Felix Kaufmann, Was kann die mathematische Methode in der Nationalökonomie leisten? (1931)

Felix Kaufmann’s 1931 journal article examines what mathematics can accomplish in economics by separating its logical validity from its empirical usefulness. Its argument moves from mathematics through empirical science and the comparison of natural and social sciences to economics. This progression makes disputes over economic method answerable to broader questions about scientific knowledge.

Hierbei wird sich ungezwungen eine Beurteilung der Argumente im Methodenstreit für und wider die Anwendung der mathematischen Methode ergeben.

English translation: In the process, an assessment of the arguments in the methodological dispute for and against the application of the mathematical method will emerge naturally.

Kaufmann treats mathematical propositions as analytic relations between symbols, not independently informative statements about reality. Geometry and probability illustrate the distinction between formal demonstration and empirical interpretation: neither a geometrical system’s suitability nor the applicability of probability calculus follows from mathematics alone. He states the governing limitation explicitly:

1. Die mathematische Methode kann aus sich heraus nicht zu Erkenntnissen über die reale Welt führen.

English translation: 1. The mathematical method cannot by itself lead to knowledge of the real world.

This limitation locates, rather than eliminates, mathematics’ scientific contribution. Substantive knowledge enters through empirical premises and the interpretation of results. Mathematical deduction cannot confer necessity on empirical laws, while induction rests on assumptions about regularities that cannot receive conclusive logical justification. Differences between reliable laws and approximate rules remain important, but do not establish an absolute division between exact and inexact sciences.

Kaufmann also rejects categorical restrictions based on the allegedly unmeasurable character of psychological phenomena. Counting does not require spatial measurement, some mathematical disciplines dispense with measurement, and physical science often measures indirectly through established empirical correlations. The decisive question is therefore not whether mathematics is permissible in economics, but under what conditions it becomes productive.

The comparison with physics requires particular care. Physical science benefits from relatively simple, systematically integrated laws with extensive applicability and high accuracy. Social life likewise contains dependable regularities, sustained by law, custom, and organized interaction. Yet precise social predictions often depend on numerous rapidly changing circumstances. An analogy between the sciences can guide inquiry without warranting the transfer of an entire explanatory system.

Die Entscheidung dieses Meinungsstreites aber erfordert die vollkommen deutliche Erfassung des Wesens der Analogie.

English translation: Deciding this dispute of opinions, however, requires a completely clear grasp of the nature of analogy.

Interpretive understanding mediates between behaviorism and categorical opposition to mathematization. Knowledge of others’ conduct relies on inner experience as well as external observation; reducing behavior to bodily movements sacrifices explanatory resources. Understanding motives is nevertheless compatible with mathematical analysis, especially when actors themselves calculate. Exchange provides favorable conditions because it numerically relates quantities of goods and money. Such compatibility does not establish that every meaningful relation admits mathematical representation.

Kaufmann’s treatment of subjective value theory exposes a related explanatory difficulty. If the strength of a need is inferred from the exchange choice it supposedly explains, it risks duplicating the observed phenomenon. Its legitimate heuristic role is to direct investigation toward the psychological determinants of choice. Conversely, dispensing with psychology and extracting laws from price curves cannot replace an empirically informed theory of action.

Economic mathematization thus confronts a tension between tractability and relevance. Simplifying assumptions may make calculation possible while weakening empirical applicability; including the relevant circumstances may make the mathematics prohibitively complicated. Kaufmann accordingly calls for scrutiny of conventions such as continuity in supply and demand curves. Such scrutiny also helps clarify foundational concepts, including free competition.

Cournot’s monopoly analysis supplies a concluding test. Given demand and cost functions, calculus can identify the profit maximum; establishing those functions remains the substantive economic problem. Technical calculation must therefore be distinguished from the discovery of economic laws. Kaufmann’s contribution is to separate computational achievement, conceptual clarification, and empirical discovery without treating them as mutually exclusive. His qualified expectation of further mathematical contributions replaces methodological absolutes with judgments about particular assumptions, explanatory tasks, and conditions of application.