One hears that the Greek philosophers Democritus and Leucippus advocated a theory of atoms. That is about all one can say for sure, because we don't have any of their writings. We know about them only because their views are described by other writers. Certainly the Roman writer Lucretius is VERY taken with the idea of atoms, which is clear proof that the idea was in the air. The idea of atoms is not a difficult one -- some would say it is a very natural one. In everyday life we don't see much compelling evidence either for or against their existence, though. I think it is quite possible that one ancient argument in their favor might have been the problem of the irrationals, not an argument that seems very natural to us, but one that would have seemed much more natural then. After all, the existence of irrational numbers seemed to suggest that the geometrical line was subdivided on an infinitely fine scale with rationals and irrationals mixed in a bewildering way, giving the line a kind of "graininess," that could suggest a kind of "atomic" structure. I have not found this argument in any Greek source, but something like this is in Galileo, who thoroughly educated himself in Greek ways of thinking, and worked in that same tradition.
The passages I am thinking of are in Galileo's last book Two New Sciences, which is a dialogue taking place over four days. (I recently found, to my astonishment, in an old German translation, a fifth and sixth day, which are never included in other editions I have seen! I was not aware of their existence until a few months ago. These last two days are entirely concerned with irrational ratios. Have they been considered not as entertaining as the first four days, and left out for that reason? I hope to find out.) Like all of Galileo's writing, Two New Sciences is lots of fun to read, and full of ideas. Three friends, Salviati, Sagredo, and Simplicio, whose names (although they were real people) suggest their roles in the conversation, discuss all manner of things. As if to counteract the popular cliche, deriving from Plato, Aristotle, Plutarch, etc. of mathematical philosophers as unworldly, abstracted idealists, our discussants meet at the Venetian Arsenal, where they admire the activities going on, shipbuilding, the handling of heavy weights by ingenious mechanical devices, etc. They analyze, for example, why a big machine is not simply a scaled up version of a small machine: if you scale up too far, the big machine will break under its own weight. Its supporting members must be made thicker than a simple scale model. This leads to a discussion of how things break, and what holds the bits of matter together that make up a solid material. Perhaps it is a kind of vacuum between the bits which holds them together? Many small vacua, each making a contribution? Here is a little of the dialogue:
Sagredo: There can be no doubt that any resistance, so long as it is not infinite, may be overcome by a multitude of minute forces. Thus a vast number of ants might carry ashore a ship laden with grain. Experience shows us daily that one ant can easily carry one grain, and it is clear that the number of grains in the ship is not infinite, but falls below a certain limit. If you take another number four or six times as great, and if you set to work a corresponding number of ants they will carry the grain ashore and the boat also. It is true that this will call for a prodigious number of ants, but in my opinion this is precisely the case with the vacua which bind together the least particles of a metal.
Salviati: But even if this demanded an infinite number would you still think it impossible?
Sagredo: Not if the mass of metal were infinite; otherwise ...
Salviati: Otherwise what? Now since we have arrived at paradoxes let us see if we cannot prove that within a finite extent it is possible to discover an infinite number of vacua... [1]
[They consider the nature of a geometrical line][2]
Salviati: ... And here I wish you to observe that after dividing and resolving a line into a finite number of parts, that is, into a number which can be counted, it is not possible to arrange them again into a greater length than that which they occupied when they formed a continuum and were connected without the interposition of as many empty spaces [i.e., a finite number of empty spaces: e.g, 3 pieces, each of length 1/3, would still have total length 1. The spaces where the divisions were made don't require any space. - MP]. But if we consider the line resolved into an infinite number of infinitely small and indivisible parts, we shall be able to conceive the line extended indefinitely by the interposition, not of a finite, but of an infinite number of infinitely small indivisible empty spaces. [This seems to be speculation about what arithmetic with infinite numbers might look like. Perhaps "infinity" x "1/infinity" would not be 1, but any length at all!]
...
Simplicio: It seems to me that you are travelling along toward those vacua advocated by a certain ancient philosopher [the reference is to Lucretius -- MP].
Salviati: But you have failed to add,"who denied Divine Providence," an inapt remark made on a similar occasion by a certain antagonist of our Academician. [ This is getting a little complicated! Lucretius' De Rerum Natura is stridently atheistic. By association, atoms and vacua came to be thought of as potentially atheistic ideas. The person Salviati refers to as "our Academician" is Galileo himself, the author! He had been smeared by "a certain antagonist," who attempted to link him to Lucretius, because of his ideas about atoms. But this remark was "inapt," because of course Galileo has no such atheistic intentions in discussing atoms and vacua.]
