Science is built of facts

        much as worldviews
        are built with truths:

            not all of them,
            nor them alone.

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

            my brain collates all that and makes a collage
            showing the world as it may well be
            when I make my next move

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

            Seeing only the future,
            tomorrow is a mystery.

      Tomorrow is another day,
      and today is mysterious: what is lying
      under the skin shed in glimpses of such creatures?

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


the unworldly.

The Way of Ways

Here are three names:

Descartes       Newton       Cantor

There being those three names, there are, of course, these two:

Descartes       Newton

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

                       Newton       Cantor

Descartes                          Cantor

So there are three ways of having some (more than one, less than all) of those three names.

Note that even if no one had ever thought about having some of those three names, there would still have been those three names; and each of those pairs of names would still have been two of them, so there would still have been those three ways of having some of them.

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

Having the red and blue ways.

Having the blue and green ways.

Having the red and green ways.

And there are, similarly, three ways of having some of those three ways. And so on endlessly:

Whenever there are three things, there will be three ways of having some of those things.

There are, in short, infinitely many ways of that kind (the word infinite comes from the ancient Greek for unending).

Presumably that means that there are all of those infinitely many ways of having some of some things (ways that exist because of the existence of all of the things that they are ways of having some of). The alternative is that there never are all of those ways, just more and more of them forever. And how could there be more and more of them forever? There does not seem to be anything like a temporal process involved in there being those ways of having some of some things:

Whenever there are three things, there will of course be each and every one of those things, and is it not in the same immediate and automatic way that there is each and every pair of those things?

It seems safe to assume that there are all of those ways.

And given that there are all of them, there will presumably be all of the ways of having some 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 (starting with the red, blue and green 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 those three 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, and so on endlessly.

And so forth, endlessly.

And so on and so forth.

Now, there certainly seem to be all of those ways:

For each and every one of those ways, there certainly seem to be all of the things that that way is a way of having some of. And of course, if each and every one of those ways does exist, then they all exist.

However, if there were all of those ways then there would also, for the same reason, be various ways of having some of the ways of that kind:

Given that there are all of the ways of that kind, there is each and every one of those ways, and each and every pair of those ways, and each and every way of having three of them, and so forth.

Those ways of having some of the ways of that kind would exist because of the existence of all of the things that they would be ways of having some of, things that are all ways of that kind, which means that 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" and been ways of that kind, had they existed. And they would have existed if there had been all of the ways of that kind, so there are not all of those ways.

So, we have a good reason why there would be all of the ways of that kind, and an equally good reason why there would not be all of those ways. A theistic solution to that problem is described in my booklet Freedom (that link opens a 26-page Google document in a new window). I suppose that if some of the cleverest atheists tried and failed to solve that problem, then that might show that an atheistic solution was unlikely, and hence that atheism was unlikely to be most logical way of making sense of the world.

Although because those atheists would not then look so clever, that might only show that those atheists were not as clever as they had seemed to be. Still, what if more and more atheists tried and failed? Would that indicate, increasingly strongly, the irrationality of atheism? Well, it would, but the cleverest atheists would almost certainly know about cost-benefit analysis. Successfully solving this problem would only show how uninteresting it was, unless the success followed a lot of attempts that failed, and why would those attempts be made? What would be the point?

Nevertheless, this problem does have a theistic solution, and no atheistic solution, and that has got to mean something (see Freedom for what I think it means).

A Tale of Two Bridges

Yesterday, on BBC Radio 4's Facebook page:

“You don’t hit schools, you don’t hit energy sources, you don’t hit bridges: those are war crimes.”

UN humanitarian chief Tom Fletcher criticises actions in the Iran war and says leaders have chosen ‘game show gambling’ over humanity by hitting civilian infrastructure.

Many years ago, when the Allied Air Forces conducted a massive campaign to destroy rail and road bridges in France and Belgium, before and after D-Day, no one called it a war crime. Has anyone ever called it a war crime?

What about Iran's partner in crime, Russia? Well, a few years ago, when the Ukrainians hit the Crimea bridge, the Ukrainian government tweeted "Sick burn," and Russia's foreign ministry reacted by saying:

"The Kiev [Kyiv] regime's reaction towards destruction of civilian infrastructure is a testament to its terrorist nature."

It is hard to exaggerate the significance, and symbolism, of seeing the bridge on fire. Ukrainian authorities said it was a legitimate target, as they vow to retake the peninsula.

That was from BBC News, 8 October 2022.

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? Because those philosophers are usually writing about mathematical models of such things, rather than the things themselves (in the interests of precision, they say), it is hard to tell.

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

❄️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 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 Odyssey Theodicy

Why, if there is a God, do bad things happen to good people? Why, given a good creator of all things, did that creator not make all creatures naturally good, in a world where only good things would ever happen to them? Well, maybe that is what God did do:

Maybe God originally created a heavenly world in which there were a wide variety of very good people, and only good people, and where only good things happened.

Those people would have been much closer to their creator than we are here. Wiser and better informed about their heavenly home than we are, about this universe, might some of them have wanted to spend some of their limitless time in a less heavenly world?
......Perhaps they thought that their relationships with each other might improve if they spent some relatively small amount of time in a world like this universe. From their heavenly perspective, it might have seemed like going camping. And it would have been safer than going camping, because God could have guaranteed that they would all end up at least as well off as they had started. If, for example, they reincarnated around the universe or multiverse, then some of their later incarnations could have been therapeutic (the fact that we cannot recall past lives does not tell against that possibility because we cannot even recall being born). Maybe some of them could dimly recall having set out on a heroic expedition, and told each other stories about how it had all gone wrong (whence the name of this theodicy). The main thing is that they would all live happily ever after in their heavenly home.
......Still, even in their heavenly home there could conceivably have been limits to the relationships that those people could have had with their creator, because their creator would presumably have been above and beyond that heavenly creation much as a story’s author is above and beyond that story. And some of those good people, in their heavenly home, might have been very clever; they might have wondered if their creator knew about a lot of very horrible possibilities, and associated virtues. They might have conceived of that possibility in an abstract way, and gone on to conjecture that spending a relatively small amount of time in a world in which their creator was less evident might be a way of developing their relationships with their creator. It would have been daunting, but not actually unsafe.
......And they might even have been able to help their creator by coming to a world like this one, where their creator can seem so very remote. It is, for example, conceivable that not even creators of universes would be able to know for sure that there were no others of their kind (such a creator would be able to think of a lot of very strange possibilities). Nevertheless, very sensitive creatures trying to sense the presence of their creator in such a world as ours could conceivable be able to sense, much more remotely (and without being able to tell the difference themselves), the existence of other beings of that kind, if there were any others.
......Indeed, there could conceivably be many other reasons why good people in a heavenly world might have wanted to spend some of their limitless time in a less heavenly world, reasons that we may well be unable to imagine, here in this universe. The main thing is that they would all live happily ever after in their heavenly home, had they originated there.

🎵Wishful thinking at dawn

🐮Eight years ago

today, there was a male bullfinch in our garden

🦆Village Life

🐤Village Life

 

✨How I wonder what you are

🥧February made me shiver

🖤Lisa Marie Presley

🧡Eight years ago today