## Sunday, July 26, 2020

### Puzzle

I find a quirky object in my attic. It is not obvious what it is made of, or even if it is natural or a work of art. But then I see that some ridges near its base are in between the imprints of fingers. Had similar ridges been found in the bark of a tree, their resemblance to a handprint would have been a coincidence. Here, they show that this object had been held during a formative process. And now I know that it is salt-glazed stoneware.
Similarly, there is something about the world that is best explained by the world having been created. That means that reality is fundamentally personal, rather than impersonal. But materialism has been ruling the academic roost for a while now. And that has made this “something about the world” seem very puzzling indeed. This book solves a puzzle that has had mathematicians and philosophers stumped for more than a hundred years.
It was not the complexity of the mathematics that stymied them, because the mathematics of this puzzle is little more than arithmetic. It was the fact that this puzzle’s solution makes sense only if there is a logical sort of God. To solve this puzzle would have meant flying in the face of the prevailing atheisms of twentieth century and contemporary academia. Such is the solution to this particular puzzle, though. So, this puzzle amounts to good evidence that there is a God. You can judge the evidence for yourselves, because it is not a matter of observation and testimony, but of self-evident truths and logic.
And there was, to begin with, a puzzle.

The numbers were not adding up, as the twentieth century began. That is, the numbers themselves were not cohering together. Mathematicians, by assuming that there are all the whole numbers, were not getting consistent results. Could that assumption have been false? But there are, quite simply, all the whole numbers that there are. And whole numbers are the simplest things in mathematics. Whole numbers are answers to “how many?” questions. How many words are there in this question? We answer that question by counting those words. It is as simple as 1, 2, 3.
As you probably know, the counting numbers are 1, 2, 3 and so forth. How many counting numbers are there? The sequence 1, 2, 3 and so forth goes on and on, without stopping at a biggest counting number, because there is no biggest counting number. (If there was, then there would be that many counting numbers in total; and there is also the fraction ½, of course. How many numbers is that in total? It is one more than that “biggest” counting number. Clearly, we can always add 1 to any counting number, and get another counting number.) So, there are infinitely many counting numbers.
Presumably there is an infinite whole number, the number of all the counting numbers. But it may not be as simple as that. The behaviour of this endless sequence is certainly not as simple as it looks. The counting numbers get bigger and bigger, boundlessly, and so most of them are unimaginably huge. However, consider any counting number. Almost all the others are an awful lot bigger than it is, which means that it is relatively tiny. And that was any counting number. So, all the counting numbers are relatively tiny. Almost all of them are huge, but all of them are tiny.
That may look like an inconsistency, but it is just typical of big numbers. The milky way is, similarly, a huge number of stars—they are so numerous that they form a milky band of starlight in the night sky—but only a tiny fraction of all the stars that there are. Still, there are infinitely many counting numbers, not just an awful lot of them. And that boundlessness does give rise to some very strange behaviour. Since you may not have seen such behaviour before, I shall quickly run through a simple example next, before turning to Galileo’s paradox (which is not only another simple example, it also introduces the core concept that is used in the main argument of this book).

Imagine that you are given a counting number. It could be any counting number. I do not know which number you have, but I am also given one. You do not know which number I have, but you could work out that (a) because so many counting numbers are bigger than yours, my number is almost certainly bigger than yours. The puzzle is that I could similarly reason that (b) because so many of them are bigger than mine, yours is almost certainly bigger. That is more like an actual inconsistency. However, even if this is an inconsistency, it is not necessarily an inconsistency in the counting numbers themselves.
We could not really have been given numbers in such a way that the counting numbers were all equally likely, because most of them are so big they cannot even sit in a file on a computer, let alone be shown to us. There had to have been some upper limit to the numbers that we could have been given. Still, if we did not know what that limit was, then whatever numbers we got, it would have been reasonable for each of us to think that that limit could have been considerably larger. You could still have thought (a), and I could still have thought (b). This puzzle—which was discussed by the French mathematician Paul Lévy (1886–1971)—is therefore a puzzle about rational expectations, rather than infinity. Nevertheless, a puzzle about infinity and physical possibilities comes from varying the original scenario in the following way.
Suppose that space is infinite, that it goes on and on in all directions. Nowadays we think that space is finite, because it began with size zero less than fourteen billion years ago, and has been expanding at finite rates since then. But could space have been infinite? For the sake of the following argument, let us assume that space is infinite.
Suppose that there is a radioactive particle every light year, in various directions. (A light year is the distance that light travels in one year.) In one direction there is one endless sequence of particles, and in another direction another. Each particle has a 50% chance of decaying during its half-life. (A radioactive substance’s half-life is the amount of time it takes for half of that substance to decay.) So, after one half-life there would an endless sequence of decays and non-decays in one direction, and another sequence in another direction, with all those decays and non-decays being equally likely.
For each sequence of particles, any sequence of decays and non-decays is a possible outcome. And all those different outcomes are equally likely. For an example of a possible outcome, all the particles up to the one that is a hundred light-years away might have decayed, after one half-life, with all the rest in that sequence not having decayed. That possible outcome is naturally associated with a hundred. And since both of those instances of the phrase “a hundred” could be replaced with any other counting number, we have some equally likely—if very unlikely—counting numbers. Lots of other associations of possible outcomes with counting numbers are possible. And the fact that those outcomes are very unlikely does not affect the following argument, because an inconsistency should be impossible, not just very unlikely.
If one of our two infinite sequences of particles decayed in such a way that it was associated, in some such way, with a counting number, then because there are so many bigger counting numbers, it should almost certainly be the case that if the other sequence decayed in such a way that it was also associated with a counting number, then that number would be bigger. But in much the same way it should also, in all probability, be smaller.
That inconsistency in the physical probabilities means that this scenario should be impossible. It indicates that at least one of our assumptions was false. And we do think that space is finite, nowadays. So, perhaps this scenario shows that space cannot be infinite. The devil is in the details; it all depends on what the alternatives are, and on just how implausible they are. But facts about the world might, just possibly, be deduced from mathematical inconsistencies. (This book is all about deducing one particularly surprising fact from a mathematical inconsistency. There are therefore some more examples of this kind of thing in chapter 1, where they will be described in more detail. To understand any proof, it helps to be familiar with simpler examples of that kind of argument.)

