Wednesday, September 05, 2018

What Do Philosophers Do?

For myself, I just notice such facts as:

(A) The overwhelming majority of professional mathematicians are not going to be wrong about what numbers are.

(B) The overwhelming majority of mathematicians assume, in their professional work, that numbers are axiomatic sets.

(C) Numbers are not axiomatic sets.

The conjunction of (A), (B) and (C) would be a contradiction, were the mathematicians of (B) not just assuming that numbers are axiomatic sets for the purposes of proving theorems from axioms, as I suspect they do. But many analytic philosophers deny (C), because of that apparent contradiction. Such philosophers also ask questions like: “Do numbers (or sets) exist? If they do, where are they? If they don’t, then what does ‘2’ refer to?”

The implication is that since numbers (or sets) are abstract objects, hence if they do exist then they exist in some Platonic realm of abstract objects, raising the question: “How is it that we can access that realm, in order to know such properties of numbers as arithmetic?” To see how stupid such questions are, one only has to ask such questions as: “Does value exist; and if so, where is it?” Clearly some things have value, but it makes no sense to ask where it is (or what colour it is); such questions hardly further the analytical task of describing accurately what value is.

A question similar to the one about numbers might be: “Do shapes exist?” Shapes are instantiated in, and abstracted from, shaped things, clearly; and similarly, whole numbers are instantiated in, and abstracted from, numbers of things. That is basically what John Stuart Mill said (in passing); it is only common sense, although his observation was jumped on by a founder of analytic philosopher, Gottlob Frege (incorrectly).

A related question is “Do possibilities exist?” It is related because whole numbers exist because of the possibility of things of various kinds. And it is itself related to the important analytic-philosophical question “Do laws of nature exist?” via David Hume, and David Lewis.

Sunday, September 02, 2018

Fissix

F is six

In ancient Greek arithmetic, alpha was used instead of our 1, beta for 2, gamma for 3, and so forth. The symbol for gamma was like a reflected L, with our letter F being digamma (an anagram of mad magi), which they used for 6. The Number of the Beast was FEX, which is the name of a man. But, this post is about physics (ancient Greek for “nature”). And in particular, an urban myth.

Urban Myth


The myth concerns a small machine with a clock-face of light bulbs, one of which is on at any given time when the machine is on. Which bulb it is that is on is determined by which one was previously on and how a small radioactive sample has decayed in the unit of time of the machine: the light will have moved one place clockwise if the sample emitted a detected particle in that time, one place anticlockwise if no such particle was detected. The unit of time is such that there is a 50% chance of a particle being detected in that time.

The machine (which is clearly from The Fury) is placed in front of a human subject, who has to try to make the light move clockwise by really wanting it to. The myth is that some physicists built such a machine and got the light to move clockwise more often than would be expected from random motion. Other physicists tried to repeat the experiment and, according to the myth, they got no positive results. The original results were explained as being random after all (quite likely because things that are truly random tend to look more structured), or as being due to methodological errors (they were physicists, not parapsychologists).

However, the original results could hardly have been undermined by similar results not being obtained with other people, who might not have had the same abilities. Further research, of an appropriate kind, would have had to have been carried out, because of the enormous implications for physics: Modern physics is based on particle physics, which is based on observations of what is essentially a scaled-up and much more complicated version of the small machine. If physicists could affect the events inside particle accelerators, via their expectations and desires, then that would throw a whole new light on particle physics, and hence the whole of physics.

Had there been something to find, then they would have found it, and physics would have changed accordingly. Now, we would have noticed, because there would hardly be this urban myth floating about had they wanted to keep it secret (the possibility of secrets arises because of the money in innovation, the world wars, the cold war and so forth). So, it must be a myth because conversely, had there not been anything to find, then that would have put micro-psychokinesis to sleep forever. Since the physicists would have wanted to be quite sure, hence they would have been quite exhaustive in their investigations. Whereas, parapsychologists still investigate micro-psychokinesis.

Evidence that there is micro-psychokinesis could therefore include, given the above, the very success of relativistic physics. The equations of relativistic space-time were developed at the end of the nineteenth century, even before the quantum-mechanical nature of moving particles had been noticed. It was an amazing discovery, and it is even more amazing that it has not fundamentally altered because it is inconsistent with quantum mechanics. But, particle physicists keep finding patterns that verify it. (Particle physicists are very proud of their understanding of the sophisticated mathematical language of relativistic physics, of course.)

Saturday, September 01, 2018

Curry's Paradox


Last year’s new SEP entry on Currys paradox followed in Haskell Curry’s footsteps by saying nothing about where our reasoning goes wrong in such informal versions of the paradox as the examples in the introductory section of that entry, the first of which was as follows:
Suppose that your friend tells you: “If what I’m saying using this very sentence is true, then time is infinite”. It turns out that there is a short and seemingly compelling argument for the following conclusion:

(P) The mere existence of your friend’s assertion entails (or has as a consequence) that time is infinite.

