Science is built of facts

      much as our worldviews
      were made out of truths.

            Not all of them,
            nor them alone.

In the world, my eyes and ears
collect glints and creaks from the world
as it was when it shed such photons and sounds.

            My brain collates and extrapolates, creating a
            collage of the world as it may well be
            when next I shift or speak;

      when I am going to catch a ball,
      my hands go to where it will be.

            Seeing only the future,
            tomorrow is a mystery.

Tomorrow is a day like today
and today is mysterious: what is it that lies
under the skin that is shed in glimpses of such creatures?

            Tomorrow will surely surprise no less
            than today, for the world no less
            than science was made for


the unworldly.

The Vanishing

Imagine an object speeding up, its speed repeatedly doubling, with each doubling of its speed taking half the time of the previous doubling. If this object does not collide with anything, it will quickly approach the speed of light. But what if there was no light speed limit, and no chance of collisions because there were no other objects?

We naturally think of space extending endlessly in all directions, so the simplest space for us to imagine is an infinite space (the word infinite comes from the ancient Greek for unending). For any finite distance (such as any counting number of miles), such a space contains places separated by that distance (that number of miles). But note that space being infinite in that sense does not mean that there are parts of it that are separated by infinite distances (distances greater than any finite distance). It does not mean that there are parts of it that are actually at spatial infinity.

Imagine an object—let us call it X—in an infinite space with no parts at spatial infinity, and suppose that X is subject to forces—let us call them F—that cause it to accelerate in a straight line by repeatedly doubling its speed, with each doubling taking half the time of the previous doubling. With only one object in the whole of space, motion relative to another object is impossible, so X just sits there, not moving at all. So, imagine another object, Y, sitting next to X.

X and Y start out together, and then X moves one mile from Y in one hour (at an average speed of one mile per hour), and then another mile in half an hour (at 2 mph), and then another in a quarter of an hour (at 4 mph), and another in an eighth of an hour (at 8 mph) and so on, until after one hour plus half an hour plus a quarter of an hour plus an eighth of an hour and so on—two hours in total—X will have travelled further from Y than any counting number number of miles. From the point of view of Y, X seems to be going to spatial infinity, and then vanishing at spatial infinity because there is nowhere it could be (without teleporting), there being no place at spatial infinity in this space.

It is easy to imagine that such endlessly increasing forces being applied to X might cause it to explode or disintegrate, at some point. So, it is conceivable that if X did survive each of those endlessly increasing forces, it would have to vanish (or teleport) at spatial infinity because of all of those forces. Of course, from the point of view of X, Y seems to be going to spatial infinity, and it is inconceivable that Y would vanish (or teleport) because of the forces applied to X at such increasingly huge distances from Y. But I don't suppose that Y would have to vanish (or teleport) if X vanished (or teleported). What if Y was also subject to forces F, though?

If X and Y were subject to exactly the same forces, then they would remain together, neither of them moving relative to the other. They would just sit there, not moving at all. Neither of them would seem to the other to be going to spatial infinity, so neither of them would have to vanish (or teleport) at spatial infinity for that reason.

Would they both vanish (or teleport) because they were both subject to such forces? Maybe. And maybe the single object in the otherwise empty space of the original scenario would similarly vanish (or teleport), even though it did seem like it would just sit there. But have we discovered that any possible object in such a simple space would, were such forces possible, have to be such that it would vanish (or teleport) if it was subject to such forces and was able to survive each of them individually? Or is it more plausible that for any such space with no parts at spatial infinity, there would be something like the light speed limit that actual space seems to have? Perhaps we have discovered the reason for there being a light speed limit.

Although there are other alternatives. Perhaps objects in simple infinite spaces do not vanish at spatial infinity because they get to spatial infinity. There would be parts of simple infinite spaces at spatial infinity if unit volumes of such spaces contained 1/0 points (as outlined in my 2005 paper) and the counting numbers were indefinitely extensible (see my 2010 post and my 2024 booklet).

