Tuesday, August 12, 2008

Philosophy of Mathematics

It's now 10 years since my interest in mathematics became an interest in the philosophy of mathematics, but I'm not much closer to knowing why mathematicians standardly assume an axiom of infinity (which is described in the second comment below)—not the historical or sociological reasons, but what philosophical (or amateur-scientific) reasons those applying standard mathematics have, for thinking that an axiom of infinity is true.
......That they have found few problems with that axiom in a hundred years, for example, is an answer in a different ball-park—cf. how medieval astronomers found few problems with assuming (what seems clear) that the stars circle (and the planets epicircle) a fixed Earth. Ultimately the interesting question was does the Earth turn? And while our being able to imagine a star in each cubic lightyear of some infinite space is at least a metaphysical reason (e.g. see Benardete's "Infinity: An essay in metaphysics"), mathematicians have naturally been moving away from their previous reliance upon such geometrical intuitions (towards greater rigour).
......Not that standard mathematicians need any very good metaphysical reasons for finding most interesting those models of arithmetic that contain such an axiom (e.g. there are benefits to having a common language, and there was originally the epistemic possibility to explore) but if they don't have any such reasons then physicists and metaphysicians should perhaps not presume that some such axiom is true. And note that the question arises given that numbers are objective (that proof aims at, but is not to be identified with, truth), so that it is hardly an answer to have included Constructive mathematics within academia.

Tuesday, August 05, 2008

Liars, Divine Liars, and Semantics

Let ‘L’ name the sentence, “This sentence is not expressing a truth.” L seems to be saying that L is not being used to say anything true. If so then, if L is expressing a truth then L is not expressing a truth, whence L is not expressing a truth. But then, L seems to have been expressing a truth—that L is not expressing a truth—after all. In short, L is a paradoxical Liar sentence.
......Letting ‘N’ denote (for brevity) the property of not expressing a truth, L seems to say that L is N, but hardly straightforwardly. If we let T be “L is N” then I can say that maybe L only seems to say that L is N because of its resemblance to T. Note that if L is expressing a truth—if it is not N—then L is N (paradoxically), but if T is not N then it only follows that L is N (relatively straightforwardly).
......On this (fairly popular and) Traditional Resolution, L is a special sort of nonsense (and hence not true), its apparent sense being due to its resemblance to T (which is simply true). A related paradox is therefore Moore’s. Let G be the sentence “George believes that S is certainly P, but S is not P.” G may be true, but if George utters G there is something odd about it (even with “George believes” replaced by “I believe), and were George omniscient he could only say G by lying.
......There are many other ways to resolve the Liar (e.g. Kripke’s, e.g. see subsection 3.2 of the recent SEP entry on Self-Reference), but this is certainly one possibility. And why should we expect there to be a unique resolution? Natural linguistic entities are usually only partially (and fuzzily) defined—maybe such incompleteness facilitates their flexibility?—and Liar sentences are relatively artificial, so it may well be that different resolutions yield different (but equally legitimate) extensions of our languages.
......Atheistic uses of Divine Liars (as follows) have therefore begged the question (so far as I can see). Let ‘DL’ name the sentence “God knows that this sentence is N.” If DL is not N—if it is expressing a truth—then God (an omniscient being, here) exists, and DL is N. So DL is N. But if DL is N then God should know that DL is N, which is what DL seems to be saying; so the atheist is tempted to conclude that God does not exist.
......A theist, on the other hand, could regard DL as some evidence that such appearances are deceptive, at least when it comes to sentences like the Liar (and to a lesser extent with Moore’s paradox), and an intelligent agnostic would be unable to make much of DL in the absence of compelling reasons to regard a different way of resolving such sentences (e.g. Kripke's) as the only correct one (not just the conventional one, as that would certainly beg the philosophical question).
......Incidentally it is usual to use non-well-founded names to define Liars (as Grim defined his Divine Liar, see second comment below), e.g. F = “F is N,” which creates additional problems for this approach (e.g. saying “F is N” is then to speak nonsense), so note that the atheist would additionally need to justify such an extension of the natural process of naming, to reap any benefit from such problems. (Prima facie it would be wise to analyse Liars and such naming seperately.)

Friday, August 01, 2008

A Platonistic potential infinity?

One would naturally think that, for any time in the future (and similarly, for any point in space, any collection in a hierarchy of collections and so forth), either it is the nth day from now, for some natural number n, or it is infinitely far into the future, so that an infinite future might be divided into the finitely and the infinitely remote, the former region being infinitely many (i.e. aleph-null) days long since otherwise there would be no nth day for some n. But maybe it is not the case that, for any time in the future, it is either the nth day (for some n) or not.
......If it was an nth day, for some n, it would be so objectively; but maybe that property, of being an nth day for some n, is sufficiently like an indefinite property (as a consequence of the endlessness of the natural number sequence, not because of anything fuzzy about units, additions or repetitions, nor because of anything specifically temporal) for it not to follow that an arbitrary future time would either be an nth day (for some n
) or not. (Such a possibility is indicated by how the natural assumption, of aleph-null, leaves us with a proper class of such cardinal numbers.)