Believe me, @graciegcunning, you sounded smart (indeed, brilliant) before, and you sound even smarter here. Your questions are ancient, deep, and fascinating. And it's vastly more important to ask good questions than to know all the answers. It's much, much harder, too. https://twitter.com/graciegcunning/status/1298804338727489536

The first thing to know about mathematics is that it is really two things: a body of knowledge, and an artistic practice.

Most people focus on the body of knowledge, and it's easy to see why. Mathematical knowledge is very powerful, and its logical justification is very clear.

But mathematicians focus on the artistic practice of mathematics. And that's because the artistic practice is what produces the knowledge.

The artistic practice is endlessly fertile; but the knowledge, when divorced from the practice that led to it, is sterile.

If you watch Simone Biles perform a floor exercise, you can't help asking yourself: how is that even possible? Forget about how she does it -- how can she even *imagine* doing the things she does? It's just so far from the kind of movement that anyone ever does in ordinary life.

The products of mathematical artistic practice are rather like a Simone Biles floor exercise. They're the endpoint of a long, long process. A huge amount of hard work goes into them, and a lot of imagination, too. And Simone Biles didn't emerge from the womb doing these things.

Even a tremendous talent like Simone Biles belongs to a larger tradition of practice. She's learned a huge amount from generations of great gymnasts who faced and solved lots of fundamental problems for her; she's built on their achievements, and added more of her own.

Mathematical practice is the same: it's built on a long tradition, and modern masters owe huge debts to their great predecessors. Even the most brilliant mathematicians alive today rely constantly on ideas and methods that are centuries, and sometimes even millennia, old.

You ask how people knew what they were looking for when the whole practice of mathematics began. I think the only really honest answer we can give is: nobody knows. The practice of mathematics is so ancient that we can only guess at the sources of its ideas and methods.

My own guess, for whatever it's worth, is that the first mathematical artists didn't know what they were looking for -- and didn't much care, either. I think at first they were probably just playing around with numbers and shapes, for the sheer fun of it. Like kids with toys.

One thing worth stressing here is the tremendous power of boredom. People will do almost anything to avoid boredom. They'll even risk their lives, stupidly. And back when mathematical practice began, life was very boring. So people needed toys. And math is filled with toys.

And mathematical toys are very good toys, because you carry them around in your mind. They cost nothing, they can't be destroyed, and no one can take them from you. Mathematicians do very well in prison, for just this reason. Even solitary confinement doesn't much scare them.

So I think, at the beginning, mathematical artists were probably just fiddling around, entertaining themselves, for no very good reason. They were just playing. And they kept right on playing, because they enjoyed it. This is an absolutely crucial point: mathematical play is fun.

If mathematical play weren't fun, we really wouldn't know anything about mathematics. The problems of mathematics are just so hard that mankind would've given up on them entirely, thousands of years ago, if it weren't so damn much fun to struggle with them. They're irresistible.

Lemme give you a quick little example of the kind of thing I mean. Suppose you try listing all the ways to write 5, using only addition. You get:

5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1

So, seven ways for 5. How about for 6? 7? 8? Is there any way to anticipate the answers?

Now, this is just playing with addition, and with whole numbers (really small whole numbers, at the moment.) There's nothing "important" going on. But as you start to fiddle around with this problem, it grows on you. You can't help noticing things. That's just how the mind works.

And one of the things you notice is that these lists get really long, really fast:

8
7+1
6+2
6+1+1
5+3
5+2+1
5+1+1+1
4+4
4+3+1
4+2+2
4+2+1+1
4+1+1+1+1
3+3+2
3+3+1+1
3+2+2+1
3+2+1+1+1
3+1+1+1+1+1
2+2+2+2
2+2+2+1+1
2+2+1+1+1+1
2+1+1+1+1+1+1
1+1+1+1+1+1+1+1

You want a shortcut!

Making lists like these isn't hard, but it sure does get boring. Frustrating, too, because at first you tend to leave things out, and only notice your mistake later. Eventually, though, you see how to guarantee that your lists are complete. And then you really want that shortcut.

And one of the things you're sure to notice is that (for example) the list for 7 has a lot in common with the list for 8. There's a sort of family resemblance between (many of) the entries on the two lists. So that suggests that maybe you can build new answers from old ones.

And the more you look back at earlier lists, and compare them with later lists, the more family resemblances you notice. There are lots of them -- so many, in fact, that it takes some care to keep track of them all. So, yes: there's work involved. But it's not drudgery. It's fun!

Figuring out the length of the list of all ways to write 100 using only addition is a serious problem! It fascinated one of the greatest mathematicians of all time, Leonhard Euler. And it took his best thinking to arrive at an answer -- which, at first, he couldn't prove correct!

