How does new math get invented/discovered?
In: Mathematics
A reasonably good analogy is the “end of meal” problem: at the end of a traditional French meal, you are faced with a terrible problem:
* When you run out of wine, you still have bread and cheese, which no gentlemen would eat without some adequate wine. So you must take more win to accompany them.
* When you run out of bread, you still have wine and cheese, which would be a shame to waste. So you must ask for more bread to accompany them.
* When you run out of cheese, you still have bread and wine. Who would say no to an additional piece of cheese in such a situation?
As the meal in the metaphor, maths is a never ending puzzle, where each time you get a new results, you can unfold its immediate consequences, but you will quickly run into the fact that something is missing for your to continue, and hopefully you have a good idea of what it is and where/how to find it.
What are the wine/cheese/bread of maths?
* Understanding of concepts. Mathematicians get better and better understanding of “how things work”, and a better and better intuition of what is right or wrong. As a simple example, after noting that 7 bags of 3 apples give the same amount of apples than 3 bags of 7 apples, you start understanding that the operation that combine 7 and 3 into 21 is actually symmetric (aka commutative), and that this symmetry is actually an important concept.
* Without this understanding, maths is a set of results that works and nobody truly knows why. It’s almost impossible to progress as the only method you have is just randomly trying new things hoping they work.
* Formalisation of maths. “How we talk and write about maths” is actually very important. You want to find ways to write maths where all the major properties that are true look “simple” and “intuitive” while things that are totally wrong look “convoluted” and “weird”. Taking again a very simple example, the fact that you can just count the numbers of digits of a number to see which one is larger is a great improvement over using words. Looking at a function through a graph showing its result, rather than a big excel table full of numbers, is also something very important.
* Better formalisation of maths is how we manage to teach to students mathematical concepts that used to be “only one person in the world understand them” when they were first discovered.
* Bad formalisation will reduce by a lot the effectiveness of research in a domain. Both by reducing the amount of new researchers taking the time to understand the domain, and by making the work harder for the specialists. And while reducing a proof from 10 pages to 1 page with a new formalisation is nice for new researchers trying to learn the domain, reducing a proof from 100 pages to 10 pages is the difference between the researcher giving up before finding the result and the researcher actually finding it.
* Better results. This one is obvious, there are questions that were conjecture before, and are now proven right or wrong. There are problems that had no method to solve them, and now have perfectly working methods. Etc. Sometimes, it comes from genius that are decades in advance in their understanding of maths, or just experts of a domain that are simply the first person ever to be that much of a specialist in this specific question. But often, it’s just the natural consequences of new formalisms and the overhaul advance of mathematical understanding, and the researcher is just let thinking “Why has no one tried this before? It just works.”
* Sometimes, you just need to unfold the results. All the tools are there, you need to pick the good one and apply it to the good problem.
* Better methods. “How do we prove maths”, “How do we think about maths”, etc. All of that is still maths. And “How do we think about how do we think about maths” is maths too. (Well, it can also be philosophy, but that’s a different approach. And it can also be computer science, but that’s arguably just a subset of maths.). As maths progresses, some of the “results” are actually better tools to do maths in general. For example, what if you’re trying to make a proof, but you don’t manage to write the proof you have in mind, and for a reason, the proof you have in mind is (literally) infinitely long. When are infinite proofs valid? (spoiler: not all of them are, that’s the difficult part). What does it even means?
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