Gerhard Tintner’s Review of Mathematical Analysis for Economists (1938)

Gerhard Tintner’s signed review of R. G. D. Allen’s Mathematical Analysis for Economists presents the textbook as a contribution to both disciplines. His praise rests on its mathematical competence, its extensive economic applications, and its usefulness for teaching. Tintner identifies the influence of G. H. Hardy’s Cambridge school, but his larger concern is the reciprocal value of the encounter: economists gain analytical tools, while mathematicians gain unfamiliar problems and a broader understanding of their subject’s applications.

Primarily an introduction to mathematical analysis for economists it should also be of interest to the mathematician for several reasons.

This reversal of the expected audience organizes the review. Rather than assessing only whether Allen makes mathematics accessible to economists, Tintner asks what economics can offer mathematics. He situates that question within a historical argument: mathematics developed in close association with physics and astronomy, but this association was contingent rather than necessary. Different scientific connections could have produced different mathematical priorities.

Whereas there is no necessary relationship, it is safe to say that we would have another kind of mathematics nowadays if its development had been closely linked with the history of, say, biology or the social sciences.

Tintner does not deny the productivity of the physical sciences. His point is that their predominance has also encouraged a certain one-sidedness, leaving some mathematically interesting problems neglected. Allen’s book supplies material for correcting this bias. Economic applications therefore matter for more than their practical usefulness: they can enlarge the range of questions mathematicians consider.

The review’s central section substantiates that claim through a sustained survey of examples. Industrial location illustrates analytical geometry; demand, revenue, cost, and indifference curves introduce functions; marginal revenue and marginal cost exemplify derivatives. Monopoly and duopoly furnish problems of maximization, while capital, interest, and elasticity illustrate logarithmic derivation. Tintner then follows the book into multivariable analysis: production and utility functions, partial derivatives, Euler’s theorem, differentials, and extrema with and without constraints. Integration appears in problems of capital valuation and durable goods, differential equations in dynamic demand and supply, and the calculus of variations in dynamic monopoly and saving.

This sequence is the review’s main evidence, not simply an inventory of topics. It demonstrates that economics can supply examples across a substantial mathematical curriculum, from elementary geometrical concepts to advanced analytical methods. Tintner’s emphasis falls on the correspondence between economic problems and mathematical operations, rather than on detailed exposition of particular economic models.

Would it not be interesting for any mathematician to supplement the well known and worn out examples from physics by some economic applications?

The pedagogical conclusion has two levels. Mathematics teachers can draw on Allen to diversify calculus and advanced-calculus courses. More ambitiously, the book can anchor a graduate course in mathematical economics. Tintner judges its applications broad enough to cover almost the whole field, while acknowledging that a few points would require supplementation, for example from G. C. Evans’s Mathematical Introduction to Economics. Such a course could serve both mathematics students interested in social problems and economics students possessing some mathematical knowledge.

The concluding recommendation extends beyond classroom use. Allen offers mathematicians an introduction to contemporary economics that Tintner regards as economically as well as mathematically competent.

Although some of the points are naturally controversial, this book will tend to give the mathematician much of the necessary background for the understanding of contemporary economic science and an idea of the treatment of economic problems at its best.

That qualification preserves a distinction between endorsing the textbook and treating every economic proposition as settled. The review’s central judgment is nevertheless strongly affirmative: Allen provides a rigorous bridge between disciplines, valuable both for learning mathematical economics and for widening mathematics’ accustomed field of application. Its distinctive conceptual move is to make the benefit reciprocal, presenting economics as a source of mathematical renewal as well as a recipient of mathematical technique.