By J. L. Austin, G. J. Warnock
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Galileo, Discorsi, 248–50. 5. 4 divided in just the same proportions. ’¹⁴ The proof of Theorem I, Proposition I in the section ‘The Motion of Projectiles’ shows that the rest of the diagram stems from the superposition ¹⁴ Galileo, Discorsi, 244. ’¹⁵ In order for the reasoning in the proof of Theorem I, Proposition I of the Fourth Day to proceed, the line abcde must mean both time and distance; it must represent time symbolically in order for the application of the results achieved in the Third Day, and it must represent distance qua displacement in order for the diagram to make sense as the icon of a trajectory, the movement of a body across a plane in space.
B. Rolf and A. George (Berkeley: University of California Press, 1967/69); Der Logische Aufbau der Welt (Hamburg: Meiner, 1928). productive ambiguity: galileo contra carnap 17 mathematics, many of its presuppositions nonetheless remain unnoticed in our discourse and thinking. I now return brieﬂy to Carnap’s classic work, in order to uncover some of these assumptions and to question them more closely. Then I will give a brief sketch of what became of Carnap and Hempel’s view of scientiﬁc and mathematical knowledge in the work of their academic children (Bas van Fraassen, Nancy Cartwright, Margaret Morrison, Ian Hacking) and grandchildren (Robin Hendry, Ursula Klein, myself), a development that will bring us back to the project of this essay in an unexpected way.
Note that in Corollary I, Galileo does not compare the interval-terms directly, but is careful to refer to them in ratios. Even if inﬁnitesimal intervals (instants and points, to use Galileo’s vocabulary) are mathematically suspect—as they surely were in the early seventeenth century—the geometry of the diagrams supports the rationality of holding that ratios between them are ‘like’ the ratios between their ﬁnite counterparts. That is, AD : DE :: AO : OP no matter what size the conﬁguration is; or, to use the other diagram, HL : LM :: 1 : 3 no matter what the size of the conﬁguration.
Sense and Sensibilia by J. L. Austin, G. J. Warnock