Richard Dedekind's "What are numbers and what are they good for?" is one of the foundational works for set theory. The search for a version of the latter that met certain formal desiderata, however, led the latter to drift a bit from what Cantor, Dedekind, and other early contributors to the field were intending. There's nothing wrong with that, but it is sometimes worthwhile to remember what the original aim was. This is particularly true since sets were not originally taken to be primitives but to be a way of formally capturing something else: thought.
Dedekind is quite explicit in his monograph. "In what follows," he says, "I understand by thing every object of our thought" (sect. 1, p. 44). One thing or object in this sense is taken to be precisely distinguishable ("completely determined by all that can be affirmed or thought concerning it") from another, and we can mark them therefore by distinct letters as labels. Because objects of thought in the sense used here are completely determined by what can be thought about them, we naturally get a criterion for identity: thing a is exactly the same as thing b when everything that can be thought about a is thought about b, and vice versa.
Objects of thought can in turn be thought-together, "associated in the mind" (sect. 2, p. 45). Dedekind calls this a system; it's the parent of what we call a set. If a, b, and c can be thought together, they form a system S; a, b, and c can then be called elements and we can say that these elements are contained in system S, or that S consists of a, b, and c. But S is something we can think about that can be exactly distinguished by all that can be thought concerning it, in the sense that S just being the thinking-together of its elements, it is completely determined by knowing exactly what is and what is not contained in S. So S can itself be a thing. Dedekind extends this to the singleton -- the system with only one element; but, while he recognizes that you could posit a system that had no elements, he sets that possibility aside in the monograph.
Dedekind also defines 'part' for systems: S is part of S' when every element in S is in S'; in this sense every system is part of itself. We can distinguish a narrower definition of part, however: S is a proper part of S' when S is a part of S' but is not the same as S'.
We can transform the elements of a system, in a sense to turn them into other elements (although it may just be a correspondence rather than an actual change); a transformation is a rule whereby for any element s there is another determinate thing φ(s), which is called its transform. A transformation of S (i.e., the new system) is similar to S if there is still a one-to-one correspondence between elements in S and their transforms.
On the basis of these definitions, Dedekind builds several important parts of mathematics, such as mathematical induction. But the key definition is that of the infinite:
A system S is said to be infinite when it is similar to a proper part of itself...; in the contrary case S is said to be finite.
In other words, if S includes a part (other than itself) that has elements that can be put into an exact one-to-one correspondence with the elements of S, it is infinite. An infinite system has proper parts that are infinite. Systems whose proper parts always have fewer elements than themselves are finite. And here we get to the most interesting argument of all, the argument that there are infinite systems, which is based wholly on the understanding of these systems as objects of thought (sect. 66, p. 64):
My own realm of thoughts, i.e., the totality S of all things which can be object of my thought, is infinite. For if s signifies an element of S, then is the thought s', that s can be the object of my thought, itself an element of S. If we regard this as the transform φ(s), then has the transformation φ of S, thus determined, the property that the transform S' is part of S; and S' is certainly a proper part of S, because there are elements in S (e.g., my own ego), which are different from such thought s' and therefore are not contained in S'. Finally it is clear that if a, b are different elements of S, their transforms a', b' are also different, that therefore the transformation φ is a distinct (similar) transformation.... Hence S is infinite, which was to be proved.
In other words, for every object of thought s, one can have the further object of thought 's is an object of thought'; we can do this one-to-one correspondence with everything about which we can think, but if we take all the explicit 'is an object of thought' thoughts, they will be a proper part of the totality of objects of thought, because there will be some objects of thought (like myself) that are not in the explicit 'is an object of thought' group. So the entirety of things that can be thought is infinite.
I find, looking at commentaries on this, that mathematicians are regularly baffled at Dedekind's 'non-mathematical' argument here. But, of course, if this is 'non-mathematical', so is all of Dedekind's set theory, since everything that could be said to be 'non-mathematical' in section 66 arises from how Dedekind defines the basic terms to begin with. But perhaps there is a legitimate sense in which this is true, and we can say that Dedekind is not doing mathematics. Rather, he is attempting to establish the existence and nature of numbers without assuming them. The only way you can do this is to start from something more basic and fundamental than number. What is more basic than number? What is a more general intelligible field out of which numbers can be drawn? Well, the obvious one is the intelligible field, all the intelligible things that can be associated with each other. More basic than the mathematical realm is the intelligible realm of which it is only a part. And the intelligible realm is infinite, which makes it possible for us to think about infinites in mathematics.
Various Links of Interest
* Rob Alspaugh, Theological Virtues
* Matthew Wills, How 1920s Catholic Students Fought the Ku Klux Klan
* Edmund Waldstein, Marx's Fundamental Insight into Capitalism
* Paul Lodge, What Is It Like to be Manic?
* Erin Blakemore, Is Emma Really the Heroine of Emma? The real answer, of course, is that she is, but Blakemore is right that Emma does have features that in other novels are associated with villainesses. It matters, though, that Emma has good taste in friends, one that gives her a way out her scheming.
* Cheryl Misak on Frank Ramsey.
* Yair Rosenberg on the interesting background to the Yiddish translation of Harry Potter.
Sir Arthur Conan Doyle, A Study in Scarlet
Susanna Clarke, Jonathan Strange & Mr Norrell
Benoît-Dominique de la Soujeole, O.P., Introduction to the Mystery of the Church