📚Current version of my book

This links to a 28,000-word google-doc

What is it like to be a bat?

You know what it is like to be a bat. To be a bat is to be a mammal like no other. You spend half the day dozing in caves, and then you all leave together. You flap about, in order to get anywhere. You find out where you are by seeing how the sounds you make come back to you. You are where all the others are. Each of you is there because everyone else is there. Everyone else is a bit batty. You know what it is like to be a bat.

Health and Safety

"Police officers are, quite rightly, furious at the government for failing to prioritise them in the vaccination schedule."

Click on that quote for more on that "quite rightly,"

Blaming the government is how our democracy works; but I was wondering, who failed to get the police vaccinated in the early days of this pandemic, when it would have done them (and therefore us) much more good?

Many months ago (here is a link to one timeline) the vaccines were safe enough to be given to thousands of volunteers, for phases II and III of the testing process (for a description of those phases see the quote below, which is cut-and-pasted from WHO). Could some of the vaccines not have been made available then, for key workers who volunteered (for the vaccines, not for participation in the trials, which were presumably randomized)? Tinkering with the timetable of the current roll-out of mass vaccinations would inevitably involve some risk to that roll-out. Why do people who failed to think this through earlier, when it would have done more good, now think that they know better than the government how to balance those risks?

Here is that WHO quote:

"An experimental vaccine is first tested in animals to evaluate its safety and potential to prevent disease. It is then tested in human clinical trials, in three phases:

In phase I, the vaccine is given to a small number of volunteers to assess its safety, confirm it generates an immune response, and determine the right dosage.

In phase II, the vaccine is usually given [to] hundreds of volunteers, who are closely monitored for any side effects, to further assess its ability to generate an immune response. In this phase, data are also collected whenever possible on disease outcomes, but usually not in large enough numbers to have a clear picture of the effect of the vaccine on disease. Participants in this phase have the same characteristics (such as age and sex) as the people for whom the vaccine is intended. In this phase, some volunteers receive the vaccine and others do not, which allows comparisons to be made and conclusions drawn about the vaccine.

In phase III, the vaccine is given to thousands of volunteers – some of whom receive the investigational vaccine, and some of whom do not, just like in phase II trials. Data from both groups is carefully compared to see if the vaccine is safe and effective against the disease it is designed to protect against."

A True Contradiction?

(a) the maths

Since adding zero to any amount does not change it, we can keep adding zeroes forever, and it will make no difference. Such additions always amount to adding zero.

We might write that as 0 = 0 + 0 + 0 + 0 + 0 + …, which can be spread out like this:

       0      =                             0                            +                             0                           +             . . .

Each 0 on the right-hand side can be replaced by 1 – 1, to give:

       0      =             (1            –             1)            +             (1            –             1)            +             . . .

In the next equation, the brackets have been removed.

       0      =             1              –             1              +             1              –             1            +             . . .

In the next equation, brackets have been put back in, in different places.

       0      =             1              +             (–1         +             1)            +             (–1         +             . . .

We now replace each (–1 + 1) with 0.

       0      =             1              +                             0                              +                         0              . . .

All those zeroes on the right-hand side add up to zero, of course. But that means that:

       0      =             1

Clearly 0 = 1 is false. So, where did we go wrong? Well, since the last equation was false, the equation above it must also have been false (the only difference between those two equations is the first equation, which was clearly true). And the next one, going upwards, 0 = 1 + (–1 + 1) + (–1 + 1) + ..., must have been false too, as each of those “(–1 + 1)” does equal zero.

Going the other way, from the first equation, 0 = 0 + 0 + ..., which was clearly true, the next equation, 0 = (1 – 1) + (1 – 1) + ..., is similarly true, because each of those “(1 – 1)” is zero.

In between those two equations, one false and one true, we have the infinite sum 1 – 1 + 1 – 1 + …, which was originally described by the Italian theologian and mathematician Guido Grandi (1671–1742).

Grandi was interested in the calculus (as described by Leibniz). And in the calculus, an infinite sum is equal to the limit of the initial finite sums as their length tends to infinity. Grandi’s infinite sum 1 – 1 + 1 – 1 + ... has initial sums that alternate between 1 and 0 = 1 – 1 endlessly (the next are 1 = 1 – 1 + 1 and 0 = 1 – 1 + 1 – 1). Since the initial sums tend to no limit, Grandi’s infinite sum is not given any value by the calculus.