Another oddity arising from the boundlessness of the counting numbers was discussed by the Italian physicist Galileo Galilei (1564–1642). It concerns the relative size of the even numbers, which are 2, 4, 6 and so forth. It is only every other counting number that is an even number. So, there are clearly fewer even numbers than counting numbers.
And yet, we can get the even numbers by doubling the counting numbers. Twice 1 is 2, twice 2 is 4, twice 3 is 6 and so forth. In that way, the even numbers can all be paired up with all of the counting numbers. And that clearly means that there are as many even numbers as counting numbers. Galileo’s paradox is that although there are clearly fewer even numbers than counting numbers, there are also, just as clearly, as many even numbers as counting numbers.
That is much more like an actual inconsistency. And now, there is nothing else that might have caused the inconsistency. However, there is not actually an inconsistency. Much as the sense in which the counting numbers are huge was not the same as the sense in which they are tiny, there are two senses of “fewer,” “as many” and “more” in use in Galileo’s paradox. The sense in which there are fewer even numbers than counting numbers is obvious enough. To get the even numbers, we take some of the counting numbers away. The other sense is of fundamental importance in mathematics. And its usefulness comes from the fact that it applies to any number of any things.
If we have five oranges and three lemons, for example, then we clearly have fewer lemons than oranges. The first sense of “fewer” does not apply, because we do not get the lemons by taking some of the oranges away. But the second sense does: We have some oranges left over after we have paired up as many of the oranges and lemons as possible. Similarly, the first sense of “as many” does not apply to a comparison of the first three even numbers with the first three odd numbers. There are just as many numbers in each group, of course. There are three of them. But that “three” comes from my description. This first sense will not give it to us. The second clearly will.
Perhaps we could use the first sense if we first renamed each of the odd numbers, calling it by the name of the number that is twice as big. But for the following reason that trick would not enable us to use the first sense to compare the totality of the even numbers with the totality of the odd numbers. If we gave the even numbers the names of the counting numbers, we would find that there are fewer odd numbers than renamed even numbers. But if, instead, we renamed the odd numbers as counting numbers, then we would find that there are fewer even numbers than renamed odd numbers. And renaming things should of course not change how many of them there are.
In short, the first sense of “fewer,” “as many” and “more” is mathematically uninteresting. The idea of a mathematically uninteresting sense of “fewer,” “as many” and “more” is admittedly a strange idea. But that strangeness might explain why this puzzle remained unsolved for so long. It was the German mathematician Georg Cantor (1845–1918) who clarified the other sense, which was soon seen to be the one that mathematicians had been implicitly working with all along. Given only that sense, mathematicians could get the whole of arithmetic, just by being logical.

