By Philip Pilkington, a writer and journalist based in Dublin, Ireland. You can follow him on Twitter at @pilkingtonphil
The influence that mathematics has had on neoclassical economics is obviously quite profound. However, when looked at in detail it appears that a certain type of modern mathematics was in fact highly suited to the direction many in the economics profession took after the work of Leon Walras – the Frenchman who founded modern neoclassical economics – appeared on the scene. So, it should not be thought that it was simply the formal tools of mathematics that transformed neoclassical economics into the obscurantist doctrine it is today. Instead it should be understood that its obscurantist skeleton was ready and waiting for its mathematical flesh.
It has been said before – and not just by the present writer – that neoclassical economics amounts to a sort of theological system that bears no resemblance to reality for the simple reason that it does not aim at reality. But the relationship between the doctrine and its formal presentation is less often talked about. In a very interesting paper entitled ‘Kaldor on Debreu: The Critique of General Equilibrium Reconsidered’ philosophers of economics Thomas Boylan and Paschal O’ Gorman deal with precisely this. We will try to highlight and expand upon their fascinating observations.
A Mathematical Crystal
The authors begin by engaging with the British Post-Keynesian economist Nicholas Kaldor’s critique of neoclassical methodology. Kaldor insisted that in order for economics to progress as a modern science the neoclassical research program had to be completely demolished. The reason for such insistence was because Kaldor was an extremely practical man. He was deeply involved in debates of British economic policy in the post-war years and in his later years, from his perch in the British House of Lords he commented on the economic policy of his day.
Yet at the same time Kaldor was a theoretical economist. Thus, it is not surprising that Kaldor should think that good economic theory should reflect the real world as much as possible. Kaldor thought that neoclassical theory did not do this at all. In one particularly stark passage he wrote that neoclassical economics was “barren and irrelevant as an apparatus of thought to deal with the manner of operation of economic forces.”
Kaldor was particularly concerned that neoclassical economic theory was, for the most part, engaged in trying to construct what he called a “mathematical crystal” that was absolutely perfect and self-contained and had no relation to the real world. Kaldor believed that science, as opposed to a sort of mathematical aesthetics, proceeded in the other direction. Yes, theory should be logically consistent, but first and foremost it should deal with the facts at hand. If it did not do this in any meaningful way then any logical consistency was only so much decoration on an otherwise inedible cake.
Before moving on it is perhaps allowing Kaldor himself to get an extended word in on the neoclassical approach that has come to dominate the economics departments.
One of the ideas I wished to convey in my first lecture was that the type of economic theory which is the core of the subject as taught in western universities — and that covers North America, Western Europe, Australia, and so on – is pretty useless and indeed harmful for developing an understanding of the laws of motion of capitalist market economies. It is expressed with a phoney kind of precision or “scientism” of a most pretentious kind, using highly sophisticated, mathematical techniques for proving propositions which have no interpretative value of real-world phenomena, for the simple reason that they are based on a priori axioms which have no relation to the conditions which can be empirically observed. All this is aggravated, not helped, by the use of mathematics.
While Kaldor did have some very specific critiques of certain aspects of neoclassical economics, we will not consider them here (the interested reader can download the above cited paper for details). Instead, we will pursue the fertile line of thought taken up by Boylan and O’ Gorman.
Debreu as Platonist
The authors focus on the work of the French economist Gerard Debreu who was the first to lay out a complete articulation of a general equilibrium theory grounded in modern mathematics. They claim, however, that it is the type of mathematics Debreu used which gives us insight into the type of theorising he was actually doing.
Debreu’s method derived from a strain of mathematical theory that was, in a strong sense, ruthlessly unpragmatic. Such a tradition reaches right back to the ancients and to the work of the ancient geometer Euclid. In his ‘History of Western Philosophy’ Bertrand Russell links this to a whole tradition of philosophy which is concerned not with reality, but instead by idealistic abstractions. This tradition – which is strongly theological and metaphysical – is generally referred to as idealism and is usually traced back to Plato. Russell explains the connection as such:
There is in Euclid the contempt for practical utility which had been inculcated by Plato. It is said that a pupil, after listening to a demonstration, asked what he would gain by learning geometry, whereupon Euclid called a slave and said ‘Give the young man threepence, since he must needs make a gain from what he learns.’
This led to a manner of viewing the world in which passive contemplation of purely logical forms was thought to be the ultimate end of all thinking. Reality was then chastised by the idealist for not living up to the Ideals of geometry and the great Ideas generated in the mind of the philosopher or the theorist. Such thinking was, of course, immensely popular with the medieval theologians and those that came after them who often applied them as a means to meditate on the infinite perfection of the Christian God. We will come back to this latter observation in a moment.
