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
First, if you’ve ever solved a mathematical problem that was not one that you were specifically *taught* how to solve, then you experienced a very small taste of what mathematical research is like. Because even if that particular question has been answered before by other people, it has never been answered before *by you* so for you it is a discovery which is a small-scale version of discovering “new” math.
You might think that is nothing like creating calculus, but no mathematical discovery comes out of thin air. Studying math gives you access to a variety of tools and ideas, and when you hit a problem that your tools/ideas are unable to solve, you modify them slightly to create new ones. Big mathematical discoveries are just particularly large or creative modifications. In the paragraph above, whatever you did to solve the “new” problem can also be used to solve other similar problems. If you were to try to “formalize” whatever trick or idea you used, that’s where new math comes from.
Since you mentioned calculus, I’ll try to use it as an example (though obviously this will have to go a bit beyond age 5). Since the ancient Greek Archimedes, we’ve been able to think of the area A inside a circle as follows: It has to be larger than the area L of any polygon inside the circle and smaller than the area U of any polygon that encloses the circle. That is, L < A < U, so L is a lower bound and U is an upper bound. By calculating areas of polygons with more and more sides, you can get L and U to be closer together, which means you can effectively compute A to any level of precision you want. This leads to a proof for why the standard formula for the area of circle is correct.
This idea makes sense for any “curvy” shape, not just circles. One thing that Newton did was *formalize* these ideas and use the formalization to help him calculate the area A for an astounding variety of shapes. In particular, he came up with the idea that A is the “limit” of those lower estimates L as those estimates become more accurate. This means that computing areas of curvy shapes boils down to being able to compute these limits, which led him to develop lots of tricks for computing such limits. In retrospect, it’s such a natural outgrowth of ancient Greek math that imho the surprising thing should be that it took so *long* for calculus to be invented.
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