Definitions

In the seventeenth century, Leibniz, a German Lutheran famous for inventing the infinitesimal calculus (following Newton's invention of a similar method), was primarily working on a logical calculus (following Descartes' algebraic innovations). Leibniz thought that a logical calculus would make the world a more peaceful place: if two people disagreed, they would not need to argue, they could just sit down with paper and pencil and compute the true answer. Leibniz hoped, for example, to use his logical calculus to show Locke that 2 + 2 = 4 is not an empirical truth but a logical truth:

Starting with the logical axiom that things are identical to themselves (expressed in symbols as A = A, with A being 2 + 2 in this example), Leibniz applied the logical rule that replacing equal things preserves equality three times (with three definitions, a definition of 2 as 1 + 1, a definition of 3 as 2 + 1, and a definition of 4 as 3 + 1) in order to obtain that arithmetical truth:

          2 + 2  =  2 + 2
          2 + 2  =  2 + 1 + 1
          2 + 2  =  + 1 3 + 1
          2 + 2  =  + 1 4

Laid out like that, the logic of that proof of 2 + 2 = 4 is certainly very clear, and Leibniz wanted to bring such clarity to all of our reasoning, by expressing in symbols all of the logical axioms and rules of inference used in logical thinking.

In the nineteenth century, Frege, another German Lutheran who wanted to make logic more like algebra and to show that arithmetical truths are logical truths, noticed that Leibniz had not shown that 2 + (1 + 1) = (2 + 1) + 1. Still, it is obvious that for small numbers of things, how many things we have does not depend upon which ones we look at first. And if we had to prove every obvious step in a logical proof, then we would never get to prove anything logically.

Logical proofs begin with descriptions that are obviously true (or descriptions that would clearly be true were the assumptions of the proof true) and then, step by logical step (each step being such that from something true it would clearly take us to something true), they take us to conclusions whose meaning and truth are thereby obvious.

Frege thought that logic should be more algebraic, though. So, he did a lot of work on using symbols to describe logical axioms and rules of inference. And he added to Leibniz's definitions by defining 1 to be 0 + 1, and by defining 0 and + 1.

In the twentieth century, Russell was infamous for proving 1 + 1 = 2 with hundreds of pages of obscure symbols. Russell wanted to make logic algebraic and to show that arithmetical truths are logical truths, and Frege's efforts had ended in failure because of a logical paradox that Russell had discovered at the start of the twentieth century (following Cantor's discovery of a mathematical paradox at the end of the nineteenth century). But Russell soon tired of mathematics. And for a hundred years, most mathematicians have implicitly defined 0 and + 1 within an axiomatic set theory. Has that enabled them to prove 2 + 2 = 4 in a rigorously logical way?

To be precise, they can prove in a rigorously logical way that 2 + 2 = 4 where those symbols have set-theoretic meanings.

Many more steps would be needed for a proof of 2 + 2 = 4 where those symbols have their ordinary meanings, and those steps could not all be as rigorously logical because axiomatic sets are algebraic models of collections whose sizes are natural numbers like 2 and 4, and such models are bound, because of such paradoxes, to be at least a little inaccurate in some rather obscure ways. It is in a much more obvious way that the symbols 2, 4, + and = get their ordinary meanings from the meanings of words like one, two, three and so forth; and those words get their meanings in much the same way as the other basic words of English get theirs. Note that while the meanings of such words are described by dictionary definitions, they are not given to them by those definitions (indeed, they could not be, because in order to define a word, other words must be used).

At the end of the nineteenth century, Lewis Carroll used a fictional dialogue to show that if we needed rules of inference to tell us how to use rules of inference, then we would never get to prove anything logically. And it is fairly obvious that if we had to define the terminology of a proof in order for it to be a logical proof, then we would only be able to prove algebraic results logically. And of course, if the logic of a logical proof had to be axiomatic, then we would have had to have had logical axioms before we had any logical proofs. In reality, we hypothesize logical axioms when we have a lot of similar proofs, all of which are clearly logical, and then test those hypotheses as logically as we can. Logical axioms are bound to be less certain than our most logical proofs, so how could deductions from logical axioms and definitions of the algebraic (rather than the dictionary) kind be the most logical kind of proof? In reality, the fact that two things and another two things make four things can be seen by counting them:

There is (to begin with the first two things) a first thing and a second thing, and then there is a third thing (the first of the other two things) and a fourth thing (the second of the other two things).

