Science is built of facts

As science is built out of facts,
worldviews are made of 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 those photons and vibrations;

      my brain collates that data, and makes a collage
      to show me the world as it may well be
      when I shift myself to speak


or to catch a ball (to catch a ball
I put my hands where it will be).

Seeing only the future,
tomorrow is a mystery.

Tomorrow will be much like today
and today is a mystery: what lies behind
the skin shed in glimpses of creepy-crawlies and

      tomorrow will surprise no less than today
      for the world, no less than science,
      was made by the unworldly

The Shape of Time

The Shape of Time

(two thousand words) will be
section 1 of chapter 4 of my book:
The Way of Things

What if there is a proof?

Even if a proof that there is a God is, as I believe it is, hidden beneath the foundations of modern mathematics, the experts will, I am sure, not want to waste their valuable time checking whether there is or not. But perhaps it would be different elsewhere.

And perhaps it could have been different. If Cantor's paradox is essentially a proof that there is a God, then anyone who could have noticed Cantor's generalized diagonal argument could have discovered that proof. And the paradox of Achilles and the tortoise is a reason to introduce infinite ordinal numbers; and Cantor did find his paradox by introducing infinite ordinal numbers. Could a similar paradox have been discovered by the mathematicians of the ancient world? Might that have been part of the motive for Plato's Form of Forms? It is too late to know now (although the experts would say that they do know that that is almost certainly not the case).

I wonder whether the work of Russell in the period 1901 to 1906 had any connection with Einstein's 1905 paper. Russell's obscure mathematics flew in the face of logic; and not too dissimilarly, Einstein's obscure mathematics survived being contradicted by empirical observations. And the powers of the world would presumably have liked to keep the truth about high energy physics secret. Still, it is for that reason impossible to know either way now (although the experts would say that they do know that that is almost certainly not the case).

Still, if God did create this universe in such a way that there was this proof, then we might expect the universe to be full of people taking themselves to be living in God's family. And if Einstein was wrong about the light-speed-limit, then that might make a difference to us.

What Do Philosophers Do?

For myself, I just notice such facts as:

(A) The overwhelming majority of professional mathematicians are not going to be wrong about what numbers are.

(B) The overwhelming majority of mathematicians assume, in their professional work, that numbers are axiomatic sets.

(C) Numbers are not axiomatic sets.

The conjunction of (A), (B) and (C) would be a contradiction, were the mathematicians of (B) not just assuming that numbers are axiomatic sets for the purposes of proving theorems from axioms, as I suspect they do. But many analytic philosophers deny (C), because of that apparent contradiction. Such philosophers also ask questions like: “Do numbers (or sets) exist? If they do, where are they? If they don’t, then what does ‘2’ refer to?”

The implication is that since numbers (or sets) are abstract objects, hence if they do exist then they exist in some Platonic realm of abstract objects, raising the question: “How is it that we can access that realm, in order to know such properties of numbers as arithmetic?” To see how stupid such questions are, one only has to ask such questions as: “Does value exist; and if so, where is it?” Clearly some things have value, but it makes no sense to ask where it is (or what colour it is); such questions hardly further the analytical task of describing accurately what value is.

A question similar to the one about numbers might be: “Do shapes exist?” Shapes are instantiated in, and abstracted from, shaped things, clearly; and similarly, whole numbers are instantiated in, and abstracted from, numbers of things. That is basically what John Stuart Mill said (in passing); it is only common sense, although his observation was jumped on by a founder of analytic philosopher, Gottlob Frege (incorrectly).

Definitive Selections?

Are definitive selections too odd?
      When we think of some things, and various combinations of them, it seems clear that all those combinatorial possibilities are there already, awaiting our consideration. And yet I am asking you to imagine that when a Creator, some such brilliant mind, considers some things, all those possibilities are blurred together (although none so blurry that it cannot be picked out); or am I?
      I am suggesting that for selections of selections of ... of selections, from some original collections, each possible selection from those will be a particular possibility only as it is actually selected by our Creator, independently of whom no collections of things would exist, were there such a Creator (as there provably is). The possible selections that make S(N) bigger than N (to use the terminology in my Cantorian diagonal argument) are those endless sequences of ‘I’s and ‘O’s that are pseudorandom; to make them, infinitely many selections have to be made, each one of which involves some arbitrarily large finite number of selections. They might be made instantaneously by our Creator, of course; and if so, then typical selections from S(N) could be made arbitrarily quickly.
      What about S(S(N)), which contains more things than infinite space contains points? Well, a Creator might be able to do all of that instantaneously. And similarly for selections from U, and UU, and maybe UUU; but still, you see how our Creator would have to do much more, and much, much more, and so on and so forth, without end. It is therefore quite plausible that for selection-collections that it would take me far more than mere trillions of pages to describe, our Creator would be unable or unwilling (and thence unable) to make all such selections instantaneously. After all, it is logically impossible for all possible selections to be made instantaneously. To will an incremental development of such abstract mathematics, as a necessary aspect of the creation of any things, might be regarded as a price worth paying for some such creations. And it is also quite plausible that were the Creator unable to do something (even as a consequence of such a choice) then that thing really would be impossible, given that the very possibility of it derives from that Creator.
      Solid things are solid; but mathematical properties related rather abstractly to their individuality can be works in progress; why not? Modern mathematics has a weirder story to tell of such matters! It is relatively straightforward to think of Creation as dependent upon a Creator who transcends even its mathematics. So, it may not be too odd to think of a Creator creating number by definitively adding units: 1, 2, 3 and so forth; is that any weirder than a Creator creating something ex nihilo? Number is paradoxical, so that the ultimate totalities of numbers are indefinitely extensible, and so numbers just do pop into existence, somehow; and what more reasonable way than by their being constructed by a Creator? What would be very weird indeed would be their popping into existence all by themselves, what with them being essentially structural possibilities rather than concrete things. It makes some sense to think of us creating them, as we think about the world around us, but there is something very objective about numbers of things. And again, if it makes sense for us to do it, then how can it be too odd to think of a Creator doing it, in a Platonistic way?
      There will be better ways to think of definitive selection, I am sure; but, it is the case that such weaselly words are the norm nowadays. For example, how can simple brute matter (just atoms, in molecules of atoms, each just some electrons around a nucleus) have feelings, such sensitive feelings as we have? How is that possible? Am I asking for a description of a possible mechanism? Perhaps; but a common enough answer is: Well, it must be possible, because we have such feelings, in this physical universe; although I don't know how sensitive we humans really are, looking at our world! Such answers are accepted by many scientific people, as they "work" on possible mechanisms!

