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.
Uneasy Money 
Great column! Unfortunately graduate economics programs increasingly consider the amount of math and likely additions thereto as perhaps the most critical accomplishment. IMHO opinion, our laboratory is actually economic history.
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Having read the argument, I think there seems to be an assertion of sameness between someone who prefers to work in “pure” or theoretical mathematics, and someone who does not understand the assumptions and limitations of bringing theory into a practical framework.
The hydrogen atom example strikes me as a bit ingenuous, in that physicists merely adopted a mathematical framework to describe an interpretation of how subatomic particles interact. To be sure, there are different models, some better than others.
A different example that I think reflects our modern reality better is how dynamic programming and optimal control theory were developed in part to aid in solving the problem of getting rockets off of the ground and into orbit. The foundations for these areas were already laid down, but were refined based on current practical need. And these same models were found to have other uses, such as the workhouse DSGE model, and in biophysics, to name two.
Schwartz is most certainly correct to acknowledge that theoretical frameworks, mathematical or not, have strict limitations and cannot describe anything that lies outside the domain created by the assumptions (or axioms). A failure to understand these limitations, or do attempt to circumvent or ignore the assumptions will predictably result in disaster.
It is not so much that mathematics is simple-minded or literal-minded, but rather that we fail to or choose not to understand how to interpret the meaning of what we have written down, and to question our choices regarding the construction of the framework.
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Keynes ― the poster boy for the weakness of the economist’s mind
Comment on David Glasner on ‘Jack Schwartz on the Weaknesses of the Mathematical Mind’
David Glasner quotes Jack Schwartz approvingly: “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.”
This worn-off cliche of the small-minded mathematician is contrasted with the flamboyant artistic scientist: “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.”
This echoes Keynes’ hallucinatory self-description of the master-economist: “The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must be mathematician, historian, statesman, philosopher—in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future. No part of man’s nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near to earth as a politician.”
Never has scientific incompetence advertised itself better. Fact is that Keynes was too stupid for the elementary algebra that underlies macroeconomics.#1
Keynes ― the trained mathematician ― stated in his General Theory: “Income = value of output = consumption + investment. Saving = income − consumption. Therefore saving = investment.” (p. 63) This is provably false. The mathematically correct relationship reads Q=I−S with Q as macroeconomic profit.#2, #3
Let this sink in: the master-economist Keynes had NO idea of profit “His Collected Writings show that he wrestled to solve the Profit Puzzle up till the semi-final versions of his GT but in the end he gave up and discarded the draft chapter dealing with it.” (Tómasson et al.) Now it holds: when the foundational concepts are false the whole analytical superstructure is false. In other words, Keynes’ General Theory is scientifically worthless.#4
That is bad enough but it gets worse: After-Keynesians did NOT spot Keynes’ blunder to this day. For example, Paul Krugman still applies IS-LM. So, not only Keynes but Post- and Anti-Keynesians alike have been too stupid for the elementary algebra that underlies macroeconomics.
Economics (Walrasian, Keynesian, Marxian, Austrian) is mathematically flawed. As Georgescu-Roegen put it: “It is difficult to contemplate the evolution of the economic science over the last hundred years without reaching the conclusion that its mathematization was a rather hurried job.” This means that economic policy guidance NEVER had valid scientific foundations. Note that the fault lies NOT with mathematics but with economists. Their proven mathematical incompetence notwithstanding, the master-economists award themselves the “Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel”.#5
Egmont Kakarot-Handtke
#1 Economics, math, pluralism, and corruption
https://axecorg.blogspot.com/2019/09/economics-math-pluralism-and-corruption.html
#2 How Keynes got macro wrong and Allais got it right
https://axecorg.blogspot.com/2016/09/how-keynes-got-macro-wrong-and-allais.html
#3 Wikimedia AXEC143 Macroeconomic Profit Law
#4 For details of the big picture see cross-references Keynesianism
https://axecorg.blogspot.com/2016/09/keynesianism-cross-references.html
#5 Economics: The greatest scientific hoax in modern times
https://axecorg.blogspot.com/2019/05/economics-greatest-scientific-hoax-in.html
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In the context of what was written, it would be very interesting to know your opinion, David, about Morishima’s support from the position of Walrasian macroeconomics – Keynes’s main point. See brief survey by Roy Grieve:
«Keynes’s understanding that the elimination of unemployment is not simply a matter of “getting
prices right” can – perhaps surprisingly – be supported by consideration of what we may call
“Walrasian macroeconomics”. The point has been made by Michio Morishima, the distinguished
Japanese economist, that, in a Walrasian-type macroeconomic model, more than flexibility of
prices is required to guarantee full employment. Even if prices are fully flexible, that may not be
enough to ensure that the system will naturally tend to equilibrium at full employment. His
argument is concerned not with the stability of equilibrium (whether the system if upset will
return to equilibrium), but with a question not normally raised – the existence of equilibrium (the
existence of a set of values at which all markets in the system can be simultaneously in
equilibrium). Let us examine first of all the characteristics of a “Walrasian macro model” and
then turn to Morishima’s (rather neglected) thesis.»
