very low priors, and at the end of the day there is such a proof. Still, were we to simply refuse to countenance the possibility of a transcendent Creator, then any such proof would become just another logical paradox; and such simple refusals are not necessarily illogical:
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, but so what? I have been assuming that it is no such thing; and even now, after thinking of this particular possibility, I still have no idea how unlikely, or likely, it really is, and so I still cannot do any better than to continue to make that assumption. Making it makes my knowledge a sort of gamble, but such is human knowledge in the real world.
And yet, where do we draw the line? If we had a proof that the tree was really an alien quasi-stick-insect, then surely that assumption would then be illogical. What if you have a very good argument for something that I really do not like; can I take that dislike to trump your argument? Surely not. My dislike can of course motivate me to believe that there is probably a fatal flaw in your argument, but I really should be bothered by the excellence of your argument. Surely I should not just exhibit my dislike, and observe that to err is human. Surely we should all assume logic. Even if it is flawed, it is our logic, and so assuming it would just be the most human error; and maybe our logic is not that bad. Let us look again at the Liar paradox:

“The assertion you are currently considering is not true.”
Let that assertion be called “L” so that: L is true if, and only if, L is not true.
Were “true” a vague predicate, L would be true insofar as L was not true,
from which it would follow logically that L was as true as not.
It follows logically that if “true” could be a vague predicate,
then the Liar paradox is actually a proof by reductio ad absurdum
that it is a vague predicate: then, and only then, is there no contradiction.

Is it only then? That is, after all, why this is a paradox. You could say that the meaning of “true” rules out truth being vague; and of course, truth itself is not normally vague, far from it: to want the truth is to want things to be made clear. But Liar sentences are deliberately constructed to be paradoxical, when they are not simply mistakes that should be rewritten to make them clearer. And consider the following, which is similarly far removed from our normal uses of language:

      Consider “It is an apple”
as an answer to the question “What is A?”
      where A is originally an apple,
but has its molecules replaced, one by one,
      with molecules of beetroot.

Originally, “It is an apple” is a correct answer (or in other words, it is true that it is an apple), but eventually it is not. If that answer must be either correct or else not (if that proposition must be either true or else not true), then an apple can be turned into something else (presumably a mixture of apple and beetroot) by replacing just one of its original molecules with a molecule of beetroot, which certainly seems absurd. It is surely possible, since it does seem more plausible, that A is, at such a stage, no less an apple than it is apple/beetroot mix; that it is, at such a stage, as much an apple as not, so that the proposition that it is an apple is as true as not; what else could it be?

Monday, March 05, 2018

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.