From: Abraham Robinson
C H A P T E R SIX The University of Toronto: 19511957 Zest for both system and objectivity is the formal logician's original sin. He pays for it by constant frustrations and by living ofttirnes the life of an intellectual outcaste. . . . The for mal logician gets little sympathy for his frustrations. He is regarded as too rigid by his philosophical colleagues and too speculative by his mathematical friends. —Hao Wang] IN MOVING from London to Toronto, Robinson did not miss a beat, mathematically. Sailing first class from Liverpool on a Cunard liner, the Franconia, the Robinsons headed for the New World in August of 1951. Although Renee was seasick most of the time, Abby worked at his usual steady pace, writing three pages a day.2 By the time they reached North America in early September, he had finished a twentyfivepage manu script. AN AXIOMATIC APPROACH TO DIMENSIONAL ANALYSIS Robinson's latest effort was designed to cast new light upon dimen sional analysis: It is shown that the main proposition of dimensional analysis, the socalled "πtheorem," can be deduced in strictly mathematical fashion from a set of simple axioms. This provides a firm foundation for the discussion of the physical assumptions involved.3 Dimensional analysis, as Robinson explained, was a very useful tool both for physicists and engineers. In his own work on fluid dynamics, the subject arose naturally because real fluids behave in complex ways. In developing satisfactory models for the behavior of fluids, deciding which variables are fundamental is naturally a basic one. Dimensional 1 Wang 1954, p. 241. 2 According to George Duff, Robinson's colleague at the University of Toronto, "when ever Robinson was writing a manuscript, he would set aside time at the end of the day. He would keep at it until he had written at least three good pages. This meant foolscap paper, and Robinson wrote in a rather small hand, so three pages was quite a lot." George F.D. Duff, interviewed in his office at the University of Toronto, November 13, 1990; hereafter referred to as Duff 11990. 3 Robinson 1951a, a twentyfivepage manuscript "On the Foundations of Dimen sional Analysis," dated "RMS Franconia, August/September 1951," RP/YUL. 186 — Chapter Six analysis serves to determine the essential relationships between variables , especially for equations that do not depend on any particular system of units of measurement, which are said to be "dimensionally homogeneous."4 One of the most important applications of dimensional analysis concerns the relation between a model and its fullscale counterpart. Wind tunnel experiments, for example, would have impressed upon Robinson the importance of predicting forces on a fullscale prototype from the results of force measurements on a particular experimental model. Dimensional analysis was controversial, however, in complex cases for which the dimensional situation was either unclear, misunderstood, or subject to different interpretations.5 The axiomatic approach, Robinson insisted, had the advantage of avoiding such debates as whether or not the concept of dimension was "fundamental." Just as in projective geometry, where an axiomatic approach allowed one to take the concept of a straight line as either fundamental , or derivable from other concepts one might prefer to take as fundamental, he found that it was "easy to formulate alternative sets of axioms in which the concept of a dimension is either fundamental or derived."6 From a purely mathematical point of view, however, he insisted that "there is nothing to choose between them," since the sets of axioms in question were logically equivalent.7 Although the axiomatic approach to dimensional analysis was obviously of interest in itself to Robinson, he took its fundamental power to lie in applications. The main practical advantage of dimensional analy4 The early works establishing the importance of dimensional analysis include Buckingham 1914 and Bridgman 1931. See also EsnaultPelterie 1948. 5 See, for example, the numerous paradoxes discussed by Garrett Birkhoff in chapter 1 of his Hydrodynamics: A Study in Logic, Fact, and Similitude. The second edition contains two chapters devoted to "paradoxes" divided between examples of nonviscous and viscous flow. Birkhoff 1960. 6 Robinson 1951a, p. 1. 7 Robinson was not always so indifferent! Several years later, his colleague at Toronto, P. G. (Tim) Rooney, remembers one of the few times he ever saw Robinson in heated disagreement, and it was over the axiom of choice. Rooney and Robinson were talking mathematics. The subject of axiomatic set theory came up and what...

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