Mathematics

/Σ/ To discover what was dearly missed in schooling, this general will serve as a playground. >What do you mean? Why talk of mathematics? >I hate mathematics. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf >Okay, but I don't know where to start. I still hate what I was put through. I'm just as lost, but I hope this is what this general will serve as. Don't be afraid to ask any 'stupid' questions here. Recount of your earlier math experiences if you wish. >Books ( which I hope to expand, depending on interest ) http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf I realized I was able to enjoy math class not because of the rigidness but because they couldn't take away curiosity. >What do these functions mean? is what I would always ask. Only when we reach Calculus II do we see the importance of Taylor Series, but that's not how our calculators and ancestors did it, past 1715. Calculators use The CORDIC algorithm (first I've heard of it, still reading about it). Still, functions of the calculator, and how they're always implied to be known, but not the discovery of it, particularly frustrated me. How did ancients do it? From what I've researched, it seems that they've made tables. >https://en.wikipedia.org/wiki/Ulugh_Beg I don't know how exactly he got them. I think it was using a particular trigonometric theory and re-inputting new discovered values. Or if you precisely draw the angle you're looking for, you can get the ratio. What about you lainon? Do you remember having some mathematical inquisition, in spite of the system?

>To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. So, what if I don't want that? What if I like math as taught in school? I like its sterility, its purity of logic and clarity of form. I like being reassured that there is an answer, and we can find it. It's the only form of security life has, the only place where that's always true (because they wouldn't give you the problem otherwise).

>>1191 """The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed.""" -- some Swiss patent clerk

>>1196 So you're saying I should join Tsukichan?

>>1197 I don't think that's his intention. You just have to give the other side a try. I do agree that the definite answer to it all was comforting, but you must realize it's only comforting in the drudgery of schooling. Where you become just another name to your teacher eventually, and the only thing left is to show the facts: you did the homework and you have the answer. Math was a way for us to rebel against the teacher. We didn't need their approval, we only needed to know we got the answer. But that satisfaction means nothing outside the institute we grew up in. It's time to give actual mathematics a chance. I wonder if there any Math graduates lurking that could give their input.

Can anyone point me to a good primer on the lambda calculus? Total beginner here.

Nice, a math thread. I wish I had something to say though. All I have to say is that it saddens me deeply to see, in lainchan, such an overwhelming amount of people say that they hate math. It demoralizes me in a strange way. I know this is a thread for those who like math, but I don't get why people go as far as to /despise/ math, and that breaks my heart.

I also have to say, I've been reading (slowly!) the Concrete Mathematics book. It is very nice, it teaches some methods of getting to a solution, and it also leads you through some processes to arrive to interesting results. It is not exactly easy to follow, you have to pay attention, but it's not actually hard either. Only requirement (up to where I've read at least!) is some essential algebra. Personally, one of the reasons I adore math is that it is entirely in the realm of platonic forms... at least the "pure" math. Another book suggestion I have, if you like the constructive side of math, is Enderton's Elements of Set Theory. Not a lot of people like set theory, but the process leading to the axiom of choice is a very interesting story.

>>1202 It's hard to find books that take you through the discovery of these formulas, or how they came up with the process precisely in history. I think I'll look for the best mathematical history book and work from there, personally. I'll check out your suggestion. >>1200 The best primers I've seen are blogposts. I'll see if I can find one for you... This is what I've found to be a great introduction. http://www.jtolds.com/writing/2017/03/whiteboard-problems-in-pure-lambda-calculus/

>>1191 Math is nice because it has definite truth, but it's also a thing of beauty that is much more than rote memorization.

>>1201 Well an answer to your question can be found on the PDF in OP. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf Simply, the way mathematics are taught in schools turns students against it.

>>1191 what is applied math? aside from statistics with its unknown unknowns there are plenty of problem spaces where you need to know what technique to apply in order to solve a problem, after which it's a matter of derivation and the computer can handle computation. take numerical analysis for example, like in OP's post, so much happens behind the scenes for a calculator to make an approximation and someone's job is to figure out how to program it in. >>1199 yes math graduate here, what you're referring to math as taught in schools is what i'd call arithmetic -- for problems of such and such form follow steps a,b,c to get an answer and show your work. real math requires creativity, yes concrete questions have specific answers but there's often multiple paths to get there and even to get to the answer in the same way as the textbook (on your own) requires you to think and understand why the author chose the approach s/he did. >>1201 arithmophobia. agree that math could be taught better in schools, common core is a nightmare. it's really disappointing how much people use a calculator as a crutch, smh at people who pull out their phones to calculate a tip but then everyone is always surprised by how quickly i can do mental math.

Anyone into algebra and category theory here? For some reason, I find those fields really amusing. Been studying 'em for about two years on my own, and I'd say that I'm getting pretty good at it. At least relative to how I used to be in the beginning.

>>1211 Do you even kan-extend?

>>1212 I do, actually. But I'd say that kan extensions and ends are the limit of my knowledge.

>>1187 A correction to the image, the term "Hindu-Arabic 15 century," makes it sound like that style of writing numbers no longer exist, but in fact it is the norm for the Arab world. Traditionally it is called Eastern Arabic numerals or Arabic–Indic numerals. The terms Hindu–Arabic numerals or Arabic numerals generally refers to the existence of nothing/zero having a representation in the numeral system. Say we have a base 10 of: a b c d e f g h i j, and we used those as symbols to express 0-9 , 'a' being 0 and so forth. This is a Arab numeral system, since there is a representation of 0. On the topic: What I found my school guilty of is the lack of focus on number theory, more explicitly stated as they never taught about different bases, or ehat base we did mathematics in.

>>1199 >But that satisfaction means nothing outside the institute we grew up in. But it was all I had going for me. It sort of became my idea of what happiness is like, and now that I don't have it, I feel like I'm not capable of being happy. >>1201 > I don't get why people go as far as to /despise/ math, I always kind of figured it was because they're stupid. >>1215 If we're complaining about the chart, the, "Ancient Chinese," row is modern Chinese in shaky handwriting. I'm sure Chinese writing in general is very old, but calling the current system, "Ancient," is misleading.

>>1191 But higher, real maths is not like that. You see, maths is really the realm of creativity and lateral thinking, not of sterile problems with trivial solutions. There are, fortunately for us mathematicians, loads of unsolved problems that will lead to new, exciting areas of math and will deepen our insight. Maths is aesthetics and pleasure!

>>1223 But I don't enjoy creativity or lateral thinking. I want a problem that I know has an answer that I can find using what I know.

>>1225 So you like doing... useless stuff? Useless calculations? I don't mean it in a mean way, don't get me wrong

>>1227 I like feeling secure. I want there to be a definite, provable right answer that I can find. I want to be able to do well by remembering things and applying them. I can't be creative. I wish I could, but I can't. I learned early on that I wasn't capable of it.

>>1228 more like you want someone to tell you that you got it right, and to to tell you well done for doing so nothing wrong with that mind

>>1231 Well ideally I could have both.

>>1228 Higher mathematics is still definite, logical proof. It's far more than just computation, and, if you are doing research, no one is telling you the answer, but you can be just as certain of the results. For every question, there is an answer. That is, either the statement being considered is true or false, or it is independent of the system in which you are working (which can also be shown- e.g. Gödel and Cohen's work on the continuum hypothesis). Of course, it's on you to show which of those is the case, but the reasoning is still just as pure and clean, if not more so-instead of simply working by rote process, one must rigorously justify their conclusions.

>>1228 School/college math = a road in a forest, all you have to do is reach the end. Higher maths = a desert. You have to find a castle, in a desert. There are no roads and nobody ever found the castle in that desert yet. You might have to be creative to reach the castle, but once you find it, it's definite. The level of 'creativity' required for that is the same as you would find in programming or any other aspect of life, really. Writing, history, geography, you need to be resourceful. This type of creativity is not necessarily artistic creativity. Either way, try you hand at things before dismissing them, Lain. Good luck.

>>1200 This is what I dig out when I need to remember how to represent basic logic or arithmetic in the lambda calculus. http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf

>>1203 >>1239 Thanks for the help guys Here is one that I found useful (it has some nice examples in scheme): http://www.cs.unc.edu/~stotts/723/Lambda/overview.html

I'm biting off more than I can chew right now; me and my friend are going to try to design a blockchain protocol from the ground up. I was thinking about formalizing the mathematics behind the verification steps and the network protocol so that maybe I can run some simulations without having to create an entire implementation. What dose /\lambda/ think of this? >>1204 >because it has definite truth Are you sure about that? >>1225 Sounds like a lot of software engineering

>>1247 >I was thinking about formalizing the mathematics behind the verification steps and the network protocol so that maybe I can run some simulations without having to create an entire implementation. >Are you sure about that? Math is just about the only thing that has real *proofs*. Everything else relies on fuzzy logic and unsaid assumptions.

>>1254 I goofed. First quote is accidental.

>>1524 >Math is just about the only thing that has real *proofs*. Everything else relies on fuzzy logic and unsaid assumptions. That's sadly not 100% true. You can't make something from nothing. If you create a formal specification of the semantics of a logic, you still rely on human "common sense" to interpret it correctly and understand what you mean (or interpret the specification of the specification language and so forth). That being said, math gives us the luxury of living on very few assumptions because we can boil it to such a simple thing as symbol manipulation.

Let's share some nice articles. Have some applied category theory.

>>1216 Not complaining, just making a correction to the image. I would not call it ancient Chinese either, sort of bull crap that the others do not have ancient prefixed to them. The more I look, the more I find wrong.

>>1256 >You can't make something from nothing. You actually can. Certain very weak axiom sets escape the Incompleteness Theorems and can thus prove their own consistency and completeness. Anyway, I'd say it's still real proof, and the point is that it's all a priori, which makes it uniquely strong.

>>1268 > Certain very weak axiom sets escape the Incompleteness Theorems and can thus prove their own consistency and completeness. [citation needed]

>>1273 https://en.wikipedia.org/wiki/Presburger_arithmetic

Best books to learn Category Theory? "Categories For The Working Mathematician" felt a bit disorganized and not very clear to me. Just don't make any suggestions that sacrifice rigour for the sake of simplicity. Also looking for suggestions on Type Theory books.

>>1275 at what point of your mathematical education are you?