That is a logical proof because it is obviously the case that counting a small number of things can tell us how many of those things there are, and it is a proof of 2 + 2 = 4 because the ordinary meaning of "2 + 2 = 4" is that two things and another two things make four things. It is because those things can be any logically possible things that 2 + 2 = 4 is a logical truth. Would that proof of 2 + 2 = 4 be more logical if it included an explanation of how counting works? Maybe, and such an explanation could always be added. But would we get a more logical proof of 2 + 2 = 4 if we modelled that proof very accurately? No, because even if that model was a very logical proof, it would be a very logical proof of a model of 2 + 2 = 4.

Numbers

The counting numbers are as simple as 1, 2, 3. But when it comes to numbers, mathematicians are the experts, and most mathematicians take set theory to be the foundation of mathematics; and set theory makes the simplest equations complicated:

According to the standard set theory, the first counting number, one (1), is the set containing the set containing nothing, where a set is a mathematical model of a collection of things; and adding one (+ 1) to a counting number results in the set containing the set that is that number and also everything that that set contains. Because mathematical proofs can be very formal, where a formal logic is a mathematical model of logical reasoning, mathematicians have some fairly long proofs that 1 + 1 = 2.

Can 1 + 1 = 2 be proved, though? Or is the meaning of “1 + 1 = 2” that insofar as there is one thing and another thing there are two things? Insofar as there is one thing and another thing there are two things because that is what the word “two” means. You will have learnt the meanings of words like “one,” “two” and “three” at roughly the same time as you learnt the meanings of words like “round,” “shoe” and “black,” which you will have done in ways that went something like this:

How many shoes am I wearing? Two

What colour are my shoes? Black

Colours are properties, of things of various kinds (stars, stamps, rainbows, hallucinations, to name but four). To say what those properties are requires some basic physics, biology and psychology. And as for individual colours, they are as hard to describe as the tastes of wines. But shapes are easier to describe (as are numbers), and shapes like round and square are properties, of things like stars and stamps. Now, mathematicians are the experts when it comes to shapes (as well as numbers). But while mathematicians have found some very strange shapes (as well as some very strange numbers), they do not say that round is the equation for a circle, an equation written in a programming language, or anything like that. Why, then, do they say that the number one is the set containing the empty set, where those sets are defined by the axioms of the standard set theory?

Well, mathematics is all about precision and proof, so when mathematicians say that set theory is the foundation of mathematics, they may just be saying that their proofs begin with set-theoretical axioms. After all, set-theoretical proofs that 1 + 1 = 2 could hardly be showing that 1 + 1 = 2; whereas, such proofs do show that set-theoretical models of arithmetic are not bad models of arithmetic. So, when mathematicians say that the number one is the set containing the empty set, they may just be telling us what their model of the number one is.

Why model such simple numbers, though? Why not use the actual numbers themselves? Mathematicians do use them (we all do), and they need to know what their arithmetic is in order to know how good their models of it are. Still, when it comes to questions like whether numbers are properties or not, mathematicians are not the experts. Such questions are metaphysical, not mathematical. Philosophers are the experts when it comes to metaphysical questions. Now, most of the philosophers who are interested in mathematics think that the counting numbers are not properties, of things like pairs of shoes. However, the first counting number, one, certainly seems to be a rather trivial property of things: each and every thing is one thing. And counting numbers bigger than one certainly seem to be properties of collections:

The numerical sizes of collections (such as pairs of shoes) are how many things those collections contain.

Collections have various properties (galaxies, for example, have shapes and spatial sizes, as well as being some rough number of stars), and while a pair of shoes is not just any two shoes (and a stamp collection is more than just some stamps), there are, in any given collection of things, things that are being referred to collectively, and it is how many of them there are that is the numerical size of that collection (and those numbers are precise, rough or variable depending upon the collection). The numerical sizes of small collections can be found by counting the things in them (which is why the simplest numbers are called counting numbers), but all collections have a numerical size (precise, rough or variable). And because there is no biggest counting number (the number one can added to any counting number), the counting numbers get bigger and bigger without end.