The only problem was that they were also getting some inconsistencies. Toward the end of the nineteenth century, Cantor constructed a relatively simple proof that, given any whole number, there is a bigger number (in the second of those two senses). Given an infinitely big collection, such as all the counting numbers, Cantor’s proof—which is known as his diagonal argument—shows that the collection of all of its subcollections is a bigger infinite collection. And there is therefore an even bigger collection (of all the subcollections of that bigger collection). In short, there are infinitely many infinite numbers. Indeed, there are too many numbers. By the end of the nineteenth century, Cantor’s diagonal argument was generating inconsistencies from the assumption that there are all the numbers.
Basically, the number of all the numbers should be the biggest number. But given that number, Cantor’s diagonal argument shows that there would be a bigger number. The puzzle now known as Cantor’s paradox concerns the collection of all the (other) mathematical collections. That collection should be the biggest mathematical collection. But Cantor’s diagonal argument shows that there would be a bigger mathematical collection, the collection of all of its subcollections. Could one conclude that there is no such collection of all the (other) mathematical collections? Cantor’s paradox will be considered further in chapter 2. This book is all about the following variant of Cantor’s original paradox, whose details are fleshed out in a very logical way in chapter 3.
Given the collection of all the counting numbers, there is a bigger collection, the collection of all the collections of counting numbers. And the collection of all of the subcollections of that bigger collection is, similarly, an even bigger collection; and so on. So, we have an endless sequence of bigger and bigger collections.
And the totality of all the things in all those collections is an even bigger collection. We can now start again with that collection: the collection of all of its subcollections is bigger still; and so on. We start again, and again; there is no end to this. But because mathematical objects are presumably timeless, there should be a totality of all the things in all the collections that can be obtained in this way. The problem is that there cannot be such a totality. If there was such a totality, the collection of all its subcollections would contain even more things, obtained in just that way.
In light of such inconsistencies, mathematicians at the start of the twentieth century naturally wondered if there was something wrong with Cantor’s diagonal argument. So, they took a long hard look at it. But the consensus was that it was perfectly logical. Naturally, academics have never stopped trying to fault it. Academics are the sort of people who would put the word “probably” in front of every other word, if you let them. However, those academics who thought that they had found something wrong with Cantor’s diagonal argument always turned out to have been mistaken. The British mathematician Wilfred Hodges wrote about such mistakes in his “An Editor Recalls Some Hopeless Papers” (pages 1–16 of volume 4 of the journal Bulletin of Symbolic Logic, in 1998). And you will be able to judge for yourselves. Elementary versions of the diagonal argument are explained in relatively plain English in chapter 2, and in greater logical detail in chapter 3.

Given that Cantor’s diagonal argument is indeed a proof, what was causing those inconsistencies? Some mathematicians wondered whether there were any infinite numbers. There are certainly infinitely many counting numbers. But it is not immediately obvious that we can think very clearly about what that means. It is an awful lot of things to think about. So, some mathematicians took the inconsistencies to be showing that we cannot think coherently about such infinitudes. Nevertheless, the process of repeatedly adding 1, starting with nothing, produces first 0 + 1 = 1, and then 1 + 1 = 2, and then 2 + 1 = 3 and so forth. So, we need only think of that process going on and on, without ever stopping, in order to be thinking about the totality of all the counting numbers.
As we think of that process going on and on without ever stopping, we may think that there never are all of the counting numbers. Some mathematicians thought so, concluding that there is no infinite number of them all, just ever-increasing but always finite numbers of them. There are infinitely many of them, but only in the sense that they are produced unendingly.
And if, for whatever reason, some whole numbers do not yet exist, then it would be false that there are all the whole numbers. It is similarly false that there are all the people that we read about in history books, because most of them are dead. Nor are there all the people that we see in movies, because most of them are fictional people. The truism that there are all the characters that there are does not make any difference to those two facts. And similarly, while there are of course all the whole numbers that there are, perhaps that is an ever-growing totality.
Or perhaps there are not really any numbers. Some philosophers have argued that the inconsistencies cast doubt over their existence. (Similarly, people can do impossible things in movies, but only because they are fictional people.) If there really are numbers, such philosophers have wondered, where are they? But such questions do not mean much to mathematicians. Two plus two equals four, not five, whether the question of where two and four are makes sense, or not. If I assume that 2 + 2 = 5, then I have an inconsistency—that assumption is not consistent with the fact that two things and any other two things make four things altogether—an inconsistency caused by the falsity of that assumption. That assumption is false howsoever those numbers are.

The big problem with it being the case that numbers are forever coming into being is the following reason why there should already be all the whole numbers that there will ever be.
To begin with, there being a whole number is, in effect, it being possible, one way or another, for there to be that many things, of some sort or another. Were that not the case—were there a whole number and it was not at all possible for there to be that many things—then what would it be a whole number of? In what sense would it be a whole number?
So, we can safely assume that if some whole number is going to exist, then it will eventually be possible for there to be that many things. That means that it will never have been logically impossible for there to be that many things, because logically impossible things cannot ever exist. To say that something is logically impossible is just to say that it is ruled out by logic alone. Logic rules out round squares, for example.
And because it was never logically impossible for there to be that many things, it was always logically possible for there to be that many things. Logically possible things are just things that are not logically impossible.
And that means that that whole number always existed. So, any whole number that will ever exist always did exist. There should already be all the whole numbers that there will ever be.

And yet, there had been found to be—in some sense—too many whole numbers for there to be all of them. Although mathematicians could make little sense of Cantor’s discovery. They thought about it, and thought about it, and they soon realised that they would need a way of working around it, while they thought about it some more.
Various philosophers with an interest in mathematics—primarily the British philosopher Bertrand Russell (1872–1970), whose first degree was in mathematics—were also thinking about these inconsistencies. But to cut a convoluted story short, mathematicians decided to change the meaning of “whole numbers” so that the “whole numbers” that they would be working with would give them consistent results.

(This is the first section of the first chapter of The Way of Things. The next section is Theory.)