>>1275 I'd say Bartosz Milewski's articles were quite helpful to me at first. They do lack rigour, but give you a great intuition for basic concepts.

>>1277 I found that to "get" cat theory it's quite important to have a good overview of mathematics as a whole, so you can start piecing together the "recurrences" that are obiquitous throughout mathematics

>>1276 Second year of my math bachelors.

>>1284 this is my advice, you are free to take it or leave it; category theory is not a standard topic to study, in the sense that I've never seen someone just take a book and self-study it with much results. To have a deep understanding of cat theory, I'd advise you to keep on studying maths, all maths, and start seeing the connections and patterns. Once you have a decent overview of maths, start looking into cat theory. At that point I think that you will find it natural and easyer (surely not easy, but easyer). Some things just need some time...

>>1275 For theoretical type theory with lambda calculus, I'd suggest http://www.cs.rhbnc.ac.uk/home/zhaohui/Barendregt92.pdf It's very rigorous and gives a broad view of different type systems with lambda calculus (even some type systems for which there is no type inference algorithms).

That Chinese row in the opening post omits the character for zero, 零. This is a fairly old usage and that row in particular is equivalent to all styles of modern Chinese anyways, traditional and simplified. I realized in my younger years, while still in the education system, that I would need to take my education under my own power if I ever actually wanted to learn anything. I've been reading nonfiction books constantly, ever since. A nice book I recommend for familiarizing oneself with the mathematical mindset is Donald Knuth's ``Surreal Numbers''; it's a small novelette set between three characters, with the primary two discovering a set of mathematical axioms and expanding upon them in their relaxation from society. It's a very pleasant work. Most anything by Knuth is bound to be good. I consider my mathematical understanding to be reasonably good. I focus primarily on mathematics related to computing and so currently find the most comfort in exact algorithmic specifications of processes, machine data, and ways to improve on these. The CORDIC algorithm mentioned is an excellent example of this and how computing is a rich field for experimentation and discovery of improved processes, especially under particular constraints. Search algorithms are another particularly interesting example. A primary issue with most mathematics as taught in school is that it's essentially mindless typographical manipulation, rather than understanding. This is easy enough to overcome for concepts such as addition and whatnot, but, as things grow more complex, less understanding and more rules come into play without any explanation of how they work. A fun way to ease into a mathematical mindset is to wonder of some property relating to numbers and attempt to see if you can prove it. It's typically easier than one would think to prove something for all integers, real numbers, etc., if one only thinks for a bit and tries. For those of you struggling to grasp a mathematical concept, I recommend simply starting from first principles. Learn what numbers are, how new types of numbers can be created, and how to define basic operations on them. Then expand from there. If you get stuck on something you reasonably know is true, but can't quite grasp the proof for it, just accept it and come back to it later. For those mathematical concepts we're all fairly familiar with, but may not actually, deeply know much about, such as π, read some history. History in particular can be an excellent way to break the potential monotony of reading more technical works constantly. The attached image is simply different notations for nothingness and whatnot that I had fun making. This is all I currently have to write, but I'd be happy to give more specific advice or help if anyone cares for it.

>>1304 Thanks for the response. I was looking at Peter Smith's gentle introduction to category theory. Apparently it is aimed at people who perhaps lack the mathematical background. >>1305 How does that compare with Thompson's Type Theory and functional programming?

>>1191 There is no place in the future for human calculators. Maybe in the past but that is no longer the case. The world now needs people who will build the frameworks of the problems that need to be solved.

>>1329 I'll look into it, I don't know it. Right about now I started reading Badiou's "Mathematics of the transcendental", just for the hell of it, and I must say, I like his philosopher's eye on the topic of category theory. If you like philosophy and ontology, check it out, it might just give you that epsilon of intuition you need to delve into more technical texts. >>1328 >A primary issue with most mathematics as taught in school is that it's essentially mindless typographical manipulation, rather than understanding. This is easy enough to overcome for concepts such as addition and whatnot, but, as things grow more complex, less understanding and more rules come into play without any explanation of how they work. Well said, I totally agree. For this reason I started studying physics in university instead of maths, because I thought that maths was too sterile. How wrong was I! In first months, where we started studying the necessary mathematics to do real physics, I understood that maths was actually my passion. I completed my physics degree, and right after that I started a maths master's degree. Never been happier!

>>1328 >I realized in my younger years, while still in the education system, that I would need to take my education under my own power if I ever actually wanted to learn anything. I've been reading nonfiction books constantly, ever since. A-fuarrrking-men. Recently I found a quote by Borges who said >At the age of six I had to interrupt my education to go to school. Sorry for the off-topic reply. To keep it back on topic, most of my mathematical education is done this way, and I believe that there are a lot of things you can teach kids without killing their potential love for math. I have a book called "Recreations in the theory of numbers" that suggests the pedagogical value of using number theory puzzles to develop a taste for mathematics. I actually plan on developing a pedagogical plan (though I know jack soykaf about pedagogy) and try to give particular lessons to kids and try to see if it works. I also dislike quite a bit the fact that calculus has such a predominant place in the mathematics field. Sure, it's useful, but ever since I learned it, I've forgot most of it. Then there are other very interesting subjects that don't have calculus as a prerequisite. Yet they seem to make you believe you absolutely need calculus for all forms of higher math. This also applies to algebra and the way it is taught. I hardly remember much about polynomials, even though it was drilled to me in middle school. Rather, computational methods (such as finding the roots of a polynomial) were drilled into me and of course I don't remember any of it. However, polynomials can be interesting in the light of linear algebra where they are treated as vectors. In my own learning of things, I feel like I should learn calculus, but I am more interested in other things such as Number Theory... and uh, other stuff I guess :P For those who want to know what this Number Theory thing is, have a book.

>>1336 Addendum. [on calculus] It is also sad to see that people for some reason that's entirely beyond me seem to think calculus is necessary for Computer Science. I mean, outside of some very specific areas of application, I can't see how calculus is possibly at all related to CS. [on polynomials] Polynomials are also really interesting in other areas, though mostly under the light of abstract algebra, which treats non numerical mathematical objects as primitives on which to perform some operations. Again, this is perhaps a general case of vectors. I think you could even go into algebraic topology without touching calculus much if you go down this path. Then Real (and Complex) Analysis turns out to be just a special case of these structures. My point is that while calculus focuses a lot on computation (computing derivatives and integrals (idk if computation is the right word, 'computare' means to count)), what is really interesting is the semantic aspects of mathematical objects. Under this light, calculus is really dull and boring (especially if you're already doing topology!)

>>1337 >I think you could even go into algebraic topology without touching calculus much if you go down this path. Not really, you'd immediately smash against deRham cohomology theory. >Under this light, calculus is really dull and boring (especially if you're already doing topology!) Topology and calculus are pretty different subjects. There are lots of intresting areas that fall under the calculus umbrella, especially at the intersections, like differential geometry. Don't throw away what you don't yet know! I understand that "engi calculus" for CS students is useless, on that I agree with you

>>1338 >like differential geometry I'd say that's in the realm of Analysis Analysis > calculus. Calculus is merely the computation of the structures/results of Analysis.

>>1343 So essentially >>1336 is talking about mindless calculations. I thought we established that maths is not like that.

any recomended books of linear/non linear optimization?

>>1344 Well, being able to quickly do certain mindless calculations in your head makes doing math a lot easier.

>>1344 >>1351 My point was that too much emphasis is placed in the merely computative and not in the general semantics. Sure, calc can be fun, but without context it simply makes no sense.

>>1351 I find that the most useful "ability" is geometric intuition, not arithmetic speed. But that's probably because of my upbring. >>1352 That's probably because you saw it in that context? My calc courses were surely quite computational, but also very deep and theoretical.

>>1351 It certainly does, but you can do fine without that too. I was diagnosed with dyscalculia in my late teens, but I've never had trouble understanding the concepts, it just takes me effort to do things that are "mindless calculations" to others.

>>1355 This and calculators exist. It's not a mathematician's job to compute. It's a common joke among mathematicians that they forget how to do elementary operations, I've had math graduates tell me "I can't multiply, you do that". I've a joke: how do you know a guy's a mathematician? When he goes to the market he is making the calculations with his fingers.

I'm still stuck at basic math and want to learn CS math, what would be a solid route to follow courses/books and can learn everything at home/library.

>>1330 >There is no place in the future for human calculators. So what exactly do you recommend? It's all very well to say, "Nobody needs the thing you're good at anymore," but unless you actually suggest something I can do about it, you're not really helping.

>>1362 be good at something else. Skills can be learned.

>>1362 Every skill can come to good use, if only for you. I'm serious, skills are valuable, whatever they are.

>>1362 Also remember that in excess the desire for certainty can be a vice.

>>1357 Khan Academy for basic math. All the material is pretty good. For advanced math, maybe give a go to Knuth's "Concrete Mathematics", great book but rather challenging, doubt you'll do well with it having little mathematical maturity. If it's too difficult, I'd go with one of the standard texts in discrete math.

>>1377 Would you care to elaborate?

Is this already dead? That's a pity... What about posting some articles? Actually discussing some maths?

>>1513 I'd say your articles are a bit too much for most of us.

let's have fun with computability and complexity, lains! This book is a really, really good intro. I'm a wee undergrad, just finished linear algebra, and I found this both understandable and challenging. Still can't do most of the exercises, I feel like they require more mathematical maturity and time than I currently have.

>>1522 forgot the file.

>>1513 It's difficult to converse when we're all so lost in this austere subject. Would you care to expand why you chose those pdfs? I'll read them over. >>1522 >>1523 I've began reading this as well - also an undergrad. Hopefully I can balance and make progress to come back to you on the exercises it presents.

Can anyone speak to the quality of Dover Publishing's math books? They have stuff like pic related. A lot of it seems to have been published a while ago, but they're evergreen topics, so who cares? If you go on Amazon, you can get paperback editions of these things for like $2-$4 a pop, so I have a bunch of them sitting on my bookshelf now.

>>1247 Why would doing that get you out of implementing a full protocol?

>>1532 Dover books are usually pretty good on most topics, especially for the price. But check the comments on amazon before you order. A lot of their math books are translated from some old soviet stuff in russian, and translated math books can be... confusing if the translator isn't fully mathy in the book's subject.

>>1529 I'm just picking some of my favorite articles.

