By Larry Smith

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Extra resources for Algebraic topology: Proc. conf. Goettingen 1984

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266). This entirely empirical definition is a definition of inertial mass. Mach argued that the mass we measure on a scale is the inertial mass. Indeed if the scale is in equilibrium it tells us that the masses on the pans via the lever of the scale produce in each other equal and opposite (gravitational) accelerations13 . If we accept additivity of mass as an experimental fact (this is not mentioned by Mach) this observation will explain why usual weighing tells us inertial mass. Mach pointed out that the constants m1 and m2 that enter into Newton’s law of gravitation F = G(m1 m2 /r 2 ) are other constants (the gravitational masses).

Kant agreed with the empiricists that we receive our knowledge of the world through our senses but he argued that our experiences are formed by a-priori forms of intuition and concepts. Space and time are such a-priori intuitions; they are described by geometry and arithmetic, respectively, and these two sciences are constructed a-priori in intuition. Also, the laws of mechanics including the law of universal gravitation are, according to Kant, a-priori in the sense that they ‘are viewed as necessary conditions of the possibility of an objective notion of true motion’ (Friedman 1992, p.

V 0 0 , q , q , . . , , q1 , q2 , . . , qr ∂q1 ∂q2 ∂qr ∂S ∂S ∂S ∂S + H − 0 , − 0 , . . , − 0 , q10 , q20 , . . 37) can be written ∂V = pρ , ∂qρ ρ = 1, 2, . . 47) ∂V = −pρ0 , ∂qρ0 ρ = 1, 2, . . 49) and ∂S = pρ , ∂qρ ρ = 1, 2, . . 50) ∂S = −pρ0 , ∂qρ0 ρ = 1, 2, . . 52) respectively. In 1837 Jacobi developed Hamilton’s methods further. Where Hamilton had assumed V or S to be solutions of two partial differential equations Jacobi pointed out that it is enough to consider one of the equations.

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