By removing the brackets, we moved from an infinite sum of zeroes, which is equal to zero, to Grandi’s infinite sum, which has no value. Adding brackets differently then took us from Grandi’s infinite sum to a sum that is one plus an infinite number of zeroes, which is equal to one.

(b) the physics

You may be familiar with the idea of a particle/antiparticle pair appearing out of the vacuum. Such pairs give rise to Hawking radiation from a black hole, but all we need to know here is that such pairs can, in theory, appear from the background fields of the vacuum. Once formed, the particle and antiparticle are moving away from their point of origin, so we might picture them moving downwards, like this: /\ (near a black hole, one of them might be swallowed by the black hole, while the other flies away from the black hole, giving rise to Hawking radiation).

Space does not seem to be infinite, but an infinite space is a physical possibility. And in such a space, an endless line of such particles/antiparticle pairs is a possibility, for all that it is highly unlikely. We might picture them like this: /\/\/\/\/\... (the zig-zag continues to spatial infinity).

The top of that zig-zag pictures a line of particle/antiparticle pairs appearing, which might be modelled mathematically by modelling each particle as +1 and each antiparticle as –1. We then get this equation:

0 = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + ...

Each (1 – 1) represents a particle/antiparticle pair appearing.

They move downwards in such a way that each antiparticle collides with the particle from the pair to the right, so that they are both annihilated. The particle at the extreme left of the zig-zag is not annihilated. The bottom of the zig-zag therefore pictures events that are modelled rather well by this equation:

1 = 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + ...

Each (–1 + 1) corresponds to an antiparticle and a particle annihilating each other.

In between those two equations, there is no mathematical sum, neither 0 nor 1. That corresponds to infinitely many particles and antiparticles just being there, in between their creation and their almost total annihilation. The highly improbable, but physically possible, appearance of this particle from an infinite vacuum is therefore so well-modelled by 0 = 1 – 1 + 1 – 1 + ... = 1, that it is essentially an instance of it. It is in a very similar way that Jack and Jill being a couple is an instance of 1 + 1 = 2.

Such equations as 1 + 1 = 2 only exist because they are such good descriptions of any collection of two things. It is the physical instantiation that ultimately justifies the mathematical equation. And of course, to say of what is, that it is, is to say something that is true. Which raises the following question.

(c) the questions

Could 0 = 1 – 1 + 1 – 1 + ... = 1 be a true contradiction?

In order to think about that question logically, should we use paraconsistent logic?

(d) my answers

Although a contradiction can be used as a description that is such a good description, it should count as a true description, that does not mean that the contradiction is true. Consider how there are two ways in which 1 + 1 = 2 is true. It is true as a description of Jack and Jill, and it is, in a different way, true by definition (of 2). Contradictions are false (as a rule). And it is not at all contradictory for there to be no particle and then, at a later time, one particle.

In order to answer that question correctly, I needed to think logically. Why would anyone think that a mathematical model of reasoning that is not a very good model of logical reasoning would help?

Murder in the Academy

Ducks are not as daft as they look. Not by comparison with philosophers. To see how daft philosophers are, you only have to consider the Trolley Problem. Here are four versions of it, called Transplant, Footbridge, Lever and Tiger.

In each version, a scenario is described. You are told what will definitely happen if you choose to do something (someone will die), and what will definitely happen if you do not choose to do it (three people will die). You have to decide what you should do.

1) Transplant

One of your friends is a doctor, and three of her patients will die within the week if they do not get transplants. One needs a heart, and the others each need a lung and a kidney. And it is not that they will probably die. They will definitely die.

You also know Mr E, whose organs are perfectly compatible with those three patients. If they had those organs, all three would go on to live long, happy and productive lives. As it is, not only are they dying within the week, Mr E does not have much to look forward to either. He works in a zoo but he does not like animals.

The question is: should you kill Mr E? If you do, you will be able to deliver the corpse to your friend the doctor, who will be able to operate successfully on her three patients. You will commit a perfect murder, and have a fake organ donor form to give to your doctor friend along with the body. You will be able to walk away with no untoward repercussions.

The answer is: No. Murder is wrong, and you should not do it. There is nothing complicated about this scenario. Almost everyone who hears this version of the problem agrees that the answer is "No." I suppose that it could be argued that, because killing in self-defense is allowed, you should be allowed to kill an innocent person in order to save three lives. But that would be daft. And you would think that a philosopher would know why. But the following version of the problem seems to confuse them.