>>1331 >Mathematics of the transcendental Took a look at this horrible book, it made me angry. Suggest an actually coherent philosophical-mathematical book, not this continental nonsense.

>>1539 What makes it so bad? Please don't call something bad without saying why you think so.

not sure if thread is still alive, but Concrete Mathematics Second Edition: https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf

how do i stop being bad at math?

i'm having to study a lot of math equations for my summer research project with neural networks and it's actually a lot of fun. i hated math in high school because i had depression in my senior year and i just slept during class and grazed by on the tests without paying attention, so i was lost the whole time

>>1640 Is that you kirbs?

>>1641 yes

>>1191 yeah lain the purity of logic and clarity of form [I]is[/I] part of the beauty. As you go along in maths you run through all the chaos and you're looking for a thing of beauty. It's enraging, confusing, and horrifying but then you see things that are beautiful. I mean proofs are all about being sure. Sounds like you'd like pure math.

>>1191 >>1196 I find this sort of debate distasteful. Trying to confine math to your standards, or your expectations of it, and detaching it from itself by trying to mistify it as some subjective from of awe and beauty... Math is what it is, it is only harmful to try and coerce people into enjoying it under your own parameters. Let people enjoy it for what they find in it, not for your abstract feelings about it which are often ill defined and meaningless to others. Really, this depiction of math seems to be a romantic idea slapped onto it. Just a bunch of nice flourishy words that add little content to it. I mean, damn, really, if the guy likes math for "its sterility, its purity of logic and clarity of form", then he is absolutely on the right path because that's what math is about in the first place, stop trying to convince him that it is like with the other arts where it is all about feeling and subjectivity because that's not how math works. Really, all this mistification of math is harmful and stupid. EOR (end of rant)

I've tried several times to learn math but it just doesn't work. I tried khan academy, but it's too easy, i can't really get focused enough to enjoy the act of learning. Do you have any recommendations? Keep in mind this is nothing more than a hobby, i just like learning things and math is one of those things i always put off because it's just not fun to learn formulas when i don't understand their purpose nor the logic behind them.

Anyone into Number Theory? It is like the Scheme of mathematics

>>1687 Yes, math is not about learning formulas but precisely about following the logic behind some reasoning that leads to a result. This also means that you should read any math book with care of being able to follow the train of reasoning therein presented, that is, to follow a proof throughout until you are convinced of it's validity. A math book is not easy to read. And you ought to have pencil and paper at hand. I just posted a book on number theory. I haven't yet read it, I am about to. But it's a reputable book so perhaps it's good. I particularly like the theory of numbers because it deals with the most elemental of mathematics, and also the ground of it all. Oh, there is a series of videos that you might like. So far it has 224 videos (split in 3 playlists). It is very intuitive and it develops from the ground up. You should be patient though, and follow through even if the first videos seem too elementary. https://www.youtube.com/watch?v=91c5Ti6Ddio&list=PL5A714C94D40392AB

>>1689 Thanks a lot, Lain. This teacher speaks in a way that is very pleasant to me.

>>1187 I am at the moment working my way through the field of discrete mathematics for a course in college and to get my programming skills to the next level. I have found the MIT course in youtube quite enlightening about the thinking behind all these formulas, concepts and how to play with them to make proofs and solve practical problems. https://www.youtube.com/playlist?list=PLcvjwyxdQk5EL0PPMawkQUtL56Re43k-l Brilliant (https://brilliant.org/) is also an epic resource to rehabilitate our math/science skills after the soykafty academic environment some of us are left to endure.

8chan has a similar thread going on I'll post pics of their book recs. you can find these books online or on libgen

>>1690 General advice: mathematics are a framework for studying reality. When you learn stuff, you're learning tools to model behaviours in real life. That's why the best way to study math imo is by practicing a lot, doing a lot of exercises until you understand how to apply theorems and when to apply them >>1697 I was really disappointed with MIT's proposed curriculum for discrete mathematics. Fairly limited for a computer science course. However it's a good way to get started!

Anyone can recommend any books on Chaos theory? My background is computer engineering

>>1952

Should I follow these steps?

>>2012 what's stewart's book?

>>2015 These books here https://www.amazon.com/Calculus-James-Stewart/dp/1285740629

>>2012 >>2016 That's very redundant, and tell me, is it your primary goal to be a god at calculus? If you want to be the best kid at calculus in your block, then go ahead. Else, I would rather just read the Spivaks in order (calulus, calculus on manifolds, and differential geometry). And if calculus isn't that much of a a priority to you, I'd advise you instead read Shilov, Dummit & Foote, and stuff like that. I wouldn't omit either Spivak or Baby Rudin though, those are great foundations on analysis. Here's a disclaimer, I haven't really read much of Spivak or Baby Rudin, but I prefer the latter. The excercises in Spivak are way too many and I get bored by the first 10 in chapter 1. Then a lot of people (like the one it that pic) seem to consider Rudin to be a hard book, though it isn't really all that hard for what I've read, best of all he is concise, of course you have to follow with care the propositions and proofs which you don't understand well, but other than that, I wouldn't preamble 2 huge books on calculus/analysis just to read this one which goes over the same matter.

>>2017 Thank you for your post, lainon. Do you recommend any pre-reading for the Spivaks books? I do not think I am good enough to get into a Calc book yet, Perhaps if the book has some intro stuff i might give it a try already.

>>2018 Serge Lang's Basic Mathematics (you can find it in libgen) is a popular one that should put you up to speed to get started with calculus.

>>2019 Thanks again. I will check it asap

Hey lains. I have had an idea for a while but I have yet to implement it. It is based on the observation that in mathematics and physics difficulty (at least to me) comes from remembering all the theorems related to a given context. As such I have had this idea of encoding theorems into a computer and later using them as a proof assistant (this already exists, I know). I'm torn between learning to use an existing proof assistant or writing one either in prolog or python. What do you guys think?

>>2023 to expand on my post and not just soykafpost I've found a couple of systems that do what I want to but they're always either very obtuse or the documentation is soykaf. I don't know why I can't just do something like this: hypoteses: 1. a+b = b+a 2. for all a in (set) there exists -a in (set) such that a+(-a) = 0 . . . /end of hypotheses proof start at #1 => a+b = b+a use hypothesis #4 on a => (sin(theta)^2 + cos(theta)^2) + b = b + 1 ... ... => meaning of life = 42

>>2023 >>2024 I know a guy who tried to do something similar. He claimed it wasn't very hard, compared to implementing normal programming languages. The reason they're so obtuse is that they're mostly used by mathematicians to do very obtuse math

>>2024 Here's the main problem: If you roll your own theorem pover, who will prove theorems with it? From your posts, it seems your ultimate goal is to make a collection of more or less all theorems that have been proven so far in formalized some proof assistant language. This will require proving most theorems for a second time inside your proof assistant which is an incredibly huge amount of work you will never possibly be able to do on your own. I highly doubt that anyone would be interested to sink their time into proving stuff in a proof assistant, quickly thrown together by someone on the internet, instead of one of the well established, well supported ones from academia. So I highly recommend against reinventing the wheel, even if building a proof assistant was easy (which it isn't).

>>2032 Yeh.. I guess you're right. My ultimate goal isn't what you said though, having all proven theorems encoded in the language is only the first step.. ;)

>>2034 Don't let you get talked down, if anything it's an interesting project that you'd learn somehting in undertaking. Your post kind of made me think about this idea. It inspired me a couple ideas: 1 I would do it as a framework for "constructive mathematics", basing it on stuff like type theory, lambda calculus, proof theory. There's extensive research on this field and pretty nice available information. It would however lie probably in the computational side of mathematics. 2. I would probably do it in scheme since it's primitives pretty much provide a minimal starting point and possibly endless extensibility. I was also thinking that The Little Prover is a fun book that can help shine a light on some of these things. Either way, the uniform structure of scheme syntax would allow for some neat things, like building new types and operators upon a set of primitives and then using them as primitives for another layer. I am thinking of the lambda cube here. Now the lambda cube shows some categories between lambda-systems, and the system could itself be used to ascertain the constructive relationships between λ→ and the intermediate constructs up to λPω.

>>2035 Damn bro.. Your post makes me think.. I mean I know I can have a lot of fun making this system, but I also want to go further (as i said in my previous post). I didn't elaborate, but the idea is to use the proven theorems to teach a neural network how to derive theorems by itself. Right now I can work on this because I'm on vacation, but they will end this month, so the progress I can gain by building a full system may be smaller than the progress I'd gain if I learn coq, which I have begun to do. Eventually I would like to create my own language, and I've thought of using scheme or another lisp, as I've read about the languages and they sound absolutely fantastic. Thank you for the kind words my man.

>>1275 Category Theory for the Sciences by Spivak. Also watch Bartosz Milewski's category theory videos on YouTube.

what is a good strategy to learn abstract algebra? i have high school math knowledge and i am willing to start from scratch if i have to.

I always ignored maths growing up, and I'm trying to make up for it now. I picked up Serge Lang's Basic Mathematics, but I'm already stumped on page 19. Based on what rule does this happen: (a+b)(a+b) = a(a+b) + b(a+b)

>>2137 the distributive property

>>2123 The book "A Book of Abstract Algebra" by Pinter is a really good. Extremely readable, often recommended.

>>2137 (a+b)(a+b) = c * (a+b) // here c is a stand in (a variable) for (a+b) c*(a+b) = c*a + c*b (distributive property) we substitute the variable (a+b)*(a+b) = (a+b)*b + (a+b)*a fin. You have to understand how to apply the axioms. First you must learn the axioms of the reals, it also helps to learn the properties of equality. If you manage this then the rest is much easier. Good luck.

>>2035 Just so you know I havent given up I'm looking for different ways to make this.

>>2024 The project has started taking shape. I'm following a tutorial to create my own lisp and concurrently sketching out the system I will call Aleph. Aleph is a language where everything is a function (programs too). Functions are seen as taking vectors from a dataspace to another dataspace through the evaluation of the propositions inside the function. for example a program to make the speakers beep would be seen as: beep (<input> button_press, <output> sound on speakers) The system will include typing. The intention is to create a mathematical programming system, using set theory. Its main goal is to allow the user to create and manipulate datasets extracted from dataspaces through the use of aleph programs.