How many counting numbers are there? Infinitely many (the word “infinite” comes from the ancient Greek for unending). Most mathematicians would say that there is an infinite number of them all; and while those mathematicians might mean all sorts of things by that word “number,” the number of all the counting numbers (in the ordinary sense of how many of them there are) would be a number that was infinitely big but of the same basic kind as the counting numbers if there was a number of them all (in that sense).

At the other extreme, zero is another precise answer to a how many question (“how many sheep are there in a field that has no sheep in it?” for example), but is zero a property? Maybe not. But then, it might seem wrong to say that zero is a number. After all, we do not use it when counting. And it is not the size of a collection. Nor is the number one, of course. And if you have only one sheep, then you do not have a number of sheep. Nevertheless, the number one is definitely a counting number. And these doubts about zero and one being numbers just go to show how similar numbers and colours are, because there is a sense in which black and white are not colours; in that sense, the colours are made out of red, yellow and blue. Orange, for example, is red plus yellow, and adding a bit of blue makes brown. There are three primary colours because there are three types of cone cells in the human eye; and counting numbers are made out of ones because collections of things are made out of things:

One thing and another thing makes two things, and adding another makes three, and writing that in symbols gives us 1 + 1 = 2 and 1 + 1 + 1 = 3.

There being three primary colours shows that colours can be counted, even though colours merge into each other in the rainbow. Clouds too could be counted, if they were distinct enough for long enough. Still, it is more obvious that you and I are two people. And even if we had never existed, we would still have been two possible people. Possible people can be counted if they can be referred to and they are distinct from each other. What about impossible people? Well, Superman is not a realistic fictional character (he flies in the face of the laws of physics), but Superman and Batman are two superheroes. I suppose that while Superman is physically impossible, he is a logically possible person, in the sense of there being no contradictory information about him in the stories about him; or, if there is contradictory information, then we could say that there are two or more versions of Superman, or different characters called Superman if the stories are sufficiently different.

Are the most basic numbers one and the sizes of logically possible collections?

Well, suppose that they are not. Suppose, to begin with, that while some name did seem to be the name of a number bigger than one, no collection of that many things was a logically possible collection. In what possible sense would that name have been the name of a number, in the sense of a precise answer to a how many question? Going the other way, suppose that while some description certainly seemed to be the description of a logically possible collection, there seemed to be no answer to the question of how many things were in that collection. Could we not say that the answer to that question was the number of things in that collection? Could we not give the number of things in that collection a name, such as nu (and a symbol, v), to make it more like two (2)? Any problems with introducing such a number would be prima facie problems with the description in question being the description of a logically possible collection.

Not too dissimilarly, my stamp collection exists, so it certainly seems to be a logically possible collection. Let us call the number of things in it v. Yesterday v was 100, but today I got a new stamp. Today, v is 101, not 100. Does that apparent contradiction mean that some logically possible collections have no numerical sizes? No, it just means that v is a variable. Insofar as “collection” means distinct things being referred to collectively, the description “my stamp collection” is obviously naming different collections (collections of different things) at different times.

If we take the word “numbers” to mean precise answers to how many questions, then numbers bigger than one do seem to be the sizes of logically possible collections. And it makes sense for such numbers to be composed of ones. So, the counting numbers, together with any infinite numbers that are similarly composed of ones, would seem to be a rather natural kind of number (although the question of the existence of such infinite numbers is even more complicated than the question of the existence of xenobiologically possible colours).

Note that when mathematicians and philosophers say that numbers are something else, they are usually thinking of wider ranges of numbers. They might, for example, be trying to say what numbers like zero, a half and pi are at the same time as they say what one, two and three are. Those six numbers are precise answers to how much questions, and mathematicians call them “real numbers” (the real numbers also include negative numbers), and there are many other kinds of numbers that mathematicians study (imaginary numbers, transfinite ordinal numbers, and surreal infinite and infinitesimal numbers, to name but four).

