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New Perspectives on Galileo (The Western Ontario Series in Philosophy of Science)

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I am summarising Wisan's chapter which shows how Galileo's philosophy of science is modelled on mathematics.

Following Archimedes, in hydrostatics "Galileo's method is explicitly modelled on mathematics and he tells us that this method 'will always make what is said depend on waht was said before, and, if possible, never to assume as true that which requires proof'" (p. 7).

"Now an important consequence of Galileo's explicit methodology is this: if one's foundations consist in true and evident principles, empirical confirmation of rigorously deduced consequences is unnecessary since true conclusions are necessarily derived from true premisses." (p. 8). For example, "in the old De Motu Galileo warns his readers that if they try to find his results in experience they will fail due to various 'accidental' effects" (p. 8). "But in astronomy the situation is altogether different" (p. 19); "For his mechanics and hydrostatics were deduced from 'true and indubitable' principles, whereas in astronomy there were no immediately evident principles" (p. 20).

Even where "true and indubitable" principles are not available, mathematics still suggests the ideal, namely the method of analysis. According to Galileo, "proceeding step by step in accordance with the 'laws of logic' for inferring causes from effects, combined with direct observation of the cause, is tantamount to or 'very little lower than' mathematical proof" (p. 34). "Above all, what Galileo sees in arguments by mathematicians as opposed to those by Aristotle and his medieval followers is that the former proceed only by small and necessary steps from clearly defined terms and assumptions which are both explicit and evident. These are the essential features of mathematical arguments and what gives mathematics its certainty." (p. 35). So for example we read in the Dialogue: "You, Salvati, have guided me step by step so gently that I am astonished to find I have arrived with so little effort at a height which I believed impossible to attain ... just as climbing step by step is no trouble, so one by one your propositions appeared so clear to me, little or nothing being added, that I thought little or nothing was being gained." (p. 36)

Only when neither of these paths are available does Galileo reluctantly resort to empiricism. For example in the Two New Sciences we read: "this idea is, I say, so new, and at first glance so remote from fact, that if we do not have the means of making it just as clear as sunlight, it had better not be mentioned; but having once allowed it to pass my lips I must neglect no experiment or argument to establish it" (p. 39). Such principles "are not merely rational for they must be confirmed by experience. This, however, does not mean confirmation of consequences, but rather the process by which the principles are to be rendered evident to the intellect. Unfortunately, the new science required principles which were not immediately evident in the sense required and could not be made so. Galileo struggled with this vexing problem for many years but never found a satisfactory solution." (p. 43).

Only one more appeal to mathematics could rescue Galileo from this predicament. "Galileo remarks that his propositions are derived ex suppositione and that even if the motions he supposed did not exist in nature, his demonstrations 'founded upon my supposition, lose nothing of their force and conclusiveness, just as nothing prejudices the conclusions demonstrated by Archimedes concerning the spiral that no moving body is found in nature that moves spirally in this way.'" (p. 44).