## AXECwiki

## Hilbert-Bourbaki-Mill and von Neumann's monster-structure

... for nothing is farther from the axiomatic method than a static conception of the science. We do not want to lead the reader to think that we claim to have traced out a definitive state of the science. (Bourbaki, 2005, p. 1274)

When we assemble the facts of a definite, more-or-less comprehensive field of knowledge, we soon notice that these facts are capable of being ordered. This ordering always comes about with the help of a certain* framework of concepts* [*Fachwerk von Begriffen* ] .... The framework of concepts is nothing other than the theory of the field of knowledge. ... If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework. ... The procedure of the axiomatic method, as it is expressed here, amounts to a *deepening of the foundations* of the individual domains of knowledge — a deepening that is necessary for every edifice that one wishes to expand and to build higher while preserving its stability. (Hilbert, 2005, pp. 1107-1109), original emphases

What are the propositions which may reasonably be received without proof? That there must be some such propositions all are agreed, since there cannot be an infinite series of proof, a chain suspended from nothing. But to determine what these propositions are, is the *opus magnum* of the more recondite mental philosophy. (Mill, 2006, p. 746)

Could all the phaenomena of nature be deduced from only thre [sic] or four general suppositions there might be great reason to allow those suppositions to be true. (Newton, quoted in Westfall, 2008, p. 642)

I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with what is going on in the original thing. (Feynman, 1992, p. 171)

From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms — the mathematical structures; and it so happens — without our knowing why — that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. (Bourbaki, 2005, p. 1276)

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However, there also occur structures entirely without application, the so-called monster-structures (Bourbaki, 2005, p. 1275, fn. 9). From the structural axiomatic point of view the subjective-behavioral axioms yielded such a formalism without application. Its sole merit was "that of showing the exact bearing of each axiom, by observing what happened if one ommitted or changed it."

Bourbaki cannot be made accountable for the conventional monster-structure. The same holds for Hilbert and, of course, Mill.

..., it was the von Neumann perspective that shaped general equilibrium theory ..., and thus reconstituted economic theory. (Weintraub, 2002, p. 78)

As student of Hilbert and proponent of axiomatization von Neumann immediately realized that Walras's mathematics was insufficient:

The so-called "mathematical" economists in the narrower sense — Walras, Pareto, Fisher, Cassel, and hosts of other later ones — especially, have completely failed even to see the task that was before them. Professor Hicks has to be added to this list, which is regrettable because he wrote several years after decisive work had been done — in principle — by J. von Neumann and A. Wald. (Morgenstern, 1941, p. 369)

Von Neumann did not realize, though, that Walras's supply-demand-zero-profit-equilibrium was mistaken in the first place and never represented any feasible economy. It was only Walras's formalism that was eventually repaired. For von Neumann general equilibrium was first and foremost a mathematical puzzle. To crack the nut he advocated the fixpoint approach. With explicit disregard of economic content the axiomatization of Walrasian economics was completed by Debreu and others. This, however, could not change the fact that equilibrium is a nonentity. What in effect had been solved was an angels-on-the-pinpoint problem. This subsequently became the core competence of standard economics. It is a curious fact that the physicists and mathematicians who came in great number with advanced tools to economics in order to help the indigenous folk (Mirowski, 1995) in effect moved the whole thing deeper into the Walrasian cul-de-sac.

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. (Georgescu-Roegen, 1979, p. 271)

Axiomatization in economics suffers from many misunderstandings (see also Crisis and Methodology: Some Heterodox Misunderstandings URL). This, clearly, is not an argument against the method but against economists.

The method of reasoning by chains of syllogisms is nothing but a transformation mechanism, applicable just as well to one set of premisses as to another; it could not serve therefore to characterize these premisses. (Bourbaki, 2005, pp. 1267-1268)

The deductive method does not prove that the premises are true. The truth of theorems and the truth of premises are entirely different questions. The deductive method guarantees that the conclusions are true if the premises are true.

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Outside pure mathematics, the profitableness of axiomatization crucially depends on the real-world domain. Not everything is axiomatizable. There is, for example, no such thing as a behavioral axiom. By borrowing from physics and mathematics, the neoclassicals got the essentials wrong from the very beginning.