Why do philosophers think that numbers (numbers like the counting numbers) are not properties (of things like pairs of shoes)? Well, some philosophers take the mathematicians at their word and are thinking about sets. And some regard logically possible collections as problematic, usually for such reasons as Russell's paradox (see Freedom for why they are wrong), even though the idea that numbers are not properties predates Russell slightly (see Brown is the Brightest Colour). And some philosophers think that numbers do not really exist, that numbers are more like the stories that we make up than things like pairs of shoes and the atoms of which they are made. Would they say the same about colors and shapes, though? Those philosophers are usually writing about mathematical models of such things, rather than the things themselves (in the interests of precision, they say), so it can be quite hard to tell sometimes.

Freedom

is the freedom to say that 2 + 2

is no more nor less than the number of ones in (1 + 1) + (1 + 1)

How could that be what freedom is?

Who would deny that that is what 2 + 2 is?

And how would their denying it get in the way of our being free?

Surprisingly, it is the experts on numbers who deny it.

A hundred years ago, academic mathematicians redefined the terminology of arithmetic in order to lose an arithmetical puzzle in that translation, because although the puzzling arithmetic does make sense if there is a God like the Trinity who geometrizes continually (see below for the details), academic mathematicians could find no other way of making sense of that arithmetic, and academia was becoming increasingly atheistic in the twentieth century, especially in subjects that used a lot of mathematics (the sciences had started to get atheistic in the second half of the nineteenth century because of an agnostic biologist, Charles Darwin).

Now, academia should not have become so atheistic, because insofar as that arithmetic only makes sense if there is a God, it proves that there is a God (that arithmetic had been discovered in the second half of the nineteenth century by a Lutheran mathematician, Georg Cantor, who thought that God had revealed it to him; perhaps it was God's answer to Darwin).

However, that proof was hidden by those mathematical redefinitions, and by related redefinitions of words like proof, logic and truth, because of a related logical puzzle discovered by an atheist aristocrat, Bertrand Russell, at the start of the twentieth century. Still, maybe this proof will not remain hidden for another hundred years (God does seem to get better results on longer timescales).

For the details of the proof, click on this link: Freedom

That link opens an eight-thousand-word Google Document called "Freedom" in a new window (and the truth will set you free).

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.

On the Hiddenness of God

I recently emailed my booklet, The Hiddenness of God, to hundreds of academic mathematicians, to see whether or not mathematicians would be interested in the proof buried beneath the foundations of their subject, and I have had some replies already. The following conversation has been edited, but it is fairly typical, in case you were wondering (as I was) what mathematicians would think of my proof.

Mathematician: Russell's paradox (and Epimenides' before him) demonstrates simply that the concept of "truth value" that many logicians had assumed to be well-defined on all statements, and which works well most of the time, must in fact have a few limitations. When we talk about truth values too loosely, plain English hides the fact that we're discussing a function from the class of propositions to the set {T,F} that may not in fact be wholly defined. It's no more mysterious than the discovery that division by 0 can't be defined except by giving up several arithmetic properties that are otherwise unproblematic. Russell simply shows a similar restriction for truth values of self-referential statements. This is well-understood.

And Cantor's theorem isn't even a paradox: it just shows that if we define an ordering by "size" on infinite sets, then the rationals and the reals are in different size classes - and why shouldn't they be? Our ability to "comprehend" either is ill-defined (this is where plain English lets us down): we do not know everything even about large finite numbers (which digit appears most often in 9^(9^(9^(9^(9^9))))?) and we know a very great deal about the real numbers, more numerous than the natural numbers though they are.

While Russell's paradox did do that, the heap paradox and the liar paradox had done it thousands of years earlier. And while Cantor's theorem is indeed not a paradox, it exists within axiomatic set theory. Cantor's paradox arises for the numbers that Cantor was working with, which were essentially the same as the numbers that we learn about at school. There is an obvious and unambiguous meaning to the word "two": two is the number of things in any collection that has as many things in it as the sum 1 + 1 has units in it.

Mathematician: I think the heap paradox is most easily interpreted as showing the axiom that one grain less than a heap is still a heap to be inconsistent. Heapiness is problematic in other ways as well. If we base our definition on general opinion, we more or less have to test it by asking an observer "is this a heap?" and the answer may depend on the observer. If we don't appeal to opinion, there's no reason not to define a heap as a thousand grains or more of sand, or sand grains piled at least five deep.

