What is the relation between proof in mathematics and observation in physics?

Question

Recently in his 2015 Hirzebruch Lecture in Bonn, Arthur Jaffe re-amplified his famous perspective that finding proof in mathematics is analogous to making experimental observation in physics. In paraphrase: One observes that a proof exists, much as one observes that a fundamental particle exists.

Accordingly, Jaffe suggests to say "experimental mathematics" for the activity of finding proof of theorems (and to say "theoretical mathematics" for activity such as partially checking the Riemann hypothesis by computation).

On first sight, this might seem to go against the grain. But on reflection, I think Jaffe has an excellent point here.

I am wondering if this kind of sentiment has not been voiced also by type theorists such as Martin-Löf. Is there decent citable literature that would expand on the close relation between, on the one hand, proof and/or truth judgement and on the other hand: observation?

Answer 1

This point of view is better reflected if we change "observation" to "experiment" in the title, mere observation is more analogous to conjecture, so it may be somewhat misleading. This is how Jaffe and Quinn phrased it in their original 1992 proposal:"we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences... We use the term theoretical mathematics for the speculative and intuitive work; we refer to the proof-oriented phase as rigorous mathematics". And they explicitly declined to use "experimental mathematics" in the sense described in the OP, to them "experimental mathematics" is part of "theoretical mathematics":

"Although the use of proof in mathematics is functionally parallel to experiment, we are not suggesting that proofs should be called “experimental” mathematics. There is already a well-established and appropriate use of that term, namely, to refer to numerical calculations and computer simulations as tests of mathematical concepts. In fact, results of computer experiments are frequently presented in a away we could call theoretical". We had a discussion of related philosophical issues in What makes something mathematics?

The proposal and responses to it from some leading mathematicians and physicists (Atiyah, Mandelbrot, MacLane, Witten, etc.) are available on arxiv. Most responses are negative, but largely on Jaffe and Quinn's idea of institutionalizing their separation between theoretical and rigorous mathematics, and making theoretical mathematics a stand alone discipline. The mere thesis that rigorous proof serves validation purposes in mathematics, just as experiments do in physics, is hardly controversial. If anything, most responders argue that the heuristic aspect is essential to the normal functioning of mathematics, and splitting off "rigorous mathematics" would sterilize it.

Martin-Löf did not participate, but another constructivist, Chaitin, did. He is largely supportive of the idea of theoretical mathematics, and uses incompleteness to argue that its methods are unavoidable in principle:

"information-theoretic approach to incompleteness makes incompleteness appear pervasive and natural... I therefore believe that elementary number theory should be pursued somewhat more in the spirit of experimental science. Euclid declared that an axiom is a self-evident truth, but physicists are willing to assume new principles like the Schrodinger equation that are not self-evident because they are extremely useful. Perhaps number theorists, even when they are doing elementary number theory, should behave a little more like physicists do and should sometimes adopt new axioms". His subsequent paper Randomness & Complexity in Pure Mathematics develops this line of reasoning further.

But if I understand the last paragraph correctly what you are looking for may be more like MacLane's vigorous defense of "rigorous mathematics", and the role of uncontaminated proof as a truth pillar in it. To him, Jaffe and Quinn give "theoretical mathematics" too much credit, "Jaffe and Quinn misappropriate the word “Theoretical” as a label for what is really speculation". In his Despite Physicists Proof is Essential in Mathematics published by Synthese he elaborates:

"The answer depends on a correct understanding of the philosophy of mathematics... mathematics is that part of science which applies in more than one empirical context... The axioms needed for this formulation must then be such as to hold in all the examples. And how is it with the consequences of these axioms? They are not established by example. They are those established by proof – rigorous proof – following the logical canons of proof. In other words, proof (and not experiment or speculation) is what is required in all of that part of science which is mathematics, and this requirement is there because of the very nature of mathematics".

Answer 2

I don't have a answer for you in the terms that are asked for, but there is an observation by Aristotle (Physics VIII.3) that you might find apposite:

As for movement, it would be strange if we failed to notice the downward motion of a stone; nor do we fail to notice that it is at rest upon the earth.

It's almost as though Aristotle is anticipating the popular critiques of Greek science in antiquity, and answering them. But the prosaic answer is more likely that he had to face similar critics in his own time and place rather than being a philosopher capable of foretelling or prophesying.

The point here is that experimental observation, in a precise and uniform manner, was a Baconian innovation. He theorized and eulogized it, but the older notion of what constitutes an observation stands - when one simply takes nature itself as a laboratory.

Answer 3

The linked paper from 1993 contains a serious critic of practices in theoretical physics at the time:

This unreliability is certainly a problem in theoretical physics, where the primary literature often becomes so irrelevant that it is abandoned wholesale. I. M. Singer has compared the physics literature to a blackboard that must be periodically erased. Physicists traditionally obtain much less benefit from the historical background of a problem, and they are less apt to search the literature. The citation half-life of physics papers is much shorter than in mathematics.

A similar critic is applied to the practice of publishing research announcements:

Some areas in the Russian school of mathematics have extensive traditions of theoretical work, usually conducted through premature research announcements. From the numerous possible examples we mention only two. [...] 1954 Kolmogorov announced [...] proofs were achieved by Arnold in 1959 for the analytic case and by Moser in 1962 for the smooth case.

[...] In 1973 the respected mathematicians Dobrushin and Minios published an announcement of that result. Two years later when no indication had come from the Russians of a proof, Glimm, Jaffe, and Spencer resumed their work on the problem and eventually gave two different proofs. A couple of years after that Dobrushin and Minios published a retraction of their original announcement.

The paper points out the devastating consequences similar practices in mathematics had in the past, and tries to propose solutions how to avoid those consequences and allow theoretical work:

Theoretical work should be explicitly acknowledged as theoretical and incomplete; in particular, a major share of credit for the final result must be reserved for the rigorous work that validates it.

Only one solution is proposed for research announcements (revealing the authors opinions):

Research announcements should not be published, except as summaries of full versions that have been accepted for publication. Citations of unpublished work should clearly distinguish between announcements and complete preprints.

I was not able to access the 2015 Hirzebruch Lecture in Bonn. It would certainly be interesting whether Jaffe believes that the problematic practices still persist, or whether he was rather reporting on the success of his proposed solutions.

Even if this may not be the answer the questioner has hopped for, the question prominently features that link to the 13 page opinion piece and a link to an unavailable lecture. Hence it should be allowed to point out that the opinion piece contains strong statements which overshadow (even so they may be true) any potential discussions about details of the suggested analogies used in the opinion piece.

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Jaffe justifies to say "experimental mathematics" for the activity of finding proof of theorems via:

A relevant observation is that most theoretical physicists are quite respectful of their experimental counterparts. Relations between physics and mathematics would be considerably easier if physicists would recognize mathematicians as "intellectual experimentalists" rather than think of them disdainfully as uselessly compulsive theorists. The typical attitude of physicists toward mathematics is illustrated by a passage from a book of P. W. Anderson, "We are talking here about theoretical physics, and therefore of course mathematical rigor is irrelevant and impossible."

This makes it clear that Jaffe is talking about sociological phenomena here. Because no comparable phenomena exists (or existed) within the mathematical community itself, there was never any need to voice this kind of sentiment by type theorists such as Martin-Löf. The close relationship between verifiability, falsifiability and meaningfulness on the one hand, and experimental observation in physics, rigor and proof in mathematics, and absence of metaphysical speculations in philosophy has been clearly voiced by proponents of logical positivism.