What does it feel like to do maths? (2016)

> Of course I wasn't the first to think of it but that didn't matter, the joy is the same.

Thank you for emphasising this key point! There's pleasure and there can be prestige in being the first to solve a mathematical problem, but there's joy and accomplishment just in solving the problem yourself, even if your approach is exactly the classical one. I am a rock climber as well as a mathematician, and I compare it to that: the first person to summit a route will get the big-name recognition, but it's still worth it for me to come along and summit myself later, even if I use exactly the same sequence as tens of people who have gone before me.

>Of course I wasn't the first to think of it but that didn't matter, the joy is the same.

Exactly right.

When I was in undergrad physics, I signed up for the intro comp-sci course (which is recommended back then but not yet required). First we learned the basic syntax of C, and in the second unit we learned about sorting algorithms. The Prof introduced us to bubble sort, selection sort, and insertion sort, and also the idea that the efficiency of a sorting algorithm was proportional to how many comparisons it required.

Being a physics student, I knew that there were physical process, like driven granular systems, that sorted objects by size without mathematically comparing anything at all. I wondered if I could beat the efficiency of all three of the algorithms by programming the computer to do something similar, treating integers like differently-sized grains of sand.

Over the weekend I first coded an algorithm I called jostle sort that sent the unsorted integers flying across a 2D array a distance equal to their magnitude, then picked them up again from left to right, top to bottom. This worked, but slowly because it had to look at every cell of the 2D array.

Thinking about it further, I imagined tipping the 2D array up from the bottom edge, so that all the integers in a column slid into a pile in the top row. I realized I only needed the count of how many integers ended up in each column, not the columns themselves. I could just initialize a 1D array of zeroes the size of the max value to be sorted, and increment the value in index n whenever I encountered n in the unsorted array. This worked really well, easily beating the other three algorithms in the criteria we'd been asked to test on our homework assignment.

I'd invented counting sort.

I would later learn that I wasn't the first, but that didn't lessen my pleasure. It's still my favorite sorting algorithm, because of that joy of discovery.

I recently read a 900 pager on Newton's life, which includes a lot of details about his mathematical accomplishments (book link: https://www.amazon.in/Never-Rest-Biography-Cambridge-Paperba...).

Key things that stand out:

+ Deep scholarship: Learn from your predecessors as much as possible; pick up their techniques, at the same time be sceptical, do not believe much, even when you're learning/mastering techniques. Ex: Newton mastered Descartes work early in his career, used that mathematics to open up Calculus, but at later stage became disillusioned with the analytic method, and converted his thought into pure geometric methods.

+ Imitate the masters before you: Newton was really one of the deepest students of the ancients. He'd ferret out obscure passages, and use them to solve his problems. Ex: He knew the intricacies of the analysis/synthesis method from Pappus, which most people barely know in the depth he knew of them.

+ Dedication to truth, desiring certainty and taking nothing less. Insanely high standards.

You solve it yourself, that is all there is to it. I have solved novel math problems, all you do is think about it until you have solved it, there are many strategies for doing that but in the end its all you thinking about different ways to solve it.

> At some point, you will hit a rock that nobody you know, even really smart people who have done this their whole lives, can jiggle. What do you do?

Keep working. Or leave it, come back after some time (hours, days, weeks, months... even years).

> Are people who do that basically experts at the little problems? Or is it like coding, where you can design a fairly large system and rely on the parts to come together one way or another? Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it?

I haven't worked too much in pure math research, but from my understanding it's a process of adding details. You have something you want to prove, and a base you start from. You have an intuition of what you need to be true in order to prove the thing, so start developing those proofs too. In the way you'll find you need certain conditions, and you might find that certain concepts reappear so you find it helpful to define them.

> Because when I read a math text, it seems really quite dense with things that require a lot of details to be correct. If it's a house of cards spanning a hundred page thesis, how brittle is it?

While technically it is a house of cards (i.e., if a proof is wrong everything that depends on it fails), most of the time you won't find such fundamental errors. Maybe a proof has some missing details that can be amended, or the same statement can be proven another way, or the theorem needs just tiny changes or extra conditions to be fixed.