Such a tradition lives on in mathematics right to this day. It is today generally associated with what have come to be called the Bourbakian School – after a group of (again) French mathematicians who wrote together under the pseudonym ‘Nicholas Bourbaki’. This tradition was passed down to Debreu and the neoclassicals through his use of Cantorian set theory – a powerful form of mathematics developed in the late 19th century. This led the higher, more theoretically-oriented neoclassicals to ultimately assert that pure theory should have absolutely no relation to reality. Indeed, in an extreme form the mathematical terms used in neoclassical economics are to be thought of as meaningless symbols – a rather strange approach when one considers that these symbols are supposed to represent real world phenomena.
Intellectual Hierarchy and Our Political Dark Age
This is, of course, not a scientific approach. It eschews empirical reality in favour of logical perfection. It thus becomes far more similar to the meditations undertaken by theologians of the Middle Ages rather than the applied logic of the scientists of the post-Cartesian era. What Debreu and his followers thus practice is a sort of Divine Mathematics. In doing what they refer to as economics they are not engaged in scientific discovery, but instead are working through logical problems in a sort of meditative pose. Their version of the old Christian idea of God they refer to as ‘the market’ or ‘general equilibrium’. This is then asserted to be the manner in which the real economy operates in the long-run. And this strange ritual is all undertaken prior to their looking at any empirical evidence at all!
The rather comical thing about this is that most higher-level theorists are well aware that what they are doing is not really any sort of theoretical economics when we understand that term in a scientific sense (modern science assuming both empirical correctness and logical rigour). It is instead a set of “what if” logical constructions that are worked through until they are thought to be perfectly consistent and harmonious. However, as Kaldor said, the higher-level theorists have not “managed to pass this message down the line to the textbook writer and to the classroom.”
Thus what we have is a strange hierarchy system in which higher-level theorists are well aware that they are engaging in mere logical games, while the lower-level theorists think that, for some unstated reason, neoclassical theory has some sort of empirical validity. The end result is that economists tend to be extremely foggy about real world problems.
This is not surprising given that their theoretical framework of choice aims at hiding reality from them rather than allowing them to approach it with a set of useful tools. Such a situation is not far away from the hierarchy of the Christian church in the Middle Ages where advanced theological scholars worked out logical deductions about God’s perfection and existence while lower-level priests disseminated this message as if it contained empirical truths. The higher-level were generally aware that they were engaged in a meditation of sorts, while the lower level thought that the higher-level had actually proved something about reality. Another term for this period is, of course, ‘the Dark Ages’. And many have concluded that the reason these ages were so dark was because the Church, which maintained authority on all intellectual activity, was engaged in idealistic reasoning constructed out of mere tautologies; just the same, of course, with neoclassical economics in our current political Dark Age.
Idealism as Formal Structure: The Mathematics of Infinity
Boylan and O’ Gorman make an extremely interesting case that we can observe this shift away from empirical reality and toward metaphysics or theology in the Cantorian set theory applied by Debreu and passed down to the neoclassicals. The authors point out that a major difference between Cantor and Bourbakian School on the one hand and their more practically-minded colleagues on the other is that while the former believe in ‘actual infinity’, the latter believe in ‘potential infinity’.
That may sound complex, but it is not very. As we all know numbers could be potentially counted to infinite – there is no “end” to the numerical chain and it could, in theory, stretch on forever. Now, more pragmatic mathematicians claim that while numbers are ‘potentially infinite’ we can never, as it were, draw ourselves back to view this infinity. Cantor, on the other hand, claimed that we could – hence why his method is referred to as ‘actual infinity’. Infinity is treated as if it actually existed as an observable or conceivable entity.
However, the idea that we – or some entity – could “zoom out” and look upon the infinite amount of numbers in existence is tantamount to positing an omnipresent God through whom we can grasp at the ungraspable. This objection was thrown at Cantor and his followers by his more pragmatically-minded colleague Henri Poincare who rejected the Cantorian system because he refused to “argue on the hypothesis of some infinitely talkative divinity capable of thinking an infinite number of words in a finite length of time.”
Some of Debreu’s proofs, according to Boylan and O’ Gorman, relied heavily on such a “theological mathematics”. That is, they could not be carried out in a finite number of steps and instead took on a timeless or Divine quality. This, of course, reflected perfectly the neoclassical obsession with an economy automatically moving toward perfection through a “Hidden Hand” mechanism. So, we can conclude that even in their formal presentation the higher-level neoclassicals are engaged in a sort of religious meditation rather than an attempt to fashion a theory which fits the real world. And that is how the idea of market perfection is squared with what Kaldor called the “mathematical crystal”.
It would be nice to think that escape from our present Dark Age would be possible by simply pointing this out to the lower-order neoclassicals, so that they could consider if such a methodology is appropriate for textbooks and students. Alas, this is not so. If the Dark Ages taught us anything it is that Heresy is considered an abominable crime. And while the neoclassicals do not today possess the methods of torture and execution held by the Church of that Middle Ages, they do possess almost total hegemony over both the university departments and the debates. And they have an extremely strong investment in the tautologies of the higher-order neoclassicals – indeed, they rely on such meditations for their Divine Authority.