What's the matter?

What is physical reality? Most obviously, it is a lot of problems, the most important of which we have in common. Physical reality is problematic for us because we are physical, and because we have feelings, sometimes problematic ones. So, it is pretty obvious that physical things can have feelings. Nevertheless, it might seem strange that physical things could have feelings: physical things appear to be composed of physical particles, and it is hard to see how any number of physical particles could possibly add up to something with feelings. Does the difficulty of seeing how that is possible mean that we are not physical? Of course not. It would make as much sense to conclude that we do not have feelings. What about the theory that ordinary physical objects are composed of atoms, could we doubt that theory in light of that difficulty? Well, we could, but it would make more sense to doubt that we are just a lot of atoms. It would make more sense for us to doubt materialism.

Does the difficulty of seeing how some large number of atoms could possibly add up to something with feelings amount to a reason for doubting the logical possibility of materialism? That might seem unlikely when you consider how little we know about what matter is. And materialism does not seem to be contradictory. So, materialism does seem to be a logical possibility.

Contradictions describe logical impossibilities: when we say that something is not a certain way, we mean that it is not the case that it is that way, so it cannot be the case that something is a certain way and that it is not that way, and so contradictions describe impossibilities. And although contradictions can also describe possibilities (such as when we say "it is and it isn't," meaning that in a sense it is, but in another sense it isn't), they describe them much less accurately than they describe impossibilities. So in short, a contradictory hypothesis is not a logical possibility; and for what other reason could we rule something out on logical grounds? It is far from obvious that there could be another reason, so it makes a lot of sense to say that something is a logical possibility when it is not contradictory.

Now, it is precisely because we do not know much about what matter is that it is, for all we know, possible that this universe was deliberately created out of nothing but its logical possibility. Of course, such a creator would certainly seem unrealistic to atheists. So note that the possibility of such a creator is not the same as the possibility that the picture of a God that atheists pick up from the religions around them is a realistic picture. Furthermore, the concept of such a creator does not seem to be contradictory.

Rational atheists should therefore have as little problem with the logical possibility of theism as they do with the logical possibility of materialism.

However, to remain atheists they must, in light of my proof that atheism is illogical, either show that the logic of my proof is flawed, or deny one of its premises. And insofar as logic is not an algebraic model of logical reasoning, there is nothing wrong with the logic of my proof. And they could hardly deny the existence of things like these words and remain rational. So, they would have to deny the other premise of my proof: the logical possibility of theism.

Definitions

Leibniz famously invented the infinitesimal calculus (following Newton's invention of the method of fluxions) while he was working a logical calculus (following Descartes' algebraic innovations). Philosophical disagreements were relatively fierce in the seventeenth century, and Leibniz thought that a logical calculus would bring peace to philosophy by enabling disputing philosophers to sit down and calculate the truth. And in particular, Leibniz disputed Locke's view of arithmetic. Locke thought that we acquired the fundamental arithmetical concepts, unity (1) and addition (+ 1), by observing a world of things. Leibniz thought that if we did not already have the concept of an individual thing, then we would be unable to see things as things. In short, Leibniz regarded 1 and + 1 as logical concepts. He thought that arithmetic is nothing but logic and definitions. For example:

Starting with the logical axiom that things are identical to themselves, which he expressed in symbols as A = A, Leibniz put 2 + 2 in place of A, and in three logical steps, each step using one of three definitions (2 = 1 + 1, 3 = 2 + 1, and 4 = 3 + 1) with the logical rule that replacing equal things preserves equality, he had 2 + 2 = 4:

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

With the logical steps of that proof of 2 + 2 = 4 so clearly laid out, it is easy to see why arithmetic might be nothing but logic and definitions. And why 1 and + 1 would not need to be defined in order for such a proof to be rigorously logical. 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.