2) Footbridge

You are on a footbridge over a bend in a railway track. Three people are working on the track on one side of the bridge. They cannot hear an approaching train over the noise of their equipment. From where you are you can see it coming from the other direction. It will pass under the bridge and then hit those three workers, who have their backs to you. It will not be able to stop in time. It will kill all three.

There is also a fat man, Mr E, sitting on the edge of the bridge, eating some sandwiches and thinking about going back to work. You happen to know that if you pushed him off, he would land on the track in front of the train and slow it enough for it to stop without killing any of the workers, although he would die instead. Again, no one would have to know what you had done. You could push him off, to save three workers, and be on your way with no one any the wiser. Should you push Mr E off?

It would of course still be wrong to murder Mr E. Imagine if you did not know that! And yet a lot of philosophers think that you should push the fat man off the footbridge. It is not that they think that murder is alright. It is that they do not realise that that is what the question is asking. That is because these problems are usually presented in the opposite order: Lever first, then Footbridge.


3) Lever

This is a lot like Footbridge. The only difference is that instead of a bridge there is a side track, onto which the oncoming train could be diverted. The train would go straight on, onto the side track, instead of going round the bend in the track, towards the workers.

You are standing by the bend, where there is a lever that controls where the train will go. You can see the workers in one direction, and the train that they cannot see (or hear) in the other. In this version, Mr E is sitting on one of the rails of the side track, eating his sandwiches and listening to some music. He is too far away for you to pull the lever, diverting the train onto the side track, and then run to him in time to warn him. And the workers are also too far away for you to warn them. So, three people will die, or just one. Should you pull the lever killing Mr E?

This is so similar to the other versions, it should be obvious that the answer is again "No." Otherwise you would be killing one person, instead of killing no one. You would not be choosing the lesser of two evils (one person dead instead of three), you would be choosing the greater of two evils: murdering someone instead of watching three people get killed by a train (while you try and fail to think of some other way of helping, presumably).

A lot of philosophers think that the answer in this case is "Yes." When people consider Lever, it seems that they naturally see it as the question "should three people die, or just one?" They tend to think that it is akin to allocating resources. Or perhaps they think that the lever should have been manned by someone who should choose to save the most lives (much as Churchill was not wrong to order the bombing of Dresden), and that they would simply be filling in for that absent person.

You might think that philosophers would cut through the natural mistakes to the actual facts of the matter. But in fact, because Lever is usually presented before Footbridge, quite a few of them carry the wrong answer over, from Lever to Footbridge. Even when they are given that version, which naturally makes people think of murder, not the allocation of resources, they continue to think that the answer is "Yes."

Some philosophers do notice that those two versions are significantly different. But then they tend to imagine that they have some evidence for a half-baked "philosophical" explanation of why they are different. Not many of them go back, from Footbridge, to revise their answer to this version; and of course, they should do that, because these two versions are not actually that different. My final version emphasizes their similarities.

By way of introducing it, consider how pushing Mr E in front of a train is a bit like pushing him in front of a tiger. If you lifted Mr E up on a platform, and then dropped him in front of the tiger, it would be like Footbridge. Would it make much of a difference if you lowered him, so that the tiger had to jump down onto him, a bit like Lever?

4) Tiger

In this version, a tiger is rushing down a tunnel towards three children. It will kill them if it reaches them. However, there is a pit between the tiger and the children, currently covered over with a board. If the board is removed, then the tiger will fall into the pit and the children will be saved. You can remove the board in time to save them. However, it was Mr E who dug that pit, and he is still in there, hiding from the tiger. If you remove the board then he will definitely be killed by it. Should you remove the board?

I would be tempted to. I could always say that I did not know that Mr E was in the pit; and I would not just be watching three children being savaged by a tiger. So, this version is a lot like Lever. Still, even a murder that I feel to be justified is still a murder. One explanation of why Footbridge is different to Lever, given by several philosophers, is that in Footbridge you are primarily killing a man, only incidentally saving some others, whereas in Lever you are primarily saving some men, only incidentally killing another.

Had it been the case that you did not know that Mr E was still in the pit, then it would have been alright to remove the board covering the pit: you would only accidentally have caused his death. But similarly, if you did not know that the children were at the end of the tunnel, then it would have been a bit cruel to remove the board. You would have been dropping the tiger into a pit for no reason, only accidentally saving some children. But in all versions of the Trolley Problem we know what will definitely happen if we choose to do something, and what will definitely happen if we do not. That gives us a relatively simple situation to consider.

