Wednesday, July 17, 2019

Saturday, March 23, 2019

My References

It is Vienna, 1894, and I am painting a goldfinch; but even in this fantasy I am painting very badly. Still, I am a prince and so everyone is pretending that I am not too bad at it. My painting is complimented, and hung up in a corridor.

Eventually the painting is found by the Nazis, who take it for a bad painting of a robin. Since it would make Hitler's paintings look good, they move it to Berlin. Everyone ignores it though. They are very busy. After the war it ends up in Boston, where someone says: "That bird was very well fed." And it was a rather plump goldfinch. But these people imagine that they are talking about a robin. Some of them assume that it was an imaginary robin, it was so badly painted. But in fact, all of them were talking about a goldfinch.

Or, do you think that they were talking about robins? How very modern! To see why they were not, suppose that we are  looking through a warped and dirty window, at a fat goldfinch in bad light. You might at first think that it is a robin. But you are looking at a goldfinch. If you point and say "that bird" then you are referring to a goldfinch.

Were you a mad man, used to seeing things that are not there, you might assume that it was your own personal hallucination of a robin that you were seeing. But even then, your "that bird" would refer to the bird that you were looking at. That is just the way reference works in our public language. And it is clearly the same if we are looking, instead, at a very bad photograph, of a very fat goldfinch.

Suppose the photograph is so bad that someone takes it for a bad picture of a robin. When she talks about the bird in the picture, she is referring to something that was photographed. And even if she thinks that the picture is a sketch of an imaginary robin, it is still actually a photograph of a goldfinch.

And we developed language on top of whatever else we already had, and so the same underlying principle would apply to the semantics of names; primarily to spoken names, and secondarily to written names like these: Mill knew that it did, whereas Russell appears not to have known.

Thursday, February 21, 2019

The Irony Age

It has always been ironic that Christians call upon saints to save them from acts of God. But nowadays you can get called "racist" for comparing Mohammed's "the Recitation" with Hitler's "my Struggle"; or for likening overdressed Muslim women to letterboxes (as Boris Johnson did), even though that is surely no worse than likening nuns to penguins, as in The Blues Brothers (hardly a racist movie). And what do we call criticizing the dresses of The Stepford Wives? Feminism? Ironically, Boris Johnson had been defending the British Muslim woman's right to dress conservatively in public, against a wave of European restrictions.

For such reasons I think of this as the Irony Age. For another example, hominids that were not human were presumably hunted by primitive tribes of humans. Many of them might have been hunted to extinction, as humans evolved; such is evolution. Neanderthals took their time becoming extinct, so they might have evolved an instinctive fear of the human form: those without some such fear may well have had a significant disadvantage. This is only a possibility. But since white people are the product of humans and neanderthals, it is conceivable that a fear of humans evolved in neanderthals and was then inherited by white people. That is, it is a reasonable scientific theory that racism was originally caused by black people (ironically). Maybe it is not a true theory, of course; but, do you find yourself wanting to say "probably not" instead of "maybe not"? Would that be for scientific, or for political reasons? Since most scientists would want to say "probably not" it would not only be politically correct, it would be scientifically correct to say that, because science is what scientists do (ironically).

So much for racism; what about sexism? Well, there is a slight difference in average height between women and men. Some of the causes of that fact may well have been sexist, but the fact itself is not. It is just a fact. Still, one of the consequences of it is that the tallest adults are predominantly male. That is mainly because of the bell curve: random variation produces values that lie on a curve shaped like a bell, most of them clustered around the middle. Put a bell curve close to another one and there is a lot of overlap in the middle, but on each extreme almost all the outliers are from just one of those curves. Again, that is not sexist, it is mathematics. Now, humans do tend, quite strongly, to look up to taller humans (pun intended); which is again probably not, in itself, sexist, but it does mean that when it comes to promotions to top positions, taller humans – who are mostly men – tend to do better. This results in part of the gender pay gap.

As a result of successful legislation against sexism, the gender pay gap is now mostly affecting older women, and those in the top jobs (who tend to be older). So the height factor might be a significant part of this gap. An irony is that trying to make things fairer by promoting women who apply for top jobs – probably taller women – over the least attractive men who apply – probably the smaller of the men – would worsen the actual unfairness of our looking up to taller humans. It is likely that seeking equality elsewhere could similarly worsen sexism (although describing too many particular cases would probably get me labelled as sexist). Another irony is that you might, in general, want to pay women more than fairness would demand, in order to get more role models, to help to combat centuries of sexism.