>>2147 It's a little late for this, but thanks! I did eventually figure it out myself, but this explanation would've been exactly what the doctor ordered.

here's a humble bump as thanks for the pdf in the OP (Lockhart's Lament). I lost the spark of curiosity way back in elementary school. It's nice to entertain the possibility that maybe it wasn't because I'm stupid and should just kms. I hope to revive that spark someday; it's painful but I won't give up.

>>2453 For now what I've been thinking is that I want to borrow from prolog where we could write statements and later derive conclusions from those statements. The system will also include a way for the user to make algebraic manipulations (though how I will implement this is still not clear). These manipulations have as a purpose arriving at a certain form for the equation (which is often what proofs are about). The system will also allow substitutions in equations. I may be experiencing a but of the dunning-kruger effect. I feel optimistic yet cautious about this project.

>>2488 you're welcome bro :) remember that simple explanations are better and examples are even better

>>2503 >>2503 High level overview of a proof in aleph: 1. user introduces hypotheses to establish a context (such as introducting reals or naturals) 2. user states what to prove so far the system has done nothing but store functional and logical statements 3. userinteracts with aleph system by composing their hypotheses calling them by name or number 4. the system only does basic bookkeeping (it checks that conclusions do follow from hypotheses) 5. ideally the user manages to make a solid proof 6. the system saves theproof for use in futuremore complex problems Sample program in aleph: q p : fun. // this is meant to be a proof of a theorem lt(q, p). / less than // x y : real. // these statements ending in . are our initial postulates. Ax, Ay( (x<y) => q(x)<q(y)). // they are assumed to be true (either as axioms or hypotheses) Ax, Ay( (x<y) => p(x)>p(y)). Theo := Lim(x->inf, p(x)) = Lim(x->inf, q(x)) ? // this is the theorem to prove. Thequestion. <calculus theorems> Thus if<hypotheses> : True then Theo : True.

>>2522 I'm posting my developements here, but I don't know if I'm derailing the thread. I do this mostly because I'm lonely and I want someone to talk to. Bonus points if it's about what I'm working in. right?

>>1200 also interested.

I just wrote a really long post criticizing Lockhart's Lament and then lost it by accidentally reloading the page. To summarize: >Despite the OP putting the essay under "I hate mathematics", that's not really what Lockhart is lamenting. He only touches very briefly upon people who describe themselves as not being math people. This isn't a fault of the essay, I'm criticizing the OP here. You could at least put a page number in the OP for that part of the essay. >Lockhart laments that schools don't get their students to invent their own problems and discover the beauty of maths >The one example he gives of his own student's work is an answer to a question he (Lockhart) gave them >Lockhart creates a character to ask his own imaginary avatar if he's really advocating abolishing the current baseline math curriculum >Lockhart's avatar: "That's not real math anyway! It's not pretty enough DURRR" >... Okay, but it's enough to prepare kids for freshman college classes on math (this response is mine, not offered by his straw man critic) and if you removed them in favor of math problem invention classes we would have even more innumerate high school grads >Oh, but you can't criticize Lockhart for being unrealistic because he's covered his own ass by admitting that all this is impossible >So what is the purpose of this essay? What are you lamenting? >If it's the fact that your fantasy is completely unfeasible, why do you frame it as an attack on education boards for not seeing the beauty of math and for twisting math into its current form? You've admitted that your alternative isn't realistic. The essay is utterly fatuous. The fact that he backtracks on the main argument, which is that kids should do real math in class, by admitting that it's unfeasible to implement and he doesn't expect any of this to happen, really makes me wonder why he bothered publishing the essay. What did he hope to accomplish by spending ten pages writing, "The education system is flawed. Here's my alternative. Oh, but it's impossible too"? Really makes you think.

How do i learn linear algebra as someone without any real maths knowledge yet is there a solid guide with books to learn up to linear algebra?

>>2726 For what kind of application to you want to learn linear algebra? You can save yourself a lot of soykaf if you define your scope correctly.

I'm humbly requesting a small list of books for someone that wants to start with programming. Is Basic Mathematics by Serge Lang a good place to start? The plan is to learn basic programming, some math to get me started with algorithms and what not and maybe a good book about general problem solving (which I have a huge problem with) Anyone read How to solve it? Will the contents in the book be good for solving programming problems?

>>2726 As someone without a great mathematical background I found it easy to learn lin. al. in my first semester by our professor's script, even if it was hand-written and badly copied. Perhaps you should just search for a nicely TeXed script from an university in your mother tongue. Imho lin. al. is way more easy to learn than Analysis or some other mathematical fields. To work with vectors and matrices is, after learning some basic formulas, very easy.

>>2606 It's his utter inability to conceive of Math as something that is fundamentally an applied art that baffles me. Yeah, it would be nice if high school aged kids could get to see some more of the beauty of math in their studies, but speaking practically, they need to be able to apply fuarrrking algebra. It won't do to have them reinventing the quadratic formula every time they have to work with a 2nd degree polynomial. It's not a totally abstract art. It's something that has to be applied in other classes if it's to have any real meaning. Otherwise, it's just pointless number games, and you're back to having people say to themselves "This is dumb. Why are we doing this again?"

>>2523 have you looked into Racket? https://racket-lang.org/ seems like the kind of thing you'd want to use for this project.

>>2761 I will, thanks. Right now I'm designing the grammar with mpc (micro parser combinators) which is a parsing library in C. The design that I have in mind is meant to be minimalistic.. though it is so ambitious that I have my doubts about that.

Hey. The Aleph system is under construction, but I no longer feel confortable sharing its developement details. If anyone is interested in working with the system, you can email me.

Still spamming the thread with Number Theory. I just found this book, looks good enough to share. Anyway, why does it seem nobody is interested in the most patrician of mathematical subjects?

>>1624 Keep being bad at it until you're not any more.

Does someone has books for differential calculus all the way to integral equations? I've been trying to find the love for these subjects but it totally fuarrrks my mind in the long run. Also any tips in working in the derivative of functions and equations in general?

>>3225 Try spivak's calculus. He has a book that builds calculus from the very basics. Also you may be interested in non-standard analysis (calculus with infinitesimals instead of with limits). [ a good book, the surreal numbers by Knuth ] It allegedly is more intuitive than calculus with limits. I'd pour more time into that soykaf if I had the time to relearn the basics. A piece of advice: read several books on the subject concurrently. Oftentimes you will find that they cover each other's shortcomings.

>>3225 One more thing. Do the excercises. They will make it very clear to you what it is that you actually understand and what you have yet to grasp.

Does someone have a book on phase spaces? They're a crazy concept. I like it.

>>3225 Most stuff like this is going to be in Python, just google numpy and differential equations

Would someone help me with the problem set 1-D ? I read the solution but I keep reaching sq(x) = ax instead of the roots of the quadratic functions. Pages 101 for the solutions and 16 for the actual exercise.

I haven't checked this thread in a while, but I was the guy who liked math sterile. >>1686 > then he is absolutely on the right path because that's what math is about in the first place, stop trying to convince him that it is like with the other arts where it is all about feeling and subjectivity because that's not how math works. Thanks man, that was actually really reassuring to me. My eyes got a little wet, but that might have more to do with the time. >>2606 > really makes me wonder why he bothered publishing the essay. What did he hope to accomplish To be fair, he did call it a, "Lament," which implies that it's less a persuasive piece and more a way to vent his sadness. >>3593 I didn't look at that problem set, but I know how to solve sq(x) = ax, if that helps. sq(x) = ax x=(ax)^2 x=aaxx 1=aax 1/aa=x x=1/(a^2)

I'm going into math next year, glad to have found this thread.

>>1965 n1

I see recommendations on this thread for learning Calculus, but what about learning the basics of Algebra and Geometry? I blanked out on math in high school due to personal problems, what are the best books for getting a good grounding in these two subjects?

tfw stressing for my diffeq test because my professor is a senile man

>>3727 >sq(x) = ax This isn't particularly relevant, but you lose a solution if you do it like that. A better way might be sqrt(x) = ax x = a^2 x^2 a^2 x^2 - x = 0 x ( a^2 x - 1) = 0 x = 0 or x = 1/a^2

lain, what are some entry level books on history of mathematics? mathematician's lament may have bedn preachy but it made me realize i hardly learned about origins of common formulas and theorems

>>4526 After reading this review of Stillwell's "Mathematics and its History" I decided to read it this holidays (along with "An Imaginary Tale: the story of sqrt(-1)"). https://www.austms.org.au/Publ/Gazette/2011/Jul11/BookReviews.pdf But it's more focused on maths than history, which may not be what you're after. One book that I *have* read, which isn't exactly what you're asking for but was an excellent and interesting read, was "The Crest of the Peacock: Non-European Roots of Mathematics" which looks at how maths was practiced throughout the (usually ancient) world. Full text is available online. http://www.ms.uky.edu/~sohum/ma330/files/Crest_of_the_peacock.pdf Looking forward to others' recommendations on this ...

>>4018 How to Solve It for Algebra, Elements for Geometry.

>>4526 >>4018 bumping because I am also interested in these two things

>Currently in high school >Took maths HL >Im OK in maths Any resources/ advice to improve my learning habits? >I want to get a 6,7 from a 4 Ty lainons

>>4798 what are your current learning habits? i assume you're talking about the IB program? can't comment on that specifically because we don't really have it in america. general advice: if you're memorizing everything, you're doing it wrong. also get used to learning from a textbook rather than lectures. if you have any long term interest, you should get comfortable reading and writing proofs. that's all academic math is once you get out of high school. pick up an undergrad textbook and see how you do.

>>4806 Im currently learning about 1.5 hours after school and then I dose of to movies.(y I should change that) >Mostly 30minutes of that time is math >I dont like memorizing but all my friends do it and get better grades than me

>>4807 Here was my method back in prépa. I was studying maths for ~3h/day after the classes so my situation was different from yours, but I see no reason it couldn't work for you to. 1. Make a list of what you have to know, typically from the simple stuff to the more complicated. Definitions, characterizations, properties, theorems. Keep it short, don't write the whole thing, just the names. (Def:real number, property:least upper bound, etc...) 2. Take just this list and a pen and a blank page. (I bought a couple dozen of draft notebooks every year just for this since it consumes a lot of paper.) For each item on the list, write everything you know about it. Write down the definitions, the properties, the theorems with both hypothesis and conclusions. Usually you can't do it all, so go back to read your notes, put them aside again, and start over. Until you can write down everything you have to know. It's important to understand that it's not about raw memorization, save maybe for the definitions, but those should be rather simple. You need to understand where the property is coming from, afterward you can't really forget it or misremember it. I don't know if your class includes all the demonstrations for the theorems ... Even if you're not required to know them, at least you must understand how they work, this will make the theorems seem more natural and easy to remember. Depending on your time, do the exercises/examples provided the teacher until you have no difficulty with them, not because you memorized the results, but because you understand every logical step.

