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Comment by marcelluspye | Hacker News Reader

>“He had formulas that enabled him to pull off his magic,” Pitt said. “And I didn’t have the formulas.”

The more I do pure mathematics, the more I realize just how important these kinds of insights are. Very often, solving a theoretical problem involves two key ingredients:

1. Rewriting your problem in a particular way, so that it is amenable to a certain suite of methods/looks like known results.

2. Apply a key bit of knowledge gleaned from intuition. This is unrelated to the formal way the problem was written down.

Sometimes, showing your intuition is true formally actually takes a lot of work. And for some proofs, looking at the problem a particular way makes the solution obvious on its own, with no need for a step 2. And other times, like this, all the technical tools in the world are no match for just knowing the right piece of information.

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I think many highly productive mathematicians are very good at organizing formulas and facts in such a way that they can be retrieved easily from memory based on context. Maybe something analogous to a hash function from computer science.

Some mathematicians seem to index facts based on geometric images, others seem to be more inclined to symbolic or algebraic statements. Whatever the representation, when confronted with a new mathematical situation they then scan quickly for matches to various aspects of the problem at hand.

Maybe to some degree my observation here is obvious. But I thought a lot about it while I was in grad school studying a book called Geometric Measure Theory by Herbert Federer. That book is enormous, and full of highly intricate technical proofs that require pulling together a large number of detailed technical facts.

The book is also very highly structured, and that led me to conclude that the text likely mirrored how Federer organized this information in his head. It reads like code for a complex but cleanly architected software system, and that's a big part of what led me from math to software development.

are you under the impression the people write books like that without referring to sources themselves? because they don't. I doubt any one knows all the proofs in a book like by heart or could reconstruct them without references for key facts.

No, I was never under that impression. I once had a discussion about the creation of the book with one of Federer's students who had also helped proofread it. The book was essentially an outgrowth of his personal and course notes, since of course he couldn't keep all this in his head.

What he likely did have in his head was an index into the contents, that's the crux of my observation. If I am very good at organizing my workshop, I can quickly grab the tools and materials needed for a particular task without breaking my flow of thought. Same basic principle applies to mathematicians and other intellectual workers, just as it does with physical trades.

People of that calibre are quite rare but they do exist. I've seen final year undergraduate courses pulled off without the lecturer once having to refer to notes. All proofs were completed in exacting detail on the whiteboard.

Sure, but those math profs have probably taught the same course 20+ times. Going through material that many times will permanently burn it into your brain.

Absolutely. And in mathematics, advanced structures and theorems are built up layers by layer upon more elementary material. A professor who has mastered presentation of undergraduate material on a topic also likely teaches a graduate course on it, and mentors students on it, and does research on it.

They can talk about their chosen topic at many levels to many different audiences, from general audience (who may provide funding to them), high school students (outreach and recruiting), university students, and peers. This flexibility is an important part of being a very successful mathematician, and you have to burn it into your brain to reach that level of fluency.

I took a philosophy class with a professor who had taught it about that many times. Talked to a guy who had taken it before me, he said that the prof has literally word-for-word memorized certain parts of the lectures because he's figured out and internalized the wording he thinks is best. Mathematicians can certainly do the same.

also they review the lecture before hand just like anyone delivering a speech does

A similar realization is that most algorithms/data structures problems posed in software engineering interviews are all about imposing the right mathematical formalisms.

If you correctly infer the _structure_ of the problem, then you're going to have an easy time solving it. If not, you'll use a great lot of time hunting for a fruitful angle of attack.

Oh please let’s not glorify those simple trick questions to be anywhere near as hard or require the degree of intuition that OP or theoretical mathematicians work on.

75% of interview questions can be solved with some form of BFS/DFS and they’re largely a hazing ritual these days. I’m saying this as someone who recently got offers from 4 of the big 5 companies

Something only takes “hard intuition” the first time it comes up. For instance, when these data structures were new, inventing them took hard intuition. The second time the same trick is used, it becomes a ‘clever application of an obscure idea’ to a new problem.

Then once the same intuitive leap has been used to solve many different problems, and gets taught in school, it becomes a “simple trick”, or even just a “standard technique”.

If this same method is useful for solving a wide range of structurally similar problems, it will go through that process, and eventually become thought of as a “simple trick”. If it is only used as a one-off for this particular proof, it will remain one man’s genius idea.

I've worked through various exams and research problems leading to a math PhD, and I see the similarity. I fail to see how someone articulating an observation about that "glorifies" the interview questions.

I've heard students complain that graduate qualifying exams are a form of hazing for people hoping to become pure mathematicians, and although I don't entirely agree with that sentiment, they do play a similar role in weeding out people based on preparation rather than ability to generate deep, novel insights.

\tangent does the formal proof make sense to a mathematician (as in, can they see it)?

Or is it more the hex of assembly of to the high level language of the intuition, in the semse that they can verify it, but not necessarily see it?

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