Morgenstern’s essay examines what it could mean to establish limits on mathematical economics. Its seven sections distinguish demonstrable impossibilities from restrictions attributed prematurely to mathematics because of inadequate economic concepts, unsuitable techniques, or deficient empirical knowledge. His defense of mathematical reasoning is consequently also a criticism of formalism that fails to advance understanding.
The opening discussion invokes impossibility results in physics and mathematics, including Gödel’s work, to show that identifying a genuine limit is itself a substantial scientific achievement.
All involve mathematical reasoning, some are, indeed, in the field of pure mathematics, which abounds in statements of prohibitions and impossibilities.
Such achievements do not justify speculative prohibitions on an evolving discipline. Neither economics nor mathematics has a foreseeable endpoint, and economic inquiry may demand mathematical instruments that have not yet been invented. Psychological motives, expectations, and qualitative evidence therefore cannot simply be declared inaccessible to mathematical treatment. The issue is how a problem is conceived, not whether it initially appears numerical.
Morgenstern also separates mathematical thought from its symbolic presentation:
Mathematics does not necessarily need symbols other than words which, up to some degree of complication, can adequately express mathematical ideas, state theorems, formulate proofs.
This distinction undermines both hostility toward mathematics and the assumption that formulas automatically improve an argument. A verbal argument may embody mathematical reasoning, while elaborate notation may conceal unresolved conceptual difficulties. The relevant standard is what an analysis establishes about economic phenomena.
The historical discussion explains how economics inherited tools whose availability could outrun their appropriateness.
The latest start of economics found a well-developed science of mechanics; indeed, Newton's crowning great work was already completed before the physiocrats began writing.
Borrowing calculus from mechanics encouraged economists to translate familiar claims into equations without always examining the conditions necessary for their validity. The assumption that equal numbers of equations and unknowns guaranteed a solution exemplifies this danger; Wald’s later existence proof exposed the substantive mathematical work that such an assumption bypassed. Morgenstern’s criticism concerns not mathematical rigor but its replacement by superficial formal resemblance.
His account of utility theory develops the positive alternative. Contrasting elaborate indifference-curve analysis with the numerical utility theory developed by von Neumann and himself, he emphasizes the conceptual importance of introducing uncertain prospects. An axiomatic treatment could then establish numerical utility up to a linear transformation and make new experimental questions possible. Progress came from reformulating preference, not merely applying more advanced machinery to an unchanged problem.
Strategic interdependence provides the essay’s strongest case for creating mathematics responsive to economic circumstances. Individuals and firms do not simply maximize under conditions independent of their actions: their outcomes depend on other agents who choose, cooperate, form coalitions, and make side payments. Morgenstern presents game theory, initially supported by von Neumann’s minimax theorem, as a response to this inadequacy of the mechanical analogy and as a fundamental departure from conventional general-equilibrium reasoning. Yet mathematical sophistication cannot compensate for an economically misleading model.
The discussions of axiomatics and disciplinary hierarchy qualify this advocacy. Axioms organize knowledge and permit deduction; they do not possess a self-evident authority superior to their consequences. Productive axiomatization requires empirical investigation and clarified concepts. Historical, statistical, experimental, and intuitive inquiry retain indispensable roles, and contributions should not be ranked by their mathematical appearance.
The conclusion turns to reciprocal development. Practical applications and electronic computation bring new opportunities for investigation, but also generate unforeseen difficulties. Accumulated rounding errors, for example, can undermine extensive numerical calculations and create new mathematical problems. Such developments explain why the boundaries of mathematical economics cannot be fixed beforehand. Morgenstern’s governing position is neither unrestricted formalization nor retreat from abstraction: economic problems should determine the tools required, and may help bring new mathematics into existence.
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