>>4811 This is the smart way of learning. Unfortunately I never could learn with just the lesson and the demonstrations, only doing exercices made me really understand. Difference between brainlets and smart people I guess.

>>4829 As long as you're able to understand you're fine. Parents are maths teachers, you have no idea how bad things can get. My mom has a student in their third year who still can't do the most basic things that you're learning in the first month of first year. (We're speaking proving-that-{(x,x,z) \ x+y+z=0}-is-a-vector-space level of basic. And he found a base for it with 3 2D vectors.) There are genuine brainlets IRL, but man is that term overused on the chans.

>>4831 yeah but I'm sure you've noticed what I mean. I might have been a bit hyperbolic but I've often noticed that the best students learn mainly from the theory whereas the average/not so good ones have to do a lot of exercices to get it. I've always been a bit annoyed at that. If you're cubing and you don't know that math then you were never cut out for prépa to be honest or even uni.

>>4832 >yeah but I'm sure you've noticed what I mean. Honestly I'm not sure that I do. I've always seen the best students spend a lot of time on exercises, since they learned the theory with ease ; while I had to spend most of my time remembering the demonstrations and could do very little exercises. Point is, don't let anyone (or yourself) belittle you for your learning methods. People can really be wired differently, and adopting a method that is suited for you goes a long way toward success. > cubing Usually people poorly adapt english words to french, I love that you're doing it the other way around. But yeah, the guy only got there because he's kinda rich and it's a private school that needs all the students and money they can get. The scary thing is he might actually end up with an engineering degree.

>>4833 private paying schools should be derezzed,brainlets should not be able to buy their degree.

>>4834 As much as I like that idea, I'd also like a pet giraffe and public transit that connects Earth and Moon. The world doesn't work that way.

>>4837 don't think it's this much of a stretch

>>4831 maybe this is an American thing, but for some reason in uni we don't do much linear algebra past the bare minimum knowledge of vector spaces and linear transformations until second or third year.

>>4856 >maybe this is an American thing It's always an american thing.

I have a question for the computer mathematicians. What is a proof that the 8 bit number x is always equal to (x&128) + (x&64) + (x&32) + ... + (x&1) ?

>>5858 Your formula is just picking out the binary digits and then putting them back together.

>>5858 for an n-digit number x, x = Σ (x & 2^k) where 0<=k<n

>>5863 Okay but how does one makes a proof out of it ?

>>5868 You can do several things. Since the proof is only for 8 bits you can literally show that it works for every single 8 bit number. Alternatively you prove it by induction, so that it is proved for any number of bits. I think induction on number of 1-bits in the number would work the best but I'm not sure.

>>5870 I've done it I'm super proud of myself right now. It was in fact really simple. Thank you for the tip!

>>5872 >>5870 Here is my proof but I'm not that happy with it I think it can be improved.

>>5874 Hey lainon, I appreciate your enthusiasm for mathematical thinking, so I hope you won't be discouraged by a bit of contructive criticism. 1) A very technical nitpick: "set of l bits" is not the term you wanted to use. Note that sets are unordered and have no repeats. "Sequence" is probably the word you were looking for. 2) A general note: there's no need to reach for formalisms if a proof can be reduced to something more abstract but still understanble and precise (like >>5861's one sentence proof). Since we don't have to go all the way down to ZFC axioms, we can also stop at any other level that is precise enough. 3) Your induction step is incorrect and you don't end up proving anything. You showed that x+1 = 1 + \sum x & 2^k, which by the inductive hypothesis is obviously true. But the formula you want to show is x+1 = \sum (x+1) & 2^k.

>>5876 Thank you - I don't mind criticism for I was pretty sure that I did fuarrrk everything up thank you for your help

>>5876 and because of your kind comment I even ended up recalling a whole part of my math class you're the best anon

>>1235 It also might be the case that there is no castle, and you have to prove that there is no castle in the desert no matter how you look. Or maybe there is a castle, but it's in another desert, and you have to find that desert before you even start searching for the castle, and you have to convince people that your desert is better than the other desert and in fact the castle in your desert isn't a facade and is the real castle in your new desert that people couldn't find in the old desert...

Noob here - how do I understand this statement in plain english ? Thanks!

>>7195 If P is a set of those numbers {p_1, p_2, ...,p_k} (finit set) then let as create a new number n_P = (p_1 * n_2 * ... * n_k) + 1 (multiply all elements of set P and add one). If I understand this correctly then this is a part of a prof that there are infinit number of prime numbers.

>>7196 Thanks! yes it is you're right

Trying to make a list of common Fourier Transforms, as well as their proofs (because some of the proofs in class were literally two lines). I know the derivations aren't *that* important, but I still think it's nice to know where the results come from. I want it to be as rigorous as possible while still being followable (is that the right word? whatevs), and I'm concerned about a step in my derivation: I say "We have an indeterminate form 0/0. We thus have a nonzero value". I was trying to motivate the fact that we have a delta function, but I'm not entirely sure this reasoning is valid.

I can't into vectors Send help

>>8131 You need to be a bit more specific here lain. Do you not understand the basic operations on vectors, or is it linear algebra you're struggling with? Or perhaps something else? Please elaborate so we can help you

>>8133 Apologies for brevity, it was partly out of frustration I'm not understanding converting them between their forms, what each vector form means, and generally visualizing them as a 3d concept I'm fine with what I've learned so far of linear algebra but the precursors aren't gelling

Any book recommendations for learning mathematical data analysis?

Whats some good reading for someone who forgot HS math?

>>8285 https://www.khanacademy.org/math

>>8205 Are you talking about statistics/"data science" or numerical analysis (aka doing calculus with actual numbers instead of symbols)?

>>8292 Both

>>8285 Axler's Precalculus

I failed Diff Eq at Uni, what books are good for not fuarrrking up the second time around?

>>8367 for ODEs, get your linear algebra up to scratch. for PDEs, you are on your own mate.

>>8377 Thanks for the useless advice

>>8367 yuefan deng

>>1532 Main thing about dover is they are cheap. In the free PDF age there isn't a point. Check this list for good books: https://github.com/ystael/chicago-ug-math-bib Sometimes dover stuff makes it on, sometimes it doesn't.

>>8291 >>8299 thanks, ill look at these tonight

>>8205 If you want an introduction to stats then casella berger is the most recommended one in english, haven't read it myself since most books on introductory math.stat. has been in swedish

>>8377 Aren't PDE's usually upper uni stuff? Like 3rd or 4th year? I'm sure he'll be better suited by then. Here's my question for the maths thread: Anyone know a place that gives good math MOOCs? Both udemy, and coursera fail me for pure math courses and I can't afford to take any more normal uni courses.

What's a good way to teach yourself higher level mathematics? I want to understand weird concepts like manifolds and non-euclidian geometry for funsies. I have taken calc, so my level starts there.

>>8536 uni calc or high school calc? if you've gotten up to sequences/series and basic differential equations, the next step is real analysis, then topology. that'll get you to manifolds. you need to get good at understanding proofs before you go any further though (if you don't already, that is). it sounds like you're interested in the analysis route, but there's also algebra, dynamics, number theory, etc. if you're really new to advanced math, you may want to try to find a book that will give you a good overview of what's out there. i had a course like that in undergrad, but i forget the name of the book we used for it.

>>8500 I always use MIT OpenCourseWare >>8536 learn basic proofs(How to Prove it and Book of Proof are good books for this. Read them all the way through and do at least half the exercises). Learn analysis, either complex or real(complex is usually easier). There are plenty of good books for that but no outstanding ones imo. Get a really good book on manifolds. Spivak is a name you should look for. Read through it, do exercises, etc.

found these excellent topology comic books http://www.savoir-sans-frontieres.com/

>>8561 Eh I'd prefer one where I can claim a cert. Looks good on the linked in.

>>8604 many colleges do math MOOCs but they still cost some money. >I'd prefer one where I can claim a cert where are you ever going to need a math cert? Almostn o college is going to recognize one, no employer will ever care.

>>8618 I want to project overwhelming intellect onto my adversaries before they ever finish reading my CV while still not shelling out for a MS degree in math?

>>8631 >I want to project overwhelming intellect onto my adversaries before they ever finish reading my CV Most people don't hire people who are so obviously smarter than they are.

>>6831 get a PhD, you pay for them with blood sweat & tears instead of money.

I've recently come to the realization that math is necessary and not bad. In order to master any kind of knowledge profession, you must know math and be good at it. I had a really bad foundation for math, starting in grade school when I wasn't very good at mental math and never made an effort to learn it. So I've been making efforts to improve my mental math before I move on. I wouldn't go as far to say that I love it, but I know new meaning can be found for it once I breach into the world of calculus more, and applied mathematics. Just tryna be a whiz at the basic stuff first before I move on. And it's pretty cool when you can do certain long multiplication problems in your head and people look at you like you're a genius

>>8899 >And it's pretty cool when you can do certain long multiplication problems in your head and people look at you like you're a genius How are you practicing?

>>8893 You mean in addition to lots and lots of money? Sadly, I have a career so I can't jump back into academia. I will if I ever hit the jackpot though. For those self learning maths: Who do you go to with questions?

