[FOM] 31st Novembertagung 2020

Tue Apr 28 10:43:13 EDT 2020

On Tue, Apr 28, 2020 at 1:18 AM Louis H Kauffman <kauffman at uic.edu> wrote:

>        Perhaps you could write "Mathematics is the collection of theorems
> derived from established axioms and the activity of creating and
> discovering new definitions and new systems of axioms.”.  I realize that I
> am opening my self to the criticism that I am confusing definitions with
> axioms. But in fact that is my point. Any axiom system, any formal system
> is a form of definition for a mathematical or logical domain.
>

He goes on to describe how K=theory and cobordism theory were created by
generalizing from the Eilenberg-Steenrod axioms for cohomology.  Those
terrific discoveries had a lot of concrete applications, such as Adams's
result on Vector Fields on Spheres
https://projecteuclid.org/euclid.bams/1183524456  I just want to add that
the Eilenberg-Steenrod axioms were also powerfully and quite deliberately
generalized far beyond topology.  Already in the 1940s Cartan and Serre,
eventually working with Eilenberg, generalized axiomatic cohomology to
complex manifolds and also to group cohomology.  Axiomatic group cohomology
in turn quickly became the basis for modern Class Field Theory in
arithmetic, leading to lots of concrete results in number theory..

When Serre turned these ideas towards proving the Weil Conjectures he was
not overly attached to the axiomatic approach,  He continued using and
developing good bits of it, but also pursued more concretely calculational
approaches he thought might work.  His friend Grothendieck was very much
attached to axiomatics -- but exactly as Lou says the point was not to
prove theorems from the existing axioms.  Grothendieck set out to find new
more general axioms for cohomology that could apply to arithmetically
defined spaces (which in fact had not been precisely defined yet when he
began, so these axioms for cohomology could not be specifically
designed for those spaces) . This led to his famous Tohoku paper
axiomatizing an idea of Abelian category, specifically AB5 category, and a
canonical cohomology theory for each AB5 category.   Within a few years
this (combined with crucial ideas from Serre) proved to be just the thing
for creating etale cohomology and settling the Weil conjectures.

Of course those are just examples I know. There are lots of others waiting
for historians, philosophers, and logicans to explore. Kauffman Is quite
right.

Colin

> For example, Eilenberg and Steenrod discovered axioms of homology and
> cohomology  theory and then these axioms were shifted to the known axioms
> for generalized homology and cohomology theories. These generalized axioms
> become the guides for understanding many new theories such as K-Theory and
> cobordism theories. So axioms evolve and we do not just deduce from axioms;
> we also look for models for those axioms. Thus, speaking pedagogically, we
> have the axioms for a group and we can deduce many properties of groups
> just from these axioms. But if you ever had a course in group theory where
> you were first introduced to groups only via the axioms and never shown any
> examples (models) of groups, then you would surely understand what kind of
> strange torture this could be to a beginner. Some of the early attempts to
> do non-Euclidean geometry were like that — investigations of a negation of
> the parallel postulate. But it was the emergence of models of non-Euclidean
> geometry that really clinched the matter. Similarly in present day
> learning, all students know the coordinate method for doing geometry, which
> is Euclidean if one uses the standard inner product. Then one does not need
> to have the Euclid axioms because they are Theorems in this model. This is
> not often pointed out to students, and it should be pointed out. But what
> are the axioms then? They are the axioms for real numbers. But still, we
> keep studying and as students perhaps we learned a construction for real
> numbers. Then the axioms for real numbers became theorems about the
> construction. And now what are the axioms? One keeps going back and in the
> usual ways we were taught, it all goes back to the axioms for set theory.
> But my point is that it is these intermediate axiom systems that are
> related to new definitions (such as generalized cohomology theories) that
> are the bread and butter of working mathematicians.
>
> And for the physicists, assumptions that certain situations involving
> summations should have integrals as generalized summations are natural
> “working axioms”. The fact that mathematicians have not yet been able to
> ground these assumptions in set theory or category theory does not mean it
> will not eventually be done. (Consider the case of infinitesimal calculus!)
> Working mathematical physicists have procedural activity with functional
> integrals and know how to check whether they are on track or off track even
> though they are strictly speaking outside of the usual foundations. In fact
> they themselves would admit that it is not fully satisfactory. These are
> situations where new axioms systems may evolve.
> Best,
> Lou Kauffman
>
>
> On Apr 26, 2020, at 11:56 PM, D. Kant <d.kant at web.de> wrote:
>
> 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> <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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