Background
Galileo (1564-1642)
- Born in Pisa
- Father was a musician
- Studied medicine but discovered and fell in love with mathematics,
left University of Pisa in 1585 without a degree
- Held posts at universities in Pisa and Padua, and taught mathematics
and astronomy
Astronomy
- Astronomy was largely what we would call astrology
- BUT: astronomy required careful observation
- AND: observations were mathematically described
- For Galileo, these two elements were an inspiration for changing mechanics,
and his basing of physics on observation and mathematical (geometrical) description
mark the beginning of modern physics
Galileo's astronomical work with a telescope
- Work in astronomy made him famous
- Copernicanism was still radical and opposed by the Church
- 1604, a "new star" appeared in the skies, and Galileo used parallax to show it
was farther away than the moon (a scandal!)
- 1610, Galileo published the "Starry Messenger" describing mountains of the moon,
additional stars, and the Jovian system.
- Jovian system suggested Copernicanism was true
- 1613, published book on sunspots
The Inquisition
- Galileo made many enemies -- he was not charitable in disagreements
- Copernicus was banned as unfit for Christians
- In 1632, G is ordered to appear before the Inquisitors in Rome
- The Inquisitors held Galileo for several weeks, trying to find something to punish him for, and ultimately made him sign a confession,
sentenced him to house arrest for life, and forbade him to write anything more.
- Galileo was pardoned about a decade ago. Presumably this means one can
without damnation read Copernicus.
The Two New Sciences
- Published in 1638 in Leyden, where the Inquisitors had no influence
- Our interest in this work is with two aspects: the application of geometry
as a formalization of some reasoning, and Galileo's pleasing discussion of the difficulty
of reasoning about infinity.
- Recall that Aristotleanism dominates still the academies at this time
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
- Salviati: based on a friend of
Galileo's, in the dialogue this character
mainly speaks for Galileo.
- Sagredo: based on a Venetian
amateur scientist, this character acts as the
intelligent lay person.
- Simplicio: in earlier dialogues,
this character was a defender of
aristotleanism and the dominant traditional
views; in this dialogue the character is more
open-minded, and can be seen as the
intelligent aristotlean.
- The "Academician" referred to is Galileo.
- In this time, "The Philosopher" means Aristotle.
A brief outline of the beginning
- 11-14. Introductory remarks, and the observation that strength of materials is not
directly proportional to their size.
- 14. Example of the collapsed beam with three supports.
- 15-16. The strength of materials.
- 16-19. The strength of rope.
- 19-21. The strength or force of void/vaccuum.
- 22. The need to measure the strength of void to determine if it is
sufficient to explain the strength of materials.
- 23-24. Description of an apparatus to measure void.
- 26+ Possibility that void explains all of the strength of materials.
- 28+ Could there be infinitely many voids in a material?
- 28-34. The geometric demonstration that there could be infinitely many infinitesimal
voids.
- 35-37. The "Soupdish."
- 37-38. Proof that the volumes of the two solids in the soapdish are equal.
- 39-44. Paradoxical features of infinities (focussing on cardinalities).
- 45. Unity as infinity.
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]).
and
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).]