And while what you said is true for "two" there are more real numbers (in the usual sense) than there are definitions in finite strings of characters... and this happens precisely at the spot we're interested in.

Plain English is good enough for the definition of "two," though; and similarly, for an arbitrary counting number (even though most counting numbers are too big for us to imagine anything about them other than that they are counting numbers). And Cantor's paradox arises for arbitrary subcollections of subcollections of [...] subcollections of counting numbers. The real numbers are complicated (and Richard's paradox is interesting) but irrelevant to Cantor's paradox. As for the answer to "is this a heap?" I think that it can depend on the observer, and that that is one of the reasons why some piles of sand are only heaps as much as they are not heaps. Insofar as they are heaps, removing a single grain of sand would make a negligible difference to that. And for such a pile, "that pile is a heap" would be true only as much as it was not true. And similarly, the liar paradox shows that there are self-referential statements that are true only as much as they are not true. So, Russell's paradox is more like the liar paradox (and the heap paradox) than Cantor's paradox.

Mathematician: I think the heap paradox is somewhat different in that it can be dealt with by saying that "well, it seems that we need to sharpen our definition of a heap. A heap will be any collection of sand numbering more than ten grains, stable, and at least a quarter as tall as it is high." That's roughly what Cantor did with infinities... a fairly small patch on existing math. The first was a paradox, and not the second, only because people had more preconceptions about heaps. Cantor's result is more a proof by contradiction, eliminating a wrong turning in an exploration of new territory. If Eubulides of Miletus had been researching novel ways to store sand (insight - we don't need a bucket!) he might have used the sorites paradox similarly. The liar paradox can't really be explained away by inventing a better liar: it needs the concept of truth that underlies all philosophy to be redefined. Similarly, Russell's paradox involved a complete revamping of basic set theory.

I don't think that the heap paradox can be dealt with by saying that we need to sharpen the definition of "heap" because similar paradoxes occur with almost all of our words (as Russell observed) and because our words simply have the meanings that they have: if we redefine what "truth" means, then we are no longer talking about the truth of our words. I suppose that Cantor's paradox is the proof by contradiction that you think it is if there is no God, but is the proof by contradiction that I think it is (a proof that there is a God) if we should not redefine what "truth" means in order to avoid an inconvenient proof.

Mathematician: It's true that if we take "Cantor's paradox" as a standalone result, rather than as the obvious (in retrospect) conclusion of his construction of sets of demonstrably different cardinality, it looks more like Russell's paradox. That's not the angle I'm used to seeing it from, but I think I see your point. Nonetheless, in Cantor's case we don't have to redefine "truth", we merely have to redefine "set" so that some things we would have naively called sets are "classes" with a smaller set of permitted construction rules. As for the relevance to God: I am not a believer, but quite happy to argue hypotheticals. I agree with Aquinas that any god that exists must be bound by the laws of logic. These are the same laws of logic that bind us: and I see no reason why using a definition of "set" that Cantor showed to be inconsistent could be a divine attribute, let alone why we should want it to be so. Aquinas says in effect that, regarding logic, what's good enough for Cantor (if Cantor is right) is good enough for God. You don't get around Cantor by supposing "theological unions" of sets that somehow differ from those of set theory (or, if you do, you must explain their properties fully and equiconsistently with ZFC or some other well-defined system).

I agree that we should be bound by the laws of logic, and I take that to mean that we cannot just make those laws up. And I am certainly not trying to get around Cantor by supposing theological unions (whatever they are). I am questioning his assumption that mathematical collections must exist timelessly. Cantor chose to believe in the existence of collections that were inconsistent, rather than give up that assumption! Mathematicians can of course use any definition of “set” and “class” that they like, but there is still the paradoxical behaviour of mathematical collections (in the logical sense) to explain. Cantor’s paradox showed that his conception of set was inconsistent, but his conception included the assumption that mathematical collections exist (insofar as such things can be said to exist) timelessly. Incidentally, although Russell found his paradox while he was thinking about Cantor’s paradox, I don’t think that Cantor’s paradox is like Russell’s paradox.