Apparently Timeless Possibilities

Apparently timeless possibilities could, possibly,
become more numerous over time, e.g. as follows:

You were always possible,
but had you never existed,
then that possibility would have been
the possibility of someone just like you.
      It could not have been
      the possibility of you in particular 
      were you not there to refer to.
Looking back now,
we can see that there was always
that possibility, of you in particular 
as well as the more general possibility,
even before you came into being.

Now, Presentism is logically possible,
and if Presentism is true then there may
originally have been no such distinction,
even though you were always possible.
      Under Presentism it could have been
      that you might not have existed.
The distinction could therefore have
arisen when you came into being.

It is therefore logically possible
for apparently timeless possibilities
to emerge as distinct possibilities
from more general possibilities.

The Signature of God

I think belief in God reasonable only if it is based on considerations available to all humans: not if it is claimed on the basis of a special message to oneself or to the group that one belongs.

Anthony Kenny ("Knowledge, Belief, and Faith," Philosophy 82, 381-97)
      So what better signature of the creator of homo sapiens than an elementary logical proof that there is a God? In my last post, I described the argument that given some things, cardinally more selections from them are possible.
      That post ended with a brief description of how that means that paradox arises: we naturally assume that each of the possible selections that such endlessly reiterated selection-collections and infinite unions would or could ever show there to be is already a possibility, that it is already there, as a possible selection; it would follow that they were all there already, that they are collectively some impossible collection of all those possible selections.
      Logic dictates that we have made some mistake; and this version of Cantor's paradox arises because we are considering combinatorially possible selections: that is why the sub-collections that define those selections were able to become so paradoxically numerous, why the paradoxical contradiction did not just show that there are not, after all, so many extra things, over and above the original things.
      My resolution begins by observing that apparently timeless possibilities could, possibly, become more numerous over time; it begins that way because if possible selections are always becoming more numerous, then we would never have all of them. A Constructive Creator could, possibly, make the definitive selections; and if that is the only logical possibility, then that is what has been shown.
      Note that serious mathematicians have taken Constructive mathematics seriously, and when constructed by a transcendent Creator the mathematics would be much more Platonic, and much more Millian. Consider, for an analogy, how God's commands could, just possibly, define ethics. And note that such creative possibilities are not that different to the Creating of mere things ex nihilo, if you think about it: how is such Creation even possible? For us, the laws of physics present immutable limits to what can be done; for a God, such laws are, metaphorically, a brushstroke.
      We live in a world of things, and numbers of things; and for us, numbers appear timeless. But logic does seem to say that such numbers are impossible. When we first think of the origin of things, we might think of things that could have been there forever, like numbers. But logic seems to say that there was originally stuff, not things; perhaps mental stuff, perhaps a God that is not exactly one thing. There would have been some possibility of things, and more arithmetic the more that God thought about that possibility.
      I should add a note about what sort of God is being shown to exist. The proof does not show that God could not have created a four-dimensional world in a Creative act above and beyond that temporal dimension. So this God might be what we call "timeless," and might know all about the future; or not. And either way, this God could always have known all of our textbook mathematics, if only because that is essentially axiomatic.

The Death of Logic

A hundred years ago (more or less) logic died.

It was either logic and a transcendent Creator,
or neither... And atheism was in the ascendant
a hundred years ago (while the Creator shown, logically, to exist
was not that of the embattled religions of those war-faring days
and these) and it still is.

Prima facie, though, logic took off at that time: it was formalized,
and we now have lots of formal logics, within Analytic Philosophy.
But, what's so logical about reacting to the Liar paradox by redefining "truth"?
And what's so logical about having each ordinal, but not having every ordinal?

We may begin with physics: nowadays we have String Theory.
Suppose we get a really good String Theory, say "S," one day.
There's no guarantee that we won't need a better theory later,
so why would we use S to redefine all our physical entities?
If the description of electrons in S was E, for example, then
we could replace "electron" with "E,"  but why should we?
A very good reason why not is that electrons are electrons.

And mathematics is a subset of the properties of possible objects:
one object and another object is (one plus one) objects, and so on.

