I was recently rereading an essay by Karl Popper, “A Realistic View of Logic, Physics, and History” published in his collection of essays, *Objective Knowledge: An Evolutionary Approach*, because it discusses the role of reductivism in science and philosophy, a topic about which I’ve written a number of previous posts discussing the microfoundations of macroeconomics.

Here is an important passage from Popper’s essay:

What I should wish to assert is (1) that criticism is a most important methodological device: and (2) that if you answer criticism by saying, “I do not like your logic: your logic may be all right for you, but I prefer a different logic, and according to my logic this criticism is not valid”, then you may undermine the method of critical discussion.

Now I should distinguish between two main uses of logic, namely (1) its use in the demonstrative sciences – that is to say, the mathematical sciences – and (2) its use in the empirical sciences.

In the demonstrative sciences logic is used in the main for proofs – for the transmission of truth – while in the empirical sciences it is almost exclusively used critically – for the retransmission of falsity. Of course, applied mathematics comes in too, which implicitly makes use of the proofs of pure mathematics, but the role of mathematics in the empirical sciences is somewhat dubious in several respects. (There exists a wonderful article by Schwartz to this effect.)

The article to which Popper refers appears by Jack Schwartz in a volume edited by Ernst Nagel, Patrick Suppes, and Alfred Tarski, *Logic, Methodology and Philosophy of Science*. The title of the essay, “The Pernicious Influence of Mathematics on Science” caught my eye, so I tried to track it down. Unavailable on the internet except behind a paywall, I bought a used copy for $6 including postage. The essay was well worth the $6 I paid to read it.

Before quoting from the essay, I would just note that Jacob T. (Jack) Schwartz was far from being innocent of mathematical and scientific knowledge. Here’s a snippet from the Wikipedia entry on Schwartz.

His research interests included the theory of linear operators, von Neumann algebras, quantum field theory, time-sharing, parallel computing, programming language design and implementation, robotics, set-theoretic approaches in computational logic, proof and program verification systems; multimedia authoring tools; experimental studies of visual perception; multimedia and other high-level software techniques for analysis and visualization of bioinformatic data.

He authored 18 books and more than 100 papers and technical reports.

He was also the inventor of the Artspeak programming language that historically ran on mainframes and produced graphical output using a single-color graphical plotter.^{[3]}

He served as Chairman of the Computer Science Department (which he founded) at the Courant Institute of Mathematical Sciences, New York University, from 1969 to 1977. He also served as Chairman of the Computer Science Board of the National Research Council and was the former Chairman of the National Science Foundation Advisory Committee for Information, Robotics and Intelligent Systems. From 1986 to 1989, he was the Director of DARPA‘s Information Science and Technology Office (DARPA/ISTO) in Arlington, Virginia.

Here is a link to his obituary.

Though not trained as an economist, Schwartz, an autodidact, wrote two books on economic theory.

With that introduction, I quote from, and comment on, Schwartz’s essay.

Our announced subject today is the role of mathematics in the formulation of physical theories. I wish, however, to make use of the license permitted at philosophical congresses, in two regards: in the first place, to confine myself to the negative aspects of this role, leaving it to others to dwell on the amazing triumphs of the mathematical method; in the second place, to comment not only on physical science but also on social science, in which the characteristic inadequacies which I wish to discuss are more readily apparent.

Computer programmers often make a certain remark about computing machines, which may perhaps be taken as a complaint: that computing machines, with a perfect lack of discrimination, will do any foolish thing they are told to do. The reason for this lies of course in the narrow fixation of the computing machines “intelligence” upon the basely typographical details of its own perceptions – its inability to be guided by any large context. In a psychological description of the computer intelligence, three related adjectives push themselves forward: single-mindedness, literal-mindedness, simple-mindedness. Recognizing this, we should at the same time recognize that this single-mindedness, literal-mindedness, simple-mindedness also characterizes theoretical mathematics, though to a lesser extent.

It is a continual result of the fact that science tries to deal with reality that even the most precise sciences normally work with more or less ill-understood approximations toward which the scientist must maintain an appropriate skepticism. Thus, for instance, it may come as a shock to the mathematician to learn that the Schrodinger equation for the hydrogen atom, which he is able to solve only after a considerable effort of functional analysis and special function theory, is not a literally correct description of this atom, but only an approximation to a somewhat more correct equation taking account of spin, magnetic dipole, and relativistic effects; that this corrected equation is itself only an ill-understood approximation to an infinite set of quantum field-theoretic equations; and finally that the quantum field theory, besides diverging, neglects a myriad of strange-particle interactions whose strength and form are largely unknown. The physicist looking at the original Schrodinger equation, learns to sense in it the presence of many invisible terms, integral, intergrodifferential, perhaps even more complicated types of operators, in addition to the differential terms visible, and this sense inspires an entirely appropriate disregard for the purely technical features of the equation which he sees. This very healthy self-skepticism is foreign to the mathematical approach. . . .

