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Additional resources for A source book in physics
Certainly, it doesn't seem as if an alien from outer space watching our lack of progress would have to conclude that our mathematical terms in such a case didn't refer to anything. The only way around this latter ob- jection that I can see is to suggest that mathematical development always tends toward a Peircean ideal (a certain perfect axiomatization) that can then be taken by definition to fix the references of mathematical terms. But Godel's theorem makes this move extremely implausible. 43 Mark Richard raised a version of this objection to me in conversation.
These can arise because of the semantic structure of the description being used by the speaker to fix the reference of the term at issue; for primary A- and `A'-mishaps with respect to the terms occurring in that description can derivatively cause mishaps with respect to the term the description is associated with. Also, secondary A-mishaps for a term can arise because of primary A-mishaps with respect to collateral information pertinent to the description being used to fix the reference of that term; secondary `A'-mishaps for a term can arise because primary `A-mishaps with respect to other terms can cause mishaps with the term at issue.
36 34 Maddy (1990, chapter 2) makes this point about set theory. 35 One might hope to hold on to the "description theory" for numbers by suggesting that individuals who cannot produce the desired axiom system defer their reference to those who can. The considerations already raised suggest this move is hopeless, but in any case surely this was false of Ramanujan, whose grasp of numbers in some ways surpassed that of anyone around him to whom he could have deferred referentially. On Ramanujan, see Snow's introduction in Hardy 1967; see also Newman 1956.