a. Having a gradual and cumulative effect
b. of a disease : developing so gradually as to be well established before becoming apparent
a. awaiting a chance to entrap
b. harmful but enticing
— Merriam-Webster Dictionary
In early topics in mathematics, one can often approach a topic from a combination of intution and first principles in order to deduce the desired results. In later topics, it becomes necessary to repeatedly sharpen intuition while taking advantage of the insights of the many mathematicians who came before — one sees much further by standing on the giants. Somewhere in the middle, it becomes necessary to accept the idea that there are topics and ideas that are not at all obvious. They might appear to have been plucked out of thin air. And this is a conceptual boundary.
In my experience, calculus is often the class where students primarily confront the idea that it is necessary to take advantage of the good ideas of the past. It sneaks up. The main ideas of calculus are intuitive — local rates of change can be approximated by slopes of secant lines and areas under curves can be approximated by sums of areas of boxes. That these are deeply connected is surprising.
To many students, Taylor’s Theorem is one of the first examples of a commonly-used result whose proof has some aspect which appears to have been plucked out of thin air.1 Learning Taylor’s Theorem in high school was one of the things that inspired me to begin to revisit calculus with an eye towards why each result was true.
I also began to try to prove the fundamental theorems of single and multivariable calculus with as little machinery as possible. High school me thought that topology was overcomplicated and unnecessary for something so intuitive as calculus.2
This train of thought led to my previous note, on another proof of Taylor’s Theorem. That note is a simplified version of one of the first proofs I devised on my own.
Much less obviously, this train of thought also led to the paper on the mean value theorem written with Miles. Originally I had thought that “nice” functions should clearly have continuous choices for mean value abscissae, and I thought that this could be used to provide alternate proofs for some fundamental calculus theorems. It turns out that there are very nice functions that don’t have continuous choices for mean value abscissae, and that actually using that result to prove classical calculus results is often more technical than the typical proofs.
The flow of ideas is turbulent, highly nonlinear.
I used to think that developing extra rigor early on in my mathematical education was the right way to get to deeper ideas more quickly. There is a kernel of truth to this, as transitioning from pre-rigorous mathematics to rigorous mathematics is very important. But it is also necessary to transition to post-rigorous mathematics (and more generally, to choose one’s battles) in order to organize and communicate one’s thoughts.
In hindsight, I think now that I was focused on the wrong aspect. As a high school student, I had hoped to discover the obvious, clear, intuitive proofs of every result. Of course it is great to find these proofs when they exist, but it would have been better to grasp earlier that sometimes these proofs don’t exist. And rarely does actual research proceed so cleanly — it’s messy and uncertain and full of backtracking and random exploration.
- It may be that the handwavy proofs involved with proving Rolle’s Theorem or the Mean Value Theorem appear equally mysterious. But many calculus courses don’t really prove these or don’t show that they’re useful (or most likely, they don’t do either).
- It wasn’t until much later, after I gained a bit of mathematical maturity, that I learned that topological ideas *do* matter.