Many hold that (P) is beyond belief (and, in that sense, paradoxical), even if time is indeed infinite.

[...]

Here is the argument for (P). Let k be the self-referential sentence your friend uttered, simplified somewhat so that it reads “If k is true then time is infinite”. In view of what k says, we know this much:

(1) Under the supposition that k is true, it is the case that if k is true then time is infinite.

But, of course, we also have

(2) Under the supposition that k is true, it is the case that k is true.

Under the supposition that k is true, we have thus derived a conditional together with its antecedent. Using modus ponens within the scope of the supposition, we now derive the conditional’s consequent under that same supposition:

(3) Under the supposition that k is true, it is the case that time is infinite.

The rule of conditional proof now entitles us to affirm a conditional with our supposition as antecedent:

(4) If k is true then time is infinite.

But, since (4) just is k itself, we thus have

(5) k is true.

Finally, putting (4) and (5) together by modus ponens, we get

(6) Time is infinite.

We seem to have established that time is infinite using no assumptions beyond the existence of the self-referential sentence k, along with the seemingly obvious principles about truth that took us to (1) and also from (4) to (5).
That may look rather formal to you, but formal logic is not even logic (it is mathematics); the above is just very well laid out. Note the two uses of modus ponens, the two sets of three steps, with the first three steps, (1), (2) and (3), all beginning “Under the supposition that”. You should note that because we cannot always use modus ponens within the scope of a supposition, e.g.:

(a) Under the supposition that modus ponens is invalid under a self-referential supposition, (A) implies (C).

(b) Under the supposition that modus ponens is invalid under a self-referential supposition, (A).

With (a) and (b) we have, under the supposition that modus ponens is invalid under a self-referential supposition, a conditional and its antecedent, but it would of course be absurd to use modus ponens within the scope of that supposition, to obtain

(c) Under the supposition that modus ponens is invalid under a self-referential supposition, (C).

There was, then, at least one step in the above argument for (P) that stood in need of some justification, i.e. the step to (3). Were no other step deficient in justification we could conclude, from the absurdity of (P), that the step to (3) was invalid. Of course, it would be more satisfying to see where precisely that step lacked justification, so presumably we need an analysis of what would in general count as justification for such a step. For now, note that in order to get to (3) we used modus ponens under the supposition that k is true, which was no less self-referential than the supposition that modus ponens is invalid under a self-referential supposition. In the step to (3) we had k being true instead of (A) implying (C), and “k is true” instead of (A).

To progress, we need to step back, I think, because I suspect that the reason why we find (P) to be beyond belief is that the above argument for (P) has exactly the same logical structure as a clearly invalid argument for the obviously false (Q):
Let your friend say instead: “If what I’m saying using this very sentence is true, then all numbers are prime”. Now, mutatis mutandis, the same short and seemingly compelling argument yields (Q):

(Q) The mere existence of your friend’s assertion entails (or has as a consequence) that all numbers are prime.
My suspicion is based on the fact that one could conceivably have a valid argument for

(S) The mere existence of “happy summer days” entails (or has as a consequence) that time is infinite.

There is a surprisingly valid argument from the existence of “happy summer days” to the probable existence of a transcendent Creator of all things ex nihilo (this links to that), and it might only take some tidying up to get to (S), because such a Creator is an omnipotent being endlessly generating a temporal dimension. Such a Creator could possibly have a logical existence proof, because of its unique ontological status. A valid argument for (S) is therefore no more implausible, a priori, than a valid Ontological Argument. It may well be that there is no valid Ontological Argument (I think that Ontological Arguments are too like the six-step arguments, in their moves from what is possible to what is actual, which are identical to moves to impossible conclusions) but, were there a valid argument for (S), then there would be an identical, equally valid argument for (P).

Anyway, a six-step argument for (Q) that is identical to the Curry-paradoxical argument for (P) would have, in place of k, some such l as “If is true, then all numbers are prime”. And is likely to be about as true as not, because (i) it is about as true as not that a contradiction follows from a statement that is about as true as not, since such a statement is about as false as not, and also because (ii) one informal meaning of is the obvious meaning of the liar sentence “is not true”, which is, if meaningful, about as true as not, according to my The Liar Proof. And of course, our logic is naturally suited to that part of our language where propositions are either true or else not true, exclusively and exhaustively. For a proposition that is otherwise, we have natural clarification procedures that enable us to construct new propositions that are more suited to logical reasoning. So, it seems likely that propositions that might be about as true as not should be ruled out from the use of modus ponens within the scope of a too-self-referential supposition (to say the least).

