Mathematicians From England

"If you hate the idea of being... told you're wrong, then you get put off math at a very early age because it's the one subject where you start being told you're wrong a lot, and... if you don't like that... you'll move off into some subject where... you can create things..."

"What if the pieces of string aren't really... string, but they're s? ...[M]aybe ...early diagnosis of Alzheimer's may come from looking at... the tangledness of brain cells that mutate... So an abstract way of telling whether it's tangled is... useful."

"[S]o mathematics comes up with abstract ways of studying these, where... it... looks like pieces of string, but how can I study them as if they were pieces of string without actually waving pieces of string around... [T]here are all sorts of practical situations where it's not practical to wave... string around."

"I've invented some new words... ingressive to replace masculine and congressive to replace feminine... Ingressive is a character trait... a behaviour... about going forward... not being waylaid about what people say... being competitive and winning. Congressive is about bringing things together and... shedding light and understanding... helping people... and maybe we are presenting mathematics at school in a very ingressive way, because it's often about being right... getting the right answer."

"Math, unfortunately is presented in this very ingressive way, despite the fact that when you get to the research level, it's very congressive."

"If we strip away the paint... the windows and the non-structural walls of this building, we'll get to the structural walls... and then this building will look a lot more like a lot of other buildings... I don't know which are the load bearing walls, and I don't need to, but it's a good thing that somebody does... [T]hat's true of all abstract structures."

"When I drew this picture it enabled me to understand the piece better, but moreover, it helped me understand why I didn't understand the piece, because the voices got wound... between each other in a way that was difficult to follow until I drew this diagram... [I]t enabled me to follow the lines of music as I was playing... which... enables me to play it better. ...This is the point of understanding the s inside things."

"We can go through life, and this is why people do believe they can go through life without needing to know any maths... [S]o when... we go "Math is really important!" They can just go, "Well, I don't... do any of those things, and I'm just fine." ...Yeah, you can be just fine, but wouldn't you like to be better?"

"[T]his is from the second Book and you can almost hear his excitement at being able to write in all these keys, that he was unable to... [do] before... [S]ometimes I think when he gets to the keys... the most far away from ones he could previously write in, things become almost simpler, because he's just enjoying the sheer joy of being in f-sharp major for the first time in his life..."

"[T]his is one of my favorite pieces of Bach... the Prelude in G minor from Book 2 of the preludes and fugues The Well-Tempered Clavier... [T]here is a huge quantity of maths in Bach. Because... the tuning system enabled him to write in every key, for the first time ever... he wrote a prelude and fugue in every key. There are 12 keys on the piano. He wrote in major and minor, so that made 24 preludes and fugues. Then he got really excited, and did it again, so that made 48..."

"So this piece... is written in , like many pieces of Bach are. So the four lines... are independent lines of music that went their way... by themselves... [E]ach... could be sung by a person."

"[I]f you know where the s are, you can... use things better, you can make things better. You can improve them. You can fix them when they go wrong."

"[W]e can try and apply logic to other areas of life... [I]t's very frustrating if we try... with the expectation that that works. ...[I]t doesn't mean we shouldn't try ...[I]t's always good to try and understand everything else according to logic, but... this is the fate of many mathematicians who... get... frustrated with the actual world, because nothing behaves the way we want... Whereas in the beautiful mathematical world everything does behave..."

"[T]hat's why we move into the abstract world of ideas, where things behave the way that we want them to... [T]hat can be a scary move... because... in the real world you get to touch things... throw things... Whereas in the abstract world... if the logic doesn't do the thing that you want... There's nothing you can do about it. That's just how it works... [T]he upside... is that if you align yourself with the way logic is supposed to work, then everything behaves the way you want it to, because everything behaves perfectly logically... [I]t is the only place where everything behaves perfectly logically."

"All our dreams can come true as long as we have the right dreams... [A]s long as we think logically, this is the world where everything behaves correctly, and... where any toy we want, we can play with, as soon as we've dreamed it up."

"The trouble with this is that nothing behaves logically."

"I'm going to declare that mathematics is the study of how things work... how logical things work... [I]t's the logical study of how logical things work. ...I don't think it's impossible to define. I think I just did it."

"[I]n order to study anything logically, we have to ignore all the pesky details that prevent it from behaving logically, and... move into the idealized world... rather than the real world of things... [T]his... is what... abstraction... is..."

"In the abstract world, as soon as you've thought of... something, you can play with it. It's there... the idea and the thing are the same... so you create things just by thinking of them... I wish I could do that for my dinner, but I can't."

"It... [abstract structure] is a beautiful thing, and sometimes all that matters is that it's a beautiful thing."

"I don't believe that one should have anything without sharing it, and that includes knowledge, money, food, love..."

"I believe very strongly in helping other people understand things. There's no point knowing things if you don't help other people know..."

"I make it my business to help everybody understand these things a bit more, because... they're very misunderstood..."