There is lots of irony when it comes to abortion. Those who are against it because it is, they believe, murder, are usually for the death penalty; this is called "ironic", ironically, by people who are usually called "clever". It is not ironic, because the death penalty is for guilty people, not babies. No, abortion is ironic because those who are for it usually say that they are defending women's right to control their own bodies. They are defending no such thing (as follows), and it is ironic that they are regarded as the scientific ones, the ones saying it how it is, the clever ones. People who are for abortion do not want women to have the right to try to abort their own baby, they want doctors to have the right to do it for them. So consider a woman who wants to kill herself (or just hurt herself). If it was about a woman's right to control her own body, then there should be a demand that doctors be allowed to help the woman to kill herself (or hurt herself, etc.) just because she wants it. To say that it is OK if she is terminally ill, but not OK if she is mentally ill, is like saying that abortion is OK if the baby or the mother was going to die anyway and the abortion is to save the life of the other (the mother or the baby respectively).

Plenty of people who wanted to kill themselves end up glad that they did not; and similarly, plenty of people who wanted an abortion end up glad that they did not have one. Less closely analogous: what about a woman's right to control her own home? Can she kill anyone who is in her home? Would the difference be that unborn babies are not people? That, I think, is the difference. If so, then it is not really about the right of women to control their own bodies, it is about whether and when an unborn baby is a person. The day before birth? A month before? A month and a day? A month and two days? There is no good scientific answer, and so the proponents of abortion on demand talk about something else, something that sounds good; which makes this a sort of populism. Ironically, the left have turned "populism" into a word that means something else, by way of their predominance in academia and the intelligent media. Actually, that is not just ironic.

Wednesday, September 05, 2018

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).

Sunday, September 02, 2018


F is six

The ancient Greeks used alpha, when doing arithmetic, instead of our one, 1, and had beta for 2, gamma for 3 and so forth. The symbol for gamma was like a reflected L, and so our letter F began life as digamma (an anagram of mad magi), which they used for 6. Since the New Testament was originally written in ancient Greek, the Number of the Beast was basically FEX, which is indeed the name of a man.

Or, is Mark Zuckerberg the Antichrist? Revelation 13:17 is:
no man might buy or sell, save he that had the mark, or the name of the beast, or the number of his name
And even this very Blogger have facebook at the top of these pages (under “More”). Even the BBC, who are not supposed to advertise (they famously say “sticky-backed plastic” instead of sellotape) show us the Facebook symbol, the ancient Greek number of his name.

Or, is the Beast Money? This post is about physics (ancient Greek for “nature” and pronounced fizzyx) though, and in particular an urban myth:

Urban Myth

The myth concerns a small machine with a clock-face of light bulbs, one of which is on at any given time when the machine is on. Which bulb it is that is on is determined by which one was previously on and how a small radioactive sample has decayed in the unit of time of the machine: the light will have moved one place clockwise if the sample emitted a detected particle in that time, one place anticlockwise if no such particle was detected. The unit of time is such that there is a 50% chance of a particle being detected in that time.

The machine (which is clearly from The Fury, which was about magical Jews, who were also exactly the sort of people who wrote the Bible, according to the Bible itself) is placed in front of a human subject, who has to try to make the light move clockwise by really wanting it to. The myth is that some physicists built such a machine and got the light to move clockwise more often than would be expected from random motion. Other physicists tried to repeat the experiment and, according to the myth, they got no positive results. The original results were explained as being random after all (quite likely because things that are truly random tend to look more structured), or as being due to methodological errors (they were physicists, not parapsychologists).

However, the original results could hardly have been undermined by similar results not being obtained with other people, who might not have had the same abilities. Further research, of an appropriate kind, would have had to have been carried out, because of the enormous implications for physics: Modern physics is based on particle physics, which is based on observations of what is essentially a scaled-up and much more complicated version of the small machine. If physicists could affect the events inside particle accelerators, via their expectations and desires, then that would throw a whole new light on particle physics, and hence the whole of physics.

Had there been something to find, then they would have found it, and physics would have changed accordingly. Now, we would have noticed, because there would hardly be this urban myth floating about had they wanted to keep it secret (the possibility of secrets arises because of the money in innovation, the world wars, the cold war and so forth). So, it must be a myth because conversely, had there not been anything to find, then that would have put micro-psychokinesis to sleep forever. Since the physicists would have wanted to be quite sure, hence they would have been quite exhaustive in their investigations. Whereas, parapsychologists still investigate micro-psychokinesis.

Evidence that there is micro-psychokinesis could therefore include, given the above, the very success of relativistic physics. The equations of relativistic space-time were developed at the end of the nineteenth century, even before the quantum-mechanical nature of moving particles had been noticed. It was an amazing discovery, and it is even more amazing that it has not fundamentally altered because it is inconsistent with quantum mechanics. But, particle physicists keep finding patterns that verify it. (Particle physicists are very proud of their understanding of the sophisticated mathematical language of relativistic physics, of course.)