* https://www.researchgate.net/publication/298972367_Price_Flexibility_and_Full_Employment_Barking_Up_the_Wrong_Neoclassical_Tree
* https://www.jstor.org/stable/10.13169/worlrevipoliecon.8.4.0480
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Inal,
Thanks for posting link to this very interesting paper.
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An excellent post on a great paper. I’m a bit late commenting but a couple of points:
First, despite your excellent summary, you understate Schwartz’s case.
Second, in the comments chiralfox, I believe, misunderstands the point of the hydrogen atom example and what Schwartz means by the literalness of mathematics
You write (emphasis mine)
“… the mathematics that physicists and other physical scientists use to express their theories is *often* inexact or approximate ”
No. Not often. Always.
Any mathematical formulation of a physical theory always and without exception involves an approximation of some kind. This is a fundamental distinction between science and mathematics and the crucial basis of Schwartz’s argument. The point has been well known to – at least 20th century – physicists, yet many outsiders, and particularly economists (eg Debreu, Lucas, Sargent, and most of their followers) seem entirely unaware of it and its implication for formal modelling of real world phenomena.
Feynman making the same point:
“… in order to understand physical laws you must understand that they are all some kind of approximation.”
“One may be dissatisfied with the approximate view of nature that physics tries to obtain (the attempt is always to increase the accuracy of the approximation), and may prefer a mathematical definition; but mathematical definitions can never work in the real world. A mathematical definition will be good for mathematics, in which all the logic can be followed out completely, but the physical world is complex and … if we have a system of discourse about the real world, then that system, at least for the present day, must involve approximations of some kind.”
“This system is quite unlike the case of mathematics, in which everything can be defined, and then we do not know what we are talking about.”
——
Chiralfox, in the comments, misunderstands the point of the hydrogen atom example.
So we start with the basic Schrodinger equation. This is a first approximation – actually a pretty decent one. But it ignores spin and magnetic and relativistic effects, so we add them. Now we have a better approximation, but it is still an approximation. So we move on to QFT. Now we have a more accurate (and complicated!) view of reality, but it is still an approximation. And this is the basic point: we can increase the accuracy of the approximation but we cannot get rid of it. And this is why a too literal use of deductive reasoning as in mathematics (esp. Bourbakism) is dangerous.
And finally a couple of additional quotes from the paper:
“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 question, central to the scientific investigation but intensely disturbing in the mathematical context – what happens to all this if the axioms are relaxed? – is thereby put into shadow.”
…
“The physicist rightly dreads precise argument, since an argument which is only convincing if precise loses all its force if the assumptions upon which it is based are slightly changed, while an argument which is convincing though imprecise may well be stable under small perturbations of its underlying axioms.”
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I came across this whilst searching for another work by my thesis advisor Jack Schwartz. A really timely article well done.
I met a few geniuses in my time and Jack was one.
I am not an economist – although I worked at Leontieff’s Institute whilst a grad student at NYU, I was trained as a physicist and Bethe was a teacher of mine and his gift was that he could convince you that you knew what he proposing via the math. Of course, once you left his loving care, you realized “wait a minute” how do I re-explain this to someone else ? Most times you could not because the math was just a descriptive synopsis inside his head and the focus was on the phenomenon and not the math.
Jack knew math and its limitations. He read everything about decidability. I worked with him on vision and he was concerned that my work in vision was just too black boxy AI, It was the beginning of NN. I changed my thesis into something a bit more mathematically explainable.
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Can we have some source references please for the main quotes?
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There are no circle in reality
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No, I understood it fine. I still disagree, and I view this as a framing or narrative issue.
