[FOM] 31st Novembertagung 2020

[Timothy Y. Chow]

Lou Kauffman brings up an excellent point that I think is often not 
addressed explicitly, namely that "axioms" play two related but distinct 
roles in mathematics.

The first role is to provide a starting point for the logical development 
of all, or most, of mathematics.  From this point of view, axioms tend to 
be interpreted as *propositions*---statements about the way things are in 
the mathematical universe, just as theorems are regarded as statements 
about the way things are.  This is the view of axioms that tends to 
dominate discussions in the philosophy of mathematics.

The second role is to isolate key *properties* of some type of 
mathematical object.  The axiomatization of homology and cohomology, 
mentioned by Kauffman, is an excellent example of this.  Nobody thinks 
that all of mathematics can be logically deduced from the axioms of 
homology and cohomology; that is not the point.  The point is that prior 
to the axiomatization, there were several known examples of homology and 
ochomology theories, and there was a desire to understand clearly what the 
similarities and differences were, and which properties followed logically 
from which other properties.  There was also interest in finding new 
examples.

Again, as noted by Kauffman, the second role is closely related to the 
role played by *definitions* in mathematics.  If I ask for the definition 
of a group, or a matroid, or a topological space, what jumps to the mind 
of the mathematician is a list of axioms.  (On the other hand, not all 
definitions in mathematics are like this.  For example, the definition of 
the Riemann zeta function is not regarded as an axiomatic definition; 
rather, that definition is perceived as playing the role of pinning down a 
unique mathematical object, as opposed to characterizing a class of 
mathematical objects.)

"Working mathematicians" are used to dealing with what I've called the 
"second role."  They deal with these kinds of axiomatizations on a daily 
basis.  On the other hand, the "first role" can actually cause confusion 
among working mathematicians.  Mathematicians are accustomed to thinking 
of propositions/theorems as assertions about the way things are, and so 
there is tacitly some expectation of *uniqueness*.  After all, things are 
the way they are; surely things can't be one way and another way at the 
same time.  I frequently encounter mathematicians who think that the 
existence of more than one model of PA or ZFC is some kind of defect. 
They would never think that existence of two groups that are not 
isomorphic to each other exhibits some kind of defect of the axioms for a 
group, but somehow they harbor the expectation that the purpose of the 
axioms of PA or ZFC is to nail down a unique object.

Anyway, my point is that this is yet another way in which the term "the 
axiomatic method" can be ambiguous.  For some mathematicians, it may bring 
to mind the practice of distilling the essence of some class of objects 
into a succinct list of properties, and that is a rather different 
activity from constructing formal proofs of theorems.

Tim