But for a hundred years, science has replaced "1" with "{{}}."
Did not the logician Frege refute Mill's description of "1"?

It turns out that he did not; and he could not have,
because 1 is, basically, Mill's 1 (and Euler's, and yours).

Set Theory mimics mathematics,
so for applications it hardly matters; but,
do the best mathematicians really believe
that 1 is nothing like Mill's 1, is really {{}}?
We all learn what 1 really is at an early age.
{{}} was chosen following Cantor's paradox,
but it also followed that logic had to be replaced.
Logically, there was that paradoxical proof; and
while the Liar paradox is nowadays interpreted
as another reason to replace truth, and its logic,
with something formal, that is not really scientific:
science pursues truth, and logic takes truths to truths
(where to say, of what is, that it is, is to speak the truth).

Doppelgangers

It seems to be logically possible for there to be an exact copy of you, say d-you, because it seems that such a thing might exist in a parallel space-time. D-you would be physically and mentally identical to you; but it would not, of course, be you. Now, we naturally assume that none of us have been instantaneously swapped with such doppelgangers. We can never have any reason to think that any of us might have been swapped; but, that is because such swapping would be undetectable, and that is why we cannot rule out the logical possibility of such swapping.

Indeed, you cannot completely rule out the possibility that you are such a doppelganger, because you would have exactly the same memories, exactly the same sense of being yourself. There would be absolutely no empirical difference; the only difference would be semantic: reference intended to be reference to you would fail to be such reference, were it to d-you, for example (and given the falsity of Functionalism, and so forth). And of course, knowledge would be lost, e.g. if I saw d-you at a bus-stop then I would not know that you were waiting for a bus. But of course, I would know that you were waiting for a bus if I saw you at a bus-stop (and you were waiting for a bus). There is no loss of knowledge caused by not ruling out the logical possibility of d-you. We simply assume that such swapping does not happen.

Note that we do not just think it unlikely (and similarly, we do not just think it unlikely that we are brains in vats, or being fooled by demons, and so on and so forth). We do not know for sure that there are no such doppelgangers, and we do not even know for sure that there are unlikely to be any (we can have no evidence for such unlikeliness). But clearly, we are assuming that there are no such things (and nothing else of that rather wide-ranging kind). That is just an obvious empirical fact about our beliefs. (We might not notice it, because being fooled by a demon would be like being a brain in an evil scientist’s vat, and a brain in a vat is like someone having a very long vivid dream; and maybe it is only highly unlikely that you are in a coma right now.)

Truth in Dreams

In one Cartesian argument for skepticism about the reality of the world, we are to assume that if we were dreaming, then were we to see hands in that dream, those would not be hands. Still, they would be dream-hands, in a dream-world, so dream-reference to them would hardly fail, or would it? If we think of someone dreaming about hands, then clearly those are not real hands; but, were this a dream (not a dream-within-a-dream, which is what our "dreaming" would then refer to), then what is meant by "real hands" within that dream would be dream-hands. You may well wonder if that would be the case, had we fallen asleep having already learnt the meaning of "real hands" in the real world. But presumably we learnt the meaning of "real hands" in this world, and if this was a dream then this world would be a dream-world. You could counter that if this was a dream, then we would still have learnt the meanings of our words in some higher realm, but as soon as we clarify what exactly we are talking about, by describing what we mean by "an external thing," we tie the meanings of our words to this world (the photo is from last year btw :-)

Lots of Misprints

I've seen quite a few misprints recently, e.g. in TV text; also top of page 159, and again on page 169, in Maddy 2017 (" 'Proof on ..." instead of " 'Proof of ..."), just before she got to Moore's reason why pointing to each of his hands was a proof that there are two hands (and hence that there are external objects, and hence an external world), which was that he could similarly prove that there were three misprints on a certain page by:

taking the book, turning to the page, and pointing to three separate places on it, saying 'There's one misprint here, another here, and another here'

Maddy 2017: 164 (Moore 1939: 147) Although of course, while that proves that there are three misprints, it does not prove that there are three misprints. And while you might agree with Moore that those were misprints, that would not amount to a proof that they were. Moore, you will recall, does not have to show that there are two hands, nor even that there are two hands, he has to show the externality (so to speak) of such things as hands, given skeptical doubts, which is more like having to prove not just assume, that it is indeed a bad thing to have lots of misprints. And of course, why would we have to prove such a thing! Ask yourself what is meant by "external world" to see for yourself how it exists by definition (and note how one gestures as one does so). And yet, it is precisely that "proof" that is challenged by skeptical doubts (as the above-linked-to review of Maddy 2017 observes).

What do Philosophers do?

I'm half-way through Maddy's 2017 (a walk through the modern history of Skepticism), where she describes a weakness of the Argument from Dreaming:

    Although we would not be knowing the world were we now dreaming in the ordinary way, we can rule that out in quite ordinary ways; and whereas we cannot rule out that we are dreaming in some extraordinary way (e.g. a life-long coma), why should we rule it out? Maybe this is a dream-world, and my hands dream-hands within it. But should the fact that I don't know much about the fundamental substance of my hands get in the way of my knowing that I'm typing this with them because they exist (whether that is in a way that is to some unknown world much as dreams are to this world, or in some other way)?