The main evidence is the operation of the human brain (how else is the mind going to influence the working of the brain?) and a few paranormal phenomena; but how odd that the equations of relativity arrived out of nowhere when wars were still being fought on horseback. We are asked to imagine how the world would look were we going at the speed of light, because physics should always look just the same. Then we find such equations. But what if bats did physics? Would Bert the bat assume that nothing goes faster than the speed of sound? What if his equations ended up being very complicated? What would that mean? That he was a batty bat?!

There was an experiment to test the speed of light in various directions. It was small enough to fit inside a plane. So imagine how sensitive it was. Then it was taken up in a plane!! (There just are not enough exclamation marks there, but I do not wish to come across as mad!) (?)

Saturday, September 01, 2018

Curry's Paradox

Last year’s new SEP entry on Currys paradox followed in Haskell Curry’s footsteps by saying nothing about where our reasoning goes wrong in such informal versions of the paradox as the examples in the introductory section of that entry, the first of which was as follows:
Suppose that your friend tells you: “If what I’m saying using this very sentence is true, then time is infinite”. It turns out that there is a short and seemingly compelling argument for the following conclusion:

(P) The mere existence of your friend’s assertion entails (or has as a consequence) that time is infinite.

Many hold that (P) is beyond belief (and, in that sense, paradoxical), even if time is indeed infinite.


Here is the argument for (P). Let k be the self-referential sentence your friend uttered, simplified somewhat so that it reads “If k is true then time is infinite”. In view of what k says, we know this much:

(1) Under the supposition that k is true, it is the case that if k is true then time is infinite.

But, of course, we also have

(2) Under the supposition that k is true, it is the case that k is true.

Under the supposition that k is true, we have thus derived a conditional together with its antecedent. Using modus ponens within the scope of the supposition, we now derive the conditional’s consequent under that same supposition:

(3) Under the supposition that k is true, it is the case that time is infinite.

The rule of conditional proof now entitles us to affirm a conditional with our supposition as antecedent:

(4) If k is true then time is infinite.

But, since (4) just is k itself, we thus have

(5) k is true.

Finally, putting (4) and (5) together by modus ponens, we get

(6) Time is infinite.

We seem to have established that time is infinite using no assumptions beyond the existence of the self-referential sentence k, along with the seemingly obvious principles about truth that took us to (1) and also from (4) to (5).
That may look rather formal to you, but formal logic is not even logic (it is mathematics); the above is just very well laid out. Note the two uses of modus ponens, the two sets of three steps, with the first three steps, (1), (2) and (3), all beginning “Under the supposition that”. You should note that because we cannot always use modus ponens within the scope of a supposition, e.g.:

(a) Under the supposition that modus ponens is invalid under a self-referential supposition, (A) implies (C).

(b) Under the supposition that modus ponens is invalid under a self-referential supposition, (A).

With (a) and (b) we have, under the supposition that modus ponens is invalid under a self-referential supposition, a conditional and its antecedent, but it would of course be absurd to use modus ponens within the scope of that supposition, to obtain

(c) Under the supposition that modus ponens is invalid under a self-referential supposition, (C).

There was, then, at least one step in the above argument for (P) that stood in need of some justification, i.e. the step to (3). Were no other step deficient in justification we could conclude, from the absurdity of (P), that the step to (3) was invalid.

Of course, it would be more satisfying to see where precisely that step lacked justification, so presumably we need an analysis of what would in general count as justification for such a step. For now, note that in order to get to (3) we used modus ponens under the supposition that k is true, which was no less self-referential than the supposition that modus ponens is invalid under a self-referential supposition. In the step to (3) we had k being true instead of (A) implying (C), and “k is true” instead of (A).

To progress, we need to step back, I think, because I suspect that the reason why we find (P) to be beyond belief is that the above argument for (P) has exactly the same logical structure as a clearly invalid argument for the obviously false (Q):
Let your friend say instead: “If what I’m saying using this very sentence is true, then all numbers are prime”. Now, mutatis mutandis, the same short and seemingly compelling argument yields (Q):

(Q) The mere existence of your friend’s assertion entails (or has as a consequence) that all numbers are prime.
My suspicion is based on the fact that one could conceivably have a valid argument for

(S) The mere existence of “happy summer days” entails (or has as a consequence) that time is infinite.

