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
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.
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