Euler also thought that 1 + 1 = 2 by definition of 2. He thought that when we have one thing and another thing, we have two things by definition of two (or rather, by definition of zwey, an eighteenth-century German word for two). And surely Euler would have known what the counting numbers are: he proved over a hundred theorems about them in the eighteenth century. Euler knew that the counting numbers (1, 2, 3, and so forth) are composed of ones (1, 1 + 1, 1 + 1 + 1, and so forth). Now, Euler conjectured that if we multiplied a counting number bigger than one by itself five times, then in order to express that fifth power of that number as the sum of other counting numbers raised to the fifth power, we would need at least five of those other numbers. However, a twentieth-century mathematician discovered a counter-example, 144, which only needs four other numbers. Nevertheless, in order for 144 to have been a counter-example to Euler's conjecture, that number had to be a number of the kind that Euler was writing about, not a number of some other kind. It could not, for example, have been composed of 144 fictional dots collectively called "144" by mathematicians who had, in the twentieth century, described infinitely many fictional dots in order to build arithmetic out of them, could it? Could a rigorously logical proof of 2 + 2 = 4 be based on fictional dots?

Surely the reason why 2 + 2 = 4 is that, firstly, 2 + 2 = (1 + 1) + (1 + 1) because one thing and another thing are two things by definition of two, and secondly, (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 because how many things we have does not depend upon which ones we look at first, and thirdly, 1 + 1 + 1 + 1 = 4 by definition of four.

The English name for how many things there are when there is one thing (of a certain kind) and another thing (of that kind) and another thing (of that kind) and another thing (of that kind) and no other things (of that kind) is four, and "1 + 1 + 1 + 1 = 4" is just that dictionary definition written in quotidian symbols. Dictionary definitions describe the meanings of such basic words, they are not, of course, how such words get their meanings. Words like four and two get their quotidian meanings in much the same way as words like pound and and get theirs (a way with which we are all familiar). And while symbols can be given algebraic meanings with algebraic definitions and axioms, quotidian symbols like 4 and 2 get their quotidian meanings in much the same way as £ and & gets theirs.


Frege completed a logical calculus in the nineteenth century; and in order to show that arithmetic is nothing but logic and definitions, he tried to use his logical calculus to build arithmetic out of logical axioms and definitions. He added to Leibniz's definitions by defining 1 to be 0 + 1, where 0 and + 1 were defined in terms of concepts that he took to be purely logical. And because associativity had been given an algebraic definition earlier in the century, he noticed that Leibniz, in his proof of 2 + 2 = 4, had not shown that 2 + (1 + 1) = (2 + 1) + 1. Now, it is obvious that 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 proof in order for it to be 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.

However, Frege thought that logical proofs should be more like calculations, in order for them to be rigorously logical. Does the logic of a rigorously logical proof have to be axiomatic though? If it did, then we would have had to have had logical axioms before we had any rigorously logical proofs. In reality, logicians hypothesize logical axioms when they have lots of similar proofs, proofs that are clearly logical, and then they test those axioms as logically as they can. And because logical axioms are therefore bound to be less certain than a lot of logical proofs, and because logic is all about certainty, rigorously logical proofs do not have to be deductions from axioms and definitions. Furthermore, Frege's attempt to show that arithmetic is nothing but logic and definitions failed because of a logical paradox discovered by Russell (following Cantor's discovery of a mathematical paradox at the end of the nineteenth century).


Russell designed his own logical calculus to build arithmetic out of, reaching 1 + 1 = 2 after hundreds of pages of algebra, where 1 and 2 got their meanings from some rather algebraic definitions and an axiom that said that there are infinitely many things. Russell wanted to show that arithmetic is nothing but logic and definitions, but how could he possibly have shown that arithmetic is nothing but logic and definitions by reaching equations of that kind after so much algebra? The question is not whether his axiom was a logical axiom, but whether his 1 + 1 = 2 was as arithmetical as it looked.

The English name for how many things there are when there is one thing (of a certain kind) and another thing (of that kind) and no other things (of that kind) is two, and that dictionary definition of "two" is what people learning the meaning of "arithmetic" take "1 + 1 = 2" to mean.

