Undergraduate Mathematics Blog

A PDEs "laboratory experience"

What is a proof?

10/22/21 - During many math (and closely related) majors' second year of study, they encounter their first proofs course. And indeed, many students (including myself) run into a brick wall when it comes to their perceived progress in math. For years, to solve a math problem was to provide (with work shown) a solution to a problem. The solution must be correct because one follows the directions the teacher had provided (in previous lectures). In proofs, that is no longer the case.

Finally, we are all asked to follow the "directions" of our own choosing. We must first decide ourselves what is valid, and use that to convince another.

Now, however, I reveal the problem that all students arrive at in their first proofs course: "How much do I need to write down for a proof?"

Indeed, no one has a good definition for what constitutes a mathematical proof. Even computer proof-checking systems such as LEAN rely on our trust that the system is unerring in its output. It so happens that every mathematician must eventually trust that something is already true, and the difficulty in proofs becomes determining what the audience trusts is already true.

That is to say, a "proof" is any argument which manages to convince its audience of its validity.

For a toddler, the proof that 1+1=2 might consist of taking one apple, then another apple, and placing them together to count "one, two". Mathematicians aren't so keen on the proof by example emphasized here. We have seen too many cases of patterns abruptly ending to be so trusting (I bring to attention the Borwein integrals).

In our proofs course, then how much should a student write to provide a proof to their professor? Well, it depends on the professor, I guess. I could take the proof from one student under one professor, and the proof (of the same result) from a student with another professor, have those two professors grade each other's students, and both would likely be disappointed. Each professor stresses different statements, implications, and grammar necessary for a proof, so that when the students are novices, there is no chance for them to "see past" their professor's style of proof into what the greater mathematical community (in this case, the other professor) may view as valid.

It is with just this perspective that we must approach our proofs to the mathematical community. When hundreds of "proofs" of the Riemann hypothesis are uploaded to the internet each year, authors are always disappointed as to the lack of excitement and fame given to their results. But in every case, while the authors have "proven" the Riemann hypothesis to themselves, what they have provided to the world, is simply their argument, not a proof. Perhaps in some cases, there is a truly valid (in the eyes of the mathematical community) argument for the Riemann hypothesis, but without a proper communication of such an argument, a proof doesn't exist.

I conclude then, with how one should approach a mathematical proof in steps:

The equivalence of the disc and shell method via integration by parts

8/31/21 - I was "lucky" enough to be asked to substitute for a Calculus II recitation today. I made a mistake, and am here to comment on how I remedied such a mistake.

The topic was on volumes of solids via revolution. Indeed, the students have already derived the relevant disc and shell method in class. Such derivations are included in every Calculus book. I'll list them both here to refresh the reader:

Disc:      Resulting from the formula for volume of a cylinder:

for rotations about the x-axis,

and

for rotations about the y-axis

Shell: Resulting from the formula for the surface area around a cylinder:

for rotations about the x-axis,

and

for rotations about the y-axis.

I was re-deriving the shell method for the class when I made the infamous mistake. Well, not perhaps a mathematical mistake, but certainly a  mistake. You see, when calculating the shell method, I decided on the "wrong" side of the curve to fill-in. This is equivalent to finding the "complementary volume" of the smallest cylinder within which the curve fits (and necessarily forces the resulting solid to have no holes). Such a choice leads to the following formula for the "wrong" shell method presented here:

for rotations about the x-axis,

and

for rotations about the y-axis.

Did I decide to stop, admit defeat, and re-derive the expression correctly? Yes. Yes I did. However, that got me thinking, and that's why I'm here... Let us examine now, the disc method around the x-axis, but in terms of

and

Then the disc method looks like

We perform integration by parts on this formula, with

so that we recover

which we identify as precisely the formula derived directly above with

Indeed such a formula is not the "standard" shell method. As mentioned, it calculates the "complementary volume" to the shell method volume, where the two together perfectly nest into the smallest cylinder within which the curve fits. The volume of this cylinder can be calculated to be precisely

Therefore, to remedy my mistake, we consider

So, basically, we could always teach the derivation of the shell method this way. But should we? Probably not.