Do We Need Sets? For Anything?
Lately I’ve been thinking about ZFC, naive set theory, and some other things; and increasingly I find myself wondering, is any of this at all important? Does set theory actually matter? For anything? What, exactly, is the point of sets? Math got a very long way before naive set theory was invented, much less ZFC. Let’s explore.
Boolean logic doesn’t need sets. In many treatments of mathematics it even precedes sets in the construction of the whole edifice.
Set theory certainly isn’t needed for arithmetic and calculation. The Peano postulates are enough. You don’t need three volumes of Principia Mathematica for what Peano did in 5 lines. Natural numbers, integers, rationals, reals, and even complex numbers seem to be well defined without considering “sets”. Same for operations like arithmetic: addition, subtraction, multiplication, division, square and other roots, exponentiation, and other operations don’t need sets.
Algebra likewise doesn’t need sets. You can find x and solve quadratic and cubic and other equations without set theory. Ditto systems of linear equations and practical uses of matrices.
Euclidean geometry doesn’t need sets. The Greeks had that figured out before anyone invented zero, much less the empty set. I’m not quite certain about non-Euclidean geometry, because I don’t know all that much about it, but I sort of doubt it needs set theory.
Trigonometry likewise doesn’t need sets. You can think of triangles and other shapes as sets of points, but you don’t have to. It works just fine without ever mentioning sets.
Moving up the ladder, calculus also doesn’t need sets. Derivatives, integrals, single and multiple variables, etc. None of that needs sets. Not even a rigorous epsilon delta treatment that avoids infinities benefits from sets. Nor do you need it for Taylor series, perturbation methods, ordinary and partial differential equations, or anything else that follows calculus. In fact, nothing I can think of in numerical analysis uses sets. No one even thinks about sets when solving practical problems.
Real analysis could perhaps use sets, but I don’t think it has to. You can define everything you want with open and closed intervals on the real line. In higher dimensional spaces, you have open balls or neighborhoods instead but that all makes perfect sense with coordinates and numbers. You don’t have to think of a ball as some sort of set.
The more mathematical end of probability and statistics uses set theory, but again I’m not sure how practically important that is. Probability and statistics is perhaps more philosophically open than any other area of modern math. We can calculate and prove till the cows come home, but even though stats is an incredibly practical and applied area of mathematics, practitioners can’t agree on what the results actually mean. If we say there is a 30% chance of snow tomorrow, is it a statement about the world, about the future, or about our knowledge? How do we add up an infinite number of events of zero probability to get a finite probability? But as long as you shut up and calculate, and don’t p-hack, probability and statistics work really, really well. Much like analysis, I don’t know that you actually need the full machinery of set theory to make sense of any of it. It’s enough to take a naive view that talks about possible events and numbers without even using the language of set theory. The word “set” is convenient in probability, but anything beyond the rough idea of an unordered group of things doesn’t seem to come into play.
The first place I think maybe sets might matter is in more advanced linear algebra. When you talk about vector spaces instead of matrices, you have to introduce fields, and that means you have to introduce groups. This brings us into abstract algebra and group theory which has practical applications for cryptography, elementary particle physics, and other areas. Groups and fields are normally defined in terms of sets. But do they have to be? I’m not sure.
Topology is also defined in terms of sets, in particular which sets are open and which are closed. However, the most practical results in topology are in metric spaces or at the very least in T2 (Hausdorff spaces). As I hinted above, these spaces can be defined in other ways that don’t involve sets. I haven’t studied topology nearly enough to be certain, but I suspect a lot of the actually useful results in topology could be defined and proved without references to sets if anyone were of mind too.
Indeed it’s worth noting that both topology and group theory were branches of mathematics that coalesced around the same time that sets were beginning to be studied as objects in their own right by Cantor and others. Were they prerequisites for groups and topological spaces? Or was it just the language mathematicians at that time liked to use?
Number theory is another field where set theory might actually be important, but I simply don’t know. Some basic number theoretic results like the infinity of primes or the uniqueness of prime factorization were well proven and understood centuries before sets were a gleam in Cantor’s eye. More recent results like Fermat’s last theorem? I don’t know. Is there a number theorist in the house?