>>8903 Math websites mostly. Lots of good mental math websites that can time you and keep track of your progress. For larger numbers, it's sometimes important to learn a few tricks. Keyword is "few". Learning even most of the number sense tricks takes a loooong time. Here's an extensive list of some tricks. I'm only electing to learn the ones that are large in scope. 1.2 is pretty useful though. https://lainchan.org/%CE%BB/res/1187.html#q8903

>>1191 SIMPLICIO: Now hold on a minute. I don’t know about you, but I actually enjoyed my high school geometry class. I liked the structure, and I enjoyed working within the rigid proof format. SALVIATI: I’m sure you did. You probably even got to work on some nice problems occasionally. Lot’s of people enjoy geometry class (although lots more hate it). But this is not a point in favor of the current regime. Rather, it is powerful testimony to the allure of mathematics itself. It’s hard to completely ruin something so beautiful; even this faint shadow of mathematics can still be engaging and satisfying. Many people enjoy paint-by-numbers as well; it is a relaxing and colorful manual activity. That doesn’t make it the real thing, though. SIMPLICIO: But I’m telling you, I liked it. SALVIATI: And if you had had a more natural mathematical experience you would have liked it even more. SIMPLICIO: So we’re supposed to just set off on some free-form mathematical excursion, and the students will learn whatever they happen to learn? SALVIATI: Precisely. Problems will lead to other problems, technique will be developed as it becomes necessary, and new topics will arise naturally. And if some issue never happens to come up in thirteen years of schooling, how interesting or important could it be?

>>8911 generally a PhD is funded either by the university or by a fellowship. You TA(and sometimes teach) classes, and write papers, to earn your keep. You get a stipend which covers tuition and is also (just barely) enough to live. If you're paying for your PhD... you probably shouldn't be doing a PhD.

>>9097 It should be obvious I don't have the math background to qualify as a doctoral student since I'm talking about math moocs. If I had a masters level math education, I wouldn't need help like that. Aren't math PHDs for people who are basicallly savant level anyway? People talk about burning out in their 20's.

i am fool, who literally failed calculus, because exam was excercises like "find asymptote" or "what kind of isosceles triangle written inside of circle is biggest?". I can solve any derivatives well, but i am bad with anything that connects "functions" and "analysis". Please give me some hints, i want to stop being fool. I actually want to learn how to think

>>9117 >I actually want to learn how to think https://en.wikipedia.org/wiki/How_to_Solve_It

>>9101 if you did well in math classes in college, and you get a good GRE subject score, you can get a math PhD. No MS required in the US, and work experience is a very big plus(they're betting on your work ethic, not necessary your prereq knowledge). I assume you've taken at least an intro proofs class since you said you can't take any more normal uni classes and you're interested in pure math. It's really not that uncommon for people to start a math PhD program without a math BS and in the late 20s. You need to self-teach yourself a lot of the stuff for the GRE(you should be an expert on calculus and have decent knowledge in algebra, analysis, and topology) but it's perfectly doable. >aren't math PhD's for people who are basically savant level anyway? I have a hard time believing I'm a savant, I just really like math.

>>2120 Thanks. Engineering graduate here. Those videos are perfect for my background.

Can you recommend some good "Math pocket reference" for basics across multiple Math/Physics (up to university level) topics? A "I always carry this with me" style of book in case you have to look something up?

>>4829 >>4831 >>4832 > difference between fools and smart people Not really, most of the time "smart" people just built a better foundation early, and have to practice their ass off too, even math professors get thrown off balance if they encounter something new, it's just less likely that they encounter something they haven't seen before, depending on their experience. You can often observe this when you see straight A+ flawless math students enter university and struggling. Even the people that are considered genius when it comes to science didn't just poop out their work while sleeping, there is a lot of damn hard work involved for them too. People learn and practice differently, and practical does not mean worse. > but man is that term overused on the chans it really is

>>2019 >>2022 Here's the PDF for Basic Mathematics

>>1274 While it is indeed complete and consistent in the strict sense of those words, It seems one could never approach all mathematical concepts with it. I'm not trying to be pedantic, but a statement such as >Presburger arithmetic cannot formalize concepts such as divisibility or prime number. ...seems to imply that one could not describe "important" stuff such as encryption, for instance.

>>1274 The problem with self-proving theories is that you can't do anything really interesting with them, and just because a theory an assert its own truth and remain consistent isn't a very convincing argument. "This statement is true" could still be true or false, even if it's consistent with everything else. This is more epistemology than math, though. We're mostly fine with having unprovable axioms, because we use so few in comparison with other forms of science and the ones we do use are so intuitively obvious that it's almost impossible to imagine otherwise. Also, we have a clear idea of *what* we assume and *why* we're assuming it. That's why mathematical proof is considered so much stronger than anything else, not because we know the axioms to be true.

>>1268 >>1274 Presburger Arithmetic can NOT prove it's own consistency and completeness. Here is some material :https://mathoverflow.net/questions/9864/presburger-arithmetic/10027#10027 >>11220 > We're mostly fine with having unprovable axioms, because we use so few in comparison with other forms of science and the ones we do use are so intuitively obvious that it's almost impossible to imagine otherwise. Also, we have a clear idea of *what* we assume and *why* we're assuming it. It seems that you miss the points of axioms, and the essence of mathematics entirely. Mathematics is not a science, in that it is not meant to explore what is the real physical world. A mathematical theory takes its root in a set of axioms, from which you can extract a minimal set of axioms which are unprovable by nature ( any axiom outside a minimal set is just redundancy). And finally, few people, even among mathematicians, know exactly what axioms their works are based on and why are they assuming it. All the fuss about the choice axiom is just an example of this mess.

>>11292 >Presburger Arithmetic can NOT prove it's own consistency and completeness. There are definitely theories that can. I read about them but forgot who had done that work, thanks for the link as it reminded me of the name Dan Willard. >Mathematics is not a science, in that it is not meant to explore what is the real physical world. debatable >A mathematical theory takes its root in a set of axioms, from which you can extract a minimal set of axioms which are unprovable by nature ( any axiom outside a minimal set is just redundancy). They are unprovable from each other. Axioms aren't even standardized, there are many different formalizations of many different axioms in most fields. Often by proving an iff correspondence with something that's easier to use in proofs. >And finally, few people, even among mathematicians, know exactly what axioms their works are based on and why are they assuming it. I'd say that most mathematicians know the axioms about whatever they're studying. Most of the time(outside of an introductory textbook), you don't build your theories up from ZFC or anything, you build them for whatever objects you want to study and then formalize later. This sometimes leads to issues(for example, when some people decided choice is not reasonable). Formalism is a nice stance to take early on when you want to ground yourself in "the real world", but once you get to more abstract things it becomes clear that the statements represent something real and immutable. Certain patterns just come up, and even if they're just results of string manipulation they're still interesting properties of that manipulation in their own right.

>>1187 Have any of you seen any textbooks for linear dynamical systems? I've been reading "Optimization Models", but haven't come across a textbook that's specifically about DT/CT Linear Dynamic Systems.

>>11302 From what I can tell, Calafiore and El Ghaoui (2014) "Optimization Models", if that's the book you're talking about, doesn't have anything to do with dynamical systems, so could you please clarify what you're looking for? What do you mean by "DT/CT Linear Dynamic Systems"? If you're looking for an introduction to dynamical systems, Hirsch and Smale is a classic. They spend a good amount of time on linear operators, which I guess might be what you mean when you emphasise linear dynamical systems? Either way, it's good to have that stuff under your belt before you go into non-linear land. https://www.elsevier.com/books/differential-equations-dynamical-systems-and-linear-algebra/hirsch/978-0-12-349550-1

>>1187 Computability theory seems so esoteric. Is there any real reward to understanding it? t. senior undergrad in discrete II

>>11354 you'll know what it means for a function to be "undecideable". You'll know how tightly bound logic and computation are. You'll understand the fundamental limitations of mathematics and knowledge.

>>11334 Yeah, my bad! I should have been more specific. The class I'm taking is all about solving systems of the form x[t+1] = Ax[t] + Bu[t] for the Discrete Time (DT) case and x'(t) = Ax(t) + Bu(t) in the Continuous Time case. The Optimization Models book, while it has a very good intro to some of the concepts such as least-norm solutions, projections, and least squares, it doesn't devote much time to the actual study of linear dynamical systems (they even mention they're not devoting much time to it... I think it's chapter 15 or so...) I guess a book that more accurately describes what I'm looking for is something in the vein of "Introduction to Dynamic Systems" by Luenberger (here: http://home.deib.polimi.it/guariso/BAC/Texts/Luenberger.pdf) I'll take a look at the book you linked though! Thank you!

>>1952 For an absolute starter, check out Aleksadar Zecevic's books, they're based off of a class he teaches, and have extensive "further reading" sections on every subject he covers

>>11534 Not him. But any tips on where to find "Truth, Beauty, and the Limits of Knowledge"? it's not on libgen nor irc.

Well, now I'm graduating college. Made this thread about one and a half years ago. I'm not as excited about math as I used to be. I feel academia has robbed me of living, in a twisted way - but I have no one to blame but myself.

>>13413 What made you stop liking maths though? Just not enough occasions for applying it?

>>13413 The school system where I am has made incredibly difficult to like any of the work I do. Not enough time for assignments that all feel like gotcha. Everything feels last minute despite me trying. I have not lost interest but its like pulling teeth most of the time.

>>13426 I think I'm too small brain to appreciate its beauty where I'm currently at. >>13433 It's always gotcha problems. Always. Seems it's hard to make anything more out of schooling.

Future CS student here. I've just finished the book "Book of Proofs" and I've got a few questions regarding math, or simply how to think. There are several questions from various exercises in that book that I find too difficult and am completely stuck on without referring to the answers. While I could argue that once I've obtained the solution, I'll know how to solve questions of the same type in the future, I feel that I'll never be capable of developing my own solution without the aids of others. In other words, I feel stupid. Any help with that?

>>13790 There's a reason why even basic theorems took years or centuries to be established and/or proven. Otherwise, calculate the percentage of solved exercises and provide examples of unsolved ones.

>>13791 Take for instance question 33 from chapter 10: >Suppose n (infinitely long) straight lines lie on a plane in such a way that no two of the lines are parallel, and no three of the lines intersect at a single point. Show that this arrangement divides the plane into (n^2 + n + 2) / 2 regions. How do I even begin to come up with a solution to this? I know I'm supposed to do it with induction but how should I event construct my inductive steps?

>>13795 >how should I event construct my inductive steps Well, the basis is always retelling of the question, so you have cases for two and three lines. Have you tried drawing them? I believe it is easier to get the idea behind the solution in general geometrical form, but arriving to the inductive one is quite fun once you get it.

