Oskar Morgenstern · 1963
This is a single-author methodological essay on the scope of mathematical methods in economics. Morgenstern’s central thesis is that fixed “limits” cannot be responsibly drawn in advance, because both mathematics and economics change through their interaction. He begins by treating negative knowledge—proofs of impossibility in physics and mathematics—as a serious achievement, but uses Gödel to warn against simple boundary claims.
Man sieht: die Lage ist gar nicht einfach und darf auch nicht als einfach angesehen werden.
English translation: One sees: the situation is by no means simple and must not be regarded as simple either.
The essay then turns the question around: the problem is not whether economics has features that exclude mathematics, but whether economists understand what mathematics is and whether they formulate economic problems properly. Morgenstern rejects standard objections based on psychology, non-quantitative data, expectations, indivisibility, or utility measurement. Such objections mistake mathematics for mere measurement or calculation.
Die Mathematik ist keine rein quantitative Wissenschaft; Messungen sind nicht erforderlich, es gibt keinen grundlegenden Unterschied zwischen einer einfachen Addition und einer Integration.
English translation: Mathematics is not a purely quantitative science; measurements are not required, and there is no fundamental difference between a simple addition and an integration.
His criticism is aimed equally at anti-mathematical economists and at empty formalism. Mathematical notation does not by itself produce insight, and many earlier uses of formulas merely translated verbal claims into symbols. The real failure was not excessive mathematics, but shallow model-building. Hence his sharp aphorism:
Nichts ist leichter, als die eigene Beschränktheit für die der Methode oder des untersuchten Gegenstandes zu halten.
English translation: Nothing is easier than to mistake one's own limitations for those of the method or of the subject under investigation.
Historically, economics borrowed from mechanics and the calculus, especially in Walrasian and Paretian equilibrium theory. Morgenstern grants their importance but argues that they framed economic agents as if they faced fixed conditions and solved isolated maximum problems. That misses the defining economic fact: outcomes depend on the actions of others. The proper object is strategic interdependence, including coalitions, bargaining, cooperation, and conflict. This is why game theory marks, for him, not a technical supplement but a conceptual break: it supplies a mathematics appropriate to social interaction rather than one inherited from mechanics.
The utility discussion illustrates the same point. Earlier ordinal utility and indifference-curve analysis became mathematically elaborate while remaining poorly fitted to uncertainty. Von Neumann and Morgenstern’s expected-utility approach is presented as a better conceptual and mathematical reconstruction because it begins from choice under uncertainty and yields a numerical utility representation through axioms. The lesson is general: one must first grasp the economic phenomenon, then seek or invent suitable mathematics.
Die wissenschaftliche Tätigkeit besteht zu einem großen Teil darin, die richtige Frage zu stellen.
English translation: Scientific activity consists to a large extent in asking the right question.
Morgenstern’s section on axiomatics clarifies that axioms are not self-evident truths but disciplined starting points chosen for their power, consistency, and empirical adequacy. Axiomatization is the mature form of mathematization, but it cannot be imposed prematurely; it depends on empirical preparation, intuition, and exact problem formulation.
Wenn eine empirische Wissenschaft zur Axiomatisierung, also zu einer gewissen Vervollkommnung gelangt, dann kann man ohne Mathematik nicht mehr auskommen.
English translation: When an empirical science attains axiomatization, and hence a certain degree of perfection, one can no longer do without mathematics.
He also warns against creating a hierarchy in which only mathematical economics counts as serious science. Historical, statistical, experimental, and theoretical work all matter; mathematical form is valuable only when it clarifies, organizes, or extends knowledge.
The final section broadens the argument. Just as natural science stimulated geometry, mechanics, analysis, and later mathematics, Morgenstern expects social science to generate new mathematics. Wald, von Neumann, game theory, linear programming, operations research, business planning, and electronic computation all show that economics is entering a new phase of practical and theoretical mathematization.
Die Naturgesetze sind in der Sprache der Mathematik geschrieben, das haben schon die Alten verstanden.
English translation: The laws of nature are written in the language of mathematics—the ancients already understood this.
The essay’s relevance lies in its balanced position: it is neither mathematical triumphalism nor methodological skepticism. Mathematics has no known boundary in economics, but its success depends on concept formation, empirical discipline, and new tools adequate to strategic and social realities.
Schon dieses Beispiel zeigt deutlich, daß es unmöglich ist, irgendwelche „Grenzen“ für die Anwendung der Mathematik festzustellen.
English translation: This example alone shows clearly that it is impossible to establish any "limits" for the application of mathematics.
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