I have for a long time inclined towards ifthenism in mathematics: the idea that mathematics discovers truths of the form "If these axioms are true, then this thing is true as well."
Two things have weakened my inclination to ifthenism.
The first is that there really seems to be a privileged natural number structure. For any consistent sufficiently rich recursive axiomatization A of the natural numbers, by Goedel’s Second Incompleteness Theorem (plus Completeness) there is a natural number structure satisfying A accordingto which A is inconsistent and there is a natural number structure satisfying A according to which A is consistent. These two structures can’t be on par—one of them needs to be privileged.
The second is an insight I got from Linnebo’s philosophy of mathematics book: humans did mathematics before they did axiomatic mathematics. Babylonian apparently non-axiomatic but sophisticated mathematics came before Greek axiomatic geometry. It is awkward to think that the Babylonians were discovering ifthenist truths, given that they didn’t have a clear idea of the antecedents of the ifthenist conditionals.
I am now toying with the idea that there is a metaphysically privileged natural number structure but we have ifthenism for everything else in mathematics.
How is the natural number structure privileged? I think as follows: the order structure of the natural numbers is a possible order structure for a causal sequence. Causal finitism, by requiring all initial segments under the causal relation to be finite, requires the order type of the natural numbers to be ω. But once we have fixed the order type to be ω, we have fixed the natural number structure to be standard.
3 comments:
I don't understand what makes the proposition (If ZFC, then continuum hypothesis) true or false under ifthenism.
What about something like this: There are possible worlds that have concrete entities and a binary relation epsilon between them such that these concrete entities together with the epsilon relation satisfy the axioms of ZFC. If all such worlds are ones where the concrete entities and the epsilon relation also satisfy CH, then your conditional is true. If they are not all like that, then the conditional is false.
The concrete entities might be angels and epsilon might be friendship, for instance.
Makes sense, thank you.
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