[FOM] From theorems of infinity to axioms of infinity

[Jeffrey Ketland]

On 9 Mar 2013, at 18:21, Michael Detlefsen <mdetlef1 at nd.edu> wrote:

> Neither does it seem plausible to say that it was simply (or even primarily) the more general shift away from the classical view of axioms (as evident or self-evident truths) to a more "hypotheticist"
> conception of them. Zermelo's axioms, after all, were supposed to form a "foundation" for set theory … that is (at least) a basis for it (them) that is secure from threat of
> further paradox. 

Mic, I think this view is plausible, a transition away from an epistemic demand for certainty, and towards a broader scientific demand for coherence, unification and explanatory power. 
There is a body K of antecedently established or accepted mathematical knowledge (cf., observational regularities & experimental data in science). A foundation for K is a coherent system which brings these pieces of knowledge together. Also, I think the related epistemic demand of having a basis that "secure from threat of further paradox" is too stringent. For how could we have secure certain knowledge in advance that our conjectures, guesses, and so on are not wrong, inconsistent, etc? So, mathematics moves into the realm of the conjectural, much like the rest of science, and its epistemic justification is, to some extent, holistic, rather than either being self-evident or certain.

> Is it perhaps that there is no answer … that is to say, is it that the only "answer" is "expediency" … that Zermelo needed infinity
> to give him the type of theory of sets he was looking for, and he saw no way to provide for the existence of infinity save that of making it
> an axiom?
> This isn't a very satisfactory "answer" to me.

I think that is the right answer, and I guess that it is unsatisfactory if one's aim is epistemic certainty. 
On the other hand, if the epistemic standards for admission into science are set too high, nothing can enter. Better to lower the bar, and instead demand that what is admitted can be gotten rid of. So, Zermelo's methodology is fine if the aim is unification and explanation.

Best wishes,

Jeff

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Jeffrey Ketland

Pembroke College, Oxford OX1 1DW
& Munich Center for Mathematical Philosophy (MCMP)

jeffrey.ketland at pmb.ox.ac.uk
jeffreyketland at gmail.com
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