[FOM] 31st Novembertagung 2020

[Louis H Kauffman]

Dear Folks,
I fail to see how the axiomatic method has been in any way harmful to mathematics.
It took a long time to realize that there was a fundamental difference between accepting axioms as truths and seeing axioms as starting points for deductive systems.
If by the “axiomatic method” we mean the (modern) point of view that a system of axioms is used as a starting point for deduction, then this view of mathematics is one of the great 
movements toward intellectual freedom. The next step in realizing that no single axiomatic system (in the above sense) can be complete (if consistent and rich enough) completes that movement into freedom of
mathematical thought. One can still search for wide formal foundations, even in the face of incompleteness. The distinction between what can be accomplished with given axioms and what may be done without them cannot be made without a clear sight of those axioms. A good example is the remarkable success of methods of functional integration in physics and related topology that is still up in the air. Here is mathematics proceeding without acceptable axioms. The research problem to find those axioms or to ground these procedures in a known system (such as ZFC) is a well-defined problem even if the methods of
functional integration appear to be ill-defined. That we have axioms allows us to see our own viewpoints in formal terms.

When a graduate student, I was at a lecture by a very intuitive topologist who shall be unnamed here. Henri Cartan was in the audience. 
He became agitated. He stood up and demanded  “But what are your axioms?”. 
It is always the mathematician’s cry “What are your assumptions?”. 
Without answers or hints at answers, there is no development of mathematics.
Axioms are the crystallizations of those answers.
Best,
Lou Kauffman

> On Apr 25, 2020, at 5:34 PM, Antonino Drago <drago at unina.it> wrote:
> 
> 24 aprile 2020 Joe Shipman wrote: 
> 
> "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?"
> 
> 
> 
> Please, take note that Lazare Carnot was a great geometer and mathematician.
> 
> Moreover, the search along millennia of a proof for the parallel axiom is an instance of harm for the progress of mathematics. The search was corroborated by an axiomatic property: since for each Euclidean theorem there exists the proof of the converse, and there exists the proof of the converse of the parallel postulate, then the proof of the postulate was believed a must.
> 
> Also Lobachevsky tried to prove the postulate; but after few years he changed attitude and developed in a non axiomatic way his new theory. 
> His invention was possible because the non axiomatized trigonometry gave an idea of space from the outside of Euclidean geometry.
> 
> The same did Goedel, who in a first time, believing in Hilbert's axiomatic method, tried to prove the completeness; but later changed opinion and obtained his celebrated theorems. 
> By incidence, it is fortunate for us that von Neumann, as he said to Goedel, the day before Goedel's first communication had realize to be unsuccessful in his last attempt for proving the completeness; otherwise no one had conceded attention to the young mathematician. In previous years Von Neumann had already publicly announced to have obtained the proof.
> 
> It was the faith in the axiomatic method that led Hilbert to state the celeber dictum: "In Mathematics there is no Ignorabimus". This dictum was first denied by Goedel's undecidability result and then by a legion of undecidabilities in Mathematics. It was the discovery of the other face of the Moon. No computer theory is possible without undecidable results.
> 
> Best regards
> Antonino Drago
> 
> -----Messaggio originale-----
> Da: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] Per conto di Joe Shipman
> Inviato: venerdì 24 aprile 2020 00:10
> A: tchow at alum.mit.edu
> Cc: FOM at cs.nyu.edu
> Oggetto: Re: [FOM] 31st Novembertagung 2020
> 
> 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
> 
>