Elaboration of a blasphemous course review

I am a pure mathematician by education, passion and marriage.

I am also a blasphemer, a heretic, and a traitor to my people.

What I’m saying is that I just finished a machine learning course full of rigorous math, and in my course report, I advised the professor to stop worrying about all that rigor. (Kind of. What I said was more nuanced, but no less disgusting.)

For the question “How can this course be improved?” I wrote:

I humbly suggest reconsidering the role of proofs and mathematical deductions in the course.

A popular view among mathematicians is that evidence increases student understanding. In my opinion, as a professional math communicator, this is wrong. Evidence is better understood than that latest Step in math work. As you know, a researcher tries to prove it after a long process of examining examples and searching for intuitive principles. Until students have gone through a parallel process, they will seldom benefit from seeing evidence.

I also believe that evidence is neither necessary nor sufficient to understand most of the materials. Sometimes when I asked the vets about the evidence in this class, I would answer their questions instead. However, I know that they understood the models better than I did!

For these reasons, I suggest moving evidence from the beginning of each topic to the end and treating it as a bracket for the work of understanding and implementing the models.

In my view, this would not mean a loss of accuracy or depth, but the opposite. It would encompass the true role of evidence (as an act of consolidation and intramathematic communication) while shifting the students’ focus to understanding the logic and limitations of machine learning methods.

Let me address my heresy.

It is common to treat evidence as some sort of statement: a careful, formal, and very detailed answer to the question “Why is this true?” According to this view, evidence 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 is based on a confusion about the word “why”.

When the arbiter of a math research journal asks why, you should respond with evidence. That’s all well and good. But when a person asks why, you should respond with something subtle, yet profoundly different: an explanation.

A proof and an explanation have a similar structure. Both show how a new statement fits comfortably into an existing structure of old statements.

The question is: what exactly is this structure?

Yes, in order to explain a new fact, it has to be wedged in your cerebrum like chewing gum.

In the case of evidence, it is the body of shared mathematical knowledge. It consists of accepted axioms, previous evidence, and agreed definitions. It’s a tower built by a community of researchers, and so should we very Be careful when adding. Nary a loose brick is certain.

In the case of the explanation is the structure already in place my personal knowledge of the world. It consists of experiences, beliefs, mental models and previous mathematical knowledge. Logic and rigor may help hold it all together, but for the most part, it’s made up of squishier things: approximations, rules of thumb, vivid examples, and vivid, memorable images.

To contrast evidence and explanations, let’s take a famous statement:

On the one hand, we can prove this by induction. Such a proof cemented the place of the statement in the body of mathematical facts. It is as truly immortal and immortally true now as any fact can be.

But does someone really feel that way? explained it?

In the meantime, here is an explanation for the same fact. Your mileage may vary, but for my part, this explanation will deepen my understanding and help me understand the truth of the statement.

Our strategy is to estimate the sum: it’s the length of the list times the average number on the list.

What is the average number? Trying to think it’s 1/2 of n2but that’s over. It’s closer to a 1/3 of n2.

So that gives us our estimate: roughly n3/ 3.

Did we prove the formula? Not at all. We made a mushy argument for something vaguely similar to the formula. But that has its own advantages. Better still, for those familiar with calculus, this sloppy formula suggests another explanatory relationship: the sum of the first n squares is very similar to the integral of x2.

Explanation is not just fuzzy evidence and evidence is not just strict explanation. They are completely different types: one logical, the other psychological.

Am I saying the evidence is bad? No! The proof is the architecture of mathematics. It’s our castle walls. Without evidence, the rain would soak our hair, the wind would extinguish our candles and the wild animals (read: physicists) would come in to eat our carpets.

Don’t give up the evidence. Just acknowledge its true purpose – or rather, its purposes. The evidence concludes and formalizes the understanding. The evidence enables generations of scientists to collaborate on a single intellectual project. The proof serves as the mathematics’ ultimate arbiter for the truth.

And as a kind of side appearance, the evidence sometimes helps explain why things are true.

But if we want students to understand math, we can’t expect proof that they are doing the heavy lifting. We need examples. We need well-chosen counterexamples. We need beautiful pictures. We need ugly pictures. We need analogies, heuristics, and loose connections to better known ideas.

We need explanations in one word.

I thought it was silly to write this course report. It was a lovely class whose primary flaw (uncritical evidence as the be-all and end-all of mathematical reasoning) is shared by approximately 99.737% of similar courses. More precisely, I’ll be shocked if my professor is at all convinced. That is the nature of the explanation: it is a gradual process, a social process, and it has to hit people where they are.

Still, I hope I’ve managed to make my blasphemy a little more readable – and maybe even a little less blasphemous.


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