My focus is on math with which you can DO something, so for instance, what if you like summing distinct squares to get a square?
42 + 62 + 102 + 142 + 862 = 882
862 + 1292 + 2152 + 3012 + 8812 = 9682
Why? Why ask why? Maybe you just like to do it for fun! That's a good enough reason for me. But how do you find them?
I can tell you a way I found with my research:
Using the techniques at that post you could make a sum of 100 distinct squares to give a square. Why do such a thing? I wouldn't know. I haven't done it, and I discovered the mathematical machinery.
That's a fun thing to be able to say too. Discovered the mathematical machinery. Being able to say that and it be true is living the dream.
So yeah, I discovered the mathematical machinery.
That's a powerful thing to be able to put up publicly. Think about it.
I like giving people mathematical tools. It's a quirk of mine.
Then ask yourself, do you really think I'm struggling to get known?
Nope. I can probably strut into any math department in the world and demonstrate half a dozen "impossible things" but there's no motivation.
Why would I bother? What does any math department in the world have that I'd want?
What can they give me?
Answer, nothing at all.
And I guess that can sound harsh, but I've tried to explain. Short of it is, I'm not a mathematician. I don't have a career in that area. Mathematicians are at best just competitors to me. I have no need nor interest in hanging out with rivals. And there really is nothing they can give me in return for spending time with them, anyway. Where thanks to the web, I don't need their recognition either.
More importantly, what can you give yourself? Love of discovery, passion for mathematics, and the thrill of playing with numbers.
Giving people the tools to pursue their interests? Now THAT is something.
The functional perspective is a great one, and can mean that years from now if I do feel like doing something with some particular math discovery of mine, I know the option is there. What might I do? Who knows. It's open ended. But without these tools, how?
For me it was often frustrating in the past to just want to play with integers and read through massively complex mathematical tomes that often wouldn't even give me a way to do that, so I couldn't get that pleasure of watching the numbers DO something interesting to me. So you do all that work to understand something you can't even really use! Why bother?
But with my own research NONE of it is that way. And I wouldn't have it any other way.
I love being able to DO things with integers. For me it is a passion.
Which gives the best benefit of a functional perspective: the math has to actually work.