Here's a thought though:

    If some higher power (maybe a UFO) replaced you with a pod-person who was exactly the same as you, physically and mentally, then the people of the world would of course not know, were they to see that person before them, that you were standing there. So, if the underlying substance of the world was such that things were frequently replaced with identical copies, in such ways (and note that we cannot even know that that is unlikely), then our references would frequently fail, and we would end up knowing a lot less about the world than we assume we do.

    We do assume that such does not happen, but that just means that, for example, it is at best epistemic luck that people know that you are there, when they see you. At worst it is knowledge by assumption, because we do assume as much; which reminds me of Wittgenstein's hinge propositions (which Maddy will be getting to shortly). Perhaps we assume that things generally continue to be the same things. Or perhaps we assume that things that look the same are the same.

    I would not say that we know such a proposition, but maybe we do thereby know propositions that depend logically upon it, such as that I have hands. Why not? Knowledge seems not to be some minimal amount of epistemic luck, but rather the sufficient reduction of certain kinds of epistemic luck, as required by one's context; and philosophy is a context with high standards. In philosophy we tend to accept the force of epistemic closure, because the standard is logic.

Much Knowledge is Epistemic Luck

In a recent post (linked to here) I observed how we simply assume that we can refer directly to the things around us: we cannot know that their substances are not changing in ways that leave their properties the same, because we can only know their properties. Were their substances changing, reference to them would keep failing (assuming that reference is direct).
     And similarly, we cannot really rule out that we are Brains In Vats: all of our evidence is compatible with our brains having been harvested by aliens (in a real world where such aliens are common) and put into high-tech vats that simulate worldly experiences. While we are unlikely to have been harvested recently (as recently noted (although note that we cannot rule out as unlikely a world where are are frequently, but not too frequently, re-vatted)) it is not unlikely (by the standards of the apparent world) that there are such aliens (what is strange is that we see no aliens).
     But of course, we can and do simply assume that there are not such aliens, that we are not currently asleep in our beds and dreaming, that all of our particles are not always being switched with identical particles, and so forth. It is upon such foundations that our knowledge of the external world is built. And of course, we are not BIVs, we are not dreaming, and so on; or at least, I do assume not. And so we do have knowledge of the external world. But, because those are assumptions, such knowledge is epistemic luck.

What Do Philosophers Do?

What do you know, for sure? Being sure that you have hands,
for example, could be justified ( you may be using them now,
to operate a phone or a keyboard), but you can hardly rule out
the following scenario, according to which you have no hands:

Your brain was recently harvested by aliens and you are now in a vat
experiencing a detailed simulation; your memories have been altered,
but for you "hand" still refers to things outside the vat. And out there,
those aliens have turned the real world into one enormous brain-farm.

You cannot rule that out,
but you can know that it is unlikely
that your brain was only recently harvested
and so you can, of course, know that you have hands.

Magical Reference

Could you have been created by a God? Is that a possibility? Physical objects are definitely possible because, for example, your body exists: it is actual so it is possible. Maybe you are asleep and dreaming now, maybe not (probably not), but if we bracket (for now) the question of what physical objects are made of (whether it is physical particles or our own thoughts in a dream or 10-dimensional strings or whatever) then they are definitely possible. And mental states are also possible, of course (your current mental state is definitely possible). But a God? Well, insofar as we do not know for sure that it is not a possibility, it is definitely an epistemic possibility. And because it is, in that sense, possible that God created you, it is similarly possible that as well as creating you, God created an exact copy of you in an identical universe, because if God could create this universe, then God could create an identical one. And it is similarly possible that a God who had made you and an identical copy of you could swap you with that other person and swap you back and do that over and over. And not just you: the same goes for everything else in this universe. Such swaps could conceivably be happening all the time, for all that we could tell just by looking around us, however closely we looked.

And this is not a problem with the possibility of there being a God. Do we know for sure that fundamental particles (or whatever substances are physically fundamental) could not be replaced by identical particles (or substances)? We naturally assume that such replacements do not occur, but they would make no difference empirically, so we cannot rule them out empirically. And prima facie, we cannot rule them out on logical grounds. They might, for all we know, be happening all the time. We do not even know that that is unlikely.

You and that hypothetical identical copy of you would not be the same person, even though there would be nothing to tell you apart. That should be obvious because the copy of you would now be in another universe. And the same goes for me, of course. You can refer to me by name (my name is in the sidebar), and that reference should be simple and direct: it should be me that you refer to, not someone else. And of course, it is me that you would be referring to. Or so we assume. The way reference works amounts to an assumption that such swaps do not occur. It is almost magical the way your referring to me means that a host of things that could, for all we know, be happening are not in fact happening.

The solution to that problem may well be the probability of there being a God.

Logic tells us that things are such that there is Cantor's paradox, and from that paradox we can get, in a purely logical way, to the probable existence of a God. Would such a God make such swaps as those considered above? There may well be good reasons why a God would not do that. So, logic may well show that such swaps are unlikely. It is something to think about...

Who's Afraid of Veridical Wool?

I have been taking an informal approach to the Liar paradox, for the following reasons. After much thought, I find self-descriptions like ‘this is false’ to be about as true as not. I am therefore beginning, with the following – previously posted – post, with the equally ancient paradoxes of vagueness. And my approach is informal because I find the precision of mathematical logic to be inapposite when there is no sharp division between something being the case and it not being the case. Although the literature on these paradoxes has become increasingly formal, following Bertrand Russell’s interest in Georg Cantor’s mathematics (at the start of the twentieth century), we do not need non-classical logic to resolve them, I think; rather, we need to focus on the context of classical logic, natural language, in which the paradoxes are expressed. Below, and temporally prior to, ‘Vagueness’, I have posted ‘Liar Paradox’ and ‘Cantor and Russell’.

