But it's not really true. At the very least, studying pure math helped me understand the first chapter of Lange's Applied Probability, and get far enough into the book to understand a much more concise solution to Buffon's Needle Problem than the one on wikipedia.
In day to day life though, knowing what measurability mean or what Borel sets are could hardly be called useful. The value of a pure math degree, for non-tenure-track mathematicians, comes ~80% from satisfying a curiosity (or what some would call intellectual masturbation).
The real value comes from learning how to think. This isn't unique to pure math though: students of a number of disciplines will likely say that it is what they learned from the course of study. Each field, however, would choose a different focus and a different kind of thought process. A philosophy student would think differently from an engineering student, who would think differently from a biology or a law student.
So what makes pure math thinking unique? I think it is several things, some of which are:
- Decoding complex notations
- Going back and forth between the "abstract" and the tangible example
- A focus on details, especially the "edge cases"
...which all come from understanding and analyzing proofs of all types.
So let's break it down.
Decoding Complex Notations
This was actually a goal of mine while studying pure math. I wanted to become comfortable with reading mathematical notations, and not be afraid of them in books and papers. Like reading other people's code, reading other people's proofs were the most dreaded task for me. Grading second year "advanced level" linear algebra was the worst.
But reading math is more difficult than just understanding and being comfortable with notations. There was a blog post from the beginning of the year about how reading code is not like reading literature. Here's a choice quote from that blog post:
We don’t read code, we decode it. We examine it. A piece of code is not literature; it is a specimen.It's really the same thing with proofs. We don't read proofs. We decode proofs. We don't really understand a proof unless we make it ours.
While reading code is still difficult for me, I think my education has helped lessen that barrier.
Abstract vs Tangible
In one of his books, Feynman talked about how during every discussion about something abstract, he would follow along with something tangible in mind. In another story that a professor had told, a grad students had proved interesting things about functions that satisfy some criteria, only to find during a presentation that the only functions that satisfied those criteria were the constant functions.
This ability to keep both the abstract and the tangible in mind is crucial for understanding mathematics, and a way of thinking pretty unique to the field (and maybe perhaps computer science as well: I sometimes joke that the math in computer science is like applied pure math).
Related to this is looking at various tangible examples, and extract the abstract features that are common to all examples. It is often easy to recognize that there is some kind of common structure, but the structure can be difficult to describe without the right set of language in your toolbox. (Again, this may apply more so to computer scientists and good programmers).
Proofs are hard because of all the edge cases that one has to consider. I haven't encountered these questions in Waterloo, but in some institutions "PODASIP" problems are typical. PODASIP stands for: "prove or disprove and salvage if possible", which basically means that for a given claim, the task is to see if the claim is true or not, and if it isn't, modify the claim so that it is (non-trivially) true.
I find that the best way to tackle those problems is to look first for counter-examples to the claim. A claim usually falls apart in extreme cases, so those were good starting points. An understanding of the extreme cases would lead either to a counter-example, or to an understanding of why the claim works, and thus a proof.
So, a pure math student learns to think like a lawyer, except not with legal language but with formal claims.
So, in summary, like some other degrees, a pure math degree is one that teaches you how to think. But all degrees that teach you how to think focuses on different types of thinking.
In hindsight, I may have preferred a degree that teaches similar thinking principles while teaching something more useful. Still, no regrets though because a lot of those "more useful" ideas are much more easily learned with the toolkits I now have available, like an understanding of abstract algebra and measure theory.