"[A]bstract theory often comes from wanting to be lazy, or... conserving brain power, conserving energy, because if you do the same thing over and over again, wouldn't you rather not... and just do it once... [T]hat's what abstract theory is there for... [T]hat applies to all sorts of aspects of life, not just sciences and programming"

"So instead, I like to show people, rather by analogy, why abstraction is useful... [A]bstraction is a process of analogy because... it's going to ignore certain parts of this... and... [that] situation, and... miraculously the two situations become the same, and then I can study them both at the same time, which saves me time, which is good because I'm... lazy."

"The work I do is totally abstract... [T]he idea is that it will help other people understand things that they can then do in the world..."

"I wrote this book because I love maths, and I love food... [S]adly most people love food more than they love maths..."

"Maths... as it's taught in school is often... boring, pointless, painful, beside the point, and doesn't show people the things that... are the most beautiful about abstract math, and what the point of it is..."

"I'd like to talk about abstract mathematics and my experience of making it... palatable to people who may have had very bad experiences of it..."

"The point is to help us. It's not there to cause people pain. The point of abstraction is to clear out the fluff in order... to see more clearly what's actually going on."

"Mathematics... helps... construct and understand arguments... too difficult for ordinary intuition. ...It is a way of eliminating ambiguity... It cuts corners, answering many questions... by showing... they're all... the same question... by abstraction: throwing out things that cause ambiguity, and ignoring [irrelevant] details... until all you have to do is apply unambiguous logical thought..."

"We will never be able to encompass everything by rationality alone... [T]his is a necessary and beautiful aspect of human existence."

"Now, if you imagine drawing a circle in the air with a , the surface you make is a over a circle. The idea is that for each point in the circle, you now have an entire vector, that is, a line given by the lightsaber at that instant."

"If you're only presented with... things you don't care about... then you won't care about having those things made easier, and so if all the problems... given are dumb... problems that don't... have anything to do with real life, then everyone.., especially young people... will immediately see that we're just talking a load of rubbish..."

"Math... is... misunderstood, and most people think it's all about numbers, and it's not... [M]ost people think it's about getting things right and wrong, and it's not. ...[I]t's more like cooking ...[Y]ou decide you're just going to fiddle around in the kitchen with some ingredients and make something, and the only thing that matters is if you like it or not. ...In the end all that matters is ...you make your own rules, and then you follow them and see what happens. ...[M]aybe you cause a contradiction ...then your whole world implodes, and that's ok. You move on to the next world."

"[T]hen I started my PhD and discovered that in higher dimensional category theory... the braids show the coherence of the structure inside some higher dimensional categories, and I didn't know this when I first drew this picture, and then I... said "Wow!" I was studying braids before I was even studying braids..."

"“Did you do what I asked you to?” “As much as I could. Have you ever tried to brief your boss, without telling her what’s going on?” “A hundred times. It’s the first rule of self-preservation.”"

"They saw every event through the distorting lens of their own paranoia."

"He was still young enough to hate looking like a fool more than anything in the world."

"“Hormones are everything, Turpin,” she said to the bird on her shoulder. “Brains are nice, and looks are nice, and logic’s even nicer; but hormones run the show. For everyone, even for me and you.”"

"“I think I’ll have a sign made for that far wall,” said Bey at last. “Indeed?” “Yes. It will say, ‘If you have nothing to do, please don’t do it here’.”"

"War was senseless. And yet war came creeping steadily closer."

"He had achieved his objective, but his little inside voice would not keep quiet. Too easy, it said, much too easy. When a difficult goal is achieved with no effort, it’s time to be suspicious."

"“I wonder why somebody would go to all that trouble to make a complete fool of himself.” “Come on, Gina, we both know why.” “Oh, I guess you’re right. Money will always do it.” Of course."

"It is remarkable that observation of the faint agglomerations of stars known as galaxies leads us, very directly and cleanly, to the conclusion that we live in a Universe of finite and determinable age. A century ago, no one could have offered even an approximate age for the Universe. For an upper bound, most nonreligious scientists would probably have said “forever.” For a lower bound, all they had was the age of the Earth."

"It’s the usual sensation mongering; the news services will say anything for an effect."

"Like many things in life, the problem I had been so sure I could solve proved more difficult than it sounded."

"Irresponsible on my part, to put pleasure before professional work? Sure—but as I get older carpe diem gets more and more reasonable as a motto for life."

"The Asteroid Belt contains everything from substantial bodies like Ceres, seven hundred and fifty kilometers across, all the way down to house-sized boulders, pebbles, and grains of sand. One good rule of thumb is that for every object of a given size, there will be ten times as many one-third that size."

"The ship climbed steadily and laboriously up, away from the plane of the ecliptic. Finally, the parallax was sufficient to move the planets from their usual apparent positions. Mars, Earth, Venus, and Jupiter all sat in constellations that were no part of the familiar zodiac. Mercury was cowering close to the sun. Saturn alone, swinging out at the far end of her orbit, seemed right as seen from the ship. Bey Wolf, picking out their positions through a viewport, wondered idly how the astrologers would cope with such a situation. Mars seemed to be in the House of Andromeda, and Venus in the House of Cygnus. It would take an unusually talented practitioner to interpret those relationships and cast a horoscope for the success of this enterprise."