It was via Russell that I came to consider the Liar paradox, having developed an interest in Cantor because of qualms about the fitness of the real number line as a model of actual continua, which developed as I did my MSc in Mathematics (at the end of the twentieth century). With this post I have come to the end of my journey; I am left wondering why our mathematics became set-theoretical, and then category-theoretical, and similarly, why our natural philosophy became the physicalism of Einstein et al, and then string-theoretical. How well, I wonder, will our democracies be able to regulate the biotechnical industries of this century? I have serious doubts, stemming from my research into physics, theoretical and empirical, and from the history of our regulation of financial industries (which are surely less complex). Still, in the absence of any interest in my research, I have been developing more aesthetic interests over on Google+...

Cantor and Russell

Georg Cantor was a brilliant nineteenth century mathematician whose discoveries led to the foundation of mathematics becoming axiomatic set theory.¹ Cantor’s paradox concerns the collection of all the sets. Now, a collection is just some things being referred to collectively, of course, and a set is basically a non-variable collection. (Sports clubs and political parties are variable collections, for example, while chess sets and sets of stamps are non-variable.) And numbers – non-negative whole numbers – are basically properties of sets. Cantor’s core result was an elegant proof that even infinite sets have more subsets than members (in the cardinal sense of ‘more’). It follows that if there was a set of all the other sets, then it would have more subsets than members, whence there would be more sets than there are sets – each subset being a set – which is impossible. And it follows that there is no set of all the other sets. That consequence is known as ‘Cantor’s paradox’; but, how paradoxical is it? Presumably {you} did not exist until you did, so why expect the collection of all the sets to be non-variable?

A more paradoxical consequence is, I think, that there is no set of all the numbers – for a proof of that consequence, see section 3 of my earlier Who's Afraid of Veridical Wool? – which is paradoxical because we naturally think of numbers as timeless, whence their collection should be non-variable. To see the problem more clearly, suppose that 0, 1, 2 and 3 exist, but that as yet 4 does not; the problem is: How could 2 exist, but not two twos? The existence of n (where ‘n’ stands for any whole number) amounts to the existence of the possibility of n objects, e.g. n tables (were physics to allow so many), and possibilities are, intuitively, timeless: For anything that exists, it was always possible for it to exist.

Nevertheless, if there are too many numbers for them all to exist as distinct numbers, perhaps they are forever emerging from a more indistinct coexistence. Possibilities are not necessarily timeless. You were always possible, for example, but that possibility would – had you never existed – have been the possibility of someone just like you. Looking back, there was always the possibility of you yourself, as well as that more general possibility; but had you not existed, there could have been no such distinction. Note that if the universe had bifurcated into two parallel universes, identical in all other respects, then the other person just like you would not have been you. And however many parallel universes there were, another would not appear to be logically impossible. So, it appears to be logically possible for apparently timeless possibilities – e.g. the possibility of you yourself – to emerge as distinct possibilities from more general possibilities.

Were objective possibilities deriving from the omnipotence of an open-theistic Creator, there would be no paradox; and it is hard to see how else numerical possibilities could vary (cf. how the main alternative to set theory is Constructivism). So, the paradox may well be a proof of the existence of God. There is much more that needs to be said, of course (although I wonder who cares); but those taking numbers to be timeless also have some explaining to do: They need to find a plausible lacuna in the Cantorian proof that numbers are not timeless; but, what they have found is more paradoxes akin to the Liar. 

Cantor’s paradox concerned the set of all the other sets because the set of all the sets would have had to contain itself as one of its own members, and we do not normally think of collections like that. But as Russell thought about Cantor’s counter-intuitive mathematics, he considered the collection of all the sets that do not belong to themselves: If that collection was a set, then it would belong to itself if, and only if, it did not belong to itself. That is basically Russell’s paradox. Like Cantor’s, it is not obviously paradoxical – it just means that there is no such set – but Russell thought of sets as the definite extensions of definite predicates, and predicate versions of his paradox are more obviously paradoxical. E.g. consider W.V.O. Quine’s version: ‘Is not true of itself’ is true of itself if, and only if, it is not true of itself. That is paradoxical because we naturally assume that ‘is not true of itself’ will either be true of itself, or else it will not. But if predicate expressions can be about as true as not of themselves, then it would follow from the meaning of ‘is not true of itself’ that insofar as ‘is not true of itself’ is true of itself it is not true of itself, and that insofar as it is not true of itself it is not the case that it is not true of itself. And it would follow that ‘is not true of itself’ is about as true as not of itself.

