What’s The Longest Proof I’ve Done?


You know what’s a question I’m surprised I don’t get asked? I mean in the context of being a person with an advanced mathematics degree. I don’t get asked what’s the longest proof I’ve ever done. Either just reading to understand, or proving for myself. Maybe people are too intimidated by the idea of advanced mathematics to try asking such things. Maybe they’re afraid I’d bury them under a mountain of technical details. But I’d imagine musicians get asked what the hardest or the longest piece they’ve memorized is. I’m sure artists get asked what’s the painting (or sculpture, or whatnot) they’ve worked on the longest was.

It’s just as well nobody’s asked. I’m not sure what the longest proof I’ve done, or gone through, would even be. Some of it is because there’s an inherent arbitrariness to the concept of “a proof”. Proofs are arguments, and they’re almost always made up of many smaller pieces. The advantage of making these small pieces is that small proofs are usually easier to understand. We can then assemble the conclusions of many small proofs to make one large proof. But then how long was the large proof? Does it contain all the little proofs that go into it?

And, truth be told, I didn’t think to pay attention to how long any given proof was. If I had to guess I would think the longest proof I’d done, just learned, would be from a grad school course in ordinary differential equations. This is the way we study systems in which how things are changing depends on what things are now. These often match physical, dynamic, systems very well. I remember in the class spending several two-hour sessions trying to get through a major statement in a field called Kolmogorov-Arnold-Moser Theory. This is a major statement about dynamical systems being perturbed, given a little shove. And it describes what conditions make the little shove really change the way the whole system behaves.

What I’m getting to is that there appears to be a new world’s record-holder for the Longest Actually Completed Proof. It’s about a problem I never heard of before but that’s apparently been open since the 1980s. It’s known as the Boolean Pythagorean Triples problem. The MathsByAGirl blog has an essay about it, and gives some idea of its awesome size. It’s about 200 terabytes of text. As you might imagine, it’s a proof by exhaustion. That is, it divides up a problem into many separate cases, and tries out all the cases. That’s a legitimate approach. It tends to produce proofs that are long and easy to verify, at least at each particular case. They might not be insightful, that is, they might not suggest new stuff to do, but they work. (And I don’t know that this proof doesn’t suggest new stuff to do. I haven’t read it, for good reason. It’s well outside my specialty.)

But proofs can be even bigger. John Carlos Baez published a while back an essay, “Insanely Long Proofs”. And that’s awe-inspiring. Baez is able to provide theorems which we know to be true. You’ll be able to understand what they conclude, too. And in the logic system applicable to them, their proofs would be so long that the entire universe isn’t big enough just to write down the number of symbols needed to complete the proof. Let me say that again. It’s not that writing out the proof would take more than all the space in the universe. It’s that writing out how long the proof would be, written out would take more than all the space in the universe.

So you should ask, then how do we know it’s true? Baez explains.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

6 thoughts on “What’s The Longest Proof I’ve Done?”

  1. I think part of the problem is that, in general, non-mathematicians don’t have much of a concept of what working mathematicians actually do. Most of the work I do that I think of as Mathematics consists of specific applications and isn’t terribly concerned with proof as such.

    That said, the most time I spent working on a proof was as an undergrad. It was a plane tiling problem involving constraints on the dimensions of the plane. I spent about a week and a half on it and only managed to prove sufficiency.

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    1. You’re right. It might also be that people don’t think much about what mathematicians do all day. I’m not perfectly clear on it myself, I must admit. But when I was a real working mathematician most of my research was really numerical simulations and experiments. There were a couple of little cases where I needed to prove something, but it was all in the service of either saying why my numerical experiments should work, or why a surprising result I’d found experimentally actually made sense after all.

      My biggest work in actually coming up with proofs might have been in a real analysis course I took as a grad student. I’d had a lovely open-ended assignment and kept chaining together little proofs about one problem to build a notebook of stuff. This was all proofs about logarithms and exponentials, so none of the results were anything remotely new or surprising, but it was really satisfying to get underneath some computation rules and work them out.

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  2. I’m more likely to be asked ‘what is the longest equation that I’ve solved’? :)

    Related to what MJ said, when I interview high-school students for undergraduate maths scholarships, I ask them what is the longest they have ever spent solving a problem. The answer is usually 15 minutes. Occasionally someone says overnight. That gives us one clue as to what non-mathematicians (albeit maths students) think it means to be good at maths and how mathematicians work (that is, solve problems relatively quickly and move on). To be fair, I don’t expect these students to answer any differently because they respond based on (1) their experience and (2) what they think we want to hear. But it is illuminating.

    I’d never heard of the Boolean Pythagorean Triples problem until earlier this week, either. I love that there are easy to understand maths ideas that I’ve never heard of. No idea how the proof works either ;).

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    1. Longest equation that I’ve solved … hm. Well, if it’s the equation I spent the longest time in solving that’s got to be something in the inviscid fluid flow that made up a lot of my thesis. The physically longest equation I don’t know. I remember shortly after starting into high school algebra at all trying to think of the hardest possible equation. Given that all I really had to work with was polynomials my first guess was just something with a bunch of variables all raised to high powers. But I also worked out that this was a boring equation. Never did work out what would be both complicated and interesting at once.

      I wonder how long non-mathematicians expect gets spent on leads that ultimately go nowhere, before a workable approach to the problem is worked out. Or if not nowhere then at least go into directions that don’t work without a lot of re-thinking and re-casting. There is a desire to show how to get right answers efficiently, for which people can’t be blamed. But the system of learning how to think of ways to get answers probably needs false starts and long periods of pondering that feel like they don’t get anywhere.

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