Simplicio: I noticed, and not without indignation, the rancor of this ill-natured opponent; further references to these affairs I omit, not only as a matter of good form, but also because I know how unpleasant they are to the good tempered and well ordered mind of one so religious and pious, so orthodox and God-fearing as you. But to return to our subject ...[Galileo's good faith is confirmed by Simplicio, who, in the dialogue, often represents the position of Galileo's opponents. You notice how cleverly and boldly Galileo addresses these surprisingly touchy matters: at the time he is writing this, he is confined under house arrest by the Inquisition for "vehement suspicion of heresy," the result of a plot against him by "certain antagonists" whose identity is still disputed.]
[They consider the notions of "infinity" and "indivisible." Note that "indivisible" translates the Greek "atom."][3]
Sagredo: ... And now let us hear something concerning the other difficulty raised by Simplicio, if you have anything special to say, which, however, seems to me hardly possible, since the matter has already been so thoroughly discussed.
Salviati: But I do have something special to say, and will first of all repeat what I said a little while ago, namely, that infinity and indivisibility are in their very nature incomprehensible to us; imagine then what they are when combined. Yet if we wish to build up a line out of indivisible points, we must take an infinite number of them, and are, therefore, bound to understand both the infinite and the indivisible at the same time. Many ideas have passed through my mind concerning this subject, some of which, possibly the more important, I may not be able to recall on the spur of the moment; but in the course of our discussion it may happen that I shall awaken in you, and especially in Simplicio, objections and difficulties which in turn will bring to memory that which, without such stimulus, would have lain dormant in my mind. Allow me therefore the customary liberty of introducing some of our human fancies, for indeed we may so call them in comparison with supernatural truth which furnishes the one true and safe recourse for decision in our discussions and which is an infallible guide in the dark and dubious paths of thought.
[He goes on to argue that a line cannot be a finite number of "indivisibles," but can only be an infinite number of them. Note the caution in the last sentence!]
Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.
Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.
I take it for granted that you know which of the numbers are squares and which are not.
Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; thus 4, 9, etc. are squared numbers which come from multiplying 2, 3, etc. by themselves.
Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not?
Simplicio: Most certainly.
Salviati: If I should ask further how many squares there are, one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.
Simplicio: Precisely so.
Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numberous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares ...
Sagredo: What then must one conclude under these circumstances?
Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. Or if I had replied to him that the points in one line were equal to the squares; in another greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each ? So much for the first difficulty...
Most mathematicians who see this passage in Galileo are quite surprised, because they associate these ideas with the late 19th century mathematician Georg Cantor. It is a little startling to see them in print nearly 300 years earlier. On the other hand, Galileo didn't really pursue the idea mentioned so casually here, and Cantor did.
To determine whether two finite collections of objects, A and B, each have the same number of objects (or cardinality), it is not necessary to count them. One can just pair up the objects of A with the objects of B, and see whether each member of A can be paired with a member of B, and vice versa. For example, if the people in an audience are each seated in a chair, and if each chair has someone in it, then there are as many people in the audience as chairs. It is not necessary to know what number it is to know it is the same number. Cantor extended this idea to infinite sets. In particular he investigated, in many examples, whether it was possible to devise a one-to-one correspondence between the members of an infinite set A and the natural numbers {1,2,3,...}, which we may call N for short. If such a correspondence exists, we can say that the infinite set A has the same cardinality as N. The same idea is expressed by saying that A is countably infinite. The idea there is that one could systematically "count" all the elements of A, assigning them sequentially to 1,2,3,...and so on forever. The assignment you make when you "count" them is the one-to-one correspondence.
We already know an example of this. Galileo pointed out that there is a one-to-one correspondence between N and the squares, {1,4,9,16, ...}. The correspondence is shown in the following table:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | ... |
| 1 | 4 | 9 | 16 | 25 | 36 | 49 | ... |
Clearly one can count the squares. Each element of N corresponds to its square, and each square to its (positive) square root, just as Galileo said. We see again that surprising fact, that an infinite set may be in one-to-one correspondence with a proper subset of itself. A proper subset (i.e., some but not all of the set N in this case) looks as if it must be smaller, but we are saying that it has the same cardinality, because there is a way to make a one-to-one correspondence.
Here is another example: the even numbers {2,4,6,8,..} also have the cardinality of N. The very fact that we can begin to list them in a systematic way shows that they are countable. Galileo said examples show it doesn't make sense to say one infinity is greater, less, or equal in comparison to another, but we are saying they are equal. All these sets are countable, which means they have the same cardinality as N.