Russell also thought of definite descriptions as names, and the English name for 111,777 – one hundred and eleven thousand, seven hundred and seventy seven – has nineteen syllables. According to Russell, 111,777 is the least integer not nameable in fewer than nineteen syllables, and Berry’s paradox is that ‘the least integer not nameable in fewer than nineteen syllables’ is a description of eighteen syllables.² Again, that is not very paradoxical; we can always use a false description as a name – cf. ‘Little John’ – and ‘John’ can name anything in one syllable. But consider the following two sentences.³ The number denoted by ‘1’. The sum of the finite numbers denoted by these two sentences. The first sentence denotes 1, so if the second sentence denotes anything, then it denotes a finite number, say x, where 1 + x = x, and there is no such number. So if the second sentence denotes anything, then it does not denote anything. But it cannot simply fail to denote, because if it does not denote anything, then the sum of the finite numbers denoted by those two sentences is 1. Since the second sentence denotes 1 if, and only if, it denotes nothing, perhaps it denotes 1 as much as not. Cf. how ‘King Arthur’s Round Table’ began as a definite description and ended up referring more vaguely.

In stark contrast, the set-theoretic paradoxes – e.g. Cantor’s – do not have resolutions akin to the present resolution of the Liar paradox: How could a collection of numbers be as variable as not? (Collections of numbers are not like collections of noses, so it could not be like Pinocchio’s nose.) Those paradoxes do have a fuzzy logical resolution, via fuzzy sets, though. And those taking L to be true and false can find it true and false that some collections belong to themselves. And those taking natural Liar sentences to be nonsensical often have a formalist take on infinite number. And of course, if the set-theoretic paradoxes have the same underlying cause as the semantic paradoxes – as Russell thought – then they should all be resolved in similar ways. But, if there are two kinds of paradox here – as Ramsey thought – then the inability of the present approach to resolve the set-theoretic paradoxes would hardly count against it. On the contrary, that inability would amount to some evidence for it, by helping it to explain the attractions of the major alternatives, especially the formal ones: A non-classical logic would be very useful were one trying to fly in the face of a mathematical proof.

Notes

1. For a detailed history, see Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton and Oxford: Princeton University Press, 2000).

2. Attributed to G.G. Berry by Bertrand Russell, ‘Mathematical Logic as based on the Theory of Types’, American Journal of Mathematics 30 (1908), 222–262.

3. Based on Keith Simmons, ‘Reference and Paradox’, in J.C. Beall (ed.) Liars and Heaps: New Essays on Paradox (Oxford: Clarendon Press, 2003), 230–252. Simmons’ version was more complicated, and omitted the crucial word ‘finite’.

What is Proof?

Over a hundred years ago, Cantor proved that the natural numbers are temporal: Assume, with Plato and against Aristotle, that they are not temporal, so that they all coexist, insofar as numbers do exist (the main thing is that we can count them, e.g. {1, 2, 3} are three numbers). Since they all exist atemporally, so do all their subsets (e.g. {1, 2, 3}), and so there is a set (an atemporal collection) of all the subsets of that set. By a simple diagonal argument (which you can google) that set of subsets is of larger cardinality than the original set. (Two sets have the same cardinality when the elements of one of them can be put into some one-to-one correspondence with the elements of the other.) And the set of all of the subsets of that set of subsets is of even larger cardinality, because the diagonal argument applies quite generally to any set and the set of all its subsets. (This gives us three equivalence classes of infinite sets, associated with three infinite cardinalities.) So, we can get cardinally bigger and bigger sets in that way. And when there is an endless sequence of such sets, the union of all of them will also be an atemporal collection, because each of those sets was implicit in the previous set, and it will have a cardinality larger than any of those sets, because each is followed in that sequence by sets of larger cardinality. And from that union we can again consider the set of all its subsets, and so on. Now, all these atemporal collections are implicit with the set of natural numbers, so they all exist (insofar as such things do exist) atemporally; but, Cantor proved that they cannot all exist atemporally: Suppose they do. Then there is a set of them all. But implicit in them is the collection of all of their subsets, which would be cardinally more of them, whereas we have assumed that we had the set of them all. So, we have assumed that the natural numbers are not temporal, and obtained a contradiction; that is a classical mathematical proof of the temporality of the natural numbers. However, most people assume that numbers are timeless, and so Cantor took himself to have proved that the totality of the numbers was indeed contradictory (akin to human reasoning being inferior to religious insight), while most of his peers replaced the natural numbers with axiomatic structures that had not been shown to be contradictory. Axiomatic set theory has been the foundation of mathematics for nearly a hundred years, but why do mathematicians throw numbers away (why take number-words to be referring to axiomatic sets) just because of an inconvenient proof? We expect others to accept the conclusions of our proofs, when we have proofs...