For a start, the mere existence of some words can entail the actual existence of something important, as when Descartes proved that he existed: I think, therefore I am. But furthermore, there is a surprisingly valid argument from the existence of “happy summer days” to the probable existence of a transcendent Creator of all things ex nihilo (this links to that), and it might only take some tidying up to get to (S), because such a Creator is an omnipotent being endlessly generating a temporal dimension. (Such a Creator could possibly have a logical existence proof, because of its unique ontological status.) And of course, were there a valid argument for (S), then there would be an identical, equally valid argument for (P).

Anyway, a six-step argument for (Q) that is identical to the Curry-paradoxical argument for (P) would have, in place of k, some such l as “If is true, then all numbers are prime”. And is likely to be about as true as not, because (i) it is about as true as not that a contradiction follows from a statement that is about as true as not, since such a statement is about as false as not, and also because (ii) one informal meaning of is the obvious meaning of the liar sentence “is not true”, which is, if meaningful, about as true as not, according to my The Liar Proof. And of course, our logic is naturally suited to that part of our language where propositions are either true or else not true, exclusively and exhaustively. For a proposition that is otherwise, we have natural clarification procedures that enable us to construct new propositions that are more suited to logical reasoning. So, it seems likely that propositions that might be about as true as not should be ruled out from the use of modus ponens within the scope of a too-self-referential supposition (to say the least).

Curry’s paradox entered into the analytic philosophy of the Forties, where the logical paradoxes were in general thought to be reasons for replacing our informal logical reasoning with formal logical reasoning (via the mathematical philosophy of formal languages), on such grounds as that (i) one would not expect primates, even highly evolved primates, to be able to reason perfectly, and (ii) the physical sciences use mathematics to get to the underlying physical laws. However, why would such primates not take themselves to be reasoning perfectly adequately; and why should I be doing mathematics when I am really doing philosophy?

Sunday, August 26, 2018

Sequence and Consequence

I add grains of sand to the same place, one by one, and eventually I have a heap of sand.

Originally I did not have a heap of sand, of course, and in between having such numbers of grains – just a few originally, and later on lots of them – there are numbers of grains with which I do not obviously have a heap, and do not obviously not have a heap.

Let n be some such number. Perhaps it is the case that when I have n grains of sand I have a heap of sand; but certainly, if that is not the case, then n grains is not a heap. So for all n, either n grains is a heap, or else n grains is not a heap. It follows logically that there must be some n, say N, such that N is large enough for N grains of sand to be a heap of sand, but (N – 1) is not large enough for (N – 1) grains of sand to be a heap of sand.

The problem is that the rather vague meaning of “heap” does not allow there to be such a number. Given any heap of sand, if I take just one grain away from it, then I would still have a heap of sand. Quite generally, if you take one unit away from any very large number of units, then you still have a very large number of units. So it must, after all, be false that for all n either n grains is a heap or else not. That is surprising, but at least we have a picture of how that can be false, with this picture of my adding grains of sand one by one. So while it is surprising, it is not beyond belief; it is not paradoxical (despite what many analytic philosophers seem to think).

It is not, then, always the case that, given some description, either that description is true or else, if that is not the case, the description is not true. It could be that there is no fact of the matter of whether the description is true or not, as when meanings are a little vague and we have a borderline case. Now, in such cases the description cannot simply be neither true nor not true, as that is just to say that it is not true while ruling out its being not true. The only thing left for us to say is that it is about as true as not. In the picture above, the sand changes from not being a heap, to being about as much a heap as not, and then more a heap than not. That is just a fact, of such matters.

One consequence of the possibility of assertions being about as true as not is that there is a relatively simple resolution of the Liar paradox.
Note that it can indeed make sense to say that a proposition is about as true as not. Consider, for another example, how if I say of some artwork that I think good “That is not good” then I am lying. I am saying something false. What would be true would be for me to say that it was good. And if some artwork seems to me to be about as good as not – and you must allow me such a possibility, because such matters are matters of opinion – then it would be true for me to say that it was about as good as not. In such a case, it might make sense (as follows) for it to be about as true as not for me to say that it was good. And if so, and if we all agreed that a particular artwork, say Z, was about as good as not, then it would make sense for “Z is good” to be about as true as not. Does that make sense? Well, were it simply true to say that Z was good, then were “Z is not good” true too, it would follow that Z was good and not good, whereas the symmetry of Z being about as good as not means that we could hardly have one true and the other not true. And if it was instead not true to say that Z was good, so that it would not be the case that Z was good, then there would be a problem with it being not true to say that Z was not good, because that would mean that Z was good.

