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 all of their terminology in order to lose some puzzling arithmetic in that translation. Why was that arithmetic too puzzling for them?

That arithmetic makes sense if we assume that there is a God who is able to change in some rather transcendental ways (a God who is more like the Trinity than a thing, and who may well be above and beyond time and changes of ordinary kinds), but it is very hard to see how it could make sense in any other way. And scientific academia was increasingly atheistic in the twentieth century.

Insofar as that arithmetic only makes sense if there is such a God, it proves that there is such a God. And the truth will set you free. But that proof was hidden by that translation. And by related redefinitions of words like truth and proof, as described here:

Freedom

That link opens a Google Doc (8,733 words, 28 pages) in a new window.

Vanish?

An object accelerates in a straight line by repeatedly doubling its speed, each doubling taking half the time of the previous doubling. This object would soon reach the speed of light; but if there was no light speed limit (and no risk of collisions because the space that it was in was otherwise empty), would it go to spatial infinity?

Well, with only one object in that space, there would be no motion relative to any other object. So, a more interesting question arises if we put another object into this otherwise empty space. Let us call these objects X and Y:

X and Y begin together, and then X travels one mile from Y in one hour (an average speed of one mile per hour), another mile in half an hour (at an average speed of two miles per hour), another mile in a quarter of an hour (at an average of four miles per hour), another mile in an eighth of an hour and so on, until after one plus a half plus a quarter plus an eighth and so on endlessly (which equals two) hours, X has travelled further from Y than any finite number of miles.

For both X and Y, the other object seems to be going to spatial infinity. Would X disappear at spatial infinity? Maybe; but why should Y vanish because of the accelerations of X at such huge distances from X? Would X alone vanish because of its accelerations? And if so, would it also vanish in the original scenario (without Y) because of those accelerations?

Maybe; but suppose that Y had accelerated in exactly the same way as X, so that they would have remained together during those two hours. Neither would have seemed to the other to be going to spatial infinity, so there would have been no reason for either to vanish. Or would they both vanish because they were both accelerating in ways that would have caused them both to seem to the other to be going to spatial infinity had they been moving in opposite directions? Would they remain together for two hours and then suddenly vanish (not suddenly explode) because of those unbounded accelerations? Or does it make more sense to suppose that such accelerations would have had to have been impossible, even in such a simple space? In that case, X would not vanish at spatial infinity in the original scenario; rather, there would have had to have been some natural limit to its accelerations. Does the idea of some such limit being part of the concept of such a simple space seem unlikely? There is another alternative:

X does not vanish at spatial infinity because it gets to spatial infinity. Such places could exist if, for example, unit volumes of space contained 1/0 points (as outlined in my 2005 paper) and the counting numbers were endlessly created by God.

Perhaps it is more likely that something like the light speed limit is a necessity even for such a simple space. If we knew more about the nature of that necessity, would that help to explain the light speed limit that actual spacetime certainly seems to have?

On the Hiddenness of God

I emailed The Hiddenness of God to thousands of mathematicians, to see if any were 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 what mathematicians would make 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.

Me: 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. And there is an obvious and unambiguous meaning to the word "two," for example: 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 (where the sense of "as many things" is equinumerosity).

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.

Me: 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 wring 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.

Me: 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).

Me: 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).

Me: 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.

Me: 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.

Me: 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). Etc.

❄️Happy New Year's Eve

🎄For the birds...

 

Saul Steinberg

📖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. 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.

That is the preface for my 8,310-word The Hiddenness of God (last updated in 2024)

🎵Wishful thinking at dawn

✨How I wonder what you are

🥧February made me shiver

🖤Lisa Marie Presley

🎁 Conceit precedes defeat

😳Plain Speaking

The Plain is a two-dimensional place, inhabited by round people and square people.

There is not much to do in the Plain, so its inhabitants argue over whether the pentagon is a circular disc with five thorns on it or a square with one corner squashed flat and the four sides pushed out slightly by that squashing.

In the three-dimensional space around the Plain, a cylindrical person called Cyril has been watching them arguing, and he decides to give them something else to think about.

As he passes through the Plain, Cyril can look like a round person or a square person, because his height is the same length as the diameter of his circular cross-section, so he pauses at various places in the Plain—sometimes looking like a round person, sometimes a square one—and says very loudly, “I am Cyril.”

The round people take the Cyrils to be a race of round and square people who can flip over to this side of the Plain from the other side. But the square people assume that “Cyril” names a single person. The square people deduce that Cyril is a round square person who is usually somewhere impossible, when he is not visiting them. And they suppose that he is visiting them now in order to show them that they were made in his image.

Should I let them know that it was me who made them all up? A square person called Martin appears and says, “I made the Plain and everyone in it.” But the other square people take him to be Cyril, and they think that he is telling them that they are right. They set out to correct the round people.

I blame myself.

💥Why you should read my book

Twenty-five years ago, as I was getting my masters in mathematics and wondering what to research further, 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 eventually found the solution that academics would not find in a hundred years. I decided to write my discovery up for a general reader with no background in philosophy, logic or mathematics, and five years later I had it down to 25,000 words.

In my book I describe and solve logical puzzles, because the only perfectly logical solution to one of the puzzles—that 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: 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. And because scientific research will progress in directions that are more realistic as a result of my book, my book could even herald the next scientific revolution. And of course, a lot of people will find it helpful to know that there is a reasonable sort of God.

🎃Have a Harrowing Halloween

💗feel good Friday

♫May's Plays

Least said, soonest men dead

First stab at a Book!

Here is a rather messy google doc: The Way of Things (it is over a hundred thousand words; I am going to tidy it up and post it section by section as I rewrite it :-)

June's Tunes 4

June's Tunes 3