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Excellent post
Except for the title
It’s not the mathematical mind that is weak
Rather it’s the charlatan application of mathematics to economics (in the context of your post)
i.e. it’s the economics mind that is weak in full context
Keynes was quite an exception. It’s noteworthy that the man who produced the great quote in your post specialized in mathematics as an undergrad before then turning to economics
That gave him perspective on how to apply his own “mathematical mind” to economics
And its clear in that context that the thinking underlying the General Theory was influenced, not by charlatan come lately calculus methods that characterize the neoclassical synthesis, but by heavy duty algebra
algebra like this, for example:
Click to access algebra-serge-lang.pdf
very different, and the type of logic development that was instrumental in his evisceration of classical economics
so please, no more weakness of “the mathematical mind”
its the content as applied to economics that is the issue – not the mathematician at source, or the “mathematical mind”, that is responsible for this
it’s the elective calculus economists, who did not have the same training or brilliance or perspective as Keynes
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“Walrasian general equilibrium analysis (GEA), he seems to be suggesting, is too far removed from reality to provide any reliable guide to macroeconomic policy-making..”
It’s not just a matter of it being too far removed from reality – GEA has no application to macroeconomics. Keynes distinguished that relative prices don’t matter in macroeconomic considerations – levels of income do.
“As a Marshallian he was less skeptical of partial-equilibrium analysis (PEA) than of general-equilibrium analysis.”
Personally, I don’t see much difference between GEA and PEA. PEA is a subset of GEA. It is GEA with even more restrictions. As much as some people say Keynes was a Marshallian partial equilibrium theorist, where in the GT is this explicit? Perhaps in chapter 2 of the GT? But I think that is a stretch also.
” 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.”
Keynes’ liquidity preference was not a demand curve in the usual sense where relative prices count.
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Chiralfox, Apologies to you and others for having left so many comments unacknowledged for the past few months. Thanks for your interesting observations on Schwartz’s essay. I think that you are expressing a similar uneasiness about how mathematics or mathematical models can be misused or misunderstood when we seek to identify mathematical abstractions with real world objects. I don’t feel able to try to parse the particular ways in which you and he are expressing that uneasiness.
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Inal, Thanks for pointing me to Grieve’s paper discussing Morishima with whose work I am only vaguely familiar. I think that there at least 4 separate issues to consider, and not having read Grieve or Morishima, I’m not sure which one they are mainly focused on. I agree that price flexibility is not enough to guarantee that a market economy in disequilibrium will find its way back to equilibrium. Price flexibility is not enough for at least the following reasons.
1. No equilibrium exists
2. Even if it exists, the interactions between markets are such that there is no algorithm by which an auctioneer could adjust prices that would converge on the equilibrium price vector. The auctioneer might never arrive at the equilibrium price vector.
3. Even if the auctioneer could get to the new equilibrium in finite time, an economy without an auctioneer (i.e., every economy that ever existed) would never get to equilibrium because trading actually occurs at disequilibrium which changes the equilibrium each time a trade takes place at disequilbrium prices, so the economy is always aiming at a moving target and never catches up.
4. Real economies don’t have a complete set of markets so it is not enough for market prices to adjust to their equilibrium values, price expectations also have to be consistent with equilibrium and there is no market mechanism that even in principle leads price expectations to values consistent with equilibrium.
I’ve discussed this in a few earlier posts such as this one
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Herman, Many thanks for your comment and helping me better understand what Schwartz was saying. I don’t think I have anything useful to add about your disagreement with Chiralfox and am happy to let your comments speak for themselves.
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John, Thanks so much for sharing your recollections of Schwartz. I had not heard of him until I saw Popper mention his article and looked for it. When I read further about him I realized what an extraordinary mind he must have had. I would be interested in reading some of his work on economics, but haven’t yet dipped into it.
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JKH, Thanks for your comment. I’m not sure that Schwartz would agree with your take on mathematics even algebra, but I agree with you that calculus can take us only so far.
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Henry, Thanks for your comment, which I think I recall your having already made more than once previously. You are free to keep making the point, but I am going to continue to disagree.
That Keynes was a Marshallian is simply a historical fact, one needn’t search the General Theory to confirm that fact. He was no doubt a rebellious and schismatic if not heretical Marshallian, but he remained one even as he rejected much of the orthodoxy in which he was trained and raised.
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