Saturday, August 18, 2018

The Modal Paradox

Here in the actual world A we have a ship, let us name it the good ship Theseus, made of 1000 planks. Our first intuition X is that the same ship could have been made of 999 of these planks plus a replacement for plank #473. In possible-worlds terms that means there is another world B where the same good ship Theseus exists with all but one plank the same as in our world A, and only plank #473 different. But then in world B one has a good ship Theseus made of 1000 planks, and by the same sort of intuition, there must another world C where the same good ship Theseus exists with all but one plank the same as in the world B, but with plank #692 different. That means for us back in world A there is another world C where the good ship Theseus exists with all but two planks the same as in our world A, but with planks #473 and #692 different, so one could have two planks different and still have the same ship. The same sort of considerations can then be used to argue that one could have three planks different, or four, or five, or all 1000. But that is contrary to our other intuition Y [a ship made of a thousand different planks would have been a different ship].
     The modal paradox resembles well-known paradoxes of vagueness, such as the heap and the bald one, for which proposed solutions are a dime a dozen — except that here what seems to be vague is the relation of identity. And the idea that ‘is the very same thing as’ could be vague is for many a far more troubling idea than the idea that ‘heap’ or “bald’ is vague. Indeed, according to many, it is an outright incoherent idea.
From John P. Burgess, "Modal Logic, In the Modal Sense of Modality" pp. 40-1.
     In the fourth line Burgess says "by the same sort of intuition," which is a relatively weak sort of thing to say, and so it might be where the paradoxical reasoning started to go wrong. Our original intuition X was that in the actual world A, Theseus could have had one plank different and still have been the same ship. We know, equally intuitively, that X coheres perfectly well (somehow) with the intuition, Y, that in the actual world A, Theseus could not have a thousand planks different and remain the same ship. Whereas the ship in B is not exactly the same as the ship in A. And the meanings of all our words are rooted in A. The further we get from A, the more vague we might expect our meanings to become.
     In the first line of the second paragraph Burgess observes that proposed solutions to the paradoxes of vagueness are "a dime a dozen" and that means, I think, that anything I might say is bound to pointlessness; but, onward and upwards. And it is certainly the case that "the very same thing" often equivocates, as in the case of the famous clay statue: if that thing is squashed then, while it is the same lump of clay, it is no longer a statue at all. So let us look at a very different formulation of the ship paradox, one that does not involve modality at all. The following is by Ryan Wasserman, "Material Constitution", §1:
[...] the story of the famous ship of Theseus, which was displayed in Athens for many centuries. Over time, the ship’s planks wore down and were gradually replaced. [...] Suppose that a custodian collects the original planks as they are removed from the ship and later puts them back together in the original arrangement. In this version of the story, we are left with two seafaring vessels, one on display in Athens and one in the possession of the custodian. But where is the famous Ship of Theseus? Some will say that the ship is with the museum, since ships can survive the complete replacement of parts, provided that the change is sufficiently gradual. Others will say that the ship is with the custodian, since ships can survive being disassembled and reassembled. Both answers seems right, but this leads to the surprising conclusion that, at the end of the story, the ship of Theseus is in two places at once. More generally, the argument suggests that it is possible for one material object to exist in two places at the same time. We get an equally implausible result by working backwards: There are clearly two ships at the end of the story. Each of those ships was also around at the beginning of the story, for the reasons just given. So, at the beginning of the story, there were actually two ships of Theseus occupying the same place at the same time, one of which would go on to the museum and one of which would enter into the care of the custodian.
For myself, I do not think that the ship in the museum was the famous Ship of Theseus, I think that what was left of that ship is now the custodian's ship. But I concede that it could be that the museum ship is legally the ship of Theseus. It would then follow that the custodian's ship was not, for legal purposes, the ship of Theseus. So I think that there are at least two senses of "ship of Theseus" in play. What we can say about those senses is another matter. Our language is inextricably rooted in the usual events of the actual world. But it could be scientific to know that there are those two senses even before our theories of such senses have become a dime a dozen. And we might find clues as to what we should be saying from related puzzles.
     There are many intuitive puzzles about identity. For just one example, suppose that the world that you are in splits into two worlds, so that you are in one while an identical person is in the other. You are the same person as the original you, of course, but so is the person in the other world. So, that other person is the very same person as the original you, who is the very same person as you, and yet that other person is not the very same person as you. So, either personal identity is not always a transitive relation, or else this scenario is impossible. And while the latter is certainly plausible, it seems to me to be the wrong sort of conclusion to draw from this scenario. Knowing whether it is the wrong sort of conclusion to draw or not may well be a prerequisite for getting anywhere with such puzzles as the modal paradox, because intuitions for "is the very same thing as" not ever being at all vague seem to be very similar to intuitions that it must always be transitive.