What is the relevance of applicability to the natural sciences in pure mathematics?

Question

I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") mathematics for applications in the natural sciences.

One very notable epiphany I've encountered recently is in relation to Oystein Ore's commentary about how mathematics has had a tremendous history in the form of "brain teasers" purely for entertainment purposes. [Number Theory and its History by Oystein Ore (Sterling Professor of Mathematics, Yale U. (McGraw-Hill, 1948), pp. 25-26]

This leads me to believe, and to much more readily accept, the idea that, culturally, the primary motivation of modern pure mathematics is simply entertainment as a kind of game and not at all for applications in the natural sciences.

In fact, it seems mathematics faculty in the great universities disdain the fact that they have to teach mathematics courses for physicists and engineers. And likewise, it seems physics textbooks often care very little about mathematical rigor. Generally, it seems physicists care more about arriving at physically testable hypotheses or standard theories than about how they arrived at those hypotheses. But physicists will engage in mathematical rigor if such rigor is within easy reach.

I have seen some literature attempting some sort of an axiomatic development of one or another physical theory. But it seems there has not been much interest in this idea. However, Euclidean geometry stands as an example of the idea of using rigorous argument to develop empirically testable, "real world" theorems.

A key motivation is the hope and hypothesis that a more mathematically rigorous theory will create far more reliable conclusions of a physically testable nature so that there is a far lesser burden on the need for (often rather expensive) experimental testing of conclusions. But is this hope and hypothesis warranted? I am thinking in particular here of what seems an excellent advanced textbook in classical mechanics that nevertheless seems to have some flaws or omissions in rigor: H. C. Corben and Philip Stehle: Classical Mechanics, 2nd Ed. (Dover, 1977). How important is such mathematical rigor from the standpoint of scientific methodology?

An expanded, advanced, well-developed theory of the relationship of mathematical rigor to the complexity and costs of experimental testing seems likely, but it seems the basic questions here need to be addressed before running over hill and dale in the development of expansive, potentially tenuous theories.

So when it comes to the applicability of rigorously derived mathematics to, say, physics, the idea seems to be simply to "play with it" and see if you might come up with some sort of interesting, testable hypotheses. It seems there is very little motivation culturally for thinking that rigor in pure mathematical theory has any bearing on developing good, testable hypotheses in physics.

That seems to be a current fact of life culturally about the relationship of mathematical rigor to credibility or plausibility of physical hypotheses which may result from such rigorous treatment.

Can anyone provide more insight into this matter or help to correct any misconceptions I may have about this?

Answer 1

What is described in the OP applies to the side of mathematics that Quine called "recreational". But there is another side, which still covers a lot of "pure" mathematics, that is tied to empirical sciences much more closely. Roughly, this is the side that supplies "representational aids" for modeling in sciences. Since we do not know in advance which modeling devices will come handy in the future the "free play" is crucial for success of this side as well. Here is from Quine's Reply to Parsons (1986), where he classifies mathematics into applied, its "rounding out", and "recreation":

"Pure mathematics, in my view, is firmly imbedded as an integral part of our system of the world. Thus my view of pure mathematics is oriented strictly to application in empirical science. Parsons has remarked, against this attitude, that pure mathematics extravagantly exceeds the needs of application. It does indeed, but I see these excesses as a simplistic matter of rounding out. We have a modest example of the process already in the irrational numbers: no measurement could be too accurate to be accommodated by a rational number, but we admit the extras to simplify our computations and generalizations.

Higher set theory is more of the same. I recognize indenumerable infinites only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g., בω or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights. Sets that are compatible with 'V = L' in the sense of Gödel's monograph afford a convenient cut-off."

There is a vast literature on this under the headline of the Quine-Putnam indispensability argument as to how much of mathematics is indispensable enough to inherit "ontological rights" from the empirical theories it services (and hence counts as "rounding out"). On this issue Quine's austere minimalism, and his cutoff at V=L in particular, has come under heavy fire, including from his own students, Parsons and Maddy. Maddy, for example, advocates maximization of mathematical entities as the most pragmatic attitude, given the role of mathematics in science, instead of Quine's minimization. And the severing of his connection between the indispensability and platonism about mathematical entities is now the mainstream view: representational aids can be indispensable without conferring any sense of real existence upon the entities they posit.

This explains the indifference (sometimes even disdain) of many mathematicians for any particular applications of mathematics they do to practical matters. If the point is to maximize representational devices it matters little if they were put to some use already, or not yet. The story of rigor fits into this picture as well. If development of mathematics is not directly controlled by empirical applicability, contra Quine, one needs separate controls to limit accumulation of errors, formalization and rigor are just that. On the other hand, in applied domains, where mathematical nonsense will likely lead to empirical mismatch, and quickly, one can afford to be lax, or such is the common attitude. This is confirmed by the phenomenon of string theory, which is currently cut off from empirical testing, and where the engagement of physicists with highly abstract parts of differential/algebraic geometry, including very intricate rigorous arguments, is far above average. Witten's Fields Medal is an early acknowledgment of that, so is the recent work of Vafa.

Leng in What's Wrong With Indispensability? even argues that, in Quine's sense, all of mathematics is "recreational", but it does not mean that it is just "entertainment", or that it is detached from reality:

"These observations lead naturally to an understanding of the relationship between mathematics and science in which areas of mathematics are used to model physical phenomena... If Colyvan is right (and I think he is) that mathematics that is not assumed by science to be true should be seen as recreational (and given some important status as such), then it follows from the modelling picture of the relationship between mathematics and science that all mathematics is recreational.

The success of this modelling should be no real surprise: many of the mathematical stories we create are created with scientific interpretations in mind. Euclidean geometry is best understood as a mathematical story whose axioms were meant to model truths about physical space. It was developed with our assumptions about real points and lines in mind, but it was always an empirical question whether it actually did provide the best model of points and lines in the physical world... Our basic arithmetic is developed so as to model our counting practice. In fact, it does this so successfully that we do not tend to see it as a model at all. Similarly, we can see our language of real numbers in mathemat ics as having been created to model our measuring practices."

Indeed, it is not hard to connect even the purest of the pure "mathematical games" to very humble and mundane roots through six (often two-three) degrees of separation, Quine's minimalism and Hardy-like disdain for applications notwithstanding (see When were the concepts of pure and applied Mathematics introduced?). This vindicates Wittgenstein's idea that mathematics, including its free play parts, "hardens" empirical regularities inherent in our practices of coping with the world. This means that we circumscribe some approximation of them in axiomatic systems and then elevate those to the exceptionless status by outlawing empirical "deviations", see Empirical Regularities in Wittgenstein's Philosophy of Mathematics by Steiner.

Some other good sources on the subject are Peressini's review of Colyvan's book The Indispensability of Mathematics (Colyvan is one of the last die-hard defenders of Quinean orthodoxy), and Azzouni's Applied Mathematics, Existential Commitment.

Answer 2

Mathematics can be crudely summarised as describing possible relationships between quantifiable values. Given that we live in a Universe in which the natural phenomena and their interrelationships are quantifiable, it is hardly surprising that mathematics can be called in to play to model them. If mathematics is capable of being extended indefinitely to describe a vast set of relationships between quantities, then only a small subset of mathematics is ever likely to be useful for physics, for example. Quine's view that the 'excesses' of pure mathematics are a 'rounding out' is opinionated nonsense, equivalent to saying that visual art is the rounding out of the diagrams we need to illustrate physics.