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).

Fissix

F is six

in ancient Greek,
and "fissix" sounds like physics,
and the external physical world is perceived in six basic ways:

Looking at it with our eyes,
hearing it with our ears,
smelling it with our noses,
tasting it with our tongues,
feeling it with our skins, and
knowing which way is up, via our inner ears.

People tend to forget about that sixth way of perceiving the external physical world because people tend to think of the sixth sense as being an ability to see spirits, or some vague sense of impending doom, or a sense of being stared at. Is there a sense of being stared at? Would that be a seventh way of perceiving the external physical world? Or a way of perceiving the external mental world? Are ghosts ectoplasmic, or psychic? I suppose that if psychologists demonstrated that there is a sense of being stared at, and if they got a good materialistic theory of how it worked (involving, say, the nature of quantum-mechanical collapses), then philosophers would say that that was indeed a seventh way of perceiving the external physical world. That is all very iffy though.

What about the sixth way of perceiving the external physical world with our inner ears? Well, "going up in the world," people say, and "feeling a bit down." People naturally associate up with good and down with bad. But physics came of age when the heavens fell under its laws, in the seventeenth century, thanks to Newton.

By pursuing empirical truth, instead of the next epicycle, Britain began the industrial revolution and built the biggest empire the world had ever seen. And the other sciences followed in the footsteps of physics. In the eighteenth century chemistry emerged from alchemy, in the nineteenth century biology became Darwinian, and in the twentieth century physics became Einsteinian. But the twentieth century was a century of cultural wars, as well as actual wars that got bigger than wars had ever been. Soviet biologists doubted the Darwinian foundation of biology. A few mathematicians even doubted the set-theoretical foundation of twentieth-century mathematics. Still, no one doubted the quantum-mechanical foundation of chemistry. And no one doubted the Einsteinian foundation of high-energy physics, which was a little odd because quantum mechanics all but refuted the four-dimensionalism of Einsteinian spacetime.

Almost all of the physical evidence supported the equations of quantum mechanics and Newton's equations, and the rest of it was evidence that could only be perceived by physicists who were assuming Einstein's equations and making very expensive observations, funded by those with the funds to fund the military, observations that could hardly be independently verified.

Physicists are still pursuing the next dark object at the cutting-edge of high-energy physics, and inventing ever more elaborate Einsteinian theories; and philosophers are still giving "empirical truth" new definitions. But if you could choose between appearing professional and being able to learn, which would you choose?

Curry's Paradox

Last year’s new SEP entry on Curry’s 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 l is true, then all numbers are prime”. And l 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 l is the obvious meaning of the liar sentence “l 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?