Mathematician: My view is that the word "exist" is not used in mathematics in the sense that Mount Everest is and Alma Cogan isn’t (as the guy on the Monty Python record put it). It's an axiomatically-defined predicate in mathematical theories and metatheories (parallel lines exist in the Euclidean plane, they do not exist in the projective plane). From this viewpoint, I don't see time/timelessness as having anything to do with mathematical existence (I suppose one could take a time-dependent Platonist view where pi really was three in Old Testament times, but that is not how I see it).

For most mathematicians nowadays, mathematical existence is indeed existence within an axiomatic structure, and for such structures it is consistency that matters. And within set theory, there is only Cantor’s theorem. But for numbers like the counting numbers and the number of all the counting numbers, and so on, it is logic that matters: such numbers are essentially properties of logically possible collections (you and I are two people, and we would have been two possible people had we never existed, and the properties of that “two” are logically prior to any axiomatic model of them). And if it is logically possible for there to be a God, then there are all the numbers (in that sense) that give rise to Cantor’s paradox. That is how I have been able to show that if it is logically possible for there to be a God then there is a God, because it is only if there is a God that such numbers could possibly be getting more numerous (and it is only in the last hundred years that mathematicians would have denied that such numbers were part of mathematics).

Mathematician: The statement that "numbers are getting more numerous" is, if not downright false, highly ambiguous. Our mathematical knowledge may encompass more numbers, but a given axiom system implies the same numbers yesterday, today, and forever, even if nobody alive at some time understands that. Furthermore I hold, with (for instance) Aquinas, that it is a logical necessity that no deity could change; so, claiming that the creation of new numbers within a fixed axiom system implies the existence of a god is true only ex falsi quodlibet. Apart from that major objection, if your argument did prove the existence of some entity X, I think (again, hypothetically) that it would fall far short of showing that this X was what was generally called "a god," let alone a specific faith's God.

The numbers in “numbers are getting more numerous” do not exist within any axiom system, but as a consequence of there being numbers of things in the world (such as us two). Axiomatic models of them are timeless, but they themselves are properties of logically possible collections of things, so it is a matter of objective fact whether they are timeless or not. And while we naturally assume that they (and logical possibilities generally) are timeless, it is conceivable that they (and some other logical possibilities) are not timeless if there is a God who is not timeless. As for your belief that if there was a God then that God would have to be above and beyond time and change, I suppose that you have a good reason for believing that, but as I do not know what that reason is, I cannot say why it is not a valid reason (and similarly for your reason for believing that X could not be called a God, unless it is the same reason). I have thought a lot about the reasons that are in the literature, and none of them are valid when it comes to the God that Cantor’s paradox shows exists (which did not surprise me because a lot of the religious believers who take God to be above and beyond time and change would also say that He is above and beyond our logical abilities).

Mathematician: You would seem to be saying that there's an argument showing, on the basis of some axiom system, that some number (call it Stigma) exists... and that at some time in the past the same argument was not valid, or was valid but did not show that Stigma existed. A fun science-fiction idea, but in reality if we pick at it, expanding the argument out to a long but finite list of axiomatic steps and going through it a step at a time, there's a step that somehow didn't work then and does now. But that step is supposedly an instance of an axiom, so the axiom set has changed. Gods whose powers vary in time (depending on who's stolen whose hammer today) are more at home in comic books than in philosophical arguments; when I said "god" I meant the sort of god that modern philosophy usually considers, whose view of the universe is in some sense ultimate and synonymous with reality. If the power of such a god were greater today than yesterday, it would have to have been less than it might have been yesterday. Which, as Spinoza would have said, is absurd.

I too meant the God whose view of the universe is the universe. And I agree that the power of such a God cannot increase, or decrease. However, the knowledge of such a creator would increase as a matter of logical necessity whenever any particular thing was created (as I show in the first “chapter” of my first email). As for your interpretation of what I was saying in terms of an axiom system, the existence of the most basic numbers (1, 2, 3 etc.) does not have to be existence within any axiomatic system, even if there is a God. The existence of such numbers could be the logical possibility of there being collections of that many things (which is why my argument is a logical argument based on Cantor’s original paradox, which he discovered before mathematicians and philosophers axiomatized numbers and collections) [...]

