Rota himself hints at methods able to complement the axiomatic method in his article:
- historical analysis
- psychological explanations
- reversal considerations
Rota blames mathematics for developments of analytical philosophy to become ahistorical and separate from psychology. Which is unfair, since mathematics was never ahistorical.
The reversal considerations are the only real alternative to the axiomatic method among the methods hinted at by Rota. His presentation is again slightly unfair:
Perform the following thought experiment. Suppose you are given two formal presentations of the same mathematical theory. The definitions of the first presentation are the theorems of the second, and vice versa. This situation frequently occurs in mathematics. Which of the two presentations makes the theory “true?” Neither, evidently: what we have are two presentations of the same theory.
The program of reverse mathematics was founded by Harvey Friedman in 1975. Rota certainly knew it, and I claim it was the motivation why he wrote that paragraph. Let me give some reasons why reverse mathematics is different from the axiomatic method:
- the axiomatic method based on ZFC is absolute mainstream mathematics, while reverse mathematics is only a special topic in mathematical logic studied by very few people
- theorems of the form "The following characterizations are equivalent: 1. ... 2. ... 3. ... " are common in mathematics, but reverse mathematics goes beyond that in also proving strict inequality between different systems
- one common way in reverse mathematics to prove strict containment between two systems is by showing that the bigger proves consistency of the smaller
- the reverse mathematics program works over a base theory which is presented axiomatically, the proofs of equivalence or consistency are axiomatic too, but the conclusions of strict containment (or inequality) go beyond the axiomatic method
However, I am not sure whether Asaf Karagila would agree that the way (carefully adapted versions of) Gödel's theorems are used as meta-theorems to prove strict containment in reverse mathematics qualifies as going beyond the axiomatic method. Even if I would elaborate it in more detail, it could still "feel like a lot of words without much meaning" to him. And maybe I would really be missing the point, by staying too close to Rota's implicit hints and simplified picture. The real point might be that Harvey Friedman (and some others) have not given up Hilbert's dream of finding a foundation of mathematics worthy of that name. Contrary to ZFC,
a base theory like Friedman's elementary function arithmetic (and it second order version RCA*0) is not arbitrary:
This "privileged" status of arithmetic statements is sort of the
reason why I protest that PA is not a weak system. I find EFA much
better in this respect, since the "mathematician in the street" with a
"reasonable" background in mathematical logic (as Scott Aaronson has
without doubts) will be in a much better position to understand the
significance of the independence results for that specific system
(like that it cannot prove cut elimination), what it entails to accept
that system (the allowed computations are no longer "feasible"), and
what it entails to go beyond that system (exponentiation is an
analytic function in complex function theory, but superexponentiation
Pardon me another quote, to really make the point that EFA is not arbitrary:
At least for fragments of arithmetic, a more explicit dividing line
between weak and strong systems has been used (in the cited chapter
"The line between strong and weak fragments is somewhat arbitrarily
drawn between those theories which can prove the arithmetized version
of the cut-elimination theorem and those which cannot; in practice,
this is equivalent to whether the theory can prove that the
superexponential function is total."
According to this dividing line, Friedman's exponential function
arithmetic (EFA) is a weak fragment of arithmetic. But if the question
is whether independence results for P != NP can be proved, then EFA
feels like a really interesting candidate, precisely because it cannot
prove cut-elimination. It would cast an interesting light on the role
of higher-order reasoning.
The psychological explanations are more challenging, at least for me. I don't know much about the role of psychology in mathematics, but I certainly agree with Rota that psychology is too important for philosophy to delegate it to the psychology department. But even in mathematics, the point of a proof is still to explain to another mathematician ("to convince him") why a certain fact is true, and psychology is not irrelevant for this. And psychology also plays a role for drawing wrong conclusions from mathematical theorems.
The historical analysis really helps set things straight in mathematics (and philosophy) in a way the axiomatic method will never be able to match. Let me again quote myself here:
But speaking of politics, one should be aware of the fact that
Cantor's (philosophy behind) set theory and his insistence that the
only real question was consistency was politically motivated
to prevent abuse of power by established mathematicians like Leopold
Kronecker. (He even founded the "Deutsche Mathematiker Vereinigung"
for that same purpose.) And it is not clear to me how much Alfred
Tarski and John von Neumann played a role in establishing first order
logic + ZFC as the undisputed foundations of mathematics. At least
Tarski had experienced the "power of the establishment" before
both had the experience of translating work from their own language
into German and later into English. So they knew the value of
established foundations for doing mathematics, as opposed to having
fruitless discussions (which would in the end be decided by the power
of the establishment).
This quote shows that the current way the axiomatic method is used in mathematics is fundamentally different from the way the ancient greeks used the axiomatic methods in Euclidean geometry. Can it even be considered to be the same method? It is the same method in the context of Agrippa's trilemma, but it is not the same method in the context of mathematical practice.
As long as you don't clearly specify what you mean by the axiomatic method, it will be very hard to explain how an alternative is different from the axiomatic method. You may see a similar issue with stoicfury claiming that science exclusively uses the scientific method, and I wondering how mathematics with its axiomatic method and medicine with (its Hippocratic oath and) its double blind studies (to handle placebo effects) were not using different methods. Here I also argued using historical analysis to point out that "mathematics and medicine are significantly older than the scientific method".