The bifurcation of motion into two fundamentally different types, one for natural motions of non-living objects and another for acts of human volition ... is obviously related to the issue of free will, and demonstrates the strong tendency of scientists in all ages to exempt human behavior from the natural laws of physics, and to regard motions resulting from human actions as *original*, in the sense that they need not be attributed to other motions. (Brown, 2011, p. 211), original emphasis

Jevons's rhetorical question points exactly to the source of neoclassical confusion:

Must not the same inexorable reign of law which is apparent in the motions of brute matter be extended to the subtle feelings of the human heart? (Jevons, quoted in Mirowski, 1995, p. 219 )

Jevons had, with the laws of motion, the archetype of science on his mind:

But it was a second and more important quality that struck readers of the *Principia*. At the head of Book I stand the famous *Axioms, or the Laws of motion* … For readers of that day, it was this deductive, mathematical aspect that was the great achievement. (Truesdell, quoted in Schmiechen, 2009, p. 213)

What Jevons did not understand was that the trinity of law, axiom, and behavior does not work because behavior is *original*, as the scientists in all ages knew well (for details see Objective Principles of Economics URL). With farfetched analogies and his shallow scientific understanding, Jevons placed neoclassical economics on poor foundations (for details see The Logic of Value and the Value of Logic URL) and there it stood, with some formal improvements, until recently.

For an outside observer without prior knowledge of habitual human reactions behavior appears at first random. This is the point to start with, not introspection. The laws of motion of the human heart are pure kitsch and remain so in the abstract form of the first derivative of an utility function. From the marginal principle follows nothing that could help to understand how the market economy works. To think of human behavior in terms of deterministic laws is an unforgivable analytical blunder. It is the structure that is deterministic (see the Period Core), not the behavior.

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... if we wish to place economic science upon a solid basis, we must make it completely independent of psychological assumptions and philosophical hypotheses (Slutzky, quoted in Mirowski, 1995, p. 362, see also Hudík, 2011).

This is the defining property of the __structural axiom set__.

References

Brown, K. (2011). Reflections on Relativity. Raleigh, NC: Lulu.com.

Bourbaki, N. (2005). The Architecture of Mathematics. In W. Ewald (Ed.), From Kant to Hilbert. A Source Book in the Foundations of Mathematics, volume II, pages 1265–1276. Oxford, New York, Ny: Oxford University Press.

Feynman, R. P. (1992). The Character of Physical Law. London: Penguin.

Georgescu-Roegen, N. (1979a). Energy and Economic Myths, chapter Measure, Quality, and Optimum Scale, pages 271–296. New York, NY, Toronto: Pergamon.

Hilbert, D. (2005). Axiomatic Thought. In W. Ewald (Ed.), From Kant to Hilbert. A Source Book in the Foundations of Mathematics, volume II, pages 1107–1115. Oxford, New York, Ny: Oxford University Press. (1918).

Hudík, M. (2011). Why Economics is Not a Science of Behaviour. Journal of Economic Methodology, 18(2): 147–162.

Mill, J. S. (2006). Principles of Political Economy With Some of Their Applications to Social Philosophy, volume 3, Books III-V of Collected Works of John Stuart Mill. Indianapolis, IN: Liberty Fund. (1866).

Morgenstern, O. (1941). Professor Hicks on Value and Capital. Journal of Political Economy, 49(3): 361–393. URL

Mirowski, P. (1995). More Heat than Light. Cambridge: Cambridge University Press.

Schmiechen, M. (2009). Newton’s Principia and Related ‘Principles’ Revisited, volume 1. Norderstedt: Books on Demand, 2nd edition.

Weintraub, E. R. (2002). How Economics Became a Mathematical Science. Durham, NC, London: Duke University Press.

Westfall, R. S. (2008). Never at Rest. A Biography of Isaac Newton. Cambridge: Cambridge University Press, 17th edition.

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See also Euclid and Newton . For details about the Period Core, which represents a structural law, see The Synthesis of Economic Law, Evolution, and History URL.

© 2013 EKH, except original quotes up