The failure of so many of philosophers to understand such simple scenarios is not an insignificant failing. Philosophers claim to be our experts when it comes to thinking logically: many of them teach Logic. Some of them, following Kant, think that it is wrong to use someone as a means to an end. That is, it is wrong to use Mr E as a brake (or to use him as big-cat-food), although it is alright if he is in the way of a diverted train (or if he happens to be in the pit you are dropping the tiger into). And it is true that in Tiger you are diverting the tiger onto Mr E, not putting Mr E in front of the tiger. But surely it is daft to think that it matters much which one falls toward the other.

If you throw a spear at someone (or pull the trigger on a gun, or a lever to divert a train) and kill her, how are you murdering her less than if you deliberately push her onto a spike? Relatively less daft is what I took a break from philosophy to spend more time looking at: ducks, on the village duck-pond. In this photo, a pair were busy making ducklings. They seemed to know what they were doing (although as it happens, the male's penis is like a corkscrew, and the female's vagina coils the other way).

What if there is a proof?

If there is a perfectly logical proof that there is a God, the experts would take themselves to be knowing that there is probably no such thing and not waste their valuable time checking the logic of that purported proof. But perhaps it would be different elsewhere. Perhaps it would have been different.

If Cantor's paradox is essentially a proof that there is a God, then anyone who could have noticed Cantor's generalised diagonal argument could have discovered that proof. Could it have been discovered by the mathematicians of the ancient world? Might that have been part of the significance of Amun-Ra, or Jehovah, or Plato's Form of Forms? It is too late to know now (although the experts would say that they know what is probably the case).

If there is such a proof, would the work of Russell in the period 1901 to 1906 have 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.

If God deliberately created this universe in such a way that there was such a 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.

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 the Cartesian argument (for Skepticism) from dreaming we are to assume that if we were dreaming, then were we to see hands in that dream, those would not be hands; but of course, they would be dream-hands in a dream-world, and so why should dream-reference to them fail? 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 refer to), then what is meant by "real hands" within that dream would be dream-hands. You might 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 were this a dream then that would be a dream-world. Might we have learnt the meanings of our words in some higher realm? But, as soon as we clarify such worries, say in some Moorean way, by describing what is meant by "external thing," we tie the meanings of our words to this world: worrying about that problem resolves that problem!

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.

50 meanings for "know"

To say that you know something is, basically, to say that you are certain of it; in effect, you are promising that what you say is true. Knowledge is important because we want, as a society, bodies of knowledge that can be relied upon. That is why, when cause for doubt is shown to us by skeptical scenarios, our natural reaction is to doubt that we did have knowledge; although of course, academics cannot conclude that they know nothing. At the other end of the scale, consider a boy sitting an exam, who is not sure of an answer but puts it down anyway, and it turns out to be correct: we say that he did know the answer. There are a range of uses of the word "know," and in between those two are all the sciences, and all their applications, and such varied uses of "know" give it a certain inconsistency. The analytic-philosophical analysis of "know" is therefore a cornucopia of papers. Continental philosophers may notice that you can only ever really know what you have yourself made up, however, because the paradigm case of knowledge is, as it has always been, that of a God: proposition P is known by subject S when S's justification for believing P guarantees that P is true. How close you have to get to that ideal, for what you believe to count as knowledge, depends upon the kind of knowledge that it is, the use that you are going to make of it, and so on. We pick up on the use of "know" as we learn English, and I for one have found that good claims to knowledge can be gambles, akin to promises, even though knowledge stands opposed to epistemic luck. More generally, we might disagree about the meaning of "know" without any of us being wrong. But one thing stands out: you either know something or else you do not. Justification, by contrast, comes in degrees; and that is so even though knowledge is basically justified true belief. That is because the justification required for knowledge is sufficient relevant justification. Note that if you think that you know something, because of some justification, but your belief turns out to be false, so that you do not know it, then as a rule your standard for sufficient relevant justification in similar cases will need to be revised.

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.

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’.

I think, so I'm iffy

"I deliberate, so the future is open," is, if you think about it, a pretty good description of a rational argument with one premise (a premise of which one can be certain). My making the effort to deliberate well (because I would blame myself if I did not) presupposes that there is, as yet, no fact of the matter of what I will be thinking.
......To make such an effort is to force the future away from a state that it would otherwise be in, of course. And for me to think of that state as already unreal would undermine my motivation to make such an effort. And of course, for me to make no such effort would be for me to care little for the quality of my thoughts, which would be irrational.
......That was a précis of my comments on a Prussian post, themselves inspired by Nicholas Denyer's 1981 defence of arguments like "I deliberate, so my will is free."