Axiomatization: Still at Keynes' juncture

My way is to begin with the beginning. (Lord Byron)


Beginnings are always difficult in all sciences. (Marx, 1990, p. 89)


There is no more fertile source of error than apparently trivial premisses. (Schumpeter, 1994, p. 269)


In fact, the history of every science, including that of economics, teaches us that the elementary is the hotbed of the errors that count most. (Georgescu-Roegen, 1970, p. 9)


If we open any book, even of mathematics or natural philosophy, it is impossible not to be struck with the mistiness of what we find represented as preliminary and fundamental notions, and the very insufficient manner in which the propositions which are palmed upon us as first principles seem to be made out, contrasted with the lucidity of the explanations and the conclusiveness of the proofs as soon as the writer enters upon the details of his subject. Whence comes this anomaly? Why is the admitted certainty of the results of those sciences in no way prejudiced by the want of solidity in their premises?  (Mill, 1874, V.4)


When the premises are certain, true, and primary, and the conclusion formally follows from them, this is demonstration, and produces scientific knowledge of a thing. (Aristotle, Analytica, Wikipedia)


For it can fairly be insisted that no advance in the elegance and comprehensiveness of the theoretical superstructure can make up for the vague and uncritical formulation of the basic concepts and postulates, and sooner or later ... attention will have to return to the foundations. (Hutchison, 1960, p. 5)


Whenever arithmetization [=axiomatization] can be worked out, its merits are above all words of praise.  (Georgescu-Roegen, 1971, p. 15)


Ein System von Gedanken muß allemal einen architektonischen Zusammenhang haben, d.h. einen solchen, in welchem immer ein Theil den andern trägt, nicht aber dieser auch jenen, der Grundstein endlich alle, ohne von ihnen getragen zu werden, der Gipfel getragen wird, ohne zu tragen. (Schopenhauer, 2016, p. 5)




What particular reality is described by a given theory can be ascertained only from that theory's axiomatic foundation. Thus, Standard theory describes the economic process of a society in which the individual behaves strictly hedonistically, where the entrepreneurs seek to maximize his cash-profit, and where any commodity can be exchanged on the market at uniform price's and none exchanged otherwise. (Georgescu-Roegen, 1966, p. 361)


Macroeconomic theory took a major turn when it was argued that models of aggregate phenomena had to be based on ‘sound micro foundations’. In other words, the basic building blocks have to be agents, each of whose behaviour is based on the classical axioms of rationality which we impose on those agents in standard theory. (Kirman, 2010, p. 507)


..., before accepting the conclusions of any economist’s model as applicable to the real world, the careful student should always examine and be prepared to criticize the applicability of the fundamental postulates of the model; for, in the absence of any mistake in logic, the axioms of the model determine its conclusions. (Davidson, 2002, p. 41)


... one can axiomatize, without being committed to a formalist reading of the axiomatic system. Axiomatization is as old as Euclid, whereas formalism is a much later development. ... Poincaré, the most outstanding mathematician at the turn of the twentieth century, while acknowledging the value of axiomatic systems, rejected Hilbert’s formalism. Similarly, Frege, the leading logician of the period, while rejecting Hilbert’s formalism, extensively used the AA [Axiomatic Approach]. (Boylan and O'Gorman, 2007, pp. 430, 432)


The axioms that are used to define “rationality” are based on the introspection of economists and not on the observed behaviour of individuals. Economists from Pareto through Hicks to Koopmans have long made this point. Thus we have wound up in the weird position of developing models that unjustifiably claim to be scientific because they are based on the idea that the economy behaves like a rational individual, when behavioural economics provides a wealth of evidence showing that the rationality in question has little or nothing to do with how people behave.  (Kirman, 2009, p. 81)


But a principle that is not universally true is false. Thus the rationality principle is false. I think there is no way out of this. (Popper, 1994, pp. 172-173)


...  General Equilibrium Theory has never justified its pricing axiom and certainly has not given any account of how wages are set when the economy is out of equilibrium. (Hahn, 1980, p. 135)


For instance, economists bend their research toward axiomatic theories that are almost embarrassing in their pre-scientific naiveté. Consider utility theory, for instance, which is now taking a drubbing at the hands of experimental psychology and neurophysiology. A scientific orientation would free us of such vestigial dogmas. (Dorman, 2008, p. 170)




Any concept can be faultily used. (Schumpeter, 1994, p. 111)


My opinion continues to be that axiomatics, like every other tool of science, is no better than its user, and not all users are skilled. (Clower, 1995, p. 308)


...  axiomatization facilitates the detection of logical errors within the model, and perhaps more importantly it facilitates the detection of conceptual errors in the formulation of the theory and in its interpretation. (Debreu, quoted in Ingrao and Israel, 1990, p. 362)


I am of the opinion that, certainly, for the purposes of research it is always necessary to combine the intuition with the axioms. (Felix Klein, quoted in Weintraub, 2002, p. 25)


If then it is the case that the axiomatic basis of theoretical physics cannot be an inference from experience, but must be free invention, have we any right to hope that we shall find the correct way? (Einstein, 1934, p. 167)


As far as economics is concerned: only if we look for nonbehavioral axioms.




