This debate took place over eighteen years in a series of publications that I shall chronicle below, starting with Edwin Robert Anderson Seligman's On the Shifting and Incidence of Taxation (SIT) in 1892, and ending sometime after 1910 with the appearance of Francis Ysidro Edgeworth's "Applications of Probabilities to Economics" in the Economic Journal (Seligman 1892; Edgeworth  1925). (1) Seligman's arguments are presented in each of the five successive editions of his master work (SIT) that were published during his lifetime. By the third edition, published in 1910, Seligman's arguments had reached something of a final mature form.
Edgeworth took an early interest in the application of mathematics to social science questions, especially the blossoming theory of probability (Mirowski 1994: 30-47). In 1881, Edgeworth published his Mathematical Psychics, in which he remarkably extended utilitarian ethical theory in mathematical form and pioneered the analysis of exchange equilibrium in directions that have had an enormous influence on our modern understanding of equilibrium (Edgeworth  1995; Creedy 1998). In the first years of his appointment (1891) to the Drummond chair at All Souls College, Oxford, he engaged Seligman in matters that at first glance seem to have mostly to do with taxation theory. Edgeworth's responses to Seligman appeared primarily in the Economic Journal, which he edited from 1891 until shortly before his death in 1926. His last response to Seligman appeared in the Economic Journal in 1910 and most curiously invoked his favorite principle of "unverified probability" from his ongoing statistical studies to put the fledgling mathematical method--as he understood it--on a more secure ground in economics (Edgeworth  1925). This debate and its methodological significance have been mentioned by Harold Hotelling (1932) and discussed by John Creedy (1986, 1998), but their importance to our understanding of the application of mathematical analysis to economics in the twentieth century has been overlooked. The purpose of this article is to emphasize this aspect of the tax incidence debate.
The arguments between these two protagonists were something like a tennis match: Seligman would make a statement in his book (SIT) and Edgeworth would publish his response with lightning speed in the form of a full-length article in the Economic Journal. Seligman would subsequently take notice of Edgeworth's arguments in time for the next revised edition of his great book, provoking Edgeworth to prepare another hasty reply to be published in the Economic Journal. On one level they debated a tiny and specialized issue in price theory--tax incidence theory. Indeed, this is the only way the debate has come down to us in the secondary literature. (2) On another level they debated the precise form economic analysis would assume throughout the greater part of the twentieth century. I am concerned here with both the substance of the taxation debate as well as the significance of the argument for the development of mathematical economics.
Edgeworth had definite ideas on mathematical economics that were extreme, uncompromising, and at times dogmatic. He bitterly complained about Seligman's use of contrived numerical examples in economics to illustrate points of far-reaching principle. The modern calculus could not bear the weight of Seligman's blasphemous numerical examples. Seligman held his ground, quite undisturbed by Edgeworth's onslaughts. His obstinacy on this issue angered Edgeworth, who, in the end, appealed to a jury of both social scientists and natural scientists to come to his aid in his battle to slay the stubborn Seligman dragon. His principle of "unverified probability" was transposed from statistical analysis to economics to try to rule out certain functional forms and admit only smooth differentiable functions into economic arguments. Edgeworth's approach was the harbinger of what became the established approach of all textbook writers in the twentieth century, making economic ideas easily subject to mathematical analysis regardless of whether the changes involved are large or small. …