Curry’s paradox entered into the analytic philosophy of the Forties, where the logical paradoxes were in general thought to be reasons for replacing our informal logical reasoning with formal logical reasoning (via the mathematical philosophy of formal languages), on such grounds as that (i) one would not expect primates, even highly evolved primates, to be able to reason perfectly, and (ii) the physical sciences use mathematics to get to the underlying physical laws. But while I have nothing against mathematics (I have a master’s degree in mathematics, 2001), I do like to do philosophy when I do philosophy (I have a master’s degree in philosophy, 2006).

Sunday, August 26, 2018

Sequence and Consequence


I add grains of sand to the same place, one by one, and eventually I have a heap of sand.

Originally I did not have a heap of sand, of course, and in between having such numbers of grains – just a few originally, and later on lots of them – there are numbers of grains with which I do not obviously have a heap, and do not obviously not have a heap.

Let n be some such number. Perhaps it is the case that when I have n grains of sand I have a heap of sand; but certainly, if that is not the case, then n grains is not a heap. So for all n, either n grains is a heap, or else n grains is not a heap. It follows logically that there must be some n, say N, such that N is large enough for N grains of sand to be a heap of sand, but (N – 1) is not large enough for (N – 1) grains of sand to be a heap of sand.

The problem is that the rather vague meaning of “heap” does not allow there to be such a number. Given any heap of sand, if I take just one grain away from it, then I would still have a heap of sand. Quite generally, if you take one unit away from any very large number of units, then you still have a very large number of units. So it must, after all, be false that for all n either n grains is a heap or else not. That is surprising, but at least we have a picture of how that can be false, with this picture of my adding grains of sand one by one. So while it is surprising, it is not beyond belief; it is not paradoxical (despite what many analytic philosophers seem to think).

It is not, then, always the case that, given some description, either that description is true or else, if that is not the case, the description is not true. It could be that there is no fact of the matter of whether the description is true or not, as when meanings are a little vague and we have a borderline case. Now, in such cases the description cannot simply be neither true nor not true, as that is just to say that it is not true while ruling out its being not true. The only thing left for us to say is that it is about as true as not. In the picture above, the sand changes from not being a heap, to being about as much a heap as not, and then more a heap than not. That is just a fact, of such matters.
Note that we do not need to model such changes with real numbers. (Cf. when we know nothing about a situation except that it is either A or B. We might think that we have 50% credence for each, but the next thing that we find out might be that B can be divided into Bi and Bii; do we then have 50%, 25% and 25%, for A, Bi and Bii, or do we have 34%, 33% and 33%?)
One consequence of the possibility of assertions being about as true as not is that there is a relatively simple resolution of the Liar paradox.
Note that it can indeed make sense to say that a proposition is about as true as not. Consider, for another example, how if I say of some artwork that I think good “That is not good” then I am lying. I am saying something false. What would be true would be for me to say that it was good. And if some artwork seems to me to be about as good as not – and you must allow me such a possibility, because such matters are matters of opinion – then it would be true for me to say that it was about as good as not. In such a case, it might make sense (as follows) for it to be about as true as not for me to say that it was good. And if so, and if we all agreed that a particular artwork, say Z, was about as good as not, then it would make sense for “Z is good” to be about as true as not. Does that make sense? Well, were it simply true to say that Z was good, then were “Z is not good” true too, it would follow that Z was good and not good, whereas the symmetry of Z being about as good as not means that we could hardly have one true and the other not true. And if it was instead not true to say that Z was good, so that it would not be the case that Z was good, then there would be a problem with it being not true to say that Z was not good, because that would mean that Z was good.