A peculiarity of the Liar paradox

Consider the following sentence: “The self-referential statement expressed by this sentence is not true.” Taking the phrase “this sentence” to refer, self-referentially, to that very sentence, the most obvious meaning of that sentence is that it is not the case that the self-referential statement expressed by that sentence is not true. But that is just to say that the statement expressed by that sentence is true, which is the negation of the obvious meaning of that sentence.
......Since the statement expressed by that sentence is both that such and such is the case and that it is not the case, which is self-contradictory, it may well follow that the statement in question is false, as suggested by Dale Jacquette (2007: ‘On the Relation of Informal to Symbolic Logic,’ in his (ed.) Philosophy of Logic, Amsterdam: North-Holland, 131–154). But other philosophers – amongst whom I would once (five years ago) have counted myself – think that because an assertion that such and such is the case is clearly different in meaning to an assertion that it is not the case, such Liar sentences do not express any proposition at all, but are rather meaningless nonsense.
......However, I argued recently that Liar statements are in fact as true as not, and that the Liar paradox is, in that sense, a typical semantic paradox (for details see my The Liar Paradox, and my On the Cause of the Unsatisfied Paradox, in the April and June issues of this year's The Reasoner respectively); whereas, the problem above seems to be unique to the Liar paradox, e.g. it does not arise with Yablo’s paradox, in which there is no self-reference. So, I am wondering how else we might address this part of the Liar paradox.
......Could the problem be due to substitution failure? Perhaps replacing “this sentence”, in the sentence in question, with a near-copy of the sentence itself – the only difference being that ‘that’ replaces ‘this – changed the proposition expressed by that sentence to its negation. Similar failures can occur with propositional attitude reports, e.g. consider the difference between “Lois believes that Clark is thirsty” and “Lois believes that Superman is thirsty”; for details see Jennifer Saul (2007: Simple Sentences, Substitution, and Intuitions, OUP). But then, Liar sentences are sentences of a very different kind; they need only involve a self-referential name, e.g. ‘L’, plus ‘is’, ‘not’ and ‘true’.
......Another possibility is that Liar statements are identical to their negations. As a rule, the negation of a proposition is a different proposition, of course; but, propositions are either true or else false, as a rule, whereas we are now looking at propositions that are as true as not. Now, an elementary part of language is the subject-predicate description, “S is P” (e.g. “that salmon is pink”), and so a simple model of truth might use strips of paper with “S is P” on one side of the strip and “S is not P” on the reverse side, for all S and P in some simple language: All the strips with non-fictional S get stuck onto the things of the world, with “S is P” uppermost if S is P and “S is not P” uppermost if S is not P. We might extend that model to include cases where S is as P as not by giving the strips a twist in the middle before sticking them down, and by including non-fictional strips with no worldly referent, such as “100 is a round number that is also a square.” And then we might think of our Liar sentence as being like a Möbius strip, the twist due to its being as true as not, and the joining of its ends being due to its being self-referential.
......In any case, this peculiarity of the Liar paradox gives us an easy answer to the Revenge problem for this resolution of the Liar paradox, which is as follows: If “what I am saying is not true” is as true as not, then what about “what I am now saying is not only not true, it is not even as true as not”? Were that about as true as not, what it said would seem false. But, what it said was that it was not at all true that what was said was not at all true, so it said not only that what was said was not at all true, but also that it was to some extent true. So if it was as true as not, then although it would indeed seem to have been false – false that what was said was not even as true as not – it should also, and to the same extent, appear true – true that what was said was to some extent true – whence it should seem to have been as true as not after all.

The Set-theoretical Paradoxes

I have another piece on semantic paradox in The Reasoner in June; but, what about the set-theoretical paradoxes? The seminal paradox of Bertrand Russell (1902: ‘Letter to Frege’, in 1967: Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, 124–5), for example, concerns the class of classes that do not belong to themselves. (In this context, classes are extensions of predicates: all and only the things that satisfy the predicate belong to the class.) Some classes – e.g. the class of humans – do not belong to themselves – the class of humans is a class, not a human – and the class of all such classes is paradoxical: it belongs to itself if, and only if, it does not. Russell conceived this paradox when thinking about the set-theoretical paradoxes, because a class is a kind of set; but, Russell’s paradox can also be expressed directly in terms of predicates:

Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. (Ibid, 125)

Note that “is a human” is a predicate expression, not a human, so it does not describe itself; the question is, what about “does not describe itself”? It describes itself if, and only if, it does not, which is paradoxical (it is widely known as Grelling’s paradox). But, we might as well say that “does not describe itself” describes itself insofar as it does not, from which it follows that it describes itself as much as it does not. So the question arises, could the class of classes that do not belong to themselves belong to itself as much as not? Classes can be like that, e.g. the class of all men would be like that if some hominids had been about as human as not. If we call one such hominid ‘Strider’, then it was about as true as not that Strider was human. Were there no such hominids, then some human would have had non-human parents.

That – the fuzzy set – resolution of Russell’s paradox coheres rather well with my preferred resolutions of the semantic paradoxes – e.g. The Liar Paradox, from The Reasoner 7(4) – but, the set-theoretical paradoxes originally arose from the mathematics of Georg Cantor, and they concerned, not classes of classes, but numbers of numbers. And of course, numbers are far from fuzzy. You and I, for example, are two people, and there is surely no doubt that we know what is meant by ‘two’ (for all the uncertainty over Strider's personhood). So, let us begin with 0, 1, 2, 3 and so forth, the products of the process of adding 1 to the previous number, starting with 0. Rather trivially, the collection of all those numbers is all of them, referred to collectively; and while some collections – e.g. stamp collections – are variable, if a collection is non-variable then we can say that it is a set (as in “a set of stamps”). On this conception, a set is some particular number of logical objects. To include the numbers 0 and 1, and so make this conception more like the standard conception (and also simplify proofs), let us also include logical objects that can play the role of singletons – sets with a single element – and Ø, the empty set. So, given that we have a set {0, 1, 2 …}, it contains some definite number of numbers, say א (aleph) of them.
......Cantor showed that every set has more subsets than it has elements, in the cardinal sense of ‘more’. (Two sets have the same cardinal number of elements when the elements of each set can all be paired up with those of the other (cf. Hume's principle).)

