How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?
In: Mathematics
It’s about seeing things in a different way (and studying math of course). You want to solve a problem, or even just see things in a different way… so you define new things and apply them (as long as they are consistent with known math) to see if something interesting pops out.
For example, Newton was trying to describe the speed of a falling object. Since gravity changes the speed of falling object, he *defined* the mathematical concept of “rate of change” (derivative) of a function. He also found out that you can *apply* this concept also if you want to compute the area under a curve.
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You start with knowing pretty much all that’s known to that point, and then, when you have really thought about it long enough, and in depth enough, you suddenly have a light bulb light up over your head and you realize you came up with something new. (Well, except for the light bulb)
* Numbers are just representations of data. There’s 1, or 2 oranges.
* Math is just the manipulation of that data. John has 2 oranges + 3 oranges
* Inventing new maths, is just figuring out **how to manipulate** that data to solve a **specific question, or type of question**. How many oranges does John have? 2+3=5!
You could say that the question has never been answered before, or you could also say the question has never been asked **properly** before. It’s never been broken down to smaller parts we can easily understand.
Like… how big is a triangle (for simplicity we’ll use a Right Angle Triangle)? How do I calculate all that area inside of it? Well I don’t know… but there must be some way to find out. So let’s see what makes up a triangle, break it down to simple parts we can already measure, parts we can use as data.
Well… we can measure the base of it, we can also measure the height, so we have that data. But now what?
Hm… well you know what’s interesting that **I** noticed? If I take another triangle of the exact same size, I turn it and put the two together, I’ve made a square! That’s kind of neat, and it happens every time.
It’s easy to calculate the area of a square… So to figure out how big a Right Angle Triangle is, I actually just need to make that a square and chop it in half. I just invented new math!
So the math for how big a triangle is, is ((The **B**ase times the **H**eight) divided by **2**). So the question for how big a triangle is, more specifically is what is ((BxH)/2)?
You can now use this new math, we’ll call it **Remote-Waste’s Amazing Triangle Triumph**, and you can praise me as the smartest person ever.
Mathematics is the process of coming up with new ideas for things where each thing has a defined set of logical behaviour or rules, then combining these things to see how the combination of rules or behaviour for the combination works out. Basically inventing new mathematics is a way of coming up with a new set of rules that you can apply to things, and seeing if the result of the combination of the new set of rules you have made up with all the things you have already done, gives any useful insight, or ability to solve problems.
Take a really simple question: if I add two odd numbers together, is the result odd or even? It sounds like a hard problem, and it is not possible to go through by trial and error. What I can do is come up with a new way of describing odd and even numbers. I can say that for any even number, a, it can be represented as double some other number, n, so a=2n. For any odd number, b, I can say it is one more than an even number, so b=2n+1. I can then go back to the original question, and look at the result of adding two odd numbers:
x=(2n+1)+(2m+1).
I can then apply normal rules to the right hand side, and say
x=2n+2m+2
and take a factor of two out:
x=2(n+m+1)
Because n, m and 1 are all integers, I can say they are some other integer, p, so
x=2p.
Because my original definition of an even number applies for any integer n, I can see that x has the form of an even number, 2n, and conclude that the sum of two odd numbers is even, for any two odd numbers.
This is a really simple result, and not one I came up with myself. The point is what happens is I use a new definition for even and odd numbers that didn’t exist at the start of the question. The definition of an even number as 2n and an odd one as 2n+1 is something that someone invented, and by using that invention, it became possible to actually answer the question for all odd numbers.
This is the basic idea behind inventing things in mathematics. You come up with a new way to describe something, be it a kind of number, a kind of process, function, operator, or whatever, and then work with that new thing you have created using all the existing standard rules for mathematics to see if you can find a path to answering the question.
I am aware that algebra is a bit more advanced than your average 5 year old, but I hope this demonstrates the idea.
It’s basically teaching a tecnique, for example in a world where people only knew addition, I could teach them multiplication, instead of doing 5+5+5 just do 5*3. This is much easier for more advanced calculations, it might not even be useful right away, but maybe in a few years some mathematician will find a use for it
It’s more like answering a question that noone asked before. Geniuses are on such another level that they ask questions noone else would ever think to ask, and since they’re the only people intelligent enough to ask, they’re the only ones intelligent enough to answer.
They ask a question and answer it themselves.
Here is a story from the 20th century. Hilbert, optimistic as usual, ask for people to find an algorithm to solve all logical problems. This is an interesting question that people want an answer to.
The issue is, that sounds impossible. Unfortunately, there is no ways to prove that it’s impossible, because the concept of algorithm have not been invented. That doesn’t mean people don’t know about algorithm at all; but more like, the ideas of algorithm is too vague. The same ways words in normal language are vague; if someone ask you “what is the precise definition of ‘porn’?” you might just answer “I don’t know, but I know it when I see it” (actual quote from a judge, btw, it’s a famous case).
For many other problems in the past, if people want an algorithm, you just give an algorithm, and everyone can check and agree that this is an algorithm. But to show that an algorithm doesn’t exist, you need to know exactly what is an algorithm. So Hilbert’s question is not a complete question, you have to fill in the definition of an algorithm to make it a complete question.
Turing and Church did this. They invented the concept of computability, to answer the question of what is an algorithm. What they invented isn’t a proof. They make concise and precise their intuitive idea of algorithm, and offer it to other people. Whether people agree or not, the key thing is, there is at least *a* definition. With this, Turing and Church had finally form a new question that nobody asked before. Then they show that the question have no answers, there are no algorithm. This is the start of computer science.
New mathematics comes about when someone has a new insight into a problem. A good example is topology. In the 18th century someone posed a question about the Prussian city of Königsberg. The city is on the river Pregel and it had seven bridges joining the two banks via a couple of islands. The question was, “Is there a path which goes over each bridge once and only once?”.
Nobody could devise such a route, but nobody could show that it was completely impossible. A mathematician named Leonhard Euler decided to try the problem. Euler’s great breakthrough was to realise that none of the usual measures made any difference to the problem – not the length of the bridges, or their distance apart, or their angles to each other. He realised that the key thing is how many paths into and out of each bridge there are: If you want to cross each bridge only once there must be as many paths out as there are in. Another way of saying this is that the total of paths in and out must be an even number (divisible by 2). Using this concept he showed that the Königsberg question had no solution – it was impossible. But he was able to go further and come up with a general rule for such problems – based on how many bridges have even numbers of entrances and exits.
Euler had distilled a certain problem into new abstract concepts: edges (the bridges), nodes (land masses) and the degrees of the nodes (paths in + paths out). This new view of the world turned out to be a very powerful tool for many different problems and developed into whole new branches of maths such as graph theory and topology.
See [Wikipedia](https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg) for a better explanation of the problem and solution.
Discover isn’t really the word. Maybe “invent.” They basically came up with extensions to the language of math so that they could describe new problems.
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