Saturday, August 18, 2018

The Modal Paradox

Here in the actual world A we have a ship, let us name it the good ship Theseus, made of 1000 planks. Our first intuition X is that the same ship could have been made of 999 of these planks plus a replacement for plank #473. In possible-worlds terms that means there is another world B where the same good ship Theseus exists with all but one plank the same as in our world A, and only plank #473 different. But then in world B one has a good ship Theseus made of 1000 planks, and by the same sort of intuition, there must another world C where the same good ship Theseus exists with all but one plank the same as in the world B, but with plank #692 different. That means for us back in world A there is another world C where the good ship Theseus exists with all but two planks the same as in our world A, but with planks #473 and #692 different, so one could have two planks different and still have the same ship. The same sort of considerations can then be used to argue that one could have three planks different, or four, or five, or all 1000. But that is contrary to our other intuition Y [a ship made of a thousand different planks would have been a different ship].
     The modal paradox resembles well-known paradoxes of vagueness, such as the heap and the bald one, for which proposed solutions are a dime a dozen — except that here what seems to be vague is the relation of identity. And the idea that ‘is the very same thing as’ could be vague is for many a far more troubling idea than the idea that ‘heap’ or “bald’ is vague. Indeed, according to many, it is an outright incoherent idea.
From John P. Burgess, "Modal Logic, In the Modal Sense of Modality" pp. 40-1.
     In the fourth line Burgess says "by the same sort of intuition," which is a relatively weak sort of thing to say, and so it might be where the paradoxical reasoning started to go wrong. Our original intuition X was that in the actual world A, Theseus could have had one plank different and still have been the same ship. We know, equally intuitively, that X coheres perfectly well (somehow) with the intuition, Y, that in the actual world A, Theseus could not have a thousand planks different and remain the same ship. Whereas the ship in B is not exactly the same as the ship in A. And the meanings of all our words are rooted in A. The further we get from A, the more vague we might expect our meanings to become.
     In the first line of the second paragraph Burgess observes that proposed solutions to the paradoxes of vagueness are "a dime a dozen" and that means, I think, that anything I might say is bound to pointlessness; but, onward and upwards. And it is certainly the case that "the very same thing" often equivocates, as in the case of the famous clay statue: if that thing is squashed then, while it is the same lump of clay, it is no longer a statue at all. So let us look at a very different formulation of the ship paradox, one that does not involve modality at all. The following is by Ryan Wasserman, "Material Constitution", §1:
[...] the story of the famous ship of Theseus, which was displayed in Athens for many centuries. Over time, the ship’s planks wore down and were gradually replaced. [...] Suppose that a custodian collects the original planks as they are removed from the ship and later puts them back together in the original arrangement. In this version of the story, we are left with two seafaring vessels, one on display in Athens and one in the possession of the custodian. But where is the famous Ship of Theseus? Some will say that the ship is with the museum, since ships can survive the complete replacement of parts, provided that the change is sufficiently gradual. Others will say that the ship is with the custodian, since ships can survive being disassembled and reassembled. Both answers seems right, but this leads to the surprising conclusion that, at the end of the story, the ship of Theseus is in two places at once. More generally, the argument suggests that it is possible for one material object to exist in two places at the same time. We get an equally implausible result by working backwards: There are clearly two ships at the end of the story. Each of those ships was also around at the beginning of the story, for the reasons just given. So, at the beginning of the story, there were actually two ships of Theseus occupying the same place at the same time, one of which would go on to the museum and one of which would enter into the care of the custodian.
For myself, I do not think that the ship in the museum was the famous Ship of Theseus, I think that what was left of that ship is now the custodian's ship. But I concede that it could be that the museum ship is legally the ship of Theseus. It would then follow that the custodian's ship was not, for legal purposes, the ship of Theseus. So I think that there are at least two senses of "ship of Theseus" in play. What we can say about those senses is another matter. Our language is inextricably rooted in the usual events of the actual world. But it could be scientific to know that there are those two senses even before our theories of such senses have become a dime a dozen. And we might find clues as to what we should be saying from related puzzles.
     There are many intuitive puzzles about identity. For just one example, suppose that the world that you are in splits into two worlds, so that you are in one while an identical person is in the other. You are the same person as the original you, of course, but so is the person in the other world. So, that other person is the very same person as the original you, who is the very same person as you, and yet that other person is not the very same person as you. So, either personal identity is not always a transitive relation, or else this scenario is impossible. And while the latter is certainly plausible, it seems to me to be the wrong sort of conclusion to draw from this scenario. Knowing whether it is the wrong sort of conclusion to draw or not may well be a prerequisite for getting anywhere with such puzzles as the modal paradox, because intuitions for "is the very same thing as" not ever being at all vague seem to be very similar to intuitions that it must always be transitive.

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, lost in pointlessness; but, the more things stay the same, the more they change.) Anyway, I also think that finding new theodicies is pretty pointless; consider this analogy: it is the first day of school, and things do not go well. And of course, you learn very little; but of course, that is no reason to have no first day of school. And the evidence that, if there is a Creator of all things, then it is an evil Creator, is a bit like that: if all of this was created by such a power, then there is very likely to be life after death (like further school days after the first, and then life after school, a life enhanced by prior schooling) because that would be better, and no less possible than this life; and so the worse this life is, the more likely there is to be life after death, if there is a God. The logic of such arguments is simple, and undeniable, and so the way the problem of evil is hyped up by mainstream analytic philosophers of religion is, clearly, pure rhetoric.

Tuesday, July 31, 2018

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.)

Thursday, July 26, 2018

An Inconvenient Proof


What follows is a logical 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, of course, but it must also be 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; 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 – although I personally do not think that the ontological argument works as a proof.

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, ‘happy’, ‘summer’ and ‘days’ are three words.
     Note that 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. Similarly, pairs of those pairs – e.g. {{‘happy’, ‘summer’}, {‘summer’, ‘days’}} – 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 ‘{‘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 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 being axiomatised; nevertheless, there are numbers of things.

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.