In view of the light speed limit that actual space seems to have, it may well be more plausible that a speed limit is a metaphysical necessity, even for such simple spaces. If philosophers interested in physics knew more about the nature of that necessity, would that help them to understand the actual light speed limit? Maybe, but knowing more about such a necessity would presumably involve finding out more about the alternatives, such as the one described in my 2005, in which there has been little interest.

☀️A Dark Vulcan

Vulcan was "discovered" by Lescarbault in 1859, in the sense that he saw something that he took to be the planet hypothesized by Le Verrier earlier that year. Le Verrier was already famous for his 1846 prediction of the existence and position of Uranus:

That prediction was based on observations of the planet Neptune. Neptune was not behaving as Newtonian dynamics predicted it would, not unless there was an unobserved planet like Uranus. A few days later, Uranus was discovered by Galle, who saw it roughly where Le Verrier had said it would be.

In 1859, Le Verrier hypothesized that observations of the planet Mercury might be similarly explained, by there being a planet between Mercury and the sun. And Le Verrier was sure that Lescarbault had discovered Vulcan.

Whatever Lescarbault had seen, it was not Vulcan. In the following decades, many observations of the absence of Vulcan were made. And while some astronomers claimed to have seen Vulcan, there seemed on balance to be no such planet. Now, the motion of Mercury was eventually explained in 1915, by Einstein. But my question is this:

Why was the balance of opinion before then not for the existence of a dark Vulcan?

Physicists only had to hypothesize the existence of dark matter, out of which Vulcan was made, in order to explain their observations. If dark matter was a very heavy, very dark form of matter, ubiquitous in the universe, then most of the dark matter in the solar system would be clustered around the sun, possibly in the form of a dark Vulcan.

Physicists do say that there is dark matter in the universe. Its existence is said to explain modern astronomical observations: the stars do not behave as Einsteinian dynamics predicts, unless there is dark matter. In other words, those observations contradict Einsteinian dynamics, to some extent. Now, Einsteinian dynamics has also been contradicted by quantum-mechanical observations of Bell's inequality, to some extent. And particle physics is increasingly reminiscent of celestial epicycles, which had, much earlier, been hypothesized to explain other astronomical observations.

Did something happen to physics in the twentieth century?

Well, science did become more of a cultural phenomenon in the twentieth century.

The God No One Wanted

(that is the new title of my booklet)

1. The Lie of the Land
introduction | expectations | descriptions

2. The Way of Things
Cantor’s paradox | set theory | the proofs

3. Proof of Probability
too many things | the shape of time | God

4. Reasonable Doubts
just bad math | deductions | explanations

5. Doubting Reason
the final straw | Russell’s paradox | truth

The Shape of Time

The Shape of Time

(two thousand words) will be
section 1 of chapter 4 of my booklet:
The Way of Things

Explanations

Explanations

(five and a half thousand words) will be
section 4 of chapter 2 of my booklet:
The Way of Things

Proofs

Proofs

(five and a half thousand words) will be
section 3 of chapter 2 of my booklet:
The Way of Things

Progress on "The Way of Things"

My booklet has been getting bigger and bigger over the past year (it is now over a hundred thousand words) but it seems ready to tidy up, so I will be posting the tidied up sections one by one and linking each post to the section titles in last July's The Way of Things, which can serve as a contents page.

Science is built of facts

for people who can see

that when they tell us and we don't learn, we are ignorant.
If I tell them and they don't learn, they assume I'm wrong.

They have everything, which makes them think that they have a duty to complain about everything.
We don't have anything, which just goes to show that we don't deserve to complain about anything?

                                                                If their physics is false,
                                                                whose problem is that?
      (if their logic is false, so that they cannot think logically
      about their own problems, then whose problem is that?

A limit of Empiricism

I set out to observe the world in a scientific way,
so I photograph the pond, and process the photo digitally,
and the sunlit willows being reflected in the shadows of other willows
show me that this part of the world at least is made of wave-mechanical string:

A true contradiction?

