Alternatives to Axiomatic Method

Question

In his article The Pernicious Influence of Mathematics upon Philosophy (see Chapter 12 of this book) Rota says (my emphasis),

The axiomatic method of mathematics is one of the great achievements of our culture. However, it is only a method. Whereas the facts of mathematics, once discovered, will never change, the method by which these facts are verified has changed many times in the past, and it would be foolhardy not to expect that it will not change again at some future date.

My question is,

Has there been any research regarding the method itself "by which these facts are verified" as has been mentioned in the previous paragraph that is not the axiomatic method? If so, can some relevant literature regarding this issue be mentioned?

Answer 1

The non-axiomatic method is common in the work of applied mathematicians. At the turn of the 20th century leading mathematicians like Felix Klein, while acknowledging the importance of axiomatisations, warned that they may be a second fiddle to other fruitful developments where such axiomatisations are not that relevant. Klein engaged a team of top-level scholars to develop a many-volume encyclopedia of applications of mathematics to fields ranging from physics to engineering. Axiomatisations are almost irrelevant for these important mathematical developments.

Felix Klein is one of the superstars of 20th century mathematics and the validity of his work is beyond dispute. Notice that the term "rigor" did not occur in the formulation of your question. Its meaning is dubious and especially at the philosophy SE could come across as naive. Many mathematicians (though by no means all) tend to identify "rigorous mathematics" with "mathematics done in a ZFC axiomatic framework" and from that point of view certainly there couldn't be any rigorous work outside the axiomatic framework, yes. But that's a rather reductive view that would probably not be shared by many editors at the philosophy SE.

Here is a useful post to consult if you think most of mathematical activity has something to do with axiomatic frameworks.

Answer 2

The process of mathematics has evolved. We have good records of pre-axiomatic mathematics, and each of these was rigorous in its time.

Mathematics started as an experimental science. The Egyptians contrived formulas out of intuitive notions of geometry, and they measured to determine the accuracy of their guesses. We have manuscripts containing formulas for things such as the frustum of a cone, that are simply wrong, and are later edited out as they proved not to be accurate.

When mathematics had proven itself adequately in predicting the positions of things in the sky, it began to be used to fix the positions of things on the earth. The Egyptians evolved trigonometry adequate for surveying out of astronomy, but only after the latter consistently worked. They experimentally imposed rigor through testing. And that mathematics was rigorous for its time.

The Pythagoreans adjusted this experimentalism into a more genuine way of looking internally to assemble logical patterns out of mathematical derivations. But they felt their methods, which relied heavily on fractions, were threatened by the existence of irrational numbers. If they had the internal assurance that modern mathematicians have, this would not have been a cause for concern. They felt that their intuitions were not really trustworthy. But the mathematics they were doing was rigorous for its time.

Calculus came into being out of an intuitive grasp of Cartesian geometry. The Real numbers and their infinite divisibility offered Newton, for instance, the intuition of numbers as the 'fluxions' of quantities. It was a notion deeply embedded in mechanical awareness, and it held together only in situations where that correlation of numerical evolution and movement held. Infinitessimal numbers could be used because they expressed the feeling that all realistic mathematical situations were highly differentiable. It took over a century to totally remove this approach from mathematics. But again it was rigorous in its time.

Now we have abstracted away the motivating physical and metrical inuitions from the vast majority of mathematics, and reduced it to axiomatics on the model of Greek geometry. We have formalized the notions that were elaborated out of more direct study into deductive systems.

We feel better about being able to cast results into an axiomatization, but the source material for modern mathematics does not arise out of the axioms, it comes, as it always has, from intuitive models, and is translated into the validating axiomatization.

Nor is it even possible for all modern mathematics. The study of the universe of all ordinals, something set theory and the related logic cannot enclose, has continued and is coming close to having a full theory. This puts this kind of math 'out there' where it is not useful to other branches. But that has not put it to an end.

So axiomatization is just the latest form Mathematics takes, and it is not even the form that all current mathematics takes. It is a powerful tool for checking results, but it is not great at generating them. So even if it remains a major method, it is not really the method, only the means of assuring the other methods stay bound together and capable of sharing ground.

Answer 3

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 is not).

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 2): "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 (http://philosophy.stackexchange.com/questions/4175/cantor-and-infinities/4287#4287) 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 (https://en.wikipedia.org/wiki/Tarski%27s_theorem_about_choice), and 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".

Answer 4

Whereas the facts of mathematics once discovered will never change

Whilst true, the relative importance of these facts at the frontier of mathematics will change; for example, group theory was once called 'gruppenpest' whereas now it's generally recognised and known as the mathematics of symmetry, though symmetry is a wider concept than this, proof: look at the concept of groupoid, which is a wider concept, and rigorously defined.

Euclid, is generally noted as the paragon of the axiomatic method; but look at the term 'geometry'; geo, of the earth, metry of measurement, and from there one can discern where mathematics as geometry began from, mensuration, or the measurement of the earth.

Feynman divided science roughly into Greek-style mathematics, and Babylonian style; it's the former that takes the lions share of attention in the discussion of the philosophy of mathematics, but the very fact that Feynman divided science into these two parts should alert us to the fact - and I am using this word deliberately and deliberatively - that the latter is important, and is understudied and under-theorised; such a study one might call this an anthropology of mathematics, the study of how men (and women) make mathematics and why.