[FOM] 31st Novembertagung 2020
Timothy Y. Chow
tchow at math.princeton.edu
Tue Apr 28 13:27:31 EDT 2020
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
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