[FOM] 31st Novembertagung 2020

Timothy Y. Chow tchow at math.princeton.edu
Thu Apr 23 13:24:26 EDT 2020

Joe Shipman wrote:

> > On the other hand, some philosophers also argue that the axiomatic 
> > view on mathematics may be harmful in that it omits fundamental 
> > aspects of mathematical practice and idealizes mathematical reasoning 
> > in an unfaithful way.
> Which philosophers? I'm interested in any references you have on this 
> topic.

Depending on what exactly is meant by "the axiomatic view on mathematics" 
(and also on who counts as a "philosopher"!), Reuben Hersh might be an 
example.  In his essay, "How Mathematicians Convince Each Other, or `The 
Kingdom of Math is Within You,'" Hersh writes (among many other things):

    One may use formal proof as a "model" of mathematicians' proof, but
    mathematicians' proof is not formal proof.  Our starting point is
    established mathematics, not some postulated axioms, and our reasoning
    is "semantic", based on the properties of mathematical entities, rather
    than "syntactic", based on properties of formal sentences.

Note particularly his claim that "Our starting point is ... not some 
postulated axioms."


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