[FOM] 31st Novembertagung 2020

[D. Kant]

Since the discussion started with our CfA, we will clarify what we talk
about. The most important point is that our claim refers to harm done to
the philosophy of mathematics and *not* to mathematics.

Original claim: “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.”

First on the notion: the axiomatic view on mathematics can be understood
in at least the following two ways:

     (1)[Mathematical activity] Mathematicians derive theorems from
established axioms.

     (2)[Mathematics] Mathematics is the collection of theorems derived
from established axioms.

One philosophical debate based on this view concerns the notion of
mathematical proof: how do mathematical proofs relate to formal
derivations in a first order system? It seems problematic to defend that
proofs are formal derivations since there are many important differences
between a proof as it is printed in a journal and a formal derivation.

Rittberg notes by referring to Buldt, Löwe and Müller: “by considering
idealised proofs we idealise away the problem, replacing it with a new
and different problem about these idealisations. These philosophers
[Buldt, Löwe and Müller] argue that the old understanding of proof
should be replaced with their more practice-oriented understanding. This
is an argument for a change in the philosophical tools we are using and
thus the authors call for a change in how epistemology is done. … I hold
that the[se] kinds of idealisations … can be harmful to our
philosophical understanding of mathematics.” (Rittberg 2015, p. 15-16)

A critical stance towards the axiomatic view is often taken by
philosophers of mathematics who care about mathematical practice. Rota
and Hersh, which were mentioned in the beginning of the discussion, are
indeed philosophers of this kind of philosophy.

However, the question of Joe Shipman concerned possible harm of the
axiomatic view to mathematics. This is another question and we do not
intend to extend the claim in this way. Though, it could be interesting
to know whether mathematicians would agree with (1) or (2) when they
characterise their own activity or discipline. And it is also
interesting to see to what extent the axiomatic method is useful for
mathematics.

Anyway, it is interesting to follow the discussion.

Best,

Deborah

References:

Rittberg, Colin (November / 2015): Methods, Goals and Metaphysics in
Contemporary Set Theory. PhD thesis, Hertfordshire. Online available:
https://uhra.herts.ac.uk/bitstream/handle/2299/17218/13036561%20Rittberg%20Colin%20-%20Final%20submission.pdf?sequence=1.

Buldt, B., Löwe, B., Müller, T., 2008, `Towards a New Epistemology of
Mathematics', Springer, available in open access.
https://link.springer.com/article/10.1007/s10670-008-9101-6

Am 26.04.2020 um 21:24 schrieb Sam Sanders:
> Dear Tim,
>
> take a look at Errett Bishop’s writings around 1967: he seems to express the idea you have in mind, namely
> that adopting a formal system for constructive mathematics would hamper its progress.
>
> Note that he changed his mind later (see “mathematics as a numerical language”).
>
> Best,
>
> Sam
>
>> On 26 Apr 2020, at 15:52, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
>>
>> On Sun, 26 Apr 2020, Louis H Kauffman wrote:
>>> 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.
>> I don't really disagree with what Louis Kauffman says here, but it makes me wonder whether *any* approach can be "harmful to mathematics."  Toward any approach whatsoever, we can adopt a take-it-or-leave-it attitude.  If it helps, adopt it; if it doesn't help, ignore it.  This way, no harm can be done.  Even sociological influences can be dismissed; suppose that Cartan's demand for axioms were actually inappropriate at that particular juncture, and that an intuitive approach without regard to axioms were the right way to make progress at that moment.  The topologist could just tell Cartan to shut up and wait his turn, and no harm would be done.
>>
>> If this is all that is meant by "the axiomatic method is not harmful to mathematics" then it verges on being vacuously true, and hence not a very interesting claim.
>>
>> Tim
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