A mathematical poem

                  12  =  3 × 4

                  56  =  7 × 8

         0 + 12  =  3 × 4

         5 + 67  =  8 × 9

📖The Hiddenness of God

As the twentieth century began, the atheist philosopher and mathematician Bertrand Russell was thinking about some puzzling arithmetic, which he correctly took to be a logical puzzle. And as he was thinking about that puzzle, he found another. Now, his answer to both puzzles was a scientific theory of logic—a mathematical model of logic—and since then, logicians have done a lot of mathematical modelling. So, logic looks very scientific nowadays. But if scientists, by thinking logically, reached an outlandish conclusion, would they think that something was wrong with logic? Or is science more logical than that?

Does that puzzling arithmetic actually amount to a scientific proof of something scientifically revolutionary?

That possibility is outlined in chapter 1 of The Hiddenness of God. The puzzle that Russell found is of a kind with two ancient puzzles—the heap paradox and the liar paradox—so chapter 1 begins with them, and chapter 2 shows why they give us no good reason to doubt the reliability of logical thinking. We should therefore think very logically about that puzzling arithmetic, which chapter 3 describes in relatively plain English, to bring out the underlying logic. Chapter 4 shows how that logical puzzle makes sense if—and in all likelihood, only if—there is a creator of all things who is above and beyond the concept of a thing but not completely above and beyond time and change.

🥳The number 23

1 + 23 = 4 × (5 – 6 + 7)
1 = 23 – 4 – 5 – 6 – 7

Furthermore, 23 is two less than 25, which is a square number;
and in two years time it will be 2025, which is another square number:

               2025 = (20 + 25) × (20 + 25)

💥Cantoring away from being Russelled

Twenty-five years ago, as I was getting my masters in mathematics, I was surprised to find an unsolved puzzle about infinity at the heart of modern mathematics. Some of my first thoughts were published in philosophy journals, so I went on to do a masters in philosophy. I got it with distinction, and by thinking laterally as well as logically I found the solution and decided to write it up as a book for a general reader with no background in philosophy, logic or mathematics. Five years later, it is down to 25,000 words.

In the book (which was 28,000 words in July, and which I will re-post when I get it below 10,000 words), various logical puzzles are described and solved because the only perfectly logical solution to one of those puzzles—the puzzle about infinity—is only a logical possibility if there is a logical kind of God. In short, my book amounts to a perfectly logical proof that there is such a God.

      A hundred years ago, the mathematical puzzle was proving to be so puzzling that mathematicians translated the whole of mathematics into a new "language" (akin to a programming language) in order to lose it in that translation. And that sea-change to academic mathematics trickled down to school mathematics in the form of the new math. Which you may have heard of, because it was quite controversial fifty years ago. The mathematicians’ responses were logical enough, but this puzzle is essentially a logical puzzle. And philosophers like Bertrand Russell responded to it by modernizing logic.

      For a hundred years, scientific philosophers have been treating logical thinking as though it was a kind of computing, as something that might be done better on a computer. By explaining these logical puzzles properly, my book will revitalize philosophy. My book may also help to defuse America’s "culture war" by making logic more interesting to religious people while simultaneously showing that atheism is not really very scientific. Indeed, it is not very progressive: how could people growing up in a world with profound problems possibly acquire enough wisdom to change their world for the better? On a more mundane note, scientific research will progress in directions that are more realistic as a result of my book, so my book could herald the next scientific revolution. And of course, a lot of people will simply find it helpful to know that there is a reasonable sort of God.

📚Current version of my booklet

This links to a 28,000-word Google document

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

Sets

Sets

(two thousand words) will be
section 2 of chapter 2 of my booklet:
The Way of Things

Russell's Paradoxes

Russell's Paradoxes

(two and a half thousand words) will be
section 2 of chapter 5 of my booklet:
The Way of Things

Impossibly Many Things

Impossibly Many Things

(one thousand words) will be
section 2 of chapter 3 of my booklet:
The Way of Things

Many Things

Many Things

(two thousand words) will be
section 1 of chapter 3 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

Infinities

Infinities
(two thousand words) will be
section 1 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.

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