That equation is a logical truth because those things can be any logically possible things. And to show that there are infinitely many logically possible things is simple enough. And if I had a crimson object and I concluded that I had a red object, that would be a logical deduction, and if I also had a scarlet object and I concluded that I had two red objects, that would be similarly logical. But what if I had no other red objects, and I defined 2 to be the number of red objects that I have, would I not have given that quotidian symbol another meaning, in addition to its arithmetical meaning? Russell's equation was an algebraic truth; for it to be an arithmetical truth, "arithmetic" would have to be redefined, and for it to be a logical truth, "logic" would have to be redefined. And what would be the point of redefining "logic"? It would hardly make it easier for the experts to be logical. For a bit of perspective, consider how male (♂) and female (♀) are biological categories. Even if it became a standard part of the biology curriculum that those two categories are (for socio-political reasons) a matter of personal choice, they would not really be a matter of personal choice, because they are (as a matter of fact) biological categories grounded in a genetic difference. The experts can of course discover new facts about their subjects, but how Orwellian would society have become if the experts changed the meanings of such basic symbols in such socio-political ways?


Hilbert turned Frege's logical calculus into first-order predicate logic (a rather algebraic model of simple logical reasoning), which became the standard logic in mathematics and philosophy in the twentieth century. And Hilbert encouraged mathematicians to define 0 and + 1 within an axiomatic set theory, in order to avoid paradoxes like Russell's. With those definitions as well as Frege's definition of 1 and Leibniz's definitions of 2, 3 and 4, mathematicians can prove in a rigorously logical way that 2 + 2 = 4 where "2" and "4" are names of axiomatic sets. Do "2" and "4" name axiomatic sets? Well, that is the standard meaning of 2 and 4 in academic arithmetic. But axiomatic sets are algebraic, in the sense that they have only the properties described, in symbols, by the axioms of their set theory. Axiomatic sets are essentially algebraic models of collections whose sizes are numbers composed of ones. And because such models are bound to be a little inaccurate in some rather obscure ways, because of paradoxes like Russell's, the steps that would be needed to take us from that rather algebraic proof of 2 + 2 = 4, where "2" and "4" are names of axiomatic sets, to a logical proof of 2 + 2 = 4 where those symbols have their quotidian meanings could not all be as rigorously logical as algebra. So, what would a rigorously logical proof of 2 + 2 = 4 look like?

Well, 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 insofar as the meaning of "2 + 2 = 4" is that two things and another two things make four things, which is the quotidian meaning of that equation. Would that proof be more rigorously logical if it included an explanation of how counting works? Maybe, and if it would then such an explanation could always be added. But would we get a more rigorously logical proof of 2 + 2 = 4 if we used first-order predicate logic and set theory to model that proof and then replaced it with that model? That model would be a rigorously logical proof: it would be a rather algebraic proof of an algebraic model of 2 + 2 = 4, and mathematics is rigorously logical. But would it be a rigorously logical proof of 2 + 2 = 4?

The Way of Ways

Thousands of years ago in ancient Greece, there were philosophers who claimed that it was impossible for there to be more than one thing. And while those philosophers certainly seem to have been crazy, the following argument that it is logically impossible for there to be more than one thing certainly seems to be a very logical argument.

The argument could begin with any three things, so let us begin with these three names:

Descartes             Newton             Cantor

Because there are those three names, there are, of course, these two:

Descartes             Newton

And there are two other ways of having two of those names:

Descartes             Newton             Cantor

Descartes             Newton             Cantor

And because there are those three ways of having two of those names, there are, similarly, three ways of having two of those three ways of having two of those names:

There is having the blue and green ways.

There is having the green and brown ways.

And there is having the blue and brown ways.

And because there are those three ways of having two of the ways of having two of those names, there are three ways of having two of those ways. And because there are, there are three ways of having two of them. And so on:

Whenever there are three things, there are three ways of having two of those things.

