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 they 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 (which is, spookily, an anagram of mad magi), which they used for 6. Now, the New Testament was originally written in ancient Greek, and so the Number of the Beast was basically FEX, which is, spookily, indeed the name of a man. But, this post is really about physics (ancient Greek for “nature”); 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) 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!

But of course, science likes materialism, not Cartesian dualism. And it likes spiritualism even less. As for relativity, we know that there are no aliens because of it. Why can nothing go faster than the speed of light? Imagine going at the speed of light, or faster, and then emitting light. And the problem with that is that we do not know how fast we are going through space. How unlikely it is that we would be going slowly. And yet, light gets emitted just the same in all directions. 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!

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?