Ramble, Part 15: Deep Water
Mar. 3rd, 2007 12:05 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowGalileo is best known, of course, as a physicist and astronomer, but he also made significant contributions to mathematics. His work on cycloids inspired several generations of successors, and he also investigated the measurement of areas in ways which heralded the coming of integral calculus. (Kepler, by the way, was also involved in this line of research.) What I want to talk about this time, though, is something he did which no one of his time picked up on - something that lay fallow until the mid-19th century.
The ancient Greeks had difficulty coming to grips with the infinite. Zeno's Paradoxes are well-known, but there was a general reluctance even in simpler areas. Euclid speaks of lines as being arbitrarily extensible; we would say that lines are infinite in length, but he seems to be thinking instead of line segments of ever-increasing length. One of Euclid's number-theoretic propositions is usually quoted as saying that there are infinitely many primes; the actual phrasing, though, is that Prime numbers are more than any assigned multitude of prime numbers - in other words, that, given any (finite) collection of prime numbers, there is a prime number not in that collection. The words (I'll concede that they are a translation, but I do not doubt their faithfulness) deny the very possibility of an infinite "multitude" of prime numbers.
The ancients, then, were willing to concede a virtual infinity (and this is a term of art) - a collection to which no finite bound could be set - but not an actual infinity. It must be conceded that they had reason for doing so, and Galileo's thoughts on the matter point to such a reason. He appears to have been the first to notice that an infinite collection could be, in some sense, "the same size" as a proper subcollection, pointing out that a one-to-one correspondence could be established between the positive integers and the set of squares of positive integers. Given that Euclid's Common Notions include the statement that "the whole is greater than the part" - and that this statement is integral to, e.g., discussions of area - it's not hard to see that this poses a serious difficulty.
Galileo also pointed out the following paradox. Consider a circular wheel, of, say, radius one foot. Nail to its center another wheel, of radius half a foot. Now nail a shelf to a wall, half a foot from the floor. Set the large wheel on the floor, with the small wheel resting on the shelf. Begin rolling the wheels along the shelf, and let the large wheel rotate exactly once. The distance it will have travelled is then equal to its circumference. The small wheel, however, has rolled along the shelf, and has rotated exactly once; hence it, too, should have travelled a distance equal to its circumference. Yet the large wheel has twice the circumference of the small, and they have travelled the same distance.
It's not too hard to resolve the paradox - I'll leave that to the reader - but the key point, once again, is the existence of a one-to-one correspondence between two collections that are, by the usual notions, of different sizes.
The problems that Galileo raised could be, and were, ignored for the time being, but when mathematicians were finally forced to confront them, they led to a philosophical schism that has yet to be settled and continues to take unexpected twists. More on that (much) later.
Previous Next
Ramble Contents
The ancient Greeks had difficulty coming to grips with the infinite. Zeno's Paradoxes are well-known, but there was a general reluctance even in simpler areas. Euclid speaks of lines as being arbitrarily extensible; we would say that lines are infinite in length, but he seems to be thinking instead of line segments of ever-increasing length. One of Euclid's number-theoretic propositions is usually quoted as saying that there are infinitely many primes; the actual phrasing, though, is that Prime numbers are more than any assigned multitude of prime numbers - in other words, that, given any (finite) collection of prime numbers, there is a prime number not in that collection. The words (I'll concede that they are a translation, but I do not doubt their faithfulness) deny the very possibility of an infinite "multitude" of prime numbers.
The ancients, then, were willing to concede a virtual infinity (and this is a term of art) - a collection to which no finite bound could be set - but not an actual infinity. It must be conceded that they had reason for doing so, and Galileo's thoughts on the matter point to such a reason. He appears to have been the first to notice that an infinite collection could be, in some sense, "the same size" as a proper subcollection, pointing out that a one-to-one correspondence could be established between the positive integers and the set of squares of positive integers. Given that Euclid's Common Notions include the statement that "the whole is greater than the part" - and that this statement is integral to, e.g., discussions of area - it's not hard to see that this poses a serious difficulty.
Galileo also pointed out the following paradox. Consider a circular wheel, of, say, radius one foot. Nail to its center another wheel, of radius half a foot. Now nail a shelf to a wall, half a foot from the floor. Set the large wheel on the floor, with the small wheel resting on the shelf. Begin rolling the wheels along the shelf, and let the large wheel rotate exactly once. The distance it will have travelled is then equal to its circumference. The small wheel, however, has rolled along the shelf, and has rotated exactly once; hence it, too, should have travelled a distance equal to its circumference. Yet the large wheel has twice the circumference of the small, and they have travelled the same distance.
It's not too hard to resolve the paradox - I'll leave that to the reader - but the key point, once again, is the existence of a one-to-one correspondence between two collections that are, by the usual notions, of different sizes.
The problems that Galileo raised could be, and were, ignored for the time being, but when mathematicians were finally forced to confront them, they led to a philosophical schism that has yet to be settled and continues to take unexpected twists. More on that (much) later.
Previous Next
Ramble Contents
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2007-03-04 12:13 am (UTC)Interesting post.
Many thanks!
no subject
Date: 2007-03-04 01:21 am (UTC)Probably, and with good reason. Concepts such as "infinite limit" and "limit at infinity" were formalized shortly before Cantor brought the question of actual infinities to a head. Given the sharply different behavior of traditional numbers and infinities, it's probably best to keep them well apart - although there've been repeated efforts to merge the two, with mixed success. I'll get to that eventually.