Galileo (1564-1642)
Galileo's astronomical work with a telescope
The Inquisition

The Two New Sciences Aristotle on math and nature
The so-called Pythagoreans make an even stranger use of principles and elements than do those who give physical explantions; for they have derived them, not from perceptible objects, but from mathematical entities, which, with the exception of those in astronomy, belong to the realm of immovables. Nevertheless, all their discussions and interest concern nature: for they explain the generation of the heavens and carefully note the events, changes, and revolutions in the celestial regions. Thus, they employ all the principles and explanations, as if they agreed with the physicists, that all being is perceptible and is contained under the vault of heaven. But their explanations and principles, as we have said, are suitable for even more exalted beings, more suited to them, in fact, than to nature. For they fail to explain motion in terms of only the limited and the unlimited, and odd and even; and they fail to tell us how, without motion or change, it is possible to have genesis and destruction or any revolution of celestial bodies. And even granting that they have shown how spatial magnitudes can come from such elements, why should some bodies be light and others heavy? On the basis of what they assume and say, they should explain perceptible as well as mathematical entities. (Metaphysics)
Elements of the dialogue: three characters, and takes place over three days A brief outline of the beginning
The argument goes something like this. Sagredo says a worker at the shipworks is foolish to suppose that supports for a ship must be of larger proportions, and not just the same proportions, to support a larger ship (that is, a ship twice as heavy may need stocks more than twice as big). Salviati points out that the worker was right. Salviati explains more about the strength of materials, but then Simplicio asks him why materials hold together at all. They agree to pursue this digression. Using the marble blocks as a starting point, Salviati considers whether the resistance to a void might explain part of the strength of materials. After Sagredo raises doubts about any glue, they set out to show that resistance to void might wholly explain the strength of materials. But many materials are stronger than the resistance to separating the marble slabs; and in fact Salviati describes how you can measure the force of the resistance to the void. But, one alternative is that there are many small voids, and the sum resistance exceeds that of a large but single void. Salviati then sets out to show how you can conceive of materials as having infinitely many voids (between infinitely many infinitesimal particles). This leads them to a discussion of the paradoxical nature of infinity.

Salviati says:
but let us remember that we are among infinities and indivisibles, the former incomprehensible to our finite undrestanding by reason of their largeness, and the latter by their smallness. (34 [73]).
the infinite is inherently incomprehensible to us, as indivisibles are likewise (38 [76]).
Our interest is: is this true?

[Rev 27 January 2010. All quotes from Galileo draw on Stillman Drake's translation (University of Wisconsin Press; Madison, WI: 1974).]