(a) the maths

Since adding zero to any amount does not change it, we can keep adding zeroes forever and it will make no difference: such additions always amount to adding zero.

We might write that as 0 = 0 + 0 + 0 + 0 + 0 + …, which can be spread out like this:

       0      =                             0                            +                             0                           +             . . .

Each 0 on the right-hand side can be replaced by 1 – 1, to give:

       0      =             (1            –             1)            +             (1            –             1)            +             . . .

In the next equation, the brackets have been removed.

       0      =             1              –             1              +             1              –             1            +             . . .

In the next equation, brackets have been put back in, in different places.

       0      =             1              +             (–1         +             1)            +             (–1         +             . . .

We now replace each (–1 + 1) with 0.

       0      =             1              +                             0                              +                         0              . . .

All those zeroes on the right-hand side add up to zero, of course. But that means that:

       0      =             1

Clearly 0 = 1 is false. So, where did we go wrong? Well, since the last equation was false, the equation above it must also have been false (the only difference between those two equations is the first equation, which was clearly true). And the next one, going upwards, 0 = 1 + (–1 + 1) + (–1 + 1) + ..., must have been false too, as each of those “(–1 + 1)” does equal zero.

Going the other way, from the first equation, 0 = 0 + 0 + ..., which was clearly true, the next equation, 0 = (1 – 1) + (1 – 1) + ..., is similarly true, because each of those “(1 – 1)” is zero.

In between those two equations, one false and one true, we have the infinite sum 1 – 1 + 1 – 1 + …, which was originally described by the Italian theologian and mathematician Guido Grandi (1671–1742).

Grandi was interested in the calculus (as described by Leibniz). And in the calculus, an infinite sum is equal to the limit of the initial finite sums as their length tends to infinity. Grandi’s infinite sum 1 – 1 + 1 – 1 + ... has initial sums that alternate between 1 and 0 = 1 – 1 endlessly (the next are 1 = 1 – 1 + 1 and 0 = 1 – 1 + 1 – 1). Since the initial sums tend to no limit, Grandi’s infinite sum is not given any value by the calculus.

By removing the brackets, we moved from an infinite sum of zeroes, which is equal to zero, to Grandi’s infinite sum, which has no value. Adding brackets differently then took us from Grandi’s infinite sum to a sum that is one plus an infinite number of zeroes, which is equal to one.

(b) the physics

You may be familiar with the idea of a particle/antiparticle pair appearing out of the vacuum. Such pairs give rise to Hawking radiation from a black hole, but all we need to know here is that such pairs can, in theory, appear from the background fields of the vacuum. Once formed, the particle and antiparticle are moving away from their point of origin, so we might picture them moving downwards, like this: /\ (near a black hole, one of them might be swallowed by the black hole, while the other flies away from the black hole, giving rise to Hawking radiation).

Space does not seem to be infinite, but an infinite space is a physical possibility. And in such a space, an endless line of such particles/antiparticle pairs is a possibility, for all that it is highly unlikely. We might picture them like this: /\/\/\/\/\... (the zig-zag continues to spatial infinity).

The top of that zig-zag pictures a line of particle/antiparticle pairs appearing, which might be modelled mathematically by modelling each particle as +1 and each antiparticle as –1. We then get this equation:

0 = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + ...

Each (1 – 1) represents a particle/antiparticle pair appearing.

They move downwards in such a way that each antiparticle collides with the particle from the pair to the right, so that they are both annihilated. The particle at the extreme left of the zig-zag is not annihilated. The bottom of the zig-zag therefore pictures events that are modelled rather well by this equation:

1 = 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + ...

Each (–1 + 1) corresponds to an antiparticle and a particle annihilating each other.

In between those two equations, there is no mathematical sum, neither 0 nor 1. That corresponds to infinitely many particles and antiparticles just being there, in between their creation and their almost total annihilation. The highly improbable, but physically possible, appearance of this particle from an infinite vacuum is therefore so well-modelled by 0 = 1 – 1 + 1 – 1 + ... = 1, that it is essentially an instance of it. It is in a very similar way that Jack and Jill being a couple is an instance of 1 + 1 = 2.

