![]() This is a descriptive law which can, however, be translated into a normative rule by saying that reciprocal displacements of weights on both sides of the fulcrum do not disturb equilibria. When applied to the “lever“, that is, to a rigid rod pivoted on a fulcrum, this is to state that the lever is in equilibrium if the weights put on it are reciprocally proportional to their distances to the fulcrum. 2 The Hölder-Mach-controversy 2.1 The Law of the LeverģIn modern terms, the Law of the Lever is a special case of conservation of angular momentum: “If the total external torque is zero, then the total vector angular momentum of the system is a constant”. I will argue that Hölder provides a pertinent defence of Archimedes’ proof against Mach’s critique. ![]() In the second part (Section 3), some key aspects of Hölder’s account of deduction in the mathematical sciences will be discussed. It will be shown that Hölder tried to refocus the discussion on the role of proof in the mathematical sciences. The paper consists of two parts: In the first part (Section 2), I shall present a rigorous analysis of Mach and Hölder’s assumptions, aims, and strategies, given that they were at cross-purposes in this discussion, and they themselves never clearly elaborated on their conflict of opinion. He attempted to explicate this concept of deduction in terms of “synthesis”.ĢIn this paper, an outline of the discussion between Mach and Hölder will be given, with particular stress on Hölder’s contributions and his epistemological thought. But unlike Mach, Hölder did not view deduction as a mere test of agreement between the premises and the conclusion: For Hölder, deduction constructively bridges a qualitative difference between the premises and the conclusion. He agreed with Mach on experience being the only source of knowledge. 1 Hölder was interested in the proof of the Law of the Lever as a case study on deduction in mathematical physics. This discussion is associated first and foremost with the name of Giovanni Vailati, though Otto Hölder was, in fact, the first to reply to Mach’s criticism. ![]() However, in 1883 Ernst Mach accused Archimedes’ proof of circularity, thereby provoking a discussion about the role of proof in mechanics, and indeed about the nature of the foundations of mechanics as a whole. The Law of the Lever thus composes the very core of rational mechanics. Shortly after its first formulation, scholars like Archimedes and Euclid were already seeking to prove it by means of deduction from general axioms and postulates. It dates back at least to Archimedes’ On the Equilibrium of Planes and possibly even to Aristotle’s Mechanics. 1 Vailati already in 1897 published two papers on the proof of the Law of the Lever and its critics [ (.)ġThe Law of the Lever was among the first laws of nature to be formulated in quantitative terms. ![]()
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