elaborations upon a blasphemous course review
I am a pure mathematician by training, inclination, and marriage.
I am also a blasphemer, a heretic, and a traitor to my people.
What I’m saying is that I just finished taking a machine learning course full of rigorous mathematics, and in my course review, I advised the professor to stop worrying about all that rigor. (Sort of. What I said was more nuanced, but no less obnoxious.)
For the question “How can this course be improved?” I wrote:
I humbly propose rethinking the role of proof and mathematical derivations in the course.
A widely held view among mathematicians is that proofs deepen students’ understanding. In my opinion as a professional math communicator, this is wrong. Proof is better understood as the last step in mathematical work. As you know, a researcher attempts a proof only after a long process of probing examples and seeking intuitive principles. Until students have gone through a parallel process, seeing proofs will rarely benefit them.
I also believe that, for understanding most material, proof is neither necessary nor sufficient. In this course, when I would ask the TAs about the proofs, I would sometimes find myself answering their questions instead. Yet I know they understood the models themselves better than I did!
For these reasons, I propose moving proofs from the beginning of each topic to the end, and treating them as parenthetical to the work of understanding and implementing the models.
In my view, this would not constitute a loss of rigor or depth, but the opposite. It would embrace the true role of proof (as an act of consolidation and intra-mathematical communication), while shifting student focus toward understanding the logic and limitations of machine learning methods.
Let me elaborate upon my heresy.
It’s common to treat a proof as a kind of explanation: a careful, formal, highly detailed answer to the question “Why is this true?” Under this view, proof is the essence of understanding, and an unproven statement is a black box. A class full of unproven statements is even worse: a shallow ditch, a clown show.
I say bah.
I say humbug.
I say this whole worldview rests on a confusion about the word “why.”
When the referee of a math research journal asks “Why?”, then you should answer with a proof. That’s well and good. But when a human being asks “Why?”, then you should answer with something subtly yet profoundly different: an explanation.
A proof and an explanation share a similar structure. Both show how a new statement fits comfortably in a pre-existing structure of old statements.
The question is: what exactly is this structure?
In the case of proof, it’s the body of shared mathematical knowledge. It consists of accepted axioms, earlier proofs, and agreed-upon definitions. It’s a tower built by a community of researchers, and we’re supposed to be very careful when making additions. Nary a loose brick is safe.
In the case of explanation, the pre-existing structure is my personal knowledge about the world. It consists of experiences, beliefs, mental models, and earlier mathematical understandings. Logic and rigor may help hold it all together, but for the most part, it’s made of squishier stuff: approximations, rules of thumb, illustrative examples, and vivid, memorable images.
To contrast proof and explanation, let’s take a famous statement:
On the one hand, we can prove this by induction. Such a proof cements the statement’s place in the body of mathematical facts. It is now as truly immortal and immortally true as any fact can be.
But does anyone really feel like this explains it?
Meanwhile, here’s an explanation of that same fact. Your mileage may vary, but for my part, this explanation deepens my understanding and helps me embrace the statement’s truth.
Our strategy is to estimate the sum: it’s the length of the list multiplied by the average number on the list.
What is the average number? Tempting to think it’s 1/2 of n2, but that’s off. It’s closer to a 1/3 of n2.
So that gives us our estimate: roughly n3/3.
Have we proved the formula? Not at all. We’ve given a squishy argument for something vaguely similar to the formula. But that has its own benefits. Better yet, for those who know calculus, this sloppy formula hints at another explanatory connection: the sum of the first n squares closely resembles the integral of x2.
Explanation is not just a fuzzy proof, and proof is not just a rigorous explanation. They are different species entirely: one logical, the other psychological.
Am I saying proof is bad? No! Proof is the architecture of mathematics. It’s our castle walls. Without proof, the rain would soak our hair, the wind would snuff out our candles, and the wild animals (read: physicists) would come wandering inside to eat our throw rugs.
Don’t abandon proof. Just acknowledge its true purpose – or rather, purposes. Proof finalizes and formalizes understanding. Proof enables generations of scholars to collaborate on a single intellectual project. Proof serves as math’s final arbiter of truth.
And, as a kind of side gig, proof sometimes helps to explain why things are true.
But if we want students to understand mathematics, we can’t expect proof to do the heavy lifting. We need worked examples. We need well-chosen counterexamples. We need pretty pictures. Heck, we need ugly pictures. We need analogies, heuristics, and loose connections to more familiar ideas.
We need, in a word, explanations.
I felt silly writing that course review. It was a lovely class whose primary shortcoming (an uncritical embrace of proof as the be-all-end-all for mathematical reasoning) is shared by approximately 99.737% of similar courses. More to the point, I’ll be shocked if my professor is at all persuaded. That’s the nature of explanation: it’s a gradual process, a social process, and it needs to meet people where they are.
Still, if nothing else, I hope that I managed to make my blasphemy a little more legible – and perhaps even a little less blasphemous.