Economics still stands at what may be called Keynes's juncture:

The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight – as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. Something similar is required to-day in economics. (Keynes, 1973, p. 16)


This insight is Keynes's most valuable methodological contribution. In practice, non-Euclidean translates to nonbehavioral. Structural axiomatization is, in a sense, the realization of Keynes's methodological program as stated above. What his followers put into practice was his political program which indeed has a weak theoretical foundation because Keynesians never came forth with the non-Euclidean axioms. Since the days when Keynes saw more clearly than his fellow economists that the 'classical' axioms had been refuted once and for all by the Great Depression economics is in dire need of new formal foundations. There is no way back before Keynes; and to stay any longer at Keynes's juncture is equally impossible. New Classicals have not got the first point, New Keynesians not the second (cf. Quiggin, 2010). Lacking correct formal foundations, both approaches are agonizing detours.


Axiomatization is the prime task of theoretical economics. Without correct axioms, no correct theory. Without correct theory, no understanding of how the monetary economy works. Without empirically corroborated understanding, no useful economic policy advice. Without effective economic advice, no good reputation: "Late in life, moreover, he [Napoleon] claimed that he had always believed that if an empire were made of granite the ideas of economists, if listened to, would suffice to reduce it to dust" (Viner, 1963, p. 1). This was long before the Great Depression's fatal dustification. However, the opinion of any political executive is in the last instance not decisive. Nor is public opinion.


Much more important than any political reputation of economics is indeed: Without correct axioms, no acceptance as science. There is no way around it, neither for Orthodoxy nor for Heterodoxy.


The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally deducible part of the theory. It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. (Einstein, 1934, p. 165)


This First Imperative of Scientific Inquiry fits structural axiomatization. Compliance with the First Imperative, though, does not guarantee that the correct set of axioms emerges automatically. The Second Imperative demands formal and material consistency. Both imperatives taken together rule conventional approaches out.



Boylan, T. A., and O’Gorman, P. F. (2007). Axiomatization and Formalism in Economics. Journal of Economic Surveys, 21(2): 426–446.

Clower, R. W. (1995). Axiomatics in Economics. Southern Economic Journal, 62(2): 307–319. URL

Dorman, P. (2008). What Would a Scientific Economics Look Like? real-world economics review, 47: 166–172. URL
Einstein, A. (1934). On the Method of Theoretical Physics. Philosophy of Science, 1(2): 163–169. URL

Davidson, P. (2002). Financial Markets, Money and the Real World. Cheltenham, Northampton, MA: Edward Elgar.
Georgescu-Roegen, N. (1966). Analytical Economics, chapter Economic Theory and Agrarian Economics, pages 359–397. Cambridge, MA: Harvard University Press.
Georgescu-Roegen, N. (1970). The Economics of Production. American Economic Review, Papers and Proceedings, 60(2): 1–9. URL

Georgescu-Roegen, N. (1971). The Entropy Law and the Economic Process. Cambridge, MA: Cambridge University Press.

Hahn, F. H. (1980). General Equilibrium Theory. Public Interest. Special Issue: The Crisis in Economic Theory, pages 123–138.

Hutchison, T.W. (1960). The Significance and Basic Postulates of Economic Theory. New York: Kelley.

Ingrao, B., and Israel, G. (1990). The Invisible Hand. Economic Equilibrium in the History of Science. Cambridge, MA, London: MIT Press.

Keynes, J. M. (1973). The General Theory of Employment Interest and Money. The Collected Writings of John Maynard Keynes Vol. VII. London: Macmillan.

Kirman, A. (2009). Economic Theory and the Crisis. real-world economics review, (51): 80–83. URL
Kirman, A. (2010). The Economic Crisis is a Crisis for Economic Theory. CESifo Economic Studies, 56(4): 498–535. DOI
Marx, K. (1990). Capital, volume I. London: Penguin Classics. (1876).

Popper, K. R. (1994). The Myth of the Framework. In Defence of Science and Rationality. London, New York: Routledge.

Quiggin, J. (2010). Zombie Economics. How Dead Ideas Still Walk Among Us. Princeton, NJ, Oxford: Princeton University Press.

Schopenhauer, A. (2016). Die Welt als Wille und Vorstellung, German Edition, Zenodot Verlagsgesellschaft. Kindle-Version. 

Schumpeter, J. A. (1994). History of Economic Analysis. New York: Oxford University Press.

Viner, J. (1963). The Economist in History. American Economic Review, 53(2): pp. 1–22. URL

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



See also Euclid and Keynes's Missing Axioms  URL and Why Post Keynesianism is Not Yet a Science  URL or URL and Walras's Law of Markets as Special Case of the General Period Core Theorem  URL


© 2013 EKH, except original quotes                                                                                                                                                                                                              up