It is not so clear that every infinite set is countable however. For example, the set of rational numbers Q, fractions of the form p/q, where p and q are integers, seems like a much larger set than the set N. After all, the subset of fractions which are actually integers seems like a very small subset. Nonetheless, the cardinality of Q and the cardinality of N are the same! That is to say, Q is countable. One way to see this is to represent the fractions in a table indexed by numerator and denominator, and then count them by slicing the table up on the diagonal. First here is the table:
| 1/1 | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | 1/7 | ... |
| 2/1 | 2/2 | 2/3 | 2/4 | 2/5 | 2/6 | 2/7 | ... |
| 3/1 | 3/2 | 3/3 | 3/4 | 3/5 | 3/6 | 3/7 | ... |
| 4/1 | 4/2 | 4/3 | 4/4 | 4/5 | 4/6 | 4/7 | ... |
| 5/1 | 5/2 | 5/3 | 5/4 | 5/5 | 5/6 | 5/7 | ... |
| 6/1 | 6/2 | 6/3 | 6/4 | 6/5 | 6/6 | 6/7 | ... |
| 7/1 | 7/2 | 7/3 | 7/4 | 7/5 | 7/6 | 7/7 | ... |
| ... | ... | ... | ... | ... | ... | ... | ... |
By slicing across the table diagonally, I mean count them in the order {1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, ...}. If you carefully find these first few elements of Q in the table, and see how they are located in successive diagonals, you will see how to count them all!
Actually someone might object that these are not all the rational numbers, because p and q are only positive integers here, and there could also be rational numbers with negative integers, and p (but not q) could be 0. So the real table should have been
| .... | ... | ... | ... | ... | ... | ... | ... |
| ... | (-2)/(-3) | (-2)/(-2) | (-2)/(-1) | (-2)/1 | (-2)/2 | (-2)/3 | ... |
| ... | (-1)/(-3) | (-1)/(-2) | (-1)/(-1) | (-1)/1 | (-1)/2 | (-1)3 | ... |
| ... | 0/(-3) | 0/(-2) | 0/(-1) | 0/1 | 0/2 | 0/3 | ... |
| ... | 1/(-3) | 1/(-2) | 1/(-1) | 1/1 | 1/2 | 1/3 | ... |
| ... | 2/(-3) | 2/(-2) | 2/(-1) | 2/1 | 2/2 | 2/3 | ... |
| ... | 3/(-3) | 3/(-2) | 3/(-1) | 3/1 | 3/2 | 3/3 | ... |
| ... | ... | ... | ... | ... | ... | ... | ... |
The ...'s mean the table goes off infinitely in all 4 directions. Your assignment will be to show how to count this infinite set. Note that the "diagonal" idea of the previous table DOES NOT WORK, because the diagonals never end -- you would never finish even the first diagonal, and that leaves all the other diagonals still to do.
There are other amazing theorems about countable sets. If you put two countable sets, say S1={a,b,c,...} and S2={A,B,C,...} together (this is called the union of the two sets), the result is still countable. For example you could count them like this:{a,A,b,B,c,C,...}.
Even the union of a countable number of countable sets is countable! You could take S1={a1,b1,c1,...}, S2={a2,b2,c2,...}, S3={a3,b3,c3,...}, ... etc., and put them all together as {a1,b1,a2,c1,b2,a3,d1,c2,b3,a4,e1,d2,c3,b4,a5,f1,...} (see the pattern?) It is clear that any given element in any of these sets eventually gets counted, and that is what is required for the union to be countable.
Show that the set of numbers of the form p/q, where p is any integer, positive, negative, or zero, and q is any non-zero integer, positive or negative, is a countable set. It may help you to picture this infinite set in an array or table.
Some people had trouble with Assignment 5, even while getting the basic idea of countable infinity. If you didn't get credit for your solution, here is a chance to make it up: (due Tuesday, October 19, in class).
Cantor proved the algebraic numbers are countable. These are all the roots of polynomials with integer coefficients. In doing this he would have thought about, for example, the set of all quadratic polynomials Ax^2+Bx+C, where A, B, and C are integers. Let us take an even smaller set, the set of all monic quadratic polynomials: that means the set of polynomials of the form x^2+Bx+C, with B and C integers (but A=1). Show that the set of these monic polynomials is countable. Note that "integers" includes negative integers.
1. Galileo, Two New Sciences, tr. by Henry Crew and Alfonso de Salvio, Dover Publications, 1954, p. 20.
2. Ibid. pp. 25-26.
3. Ibid. pp. 30-32.