>>13795 >I know I'm supposed to do it with induction Different Lain. You do not have to use induction for this. It is less elegant since it requires the formula to be handed to you to begin with. You can proceed by direct computation: - Let the number of lines be n. - Disregard for a moment the half-open infinite segments that connect to a single intersection point, but notice that there are 2n of them. - You are left with a connected planar graph. - The condition on the lines allows you to express the number of vertices as a function of n. See if combinations might help. - The same condition on the lines and the placement of the vertices allow you to write the number of edges as a function of n. - Recall the formula linking the number of vertices, edges and faces. Compute the number of faces as a function of n. - Recall the 2n half-open infinite segments that we put aside earlier. Notice that they slice the exterior face circularly into as many slices as their count. - Adjust the number of faces computed above by subtracting 1 and adding 2n, to account for replacing the exterior face with its 2n slices. - Take a moment to savor the beauty of math when all the pieces fall into place.

>tfw like mathematics but deeply worried that if I continue to pursue an education in it, that I'll hate it Going to school always just seems like a painful obligation Do mathematics on my own time seems so much more fun. but you can't get work without getting ~degree~

>>13804 well, doing basic math on your own is really fun, but if you want to study really abstract things it is *really hard* to learn on your own. The learning curve for most math is forgiving at first, then becomes impossibly steep after a little bit. I study for easily 6 hours a day every day and I'm just barely keeping up. It's worth it though!

Should I stick with Books or Classes such as Khan Academy or MiT OC when studying math?

>>13853 also look at lecture notes. Many university professors publish free lecture notes online and some of them are very good.

>>13854 Lecuture notes seem great. I remember when I was studying some Algorhitms class from MIT OCW, and I downloaded the notes and it was amazing. I wish I could into taking notes, mine is like kicking dead whales down the beach.

>>13790 So you thought math was about getting to results? Proving them is the hard part, and in proving them you understand how to navigate logic, which leads you to other results. But coming up with new proofs and proof schemes is the pinnacle of human ingenuity, no wonder you can't, you glitterboy

I don't know if this fits the thread's purpose but i'm currently strugling with some of David Kincaid's Numerical Analysis excercises, especially the part about floating point arithmetic, i would appreciate a little guidance.

does lainchan has mathjax support?

>>1214 I hope you've read cofriend (soon to be published in a much extended and refined version by CUP, yay!)

Reviving this old thread to share a blog a found a while ago it is good imo https://jeremykun.com/ and since lainons here are sharing eduacational-like resources then this one too -> https://awalterschulze.github.io/blog/post/neglecting-math-at-university/

>>16675 >>16675 Great find on the jeremykun thanks, some quite high level math in there as well. To contribute here's a real analysis cheatsheet I made. I have others too if people like it

I'll simply leave it here.

>>16802 Ted's a man before his time. Shame the route he had to go.

>>16802 he forgot another reason to do mathematics: as a method for gaining deeper understanding of the world. In my view, that's the most important thing a human can do, once their material and emotional needs are taken care of. I think Ted was just (rightly) disgusted with the modern mathematical community, and identified it with mathematics itself.

>>16802 oddly, mathematics is one of the hardest things to automate. The holy grail of AI research has always been a machine that makes mathematicians obsolete, and it turns out they're making everyone else obsolete.

>>16804 this

>>16805 yeah, i don't know where he got that from

Hello, has anyone seen any 'popular' linear logic materials?

>>1228 >I can't be creative. I wish I could, but I can't. I learned early on that I wasn't capable of it. Captive elephants learn at a young age that they cannot escape the rope tied to a small wooden post. They never unlearn that, however manifolds more force than necessary to pull it right out of the ground they gain as they grow larger. Learning things about yourself is good, carving that into a rock of things you believe to be a constant of yourself is harmful to the person you could be.

>>1191 enlightened

Assume I have a function (an automata really), whose output is defined by the angle theta: θ = 2arctan(e^u) - π/2 Where, for e^u = cot(π/8) = 1 + √2 sinθ = cosθ = 1/√2 This is the stable function. Instead, for e^1 = e, sine and cosine generate the critical point sinθ = ln(1+√2)/2 and the global maxima, cosθ = ?? = cos(2arctan(e) - π/2) respectively. Can I simply derive the global maxima from the critical point, as its derivative? What does the proper representation of the maximum?

>>18287 >Can I simply derive the global maxima from the critical point, as its derivative? What does the proper representation of the maximum? Nevermind I've found it.

Is this infographic any good?(Feel free to resubmit the infographic without the unnecessary character edited in.)

>>18837 i haven't actually read anything on there, so i won't bullshit you. it seems fine, but it definitely is putting a lot of emphasis on logic, set theory, and proof writing. that could be a good thing if the fundamentals really interest you, but you don't necessarily need all that to progress. once you can read and write proofs you're fine. it's also bee-lining calculus and totally neglecting algebra. going straight to analysis after all the set theory stuff makes sense if that's what you want to do, but for most people i'd suggest a course in discrete math first (note that "discrete math" is what a lot of colleges/textbooks call it but it is somewhat ill-defined. sort of like "precalculus" in that it's just kind of a grab bag of concepts that you're going to need later; an education curriculum not an actual field of study). for what it's worth the typical american undergrad curriculum looks like this: >precalc (usually done in high school) >differential and integral calculus (usually done in high school, but almost always repeated freshman year) >sequences and series (often called "calc 3") then you usually take a course called something like "intro to mathematical reasoning" which teaches you basic logic, set theory, and how to write proofs. after that the most important courses are real analysis and algebra. the remaining courses can be fairly neatly divided into two categories: the ones where algebra/analysis aren't a prereq: >"discrete mathematics", often including set theory, logic, more on sequences >multivariable calc >linear algebra >differential equations >graph theory >combinatorics >basically any course with "applied" or "numerical" in the title and the ones where they are a prereq: >topology >dynamical systems >complex analysis >basically any course without "introduction to..." in the title this isn't the best way to learn math but it helps to know what other people are doing.

>>18837 spends way too much time on set theory and proofs and introduces calculus far too late, and also recommends you read analysis before calculus with sounds boneheaded to me. You only need one "intro to proofs" book, and it should probably come after you are very familiar with basic math. I would recommend Basic Mathematics, then How to Solve It, then How to Prove It, then Spivak's or Apostol's Calculus. Both are top-teir calculus books. Then read Strang's Introduction to Linear Algebra. Linear algebra is one of the most important things that every student should know, you need to read that book cover to cover and do all the exercises. From there it really depends on what you want to do with math... if you're interested in CS, pick up one of the billions of logic books, then move on to Sipser's Introduction to the Theory of Computation. Sipser's is one of the best textbooks I have ever read. Then maybe Concrete Mathematics, but that's really more like a reference. If you want to do more 'pure' math, you will want to learn vector calc and differential equations. I just grinded through those courses in university so I don't really know what a good textbook for either is. I have heard V. I. Arnold's Ordinary Differential Equations is good. Then you need to read a topology book(Armstrong's Basic Topology is excellent), an abstract algebra book(Herstein's Abstract Algebra is decent), and an analysis book(also a course I just grinded through -- no recommendation. I hear Baby Rudin is a gauntlet). From there you will be in a good position to start looking for books on topics that interest you. If you feel ready to tackle really hard stuff, Dummit & Foote is unbeatable for abstract algebra. You could easily spend a whole year working on those problems though.

>>18840 >analysis before calculus with sounds boneheaded to me it's unconventional but it's not that unreasonable. i wouldn't bother getting too invested in learning calculus until after analysis because it's not like anything in calculus really makes sense until you know what "convergence" means. that being said, everyone learns introductory calc before analysis so most analysis books will be written with that assumption in mind, so i think it's at least worth getting some basic familiarity with the basic concepts before going straight to analysis.

>>18840 also, >differential equations i'm of the opinion that differential equations should really be taught after a serious linear algebra course (i.e. one that starts off with vector spaces, not the abridged version they teach engineers). >abstract algebra just going to throw out a recommendation for fraleigh's book. imo it's one of the best organized math textbooks out there, and it hits the right balance of touching all of the important sub-fields while going just deep enough to capture the core concepts without overreaching for an introductory course. i still reference it in grad school sometimes. >baby rudin the first few chapters are great but i personally prefer learning analysis from a more topological perspective. rudin really hammers in the details of real analysis in particular but the more general facts about continuity, compactness, completeness, etc. can get lost in the weeds. it's good if you specifically want a deep understanding of real analysis but these days real analysis is mostly taught for the same reason that calculus is taught before analysis: so that you have something concrete to hold on to when you get into the more abstract stuff, and i feel like baby rudin isn't the best choice for this purpose. also open neighborhood proofs are so much more elegant than epsilon-delta proofs

I don't think so. First off, it's fuarrrking redundant, why would you read 3 books on elementary mathematics? And 2 books on proofs? It also makes little sense, imo, to read about proofs before seeing them in use. Reading a whole book on logic and one on axiomatic set theory is overkill. Books about mathematics are not read like a novel, takes time and effort. If you start reading a bunch of what amounts to legalese you'll get annoyed long before you get to the main thread. Now my second and more important argument is that each person has their own goals in studying mathematics, and a different way to learn. So first off tell use, what do you want to learn maths for? You can aim for logic, ofc, but also geometry (say for computer graphics), analysis, number theory, etc. And some books are better for some people, at least in the initial stages. My personal suggestion for a first book would be Knuth's Concrete Mathematics. It'll give you a taste for how it's done though it's not as rigorous. Spivak's calculus is also a good one, but be sure you remember your basic algebra. I am also a beginner, and I've looked into many books, am personally reading Baby Rudin among others, but that's just me and doesn't mean I recommend it for anyone starting out. It just happens to fit my own personality and goals for learning.

>>18844 Mama mia, forgot to quote >>18837

>>18842 >i wouldn't bother getting too invested in learning calculus until after analysis because it's not like anything in calculus really makes sense until you know what "convergence" means. The reason why Apostol's and Spivak's books are so good(specifically Spivak) is because they introduce that kind of thing. Spivak starts with constructing the real numbers and a rigorous definition of convergence.

What is special about cos^2(x) = x x = 0.641714 I found it as the root of my automaton.

Is mathematics a good major? I'm torn between mathematics and computer science. Can mathematicians, computer scientists and software engineers can give advice?

