How does new math get invented/discovered?

In: Mathematics

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People come across a problem they can’t solve. They then explain that problem to others. Somebody discovers a solution for it. From that solution they create a step by step guide. Now the real test of whether or not that is a correct solution or just something that happened to work is whether or not the same result under the same rules are achieved regardless of the number applied.

So a² x b² = c² would have come about in a similar fashion to someone noticing that there always seemed to be a size relation between the longest end of a triangle and the other two shorter ends. Then Pythagorean comes around and declares that his formula explains this ratio for non-right triangles. In the entire time since his claim, not one person has ever been able to produce a non-right triangle that doesn’t follow that this formula.

Nothing changed. We’ve just evolved from trying to explain triangles to trying to understand the nature of black matter. Same thing though. It’s all just seeking a solution to explain a problem.

People solving problems, people making new problems, people forming theories and definition, constructing new objects, and so on.

Where do people get problems from?

1. External motivation. Some real world problems get modeled into a math model. Many questions in differential equation started with this.

2. Intrinsic interest. Sometimes people just get curious. This usually applies to many basic question in number theory.

3. Conjecture. People suspected that something might be true, based on heuristic reasonings, and want to confirm or deny it.

What about theories and definition and new objects?

1. Definitions can be motivated by a problem being solved (gives a simple name to a complicate-to-describe idea, or constructing a new object), or part of a theory. Occasionally it’s also an attempt at solidifying an intuitive but imprecise concept. Definitions might persist beyond the solution of a problem if it turned out to be useful enough, for example it might define a new object with helpful property.

2. Theory comes from attempt at consolidating or generalizing previous math knowledge. They set up a framework, defining new objects, and conjectures about them.

A big part of the process of developing new maths is generalisation. Often someone will notice that multiple mathematical systems have some features in common and they will try to develop a framework that includes all of these systems. For example, suppose you notice that two simple models of unrelated real-world systems – a model of radioactive decay and a model of population growth under limited resources – both have something in common: over time they both get closer and closer to a fixed point in which nothing changes any more (maybe the amount of the radioactive substance approaches zero, while the population approaches the maximum that can survive on the available resources). You might try and develop a theory of all systems which approach a fixed point like this. This tends to be very fruitful as it helps you understand what exactly it is about these models that results in this behaviour, what would need to change for them to do something different, and what details of the models are irrelevant to this behaviour and can be safely ignored. And once you’ve developed this generalised theory of systems that converge to fixed points, it’s very straightforward to apply it to some new model that comes along: you don’t have to do everything from scratch, you just have to work out where this new model fits in your theory.

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?

To really get a feel for this it’s probably best to go looking for the specific stories of how specific new concepts were invented, because it’s a little different every time. Sometimes math techniques are invented in order to try and solve a specific real-world problem.

Calculus, for instance, was first invented for the purpose of understanding planetary orbits and figuring out how gravity worked. Newton and Leibniz both, were looking for ways to understand the motion of things whose speeds were always changing, and changing by an amount that’s *also* always changing. It turned out to be useful in a ton of other ways too.

Early geometry was, in great part, a science developed for the purpose of mapping and measuring the Earth. Hence the name geo=earth, metry=measurement.

Some math just kinda suggests itself as a by-product of the practical stuff we do all the time. The ideas of prime and composite numbers and divisibility, for instance, would arise naturally from an everyday problem like “can we share x apples fairly among y people?” … “Oh jeez weird, it seems like 23 apples can’t be shared fairly no matter what, *unless* you have exactly 23 people. Sharing 24 apples yesterday worked great, so what the heck is wrong with 23?”

Sometimes math is discovered by mathematicians exchanging guesses about what they think might be true, called conjectures, and trying to prove each other right or wrong. A famous one called Goldbach’s conjecture was the result of a pen-pal correspondence between Goldbach and Euler.

Sometimes new math is discovered through play. Sometimes mathematicians – or just regular folks doodling with pencils – just idly put different mathematical structures together, just to see if they add up to anything interesting, and sometimes they find patterns which they weren’t expecting and don’t know how to explain.

Some math is just kind of inevitable, and impossible to avoid. Numbers like pi and e, for instance, show up in *so many places* that we basically couldn’t have *not* discovered them.