Your phrase "or even what it means to understand that thing" strikes a chord with me. As someone studying math and computer science on my own, without anyone around who shares my research interests, I have to figure out from first principles not only how to gain an understanding of things but define what understanding it even means.

For example, I have recently been pursuing the idea of identifying data streams with bijections with the natural numbers, and higher-order operations on these bijections. The bijection is arbitrary, so the order assigned the sequence "doesn't matter" (the subject of study happens to be associative and commutative in its pure form). However I ran into an inconsistency with this when modeling products of streams, and came to an epiphany: yes, the order (the specific bijection) is arbitrary BUT it is fixed: that is, you can pick any order, but it has to be committed to a priori. (An intuitive analogy would be to consider the difference between a card game where the dealer shuffles the deck before the game and deals from the top, versus a card game where the dealer selects the cards he wants to give you as the game proceeds.)

I felt like that was a moment when I understood what it means to understand this thing I'm studying which is so abstract. I saw it as a fork in the road between mathematics and mysticism: prior to that realization I had been studying mysticism, "assume the (nondeterministic) computer 'magically' chooses the best next item in the sequence to pursue" versus "assume the sequence is a priori in the best order to search (but that order is fixed)". The former is mysticism because it asserts something exists without pinning it down. The latter is mathematical because, although it describes something abstract, the properties of the thing it's describing do not shift during the course of working with it.

I think Davis and Hersh say this. Ruben Hersh "what is mathematics really" felt like it was opening the door to the sausage factory.

Well,

It is interesting that a lot of great mathematicians have believed in truth beyond the sausage factory. Not that the reality isn't a sausage factory but belief in an ideal (truth, beauty) seems to often be what keeps them going.

Kurt Godel proved that the machinery of mathematics couldn't create a complete truth system accessible through proofs (which are ultimately a mechanical process) but he definitely was on the side of ideal-truth when he said "Either mathematics is too big for the human mind, or the human mind is more than a machine."

Edit: And Grigori Perelman was certainly famous for rejecting what he saw as the sausage factory of mathematics.

Brilliant book that. The way he explains how a polished rigorous proof hides the process(compared with frontend and kitchen views of a restaurant) by which a mathematician has arrived at it changed my perception of how mathematics is done.

I notice the same phenomena in programming too. The final piece of code that works doesn't reflect all the failed attempts at solving a problem. This is why I tend to not delete/clear cells at all from my jupyter notebooks. :) It is always informative/fun to refer to them at a later point in time to relive that problem solving experience.

PS: In case you haven't, you should try Proofs and Refutations by Lakatos.

Edit: Fixed name of the book mentioned and added some text.

A big portion of Lakatos’s book is online, https://math.berkeley.edu/~kpmann/Lakatos.pdf

Do you mean "proof and refutations" ?

Ah yes, thanks for correcting me :)

To those like me, not familiar with the term "sausage factory":

"An unpleasant process, especially one that is hidden from public view, that is used to produce a widely consumed product" - Urban Dictionary

I feel like the article does a good job of addressing this when it talks about the role of "ugly" proofs. It might deserve more emphasis on this; that all pioneering mathematics starts with this, rather than assigning it a space as a "stepping stone" to the "book proof". But it is true that "cleaning up" after the pioneers is a difficult and interesting part of mathematics, that serves to make it easier for new pioneers to get to the end of the paved trail.

> Q: There’s a famous quote from the mathematician G. H. Hardy that says, “There is no permanent place in the world for ugly mathematics.” But ugly mathematics still has a role, right?

> A: You know, the first step is to establish the theorem, so that you can say, “I worked hard. I got the proof. It’s 20 pages. It’s ugly. It’s lots of calculations, but it’s correct and it’s complete and I’m proud of it.”

> ...

> To do these short and surprising proofs, you need a lot of confidence. And one way to get the confidence is if you know the thing is true. If you know that something is true because so-and-so proved it, then you might also dare to say, “What would be the really nice and short and elegant way to establish this?” So, I think, in that sense, the ugly proofs have their role.

I am also a professional. Sometimes your job deals with the pretty marigold and sometimes with the nasty slug.

Reality has its thing.

And sometimes you need a jeweler’s tool and other times just your fists.

You sound like a jaded professional, but keep in mind most people are not working mathematicans and never will be, so anything they hear from the pros they're going to take at face value. When it comes to the general layperson, I think popularizing the idea that math is all "months of flailing around, trying to understand one thing" is more unhelpful than portraying it as the pursuit of beauty and simplicity, even if it's more true on a practical level. If you bring up the word beauty in reference to math, most people are a little surprised you even would link the two concepts, let alone get the impression that pursuing clever little proofs is all there is to it. Better to start by giving people get the idea they might enjoy it, and let them find out it's not all about elegant proofs in undergrad or whatever.

I highly disagree. They’re not a jaded professional. It's exactly what a professional goes through. Part of the beauty is in the “months of flailing around, trying to understand one thing”. It’s that “Aha!” and “Eureka!”. The history mathematics is riddled with this. And what if they don’t see this beauty? They’ll give up and not realize they have to persevere.