Such equations as 1 + 1 = 2 only exist because they are such good descriptions of any collection of two things. It is the physical instantiation that ultimately justifies the mathematical equation. And of course, to say of what is, that it is, is to say something that is true. Which raises the following question.

(c) the questions

Could 0 = 1 – 1 + 1 – 1 + ... = 1 be a true contradiction?

And in order to think about that question logically, should we use paraconsistent logic?

(d) my answers

Although a contradiction can be used as a description that is such a good description, it should count as a true description (as when we say that something is and isn’t a certain way, meaning that it is that way in one sense but not in another, or that it is that way about as much as it is not), that does not mean that the contradiction is true. Not too dissimilarly, there are two ways in which 1 + 1 = 2 is true. It is true as a description of Jack and Jill, and it is, in a different way, true by definition (of 2). And it is, in any case, not at all contradictory for there to be no particle and then, at a later time, one particle.

Why would anyone think that a mathematical model of reasoning that is not a very good model of logical reasoning (because if something is not the case, then it cannot also be the case: it not being the case means that it cannot) would help them to think logically?

The Way of Things

1. Introduction

    the lie of the land | low expectations

2. The Way of Things

    infinities | sets | proofs | explanations

3. Extraordinary Evidence

    many things | impossibly many things

4. A God Hypothesis

    the shape of time | authorial authority

5. Exceptional Logic

    heaps | Russell's paradoxes | truth

In times of uncertainty...

Time spent in nature is linked to lower stress, restored attention, a balanced nervous system, increased levels of cancer-fighting “natural killer cells”, the activation of neural pathways associated with calm, and decreased levels of anxiety and depression. Phytoncides (compounds emitted from trees and plants), relaxation, stress reduction and awe are known to enhance immune function.

Lucy Jones, In times of uncertainty, let nature be your refuge (The Guardian, Friday 20 March 2020)
I took the photo below, of a collared dove in a cherry tree (on a road in my village), on 20 March 2015:

How to Turn Matter into Antimatter

1) Turn matter into electricity, using a nuclear power station.

2) Turn that electricity into light of a particular frequency.

3) Those photons decay into particle/antiparticle pairs.

Fissix

F is six

in ancient Greek,
and "fissix" sounds like physics,
and the external physical world is perceived in six basic ways:

Looking at it with our eyes,
hearing it with our ears,
smelling it with our noses,
tasting it with our tongues,
feeling it with our skins, and
knowing which way is up, via our inner ears.

People tend to forget about that sixth way of perceiving the external physical world because people tend to think of the sixth sense as being an ability to see spirits, or some vague sense of impending doom, or a sense of being stared at. Is there a sense of being stared at? Would that be a seventh way of perceiving the external physical world? Or a way of perceiving the external mental world? Are ghosts ectoplasmic, or psychic? I suppose that if psychologists demonstrated that there is a sense of being stared at, and if they got a good materialistic theory of how it worked (involving, say, the nature of quantum-mechanical collapses), then philosophers would say that that was indeed a seventh way of perceiving the external physical world. That is all very iffy though.

What about the sixth way of perceiving the external physical world with our inner ears? Well, "going up in the world," people say, and "feeling a bit down." People naturally associate up with good and down with bad. But physics came of age when the heavens fell under its laws, in the seventeenth century, thanks to Newton.

By pursuing empirical truth, instead of the next epicycle, Britain began the industrial revolution and built the biggest empire the world had ever seen. And the other sciences followed in the footsteps of physics. In the eighteenth century chemistry emerged from alchemy, in the nineteenth century biology became Darwinian, and in the twentieth century physics became Einsteinian. But the twentieth century was a century of cultural wars, as well as actual wars that got bigger than wars had ever been. Soviet biologists doubted the Darwinian foundation of biology. A few mathematicians even doubted the set-theoretical foundation of twentieth-century mathematics. Still, no one doubted the quantum-mechanical foundation of chemistry. And no one doubted the Einsteinian foundation of high-energy physics, which was a little odd because quantum mechanics all but refuted the four-dimensionalism of Einsteinian spacetime.

