i love nerding out about math, so i'm gonna dive into this twitter topic of the day with another fun math thread!! this girl is asking some really good questions!

(and yes, because it is necessary on this hell website: i have two postgraduate degrees in mathematics, so sit down) https://twitter.com/graciegcunning/status/1298804338727489536

(this may or may not be a bad idea since, last time i tried to talk about math on twitter dot com, an army of angry alt-right boys descended on my timeline to defend "2+2=4", the last bastion of justice in this darkest of timelines. but fuck it! i like talking about math. )

there's some good ideas being discussed in this video:

- is math real?
- why would people have needed (advanced) math in ancient times, before modern technology?
- how do people come up with math?

let's go through these one by one!

first of all... is math real? well, if you're an angry twitter bro you might say "yes, duh" and move on, but brighter people know to ask questions. the philosophy of mathematics kind of revolves around this question, in a way. for a cursory read: https://en.wikipedia.org/wiki/Philosophy_of_mathematics

if i claim, say, that "2+2=4", is that *real*? obviously, if i have two real apples and two other real apples and put them in a pile, i now have four real apples. sure. but the numbers *themselves*, and the equation that connects them, are those "real"?

there's some different schools of thought on this, but the two major ones are, roughly, platonism and formalism. to put it briefly: platonism says "yes, math is real" and formalism says "no, math is not intrinsically real (although it may happen to be practically useful)".

platonism is named after the philosopher plato, who believed that abstract concepts are real (although not tangible, physical things we can touch or see). the idea of the number "5", for example, is part of the world, beyond any specific pile of five things.

you may recall plato's famous allegory of the cave, or words like "platonic ideal". these all stem from this worldview: that abstractions are real and exist, in some way. the *idea* of a "chair" or a "book" is a thing we can understand, independently of any specific object.

if every single book in the world was destroyed, all across the universe, wouldn't the *concept* of a book still exist in some way? and conversely, even if every book in the world had a different color, shape, size, texture, material - wouldn't i still recognize them as "book"?

so this is the platonic approach to mathematics, then. mathematical ideas and theorems are *real*, they're just a different kind of real than tangible things we can touch or feel or smell. we can't discover them through our senses, but we can try to do so with our thinking.

the problem with this - which is what prompted other major schools of thought to emerge, like formalism - is that it... sounds a bit made-up, right? i mean it's hardly *scientific* to talk about abstract intangible stuff that lives in Fantasy Idea Realm. not very rigorous, hey.

so then there's the formalist approach, which many mathematicians will resort to if they're tired or don't feel like prattling on about philosophy all day, haha. formalism holds that math is literally just squiggles and symbols that we manipulate according to certain rules.

formalists feel no need to claim that 5 is "real". 5 is just a symbol, and so is +, and so is 2. and when we write "2+2=4", we have a set of rules that say this is correct; when we write "2+2=5", our rules say that it's incorrect. it doesn't really mean anything more than that.

mathematics, then, is just the process of figuring out what things are correct/incorrect under a certain system of rules, or "axioms". start with some rules, deduce what follows from those. math is, in some sense, nothing more than a *game* in this view. but...

... then that obviously raises the question of "how come math happens to be a really useful tool for describing reality, then"? because of course, if you take this meaningless game of symbols and squiggles and rules, and apply it to real life, useful results tend to emerge!

i'm not an expert on philosophy or history, so i will leave it to brighter people than i to continue this discussion. but suffice it to say: there are profound questions we can ask about whether math is real or not. gracie is right to ask them!

moving on to the next question: what did people in ancient times need math for? they didn't have the kinds of technological problems to deal with that we have today, so why would they bother with algebra?

well, for one, math is useful for loads of things that humans have had to worry about since the very beginning. how do you build a house or a wheel, what are the right dimensions to make it all fit together? how long is daytime/nighttime, and why does it change over the year?

how far is it from this town to the next, and actually, how big is the earth? if i want to draw a perfect hexagon to honor my lord Dionysus but all i have is a compass and straightedge, how do i do it? (okay, that last one i just made up. but still.)

all the way back in ancient greece, a guy named Eratosthenes ( https://en.wikipedia.org/wiki/Eratosthenes) used math and shadows to estimate the circumference of the earth! (no, people back then didn't all think the earth was flat, get outta here with that shit.)

so there's actually loads of things people could use mathematics for even back in ye olden days. but more importantly - why did they bother!? clearly *some* of these things have practical uses, but not all of them.

well, the answer was the same back then as it is today: a lot of the time, mathematicians are driven by curiosity, fun, and passion!

certainly a lot of mathematical discovery is driven by practical needs, but you'd be surprised how often math is completely useless at inception.

when people figure out things like "how many ways can i shuffle a deck of cards" or "can i invent some kind of arithmetic where 2+2=5", they don't do it because it makes money or solves a pressing need - they do it because they just like solving the puzzle, figuring things out!

in some ways, math is often a toolbox, where the mathematicians find solutions to problems that nobody actually has - just for funsies. and then, much later, somebody shows up and actually *does* have that problem, and we've made their life a bit easier. hopefully. hehe!

there's a fantastic quote by richard feynman on the matter - it's too long to recite here, so i'll put it in a screenshot. it's quite accurate, in my experience.

and this kind of brings us to the last of those questions: how do people even come up with new math? well, ironically, the way we do it is often by doing exactly what gracie does in this video: asking questions! *especially* about things that might not seem to matter.

you can't be a good mathematician if you look at a puzzle or a curious stray thought and go: why would anyone ask this? isn't it silly to wonder about this? but you *can* be a good mathematician if you choose to examine things!

so the way math discoveries happen is, well... i mean, obviously a lot of experimenting, thinking, discussing, computer simulating, trial and error, coffee-induced nightmares poring over thick tomes of forbidden knowledge...

... but more than anything, it stems from noticing a puzzle and deciding to solve it. and you'd be surprised how many puzzles there are out there! let me end on a fun note and tell you about an unsolved problem in mathematics: the map-folding problem.

so the map-folding problem is really just the question: "how many different ways can you fold a map with a certain number of creases?" https://en.wikipedia.org/wiki/Map_folding

basically: you know how, if you have a map or a pamphlet or other large sheet of paper, and it has some creases in it, you can fold it together in different ways? whenever i get a museum pamphlet or something, i always struggle to fold it back the way it was originally, haha.

so how many *different* ways are there to fold together a map with creases in it like this? well, we know the answer up to 7x7 creased maps. that's it. in general we have no idea.

cool, right? a simple problem to explain, and yet mathematicians are stumped! someone out there is thinking about this right now, and someday they *will* figure it out. then we'll all ask: "what the hell is this useful for"? and, much later, we'll understand to thank them for it.

thanks for reading this thread, if you made it all the way to the end. i'm glad you shared in my nerdy math rambling. never be afraid to ask questions.