Jeffrey Kaplan doesn't just explain a mathematical curiosity; he dismantles the very foundation of how we define reality through logic. In a brisk, eight-minute journey through the history of mathematics, Kaplan argues that the elegant system built to prove numbers are objective truths actually collapses under its own weight, leaving the question of "what is a number" permanently unsettled.
The Quest for Objective Truth
Kaplan begins by contrasting the subjective view of Immanuel Kant, who saw math as a construction of the human mind, with the rigid "logicism" championed by Gottlob Frege and Bertrand Russell. The goal was ambitious: to prove that arithmetic could be reduced entirely to logic and set theory, making mathematical truths as undeniable as the laws of physics. Kaplan writes, "According to logicism mathematics is a branch of logic and the most basic type of math which is arithmetic could be reduced to just first order basic logic and set theory." This framing is crucial because it sets the stakes: if the foundation cracks, the entire edifice of science, which relies on these numbers, faces a philosophical crisis.
To build this case, Kaplan quickly defines a set as a "collection of objects," noting that these objects need not be physically close or even related. He uses a vivid, if slightly absurd, example: "Another set could be the set of lebron james the four-time nba champion and the top half of the eiffel tower these things have nothing to do with each other really but we have a set with those two things in them." This approach effectively demystifies the abstraction, showing that set theory is simply a way to group anything we can imagine. However, by relying so heavily on "naive set theory"—the ordinary language version of the rules—Kaplan sets the stage for a collision between human intuition and formal logic that many mathematicians hoped to avoid.
The Rules of the Game
Kaplan walks the reader through eleven specific rules that govern how sets behave, from the idea that order doesn't matter to the concept that a set can contain other sets. He explains that "the union of any two or more sets is itself a set," meaning if you combine the set of all cats and the set of all dogs, you simply get a new, larger set. The logic seems sound until he introduces the concept of self-reference. "Sets can contain themselves," he notes, pointing out that while the set of all cats does not contain itself (because it is a set, not a cat), the set of all sets must contain itself.
This is where the argument tightens. Kaplan describes the moment in 1901 when Russell realized that if you can make any set you want, you can make a set of all sets that do not contain themselves. He writes, "think about all the sets that do not contain themselves... the set of all cats doesn't contain itself because it itself is a set not a cat." The paradox arises when you ask: Does this new set contain itself? If it does, it violates its own definition; if it doesn't, it fits the definition and must be included. Kaplan's delivery here is masterful, stripping away the jargon to reveal a logical loop that "blows the whole thing up."
"Russell himself and many other mathematicians thought that they could solve this paradox but i argue that they can't and they don't."
This is the piece's most provocative claim. While standard textbooks often present the resolution (axiomatic set theory) as a clean fix, Kaplan insists the problem remains unresolved in a philosophical sense. Critics might note that modern mathematics has largely moved past this by adopting stricter axioms that simply forbid the creation of such paradoxical sets, effectively sidestepping the issue rather than solving it. Yet, Kaplan's insistence that the intuition of the paradox remains valid challenges the reader to accept that our logical tools may be inherently flawed when pushed to their limits.
The Bottom Line
Kaplan's commentary is a rare gem that treats a 120-year-old mathematical puzzle as an urgent, living problem rather than a historical footnote. Its greatest strength is the refusal to let the reader off the hook with a technical fix, forcing an acknowledgment that the definition of a number is still philosophically fragile. The biggest vulnerability is the dismissal of formal axiomatic solutions, which, while perhaps not satisfying to a philosopher, have allowed science and technology to function perfectly well for a century. For the busy reader, the takeaway is clear: the certainty we place in mathematics is a triumph of utility, not necessarily of absolute truth.