51 quotes found
"Infinity is a Loch Ness monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral."
"Abstraction is about digging deep into a situation to find out what is at its core making it tick. Another way to think of it is about stripping away irrelevant details, or rather, stripping away details that are irrelevant to what we're thinking about now. These details might well be relevant to something else, but we decide we don't need to think about them for the time being. Crucially, it's a careful and controlled forgetting of details, not a slapdash ignoring of details out of laziness or a desire to skew an argument in a certain direction."
"... I wish we could educate people not just to be able to do things and know things, but to be nicer humans — and to get their self-esteem not from being better at something than someone else — but from how much they are able to help someone else do something."
"...The idea of math is to look for similarities between things so that you only need one "recipe" for many different situations. The key is that when you ignore some details, the situations become easier to understand and you can fill in the variables later. This is the process of abstraction."
"I am going to show that mathematics... is precisely "that which is easy.""
"[S]omething is easy if it is attainable by logical thought processes.... without having to resort to imagination, guesswork, luck, gut feeling, convoluted interpretation, leaps of faith... drugs, violence... [etc.]"
"We will never be able to encompass everything by rationality alone... [T]his is a necessary and beautiful aspect of human existence."
"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..."
"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."
"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..."
"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..."
"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."
"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..."
"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."
"[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"
"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 believe very strongly in helping other people understand things. There's no point knowing things if you don't help other people know..."
"I don't believe that one should have anything without sharing it, and that includes knowledge, money, food, love..."
"I make it my business to help everybody understand these things a bit more, because... they're very misunderstood..."
"I wrote this book because I love maths, and I love food... [S]adly most people love food more than they love maths..."
"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."
"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."
"The trouble with this is that nothing behaves logically."
"[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..."
"[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."
"[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..."
"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."
"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."
"[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..."
"[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..."
"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."
"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."
"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."
"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?"
"[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."
"It... [abstract structure] is a beautiful thing, and sometimes all that matters is that it's a beautiful thing."
"[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..."
"[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."
"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."
"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."
"I'm not interested in being right... I'm not interested in winning... but I hate losing, and I don't like being wrong. ...But if it's a situation when nobody is going to lose because we're all trying to understand something together, then there's no risk of losing, and... we can all gain from it."
"I'm not interested in playing sport... because I hate the idea of losing, and I'm not interested in winning, so there's no upside and there's only potential downside..."
"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..."
"Math, unfortunately is presented in this very ingressive way, despite the fact that when you get to the research level, it's very congressive."
"[T]here are many fields where we use ingressive means to filter people, despite the fact that congressive characteristics would be more useful... and that's the thing that I would like to change."
"A lot of programs for good mathematicians at a young age are... competitions and... problem solving... and Olympiads, and that is very ingressive..."
"I love to focus on how to make mathematical activities more congressive... so when I'm teaching art students I do a lot of activities where there is no right and wrong... [W]e're not trying to achieve an answer, we're trying to explore... [W]e build something. It's more craft-like. ...[M]aybe we're trying to build s, but it doesn't really matter if you didn't... [A]long the way we discover how triangles fit together and... how versatile an equilateral triangle is... and that you can make all sorts of shapes... and some... are Platonic solids... [E]veryone can explore... and... in the world of , this is... "low floor/high ceiling" activities where there's a very low floor to entry and a very high ceiling, so if someone really does want to go far they can, but... there's no real failure, because everyone has done something. ...[I]f we do more of that, then we will stop putting off congressive people from mathematics."
"The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions. As teachers and professors of math, we should welcome all questions and understand that answering questions is what helps students learn. ... “Is Math Real?” treats mathematical topics in a unique and original way. Discussions on number patterns, Platonic solids, math history, ethnomathematics, and mathematical structures presents the reader with a plethora of ideas on how one can envision mathematics."
"How to Bake 𝜋 is a success at explaining what mathematics is and how it is done, using simple, appealing language. It should be a rewarding read for mathematicians and nonmathematicians alike. ...[T]eachers will find plenty to borrow for... classrooms... Cheng frequently strips away technical details in order to show the big picture... [T]he book’s frequent digressions... topology, Arrow’s theorem, fair-division problems, s, the Poincaré Conjecture, the Riemann Hypothesis... [etc.]"
"… By relating personal stories, historical examples and mathematical analogies, Cheng explains how, when we rely on simplistic concepts like female and male, and the crusty logic that accompanies those concepts, we cannot have good conversations. As Cheng puts it: “If we object to the idea that ‘men are better,’ it’s not that helpful to declare instead that ‘women are better.’ It pits men and women against each other and sets up a prescriptive framework rather than a descriptive one.” She motivates us to strip away consistent triggers for dumb fights that lead nowhere. What would she have us strip away? This is where Cheng becomes a logician. She wants to carefully think through our associations with the word “success” as they relate to gender."
"As a category theorist, Cheng researches relationships. She uses this focus on relationships to address the problem of the divisiveness of arguments around gender equality. She abstracts the ideas and reframes the discussion based on relevant character traits that she demonstrates do not have to be linked to gender. She looks for assumptions that have been made, seeks to discard them, and discovers fundamental relationships. In order to better articulate these relationships, she invents new terminology as a way of preventing futile divisive arguments. These new terms are ingressive and congressive. She defines ingressive behavior as “going into things” where the focus is on the self and is more competitive, individualistic, and adversarial. She defines congressive behavior as “bringing things together” where the focus is on community and is more collaborative, interdependent, and cooperative. She gives many examples to illuminate her definitions. ... Cheng is deeply interested in making mathematics accessible to everyone."