And this brings me to your second question: once mathematical artists arrive at answers they believe in, how do they know for sure that they're right? Well, let's be clear: there are two answers here. And the first, simplest answer is: they conduct tests. Lots and lots of tests.

A general formula (which, by the way, is not the only sort of answer that mathematicians seek) says that a huge (usually, an infinite) number of things are true. So you check the ones you can check, and see if they're true. If they are, then you're more confident in your answer.

Leonhard Euler was extremely good at this kind of spot-testing, and he did it constantly, whenever he was in doubt. For him, it was often good enough. Not ideal, because he knew that the range of his tests was limited, and that his answer might fail somewhere beyond that range.

But Euler was often sufficiently convinced by encouraging results from extensive spot-testing. And so have many other great mathematicians been, despite the limitations of the method. It isn't perfect, but if you do it cleverly and with great care, it's actually pretty good.

I must add, though, that Euler also knew of another, (much) better way to be certain of the correctness of mathematical statements. Euler knew that it is sometimes possible to prove things -- and that proof effectively guarantees correctness. Proof pretty much removes all doubt.

Now, on the subject of proof (what it is, how it works, why it's so persuasive, what doubts remain even after a proof is given) there's so much to say that I doubt I could do an adequate job, no matter how much I tweet. Proofs, and the art of proving, are endlessly subtle things.

But I do want to stress this one point about proof: it is almost always the endpoint of a long, intricate process, in which often the most important and difficult work goes into figuring out the right way of thinking and speaking about the phenomenon you're trying to understand.

Devising a proof often boils down to building an exquisitely specialized language, whose words perfectly express certain crucial ideas -- ideas that take a long time just to recognize, and longer still to formulate explicitly. It's often a painstaking struggle. But it's worth it!

And it's worth it not only because the final product of that struggle removes all doubt. It's worth it because the struggle itself is often profoundly illuminating. The best proofs actually show you *why* a mathematical conclusion is correct. They don't just provide certainty.

Here's a helpful analogy: a proof is like a plot. A plot makes a story fit together, so that all the events it includes become meaningful, in the end. A proof makes a mathematical result fit together, so that all the elements it includes become meaningful, in the end.

The best proofs explain *why* things are true, and it's this *why* that conveys meaning. Mathematicians, like readers (and writers) of stories, want a sense of meaning. Nobody's happy with a view of Life as "just one damn thing after another." We all want it to mean something.

One of the greatest joys of a mathematician's life is that, sometimes, mathematical discoveries really do mean something. Sometimes, mathematicians really do manage to expose the underlying meaning of the patterns that fascinate and tantalize them. Sometimes, it all makes sense.

You've asked why everyone is being mean to you on Twitter. Probably there are many answers to this question; but one of them is that so very few persons have received a proper mathematical education. None of us who have could possibly imagine being mean to you. You thrill us.

But mathematically ignorant people are often strangely deluded. Those who are mean to you on Twitter mistakenly imagine themselves knowledgeable enough to deride your questions. But believe me: no one is sufficiently knowledgeable to deride your questions. They're genuinely deep.

You've also asked why a physicist who's followed by Barack Obama would retweet you. And by now I'm sure you can anticipate my answer: this happened, and will keep on happening, because all mathematically educated persons recognize your questions as both deep and admirable.

And, frankly, we're thrilled to see you posing these questions with such obvious sincerity and directness. We mathematicians admire directness and sincerity, and abhor pretense and self-indulgent obscurity. You are obviously our kind of person, and we can't help liking you.

You've also asked whether your fifth question was the fifth on your list -- and this is also sure to endear you to all mathematicians everywhere. Because mathematicians know what most people don't imagine or guess, which is that counting, properly understood, is incredibly hard.

Indeed, it's not going too far to say that counting, properly understood, is one of the hardest things humans are (just barely) able to do. Our (alas, too few) successes with really difficult counting problems are among the greatest achievements to which our species can point.

You've also asked whether anyone is going to post your video on Twitter -- and to this I'd just like to say, personally, that I'm awfully glad this happened. It delights and encourages me more than I can say to see you enjoying a viral success with your questions. You rock.

Finally, you've asked why only dumb people are disagreeing with you, while physicists and mathematicians are applauding. And here's my response: I'm overjoyed that you recognize your critics as dumb people. I don't mean to encourage condescension. But you're right to ignore them.

Please, @graciegcunning, never allow those who've been mean to you on Twitter about all this to undermine your natural curiosity or sense of self-confidence. These are both precious virtues which are obviously serving you well. And if you'd ever like any help from me, just ask.

I would consider it an honor to be permitted to try to make mathematics and its practice more accessible and meaningful to you, and, above all, to make these things an enduring source of joy and satisfaction in your life. Please don't hesitate to ask; I'd really be delighted.