Schwartz’s Pernicious Influence of Mathematics
Jack Schwartz The Pernicious Influence of Mathematics on Science
In Nagel, Suppes & Tarski (Eds.) Logic, Methodology and the Philosophy of Science 1962.
Also Kac, Rota & Schwartz Discrete Thoughts: Essays in Mathematics, Science and Philosophy 1992.
“… The mathematician turns the scientist’s theoretical assumptions, i.e., convenient points of analytical emphasis, into axioms, and then takes these axioms literally. This brings with it the danger that he may also persuade the scientist to take these axioms literally. …
The literal-mindedness of mathematics thus makes it essential, if mathematics is to be appropriately used in science, that the assumptions upon which mathematics is to elaborate be correctly chosen from a larger point of view, invisible to mathematics itself. … That form of wisdom which is the opposite of single-mindedness, the ability to keep many threads in hand, to draw for an argument from many disparate sources, is quite foreign to mathematics. This inability accounts for much of the difficulty which mathematics experiences in attempting to penetrate the social sciences. …
Related to this deficiency of mathematics, and perhaps more productive [of] rueful consequence, is the simple-mindedness of mathematics—its willingness, like that of a computing machine, to elaborate upon any idea, however absurd; to dress scientific brilliancies and scientific absurdities alike in the impressive uniform of formulae and theorems. … The result, perhaps most common in the social sciences, is bad theory with a mathematical passport. …”
Schwartz quotes Keynes:
“It is the great fault of symbolic pseudomathematical methods of formalizing a system of economic analysis…that they expressly assume strict independence between the factors involved … Too large a proportion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.”
“The intellectual attractiveness of a mathematical arguments, as well as the considerable mental labor involved in following it, makes mathematics a powerful tool of intellectual prestidigitation—a glittering deception in which some are entrapped, and some, alas, entrappers. …
… Typically, mathematics knows better what to do than why to do it. Probability theory is a famous example. An example which is perhaps of far greater significance is the quantum theory. The mathematical structure of operators in Hilbert space and unitary transformations is clear enough, as are certain features of the interpretation of this mathematics to give physical assertions, particularly assertions about general scattering experiments. But the larger question here, a systematic elaboration of the world-picture which quantum theory provides, is still unanswered. Philosophical questions of the deepest significance may well be involved. Here also, the mathematical formalism may be hiding as much as it reveals.”
The emphasis above is mine.
As Popper says (Objective Knowledge), “the role of mathematics in the empirical sciences is somewhat dubious in several respects.” To take Keynes’ quote, we may distinguish between mathematics proper and pseudomathematics. Mathematics proper is a discipline in its own right, separate from its application domains, but supplying material for them to use as tools. Nothing in the above seems to be critical of such mathematics ‘as such’.
Diabetics might criticise those who make cakes, but the problem is not with the cakes ‘as such’. It would be a problem if the cakes were inadequately labelled and shoppers inadequately informed, or if there were no suitable alternatives available. In the case of mathematics proper the traditional situation has been straightforward: mathematics comes certified as mathematics: for anything else, buyer beware. Thus the whole task of the scientist has traditionally been to determine which equations are appropriate and how to interpret. If there is a lack of appropriate mathematics it may be that there has been inadequate funding, or such developments have been discouraged. Most bakers would make cakes for diabetics, if they had an outlet. The rewards and constraints on mathematicians are largely determined by funding, and hence by the fashions and supposedly practical needs within the user communities. Are scientists, for example, challenging mathematicians to develop tools to support the handling of uncertainty and quantum concepts?
Where one has an equation such as:
effect = resources*skill
does anyone think that these are properly mathematical? If not, should it be given to a proper mathematician without explanation?
My own experience is that typically less experience users of mathematics share many of the views that Schwartz is critical of above, with more experienced users having a broader view. They do not necessarily talk mathematics but do have their own ways of understanding these things, such that one can readily develop a dialogue. Thus the traditional picture has been of mistakes at the junior level corrected at the senior level.
In my own work I have noticed some worrying trends:
- Less experienced users often seek to justify their attitudes on the grounds of some local or short-term success.
- The spread of computerization has made limited technical successes seem more impressive and longer experience seem less relevant.
- Many organisations are under pressure to perform in the short-term, thus valuing understanding of long-run issues less.
- In some cases this has lead or contributed to relatively inexperienced people getting into senior positions, with no insights into long-run issues. (‘No-one is minding the shop’.)
Thus the UK became trapped in a high-growth high-risk financial ‘bubble’, with no way out. This raises the question of whether finance was corrupted by mathematics proper, by pseudo mathematics or by the inability of its pay-masters to face up to reality. Similarly, much of what we call science is conducted as part of product development for which the harsh clarity of science proper (properly based on mathematics proper) may not be acceptable. Sometimes pseudomathematics can support a desired pseudoscience which gives ‘results’. I agree that there is a problem, but who to blame? And more importantly, who should be doing what about it?
I would argue that the problems identified in the above paper are a symptom of the relationship between mathematics and science being blurred, with a pernicious influence on both. Similarly for pseudo-sciences, such as economics. I do not think that improvements in mathematics alone would make much difference. What seems to be needed is a greater appreciation of the proper role of mathematics in science and similar endeavours, emphasising its role in criticism as well as computation.