Process over state: Math is about proofs, not theorems.

August 10, 2019 by Artem Kaznatcheev

A couple of days ago, Maylin and I went to pick blackberries along some trails near our house. We spent a number of hours doing it and eventually I turned all those berries into one half-litre jar of jam.

On the way to the blackberry trails, we passed a perfectly fine Waitrose — a supermarket that sells (among countless other things) jam. A supermarket I had to go to later anyways to get jamming sugar. Why didn’t we just buy the blackberries or the jam itself? It wasn’t a matter of money: several hours of our time picking berries and cooking them cost much more than a half-litre of jam, even from Waitrose.

I think that we spent time picking the berries and making the jam for the same reason that mathematicians prove theorems.

Imagine that you had a machine where you put in a statement and it replied with perfect accuracy if that statement was true or false (or maybe ill-posed). Would mathematicians welcome such a machine? It seems that Hilbert and the other formalists at the start of the 20th century certainly did. They wanted a process that could resolve any mathematical statement.

Such a hypothetical machine would be a Waitrose for theorems.

But is math just about establishing the truth of mathematical statements? More importantly, is the math that is written for other mathematicians just about establishing the truth of mathematical statements?

I don’t think so.

Math is about ideas. About techniques for thinking and proving things. Not just about the outcome of those techniques.

This is true of much of science and philosophy, as well. So although I will focus this post on the importance of process over state/outcome in pure math, I think it can also be read from the perspective of process over state in science or philosophy more broadly.

It is tempting to switch from blackberries to fish and summarize the focus on process over outcome with the proverb attributed to Anne Isabella Thackeray Ritchie: “Give a man a fish and you feed him for a day; teach a man to fish and you feed him for life.”

So in the context of mathematics, it is tempting to adapt the above as: “Give a mathematician a theorem, you satisfy her for a day; teach a mathematician a new proof technique and you satisfy her for life”.

But both of these are too outcome-focused for my taste. Too grounded in fish. They both put the outcome state as primary and only value the process in that it achieves the state. I don’t think that this is the case in pure mathematics.

Let’s return to the opening machine that replied with perfect accuracy if a given statement was true or false. In Lady Ritchie’s metaphor, this would be like the supermarket or sushi bar around the corner: you can easily get food anytime without having to know how to fish.

Yet sometimes people with easy access to supermarkets or sushi bars still like to go fishing. They don’t go fishing because they need to eat some fish, they go fishing because they like to go fishing.

I think this is also the case for mathematicians.

In many cases, mathematicians prove theorems not because they need to know the truth-value of those statements but because they like proving theorems. But this like isn’t a purely meditative one, they don’t want to reprove the same theorem over and over again. They want to find new ways to prove new kinds of theorems.

Mathematicians are after tricks, not theorems.

Recently, an interesting discussion was started on /r/math by /u/Junglemath on horrible words that should never be used anywhere in math. In particular, they were concerned about words like ‘obviously’, ‘clearly’, ‘trivially’, ‘easy to see’, ‘doesn’t warrant a proof’, and ‘left as an exercise to the reader’. If our goal is to establish the truth of mathematical statements then these words have no place in mathematics. They are just invitations to make mistakes. The thinking goes: if something is obvious then just write down the proof.

But our primary goal isn’t to establish the truth of mathematical statements.

If you are writing a paper for other mathematicians then it is important to establish what new trick you develop. And there is no reason to bore the reader with applications of existing and well-known tricks. Good taste and insight in mathematics come from recognizing when some approach or technique is novel or likely to be widely relevant.

This brings us back to the Reddit discussion on horrible words.

In this context, words like trivially, clearly, obviously, and the omission of proofs is justifiable. It helps us focus our attention on the heart of an idea. On the new gem that powers the proof. As such, seeing those words in textbooks or lectures is the author or instructor training us on good taste. It is why it is okay for an instructor to skip steps in proof with ‘obviously’ but not as good for the student to do so in their writing. Although I still try to teach this to my students. I encourage them to write a proof sketch drawing my attention to the ‘central’ parts of the proof before they embark on giving a fully fleshed out proof. This gives them an opportunity to identify which part of the proof is the central idea and which parts are routine applications of straightforward techniques.

Within the myth of genius of mathematics, there is the expectation that great mathematicians simply see the truth or falsehood of mathematical statements. That things are trivial for them that are not trivial for us. As such, it might seem like it would be impossible to follow somebody like Terry Tao when he writes that ‘this part is trivial’. But I think this is a mistaken view of math.

One of the things that makes Terry Tao a great mathematician — alongside developing new ideas — is the ability to identify which ideas are old and which ideas are new. A good mathematician can identify which parts of their proof are non-standard and new (thus, not trivial) and which parts are rehashing of standard (sometimes very complex) techniques. Hence, the reason it is useful to know that Tao finds trivial vs non-trivial is not that it highlights his ability to see the truth of statements like some sort of oracle of Los Angeles. What matters is that Tao can point us to which techniques are new (and thus we should learn and incorporate into our own mathematical tool-set) and which techniques are applications of things we already know how to do. This is very useful information.

Thus, when used correctly, ‘trivially’, ‘clearly’, and ‘obviously’ are great signposts. As long as you are using them to highlight new ideas and processes instead of to establish the truth of statements.

As for the machine that establishes the truth or falsehood of any statement. Apart from being impossible (thank you, Halting Problem), it would be rather unfortunate. We understand why something is true not through the truth value of a statement but through the building and thinking about proofs. For me, this hypothetical machine would fall under machine learning without understanding. Unless, of course, it produced not just truth statements but human-readable proofs that can pass Ganesalingam & Gowers’ Mathematical Turing test. But in that case, we might as well call that machine another mathematician and welcome it into our community with open arms.

I think that similar sentiments can also be found in science and philosophy more broadly. I especially like how this sentiment is turned in on itself in theoretical computer science and parts of math to find the limits and barriers to our best proof techniques.

It is certainly fun to have useful reliable knowledge come out as the outcome state of our process of inquiry. Just as it might be great to eat that bass the fishers caught on their fishing trip, or fun for Maylin and me to enjoy our jam. And there might also be practical disciplines that only focus on producing useful knowledge — much like a fishing boat or a commercial farm feeds a supermarket. But I suspect that for most scientists and philosophers, just as with mathematicians or recreational fishers, it is the process and not the outcome that matters most. Reliable knowledge as an outcome state is just a wonderful bonus.