Let S be any set, and let P (for ‘powerset’) be the set of all its subsets (including Ø and S). If S and P had the same cardinality, then there would be one-to-one mappings from S onto all of P, so let M be one such mapping. Let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it, then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. So, D is contradictory, and so there is no such M. So S and P do not have the same cardinality, and since P contains a singleton for each element of S, P is bigger than S.

So, {0, 1, 2 …} has beth-one subsets, where beth-one is bigger than aleph, and the set of all those sets has beth-two subsets, and so on. If that endless sequence of bigger and bigger sets is a non-variable sequence, then there is a union – a set of the elements – of all those sets, which is even bigger, with בω (beth-omega) sets. (Omega is the ordinal number of the sequence 1, 2, 3 and so forth.) And that union has בω + 1 subsets, and so on: for any such set there is the set of its subsets, and for any endless sequence of such sets there is, if it is a non-variable sequence, its union. In total, there is a sequence of sets – and a corresponding sequence of numbers, the sizes of those sets – which must be variable; were it not, we would have moved on from that ordered set of sets to its union, and thence to the subsets of that union (and so on). But of course, it is paradoxical that our total sequence of numbers is variable – is of necessity growing forever – because few of us think that numbers that do not already exist could suddenly appear. Suppose, for example, that the number 101 had not always existed; would that not mean that there was once a time when there were no such possibilities as, for example, the possibility of 101 Dalmatians? And note that this paradox cannot be resolved as Russell’s paradox was resolved above, because the idea of something being as variable as not is nonsensical.

Nevertheless, the intuition that numbers are atemporal is not unquestionable, because new possibilities can be constructed out of more general possibilities. You were always possible, for example, and yet the possibility of you in particular was only distinct from the more general possibility of people just like you once you existed (to be directly referred to). And it is not too odd to think of arithmetic as constructed from such logical concepts as those of possibility and class. E.g. the obvious meaning of “2 + 2 = 4” is that if we had two things of some kind, then if we got another two of that kind we would have four. So, it is conceivable that, while 101 Dalmatians were always possible, there was once a time when that possibility only existed as part of a more general possibility (of bigger numbers). Such constructivism can be defended atheistically – e.g. see George Lakoff and Rafael E. Núñez (2000: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, New York: Basic Books) – and theistically, e.g. see Paul Copan and William Lane Craig (2004: Creation out of Nothing: A Biblical, Philosophical, and Scientific Exploration, Grand Rapids, MI: Baker Academic).
......Whether we are atheists who believe that the human brain evolved in a finite world, or theists who entertain divine ineffability and infinitude, we would have such reasons to doubt that we could ever be justifiably sure about the nature of infinity, though. And another reason why we should keep an open mind about this is that, while it is clearly counter-intuitive to think of the finite cardinal numbers as temporal, it is, if you think about it, no less counter-intuitive to think of them as atemporal. E.g. the arithmetic of such numbers as א and ω is very different to that of the finite cardinal numbers, whence the theoretical behaviour of that many objects is counter-intuitive. Hilbert’s famous Hotel can be built upon Galileo’s paradox, for example. And the difference between cardinal and ordinal arithmetic gives rise to the counter-intuitive behaviour of my quasi-supertask (2003: ‘Infinite Sequences, Finitist Consequence’, British Journal for the Philosophy of Science 54, 591–9). And the infinite set of the finite cardinal numbers covers the whole range of the finite (in units), and yet every one of those numbers is infinitely far from infinite, whence Lévy’s paradox. For more examples, see José Benardete (1964: Infinity: An Essay in Metaphysics, Oxford: Clarendon Press), and Peter Fletcher (2007: ‘Infinity’, in Dale Jacquette, Philosophy of Logic, Amsterdam: Elsevier, 523–585).
......Intuitively, numbers are timeless; but while it is certainly possible that there is a set of all the finite cardinal numbers, it is also possible that there is not. Both possibilities are counter-intuitive, so both can be supported in ways that would seem compelling were it not for that ‘both’. So, one might think that modern mathematics would have been based on results that follow, not just from one, but from both possibilities. However, such is not the case. Now, the ubiquity of the standard real number line might be explained by its being easy to use, simple and familiar, but there is a similar bias towards assuming that there is a non-variable collection {0, 1, 2 …} in such fundamental research areas as theoretical physics and pure mathematics, which is puzzling. For clues, see Ivor Grattan-Guinness (2000: The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel, Princeton University Press), and Peter Markie (2013: ‘Rationalism vs. Empiricism’, in Edward N. Zalta, The Stanford Encyclopedia of Philosophy).

It might be objected that there is no real puzzle because {0, 1, 2 …} is not an informal set in pure mathematics, but is an axiomatic set defined by means of a formal logic. However, that would be to ignore, not to explain, the mystery. We know perfectly well what the cardinal numbers 0, 1, 2, 3 and so forth are; if we had some axioms that did not describe them, we would not throw those numbers away and start using those axioms instead, however nice their formal properties were. To do so would hardly be scientific.
......Perhaps I should add that we do not get a third kind of set-theoretical paradox from the axiomatic conception. Paradoxes arise when we have contradictory beliefs, and formal structures have no intrinsic meaning; formal axiomatic sets only give us mathematical models of set-theoretical paradoxes. So, while it is true that paradoxes can be avoided if we use formal sets, we did not really resolve the set-theoretical paradoxes by moving from naïve set theory to axiomatic set theory.