Schwartz, in other words, is noting that the mathematical equations that physicists use in many contexts cannot be relied upon without qualification as accurate or exact representations of reality. The understanding that the mathematics that physicists and other physical scientists use to express their theories is often inexact or approximate inasmuch as reality is more complicated than our theories can capture mathematically. Part of what goes into the making of a good scientist is a kind of artistic feeling for how to adjust or interpret a mathematical model to take into account what the bare mathematics cannot describe in a manageable way.

The literal-mindedness of mathematics . . . 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. The single-mindedness of mathematics reinforces this conclusion. Mathematics is able to deal successfully only with the simplest of situations, more precisely, with a complex situation only to the extent that rare good fortune makes this complex situation hinge upon a few dominant simple factors. Beyond the well-traversed path, mathematics loses its bearing in a jungle of unnamed special functions and impenetrable combinatorial particularities. Thus, mathematical technique can only reach far if it starts from a point close to the simple essentials of a problem which has simple essentials. 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. The inability accounts for much of the difficulty which mathematics experiences in attempting to penetrate the social sciences. We may perhaps attempt a mathematical economics – but how difficult would be a mathematical history! Mathematics adjusts only with reluctance to the external, and vitally necessary, approximating of the scientists, and shudders each time a batch of small terms is cavalierly erased. Only with difficulty does it find its way to the scientist’s ready grasp of the relative importance of many factors. Quite typically, science leaps ahead and mathematics plods behind.

Schwartz having referenced mathematical economics, let me try to restate his point more concretely than he did by referring to the Walrasian theory of general equilibrium. “Mathematics,” Schwartz writes, “adjusts only with reluctance to the external, and vitally necessary, approximating of the scientists, and shudders each time a batch of small terms is cavalierly erased.” The Walrasian theory is at once too general and too special to be relied on as an applied theory. It is too general because the functional forms of most of its reliant equations can’t be specified or even meaningfully restricted on very special simplifying assumptions; it is too special, because the simplifying assumptions about the agents and the technologies and the constraints and the price-setting mechanism are at best only approximations and, at worst, are entirely divorced from reality.

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. Unfortunately however, an absurdity in uniform is far more persuasive than an absurdity unclad. The very fact that a theory appears in mathematical form, that, for instance, a theory has provided the occasion for the application of a fixed-point theorem, or of a result about difference equations, somehow makes us more ready to take it seriously. And the mathematical-intellectual effort of applying the theorem fixes in us the particular point of view of the theory with which we deal, making us blind to whatever appears neither as a dependent nor as an independent parameter in its mathematical formulation. The result, perhaps most common in the social sciences, is bad theory with a mathematical passport. The present point is best established by reference to a few horrible examples. . . . I confine myself . . . to the citation of a delightful passage from Keynes’

General Theory, in which the issues before us are discussed with a characteristic wisdom and wit:“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 and lose all their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume they all vanish. Too large a proportion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the initial assumptions they reset on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentions and unhelpful symbols.”

Although it would have been helpful if Keynes had specifically identified the pseudomathematical methods that he had in mind, I am inclined to think that he was expressing his impatience with the Walrasian general-equilibrium approach that was characteristic of the Marshallian tradition that he carried forward even as he struggled to transcend it. Walrasian general equilibrium analysis, he seems to be suggesting, is too far removed from reality to provide any reliable guide to macroeconomic policy-making, because the necessary qualifications required to make general-equilibrium analysis practically relevant are simply unmanageable within the framework of general-equilibrium analysis. A different kind of analysis is required. As a Marshallian he was less skeptical of partial-equilibrium analysis than of general-equilibrium analysis. But he also recognized that partial-equilibrium analysis could not be usefully applied in situations, e.g., analysis of an overall “market” for labor, where the usual *ceteris paribus* assumptions underlying the use of stable demand and supply curves as analytical tools cannot be maintained. But for some reason that didn’t stop Keynes from trying to explain the nominal rate of interest by positing a demand curve to hold money and a fixed stock of money supplied by a central bank. But we all have our blind spots and miss obvious implications of familiar ideas that we have already encountered and, at least partially, understand.

Schwartz concludes his essay with an arresting thought that should give us pause about how we often uncritically accept probabilistic and statistical propositions as if we actually knew how they matched up with the stochastic phenomena that we are seeking to analyze. But although there is a lot to unpack in his conclusion, I am afraid someone more capable than I will have to do the unpacking.

[M]athematics, concentrating our attention, makes us blind to its own omissions – what I have already called the single-mindedness of mathematics. Typically, mathematics, knows better what to do than why to do it. Probability theory is a famous example. . . . Here also, the mathematical formalism may be hiding as much as it reveals.