There is no end to this, no final three ways of this kind (because for any three things, there are three ways of having two of them), so there are infinitely many ways of this kind (the word infinite comes from the ancient Greek for unending). Of course, infinitely many things is an awful lot of things to have, given only three names; are there really all of these ways? Well, those names clearly exist, and because they do, there are clearly three ways of having two of them. And because there are those three ways, there are, similarly, three ways of having two of them. And so on endlessly:

Each and every way of this kind exists because of the existence of the three things that it is a way of having two of.

Could these ways of having some of some things (ways that exist because of the existence of those things, starting with those three names and continuing in the endless sequence described above) be coming into existence endlessly? Could there be more and more of them forever, without there ever being all of them? Well, although it takes time for us to read descriptions of these ways, there does not seem to be anything like a temporal process involved in there being these ways of having some of some things. Whenever there are three things, there will of course be each and every one of them, so the existence of each and every way of having just one of those things is immediate and automatic, and there being each and every way of having two of those three names certainly seemed, at the beginning of this post, to be similarly immediate and automatic; and if, whenever there are three things, the existence of each and every way of having two of those things is immediate and automatic, then there will be all of those ways of having some of some things. So, presumably there are all of them, and so there will presumably be each and every way of having some (more than one, less than all) of them:

Each way of having some of them is a way of having some things all of which we are assuming there are.

And presumably there will, for the same reason, be even more ways of having some of those ways. And even more ways of having some of them. And so on endlessly. And given that there are all of those ways, there will presumably be even more ways of having some of them, and even more ways of having some of those ways, and so on endlessly. And so forth, endlessly. And given that there are all of those ways of having some of some things (starting with the blue, green and brown ways of having some of those three names), there will presumably be even more ways of having some of them. And even more ways of having some of those ways. And so on endlessly. And given that there are all of those ways, there will presumably be even more ways of having some of them, and so on endlessly. And so forth, endlessly. And so on and so forth:

For each and every one of these ways of having some of some things, that way exists because of the existence of all of the things that it is a way of having some of.

And of course, if each and every one of them exists, then they all exist. However, if there were all of them, then there would also, for the same reason, be all of the ways of having some of those ways: given that there are all of those ways, there is each and every one of those ways, of course, and each and every pair of those ways, similarly, and each and every way of having three of them, and so on endlessly, and also each and every way of having infinitely many of them, similarly. Those ways of having some of the ways of that kind would exist in addition to the ways of that kind, so presumably they would not be ways of that kind (and a precise mathematical reason why they could not all be ways of that kind is described in the Google document linked to below).

But those ways of having some of the ways of that kind would all exist because of the existence of all of the things that they would be ways of having some of, and those things would all be ways of that kind, so those ways of having some of the ways of that kind would all have fallen under the scope of that "and so on and so forth," if they existed, and would therefore be ways of that kind, if they existed (and while the phrase "and so on and so forth" is rather vague, the terminology in that Google document is much more precise).

And they would exist if there were all of the ways of that kind; so, as well as the rather logical reason why the ways of that kind would all exist, there is an equally logical reason why they would not all exist: if they did, there would be things that would be what they could not be.

A solution to that logical puzzle is described in my booklet Freedom (that link opens a 26-page Google document in a new window). In that booklet I describe how such logical versions of Cantor's mathematical paradox show that there is a God because only a God would be able to create such things as those ways in a never-ending but not necessarily temporal process of creation.

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) [...]

📖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.

💥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

Authorial Authority

Authorial Authority

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

Truth

Truth

(three thousand words) will be
section 3 of chapter 5 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

Heaps

Heaps

(two and a half thousand words) will be
section 1 of chapter 5 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

Low Expectations

Low Expectations
(five and a half thousand words) will be
section 2 of chapter 1 of my booklet:
The Way of Things

The Lie of the Land

The Lie of the Land

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