[FOM] 31st Novembertagung 2020

[Joe Shipman]

Hersh is saying that the axiomatic method doesn’t capture the essence of what mathematicians do, I I am not arguing with that.

I have been trying to ask a specific question here, and I keep getting off-point answers. My question is how the axiomatic method is “harmful” to MATHEMATICS. In what ways have mathematicians been harmed, not by having the axiomatic method available as a tool which they may or may not choose to use, but by being either hampered in their regular work by an insistence upon axiomatics, or denied insight or blocked from progress due to a subject having been axiomatised in an unfortunate way?

I expect there are examples of this, but I want examples of THIS, rather than general criticism from outside mathematics that professional mathematicians may ignore.

— JS

Sent from my iPhone

> On Apr 23, 2020, at 5:19 PM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> 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."
> 
> Tim