Almost all of the physical evidence supported the equations of quantum mechanics and Newton's equations, and the rest of it was evidence that could only be perceived by physicists who were assuming Einstein's equations and making very expensive observations, funded by those with the funds to fund the military, observations that could hardly be independently verified.

Physicists are still pursuing the next dark object at the cutting-edge of high-energy physics, and inventing ever more elaborate Einsteinian theories; and philosophers are still giving "empirical truth" new definitions. But if you could choose between appearing professional and being able to learn, which would you choose?

The Signature of God

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Death of Logic

A hundred years ago (more or less) logic died.

It was either Logic and a transcendent creator, or neither,
and atheism was in the ascendant a hundred years ago,
while the God that could be shown (in a logical way) to exist
was not that of the embattled religions of those war-faring days
so it was not even a contender.

Prima facie, logic took off at that time.
We now have lots of formal logics, and they all look very rigorous
because they are very mathematical. They look very scientific.
But, what's so logical about reacting to the Liar paradox by redefining "truth"?
And what's so logical about having each ordinal but not having every ordinal?

Suppose we get a really good String Theory, say "S," one day.
There's no guarantee that we won't need a better theory later,
so why would we use S to redefine all our physical entities?
If the description of electrons in S was E, for example, then
we could replace "electron" with "E," but why should we?
A good reason why we should not is that electrons are electrons!

And arithmetic is a subset of the properties of possible objects:
one object and another object is one-plus-one objects, and so on.
But for a hundred years, science has replaced "1" with "{{}}."
Did not the logician Frege refute Mill's description of "1"?
It turns out that he did not. And he could not have,
because 1 is, basically, Mill's 1 (and Euler's, and yours).

Set Theory mimics mathematics,
so for applications it hardly matters; but,
do the best mathematicians really believe
that 1 is nothing like Mill's 1, is really {{}}?
We all learn what 1 really is at an early age.
{{}} was chosen following Cantor's paradox,
but it also followed that logic had to be replaced.
Logically, there was that paradoxical proof; and
while the Liar paradox is nowadays interpreted
as another reason to replace truth, and its logic,
with something formal, that is not really scientific:
science pursues truth, and logic takes truths to truths
(where to say, of what is, that it is, is to speak the truth).

Is Logic Necessary?

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

Logic Needs That Hypothesis

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

Why was the Big Bang not a Black Hole?

A documentary about what happened before the Big Bang was repeated on the BBC last night, and it got me wondering: why did anything happen after it? Why was the Big Bang not a Super-Duper-Massive Black Hole? There was all this matter, all the matter in the universe, in this tiny, tiny space; so why the explosion? Why an inflationary explosion? And why is the universe still accelerating? Did dark energy make it all happen? Dark energy sounds like a physics of the gaps!
......I had already been wondering why an amount of antimatter equal to the observable matter of the universe would not be in the form of an uncollapsed standing wave (like electron shells around atomic nuclei). The popular theory of where all the antimatter went is that there was originally a lot more extra matter and an equal amount of antimatter which annihilated each other. But that would just create a lot of heat and light, none of which could escape a Black Hole. But, were the antimatter in a standing wave, then the uncollapsed antimatter suffusing the primordial atom would make it effectively massless, so there would be no Black Hole, while the repulsive force between the matter and the antimatter would cause an explosive expansion. Furthermore, the appearance of dark matter would be explained; while the standing wave would enforce a certain uniformity, much as the inflationary period is supposed to have done.
......I have not heard of any such theory, so that thought is not even philosophy of physics, but listening to the physicists in that documentary made me wonder whether there might be such a theory. The things they were saying were pretty off the wall (according to each other).