>>18855 Nothing. Just that x = 0.641714 is a fixed point for the function f(x) = cos^2(x). Just like 1 is a fixed point for the function f(x) = x^2 or x = 0.739085 is a fixed point for the function f(x) = cos(x). Just look up fixed point(it's extremely important for solving nonlinear functions).

>>18857 Thanks.

>>18856 math major who doesn't know how to code: you better be really fuarrrking good at math math major who knows how to code: hard to convince people/robots that you're qualified for anything but once you do life is a breeze specialist "applied mathematician": you will be showered with money if you do neural net soykaf computer science or software engineering(some places there is no practical difference): your classmates will be insufferable but job prospects are ok

>>18856 >>18859 oh btw this is just the job prospects. I find math very satisfying. But if you choose math, take at least a few programming classes or teach yourself to code

>>18856 Did a maths degree, regret not doing a joint honours maths and compsci degree, hated my final years fuarrrked my degree as a consequence. Don't worry about jobs if you're doing a science like this then you'll be fine, you just need something you enjoy to help you flourish. Maths is king if you have time and motivation though, you can do anything once you know all the basics, but it is hard. Theoretically I could turn to pretty much any other field from philosophy to chemistry with the knowledge I've learned from a maths degree, you couldn't really do the same as a computer scientist though. I'd recommend doing a joint degree personally. Learn python today (right now) if you don't know how to code though.

I participated of math olympiads three times as a kid but when I asked my teachers what to study they always said "relax, what we have here is enough". Well, it turns out it wasn't, and that massively demotivated me. After some time of collecting resources and trying to get my soykaf together overall, I'm with Velleman's How to Prove It. I plan on doing Stewart next and later Spivak(which I already had some contact with but couldn't go through with, it was then that I saw Velleman's book being recommended). I'm late to the game and this makes me very anxious about learning fast, but I know these things take time. I can't dedicate full-time to it either, which is the main reason why I'd have to wait a while to do Spivak even if I had already finished Velleman and Stewart, but I guess learning to work with what you have is a skill.

>>18837 >(Feel free to resubmit the infographic without the unnecessary character edited in.) That's just /sci/ culture.

>>18868 >Dumit and Foote abstract algebra This is cool if you want to torture yourself, The book's a reference, not really a textbook. If it's for an introductory course I would suggest instead Stillwell's Elements of Algebra. Also lambdamod, wtf? What the fuarrrk difference does it make to have a cartoon character in there? That's fuarrrking stupid. Especially in chan culture. Suck the fun out of all things. Are you also going to ban the Touhou character holding SICP? Perhaps you shouldn't have chosen LAINchan for your dry academic paradise.(Complain in /q/.)

>>18868 >going straight from calculus to Dummit & Foote this infographic should be called "how to burn out quick". D&F is a very difficult book, it's mostly used as a reference in 1-year graduate level algebra courses. Not suitable for babby's first algebra. I love it, one of my favorite textbooks, but if I tried reading it right after I finished calculus I probably would have given up on it and switched to a different book. >>18869 lmao. pic related

>>18840 I've seen How to Solve It being criticized several times, but since I don't have enough experience to know myself I remain in doubt whether those were imageboard edgelords or if they had a point.

>>18855 >>18857 >>> import math >>> def banach (f, seed, diff): ... absol = abs ... fx = f (seed) ... count = 1 ... while True: ... next = f (fx) ... count += 1 ... if absol (next - fx) < diff: ... return (count, next) ... fx = next ... >>> f = lambda x: math.cos (x) ** 2 >>> banach (f, 0, 10**-3) (144, 0.6412273438135871) >>> banach (f, 0, 10**-6) (309, 0.6417148573805437) >>> banach (f, 0, 10**-9) (474, 0.6417143703868572) >>> banach (f, 0, 10**-12) (639, 0.6417143708733682) >>>

>all these books >so little time ;_;

>>18880 It goes by faster than you think since a lot of books the first 1/4 is review Just focus on a single book, when you're done do another. In a year you'll be surprised how much you finished Personally I wage slave Terrence Tao's Analysis I because it starts at absolute scratch from Chapter 2 on.

>>18892 >Terence Tao What do I need to master this? https://arxiv.org/abs/0708.2895

So, if I'm getting this right, the broadly-general flow of self-teaching math is roughly: 1.) Set theory, logic, and proofs 2.) Relearning basic HS mathematics in proper context 3.) Calculus 4.) Delving into other fields of interest I'll work through the Book of Proof (Hammock), and Basic Mathematics (Lang), and pick up a calculus book after I've finished those two. Should I read Hammock or Lang first though? My goal is to have a stable foundation to build off of so I can learn the necessary math for CS, computer engineering, and just pure math in general. Especially when it comes to stuff involving number/chaos theory. I have a huge appreciation and fascination with it, but I just can't hack it mathematically (makes sense, I never had any foundation to work off of to begin with) and am really eager to get into the nuts and bolts of why this stuff works, and the proofs for it. Does this look about right though? These are just my thoughts on a first day delve into figuring out what I should learn.

>>18921 >Should I read Hammock or Lang first though? Read them together, that may work better. On a side note, I have never read books on how to prove theorems, I learned proof by reading Analysis(Rudin's book) and Algebra(Herstein's book) and doing exercises. But I can understand that a book like Hammock's one can help.

>>18894 Probably just random matrix theory. So you learn real analysis properly, general topology, measure theory, probability theory, and then pick up Tao's book on random matrices. You can always just email Tao and ask, tho. Although I'd recommend learning all the above before doing that. >will he really answer He looks like the type who would. As long as you don't send him a five page long essay or something.

I have a question about Calculus amd Analysis. How different are they as subjects? As I understand it, calculus is just computational methods of analysis, finding derivatives and integrals, among other things. Is that accurate? Can I learn that directly from analysis? I am more in for theory, and calculus books are super tedious for they usually assume no technical maturity and are long winded. I am starting Baby Rudin, and while it's dry, it goes to the point and is challenging from page 1. And rigorous. Not so with calculus textbooks where inevitably they explain a lot of things that a more 'mature' book merely states formally. Is there an analysis book that touches computation, or a calculus book that doesn't wastemthe reader's time explaining sets, functions, and limits? Rudin is pretty concise in his treatment of neighbourhoods, I don't mean to desestimste the required concepts, but to have them laid out simply so I can move on at my pace.... Thanks alice.

>>18936 >Can I learn that directly from analysis? Yes, you can. However, you are going to learn the theorem that allows you to do what you do in Calculus. As an example, the derivative is defined as: f'(x) = [f(x+h) - f(x)]/h as h -> 0 guaranteed that such limit exist. Now, while that works perfectly, if I were ask you to take the derivative of: f(x) = cos(( sin(x)²) I'm pretty sure you'd have a hard time doing it by the definition and even using the chain rule theorem requires you to still apply the definition in cos(x), sin(x) and x². My point is: you can learn Calculus using an Analysis book, however, doing Calculus textbook teaches you how to solve those exercises fast. You are also not going to learn optimization in an Analysis book, though. You'd in a Calculus book, but it is better if you use a numerical analysis one, since it's mostly aimed to what you can do with a computer so you can also do some cool programs on the side. If you want a computation book, take any numerical analysis, however, most of them assumes you can do your Calculus stuff(multivariable too).

>>18936 Oh yeah, I forgot to say. If you want a good Calculus book(single variable) that teaches rigor, you can try Spivak's book. It's more like an intro to analysis than anything, but still has computations.

>>18936 I just realized I might seem too arrogant with this post. No. I don't mean that. It's just that I have no attention span to read again all the basic stuff all over again. Even spivak elaborates a lot and throws a wall of excercises before even starting. I started reading Apostol and it started on a high note, in the introduction itself. But then it went all the way back down to sets, and functions.... That's my problem, my attention span goes away if I'm just reading basic review, and calculus books seem to be all aimed at high school students (understandably, but not what I need)

>>18940 >But then it went all the way back down to sets, and functions You could just skip that. A lot of math textbooks(even at grad level) has a short first chapter to recall some set theory(and other stuff from under or even grad level), it's just there because there is a chance that you have not deal with those for a long time, otherwise, you can just skip it.

>>18936 Calculus could be seen as applied analysis, sure. The majority of any calculus book will involve proving a few simple facts about derivatives and integrals, and the Fundamental Theorem of Calculus, Taylor series, and then hundreds of exercises. You can skip to the important parts if you want, there's no restriction that you have to read a textbook in order(in fact very few of them are designed for that). The most terse, rigorous, and 'mature' introduction to calculus that I can think of is Spivak's Calculus on Manifolds. It develops calculus in the fullest possible generality. You can't expect to learn calculus(or any field of math, for that matter) without hours and hours of practice doing repetitive exercises. That is the main advantage of those cumbersome books, is the exercises.

>>18936 calculus is a sub-field of analysis. broadly speaking, calculus is the study of differentiable and integrable functions, and analysis is the study of limits.

>>18948 on second thought, sub field is maybe not the best way of putting it. analysis is to calculus what topology is to analysis.

>>18936 What you could do, lainon, is to learn Analysis by any Analysis book you want and then downoad Demidovich's Problems in mathematical analysis. It has a lot of applied exercises. So you avoid reading soykaf like a Calculus book, learn what is important(theorems and so on in Analysis) and then go do as much as you want of applied exercises. By the end of it, you know even more than someone that read a calculus book.

>>18938 >>18939 >>18942 >>18943 >>18957 Well thank you for the advice. I realize it may be a case of Dunning-Krueger on my part, or plain laziness. But in reading higher math books there seems to be little use for actually taking derivatives or integrals. Wherever I see the symbol it's often the end result in itself.

>>18965 often you have to know when an integral exists or something. It is also definitely true that if you investigate some particular mathematical object, you will have to do an integral from time to time, even if you are very far away from applications

>>18965 >But in reading higher math books there seems to be little use for actually taking derivatives or integrals Look up Symplectic/Compact Geometry. Knowing if the manifold you are studying is integrable or not is extremely important, more so if taking the derivative(after all, we are dealing with differential forms).

>>18967 Contact, not compact.

how to solve logic/mathematical logic properly? It seems to me that all solutions are random algebraic manipulations and I can never follow the same line of reasoning

>>19002 if you mean "start from a string in symbolic logic and deduce whether it is true or false" it is really just random algebraic manipulations. The only way to get good is to practice.

I've just bought this. What should I expect?