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A Programmer's Introduction to Mathematics (jeremykun.com)
937 points by chocolateboy 13 days ago | hide | past | web | favorite | 214 comments





Do the exercises have solutions? The most annoying things about math books is the lack of solutions. A beginner absolutely needs to know whether or not their solutions are correct.

The “the reader should know if they are correct” logic doesn’t apply here. A beginner could easily have faulty logic and fool themselves into thinking their solutions are correct.

I usually don’t buy math books without solutions if I’m self-studying. Would like to know if solutions are provided in this book. If not, I won’t consider buying it.

If this book doesn’t make the cut with a solution manual, does anyone have recommendations on an intro to proofs book with one?


Looks like no:

"No solutions, sadly, but maybe that would be a good idea for a GitHub repo associated with the book…"

https://jeremykun.com/2018/12/01/a-programmers-introduction-...



Agree. Actually, studying a good solution even if you have one of your own is one of the best ways to learn mathematical tricks of the trade, so to speak - just as it's a great way to learn coding: one learns from the master (as one should) and not just "from the book."

I also champion this method; I used solutions manuals as guides until I could internalize and reproduce the logic on my own. I was a particularly dense student though.

being told a solution can sometimes lead to "rote" learning - where you learn a particular way of solving the problem, rather than applying creative thinking.

Also, if you can't prove a solution correct, then you haven't solved it!


>Also, if you can’t prove a solution correct, then you haven’t solved it!

This is the type of thinking I’m talking about. A beginner can think they “proved a solution correct” but have a subtle or even blantant error that isn’t obvious to them.

Also, if reader wants to just skip to the solutions then that’s their fault. But don’t let such people rob the student that put serious effort into their work from seeing the solution.


Is this a feature or a bug? If it takes you two weeks to apply creative solutions and it take me one week to apply a “rote” solution, what is the benefit?

The idea is that it might take you longer to learn, but when you are applying it in the real world and hit real world problems that are messy, you’ll be a lot faster

That’s the idea. In practice does it actually work out like that?

The main reason to study mathematics is to build and fix your intuition, hence studying rote solutions is a waste of time. It might help you pass the class but it wont help you much at all in other parts of life.

For example, at work nobody will care if you have rote memorized a solution or not since they will have already done the math, you will just apply formulas others have came up with. In order to do anything important with the math (not just solving school problems) you need to have intuition for it.


I’d argue that a true beginner shouldn’t trust their intuition unless they have the rigor to prove it is correct. True beginners may think they have solid intuition and a rigorous proof, but without outside validation, may have a blantant or subtle error they cannot see. The point is not to study rote solutions, but to check correctness.

Edit: To your reply below, I’m talking about someone that is self-studying and has no access to a teacher. The suggestion the beginner (without a teacher) should “know” whether or not their solutions are correct is something I disagree with.

I do agree with there are multiple ways to prove something and such a beginner may think their proof is incorrect based on a provided solution, when it may be correct, just different. This is why an instructor is valuable.


> I’d argue that a true beginner shouldn’t trust their intuition unless they have the rigor to prove it is correct.

Yes, which is why you want a teacher until you reach that stage.

> The point is not to study rote solutions, but to check correctness.

Looking at others solutions can in no way prove that your solution is incorrect so I am not sure how those would help you. However having access to others solutions often makes students doubt their own solutions if they don't look very similar, this hampers their growth since they learn to distrust intuitions which are actually correct.


It might not prove that your solution is wrong, but it can help show that your solution is correct (if you trust the provided solution), and if you made a silly mistake in your solution then seeing someone else's solution might well help you spot it.

Yes, which is why you want a teacher until you reach that stage.

Not everyone has that luxury.


I absolutely agree. From my experience this is also true for the best professional mathematicians, it seems that generally they have picked a very large toolbox of memorized "tricks" that they can effectively (and intuitively) apply, rather than approaching every problem as if it were an entirely new challenge.

> The most annoying things about math books is the lack of solutions.

To me the most annoying thing about math books is hand-waving, lack of rigor, and unexplained notation.

At least in programming, everything is formal and I can figure out the entire problem by looking at the source.


I think you aren't buying very good math books then. I find the exact opposite: the thing about math books I have read is that they overemphasize rigor at the expense of intuition. Everything is painstakingly illustrated in such great detail that I sometimes see the trees and lose sight of the forest. I feel as if reading proofs and doing problem sets in math books is just manipulating symbols in well-known ways without really understanding intuitively why something must be true. For example my introduction to metric spaces started by defining the characteristics of a certain function d without explaining how this could be thought of as a generalization of distance.

On the other hand, many programming stuff is ruefully hand-waving and lacks rigor. They might present important algorithms in pseudocode; even when they present in real code, the precise semantics of the real code is often underspecified and vaguely described in English. I mean take a language; how often do you see in the language specification the semantics of the language defined rigorously, using operational or denotational semantics? PL nitpicking aside, how many programmers think a piece of code must be correct because they pass a few test cases, without ever giving a proof?

I'm of course not saying the lack of rigor in programming is bad. Perhaps 95% of the software we are building isn't mission-critical and relying on intuitions is fine; we ain't got no time to prove every piece of code we write. But my point is your observation really does not match mine.


> I feel as if reading proofs and doing problem sets in math books is just manipulating symbols in well-known ways without really understanding intuitively why something must be true.

I have a few thoughts about this. The first is that it is largely up to the learner. Sure, you can just symbolically manipulate things with no real understanding, but no one says you have to stop there (and arguably that's what you should be working on when you read the proof).

The second is that a lot of times the intuition comes later. There was a joke in my school (and I presume others) that you learn algebra in calculus, calculus in DEs, etc. It takes a while to develop that intuition and sometimes just doing it frequently is part of what it takes. I think of it like learning music theory but not really getting it until you've gotten proficient at an instrument.

The third is that some math is just like that. There might be someone somewhere who has some type of intuitive understanding of it, but they are an outlier.

That said, the best learning I've done in mathematics is to take a book without answers in the back and work through it with other people. It's difficult, but very rewarding to really grok the topic and be confident in your answer; and having people you have to explain your solution to helps tremendously.


But at least all imperative procedural steps are clearly understandable from source code given in many programming books. Where as maths books routinely leave out many steps in proofs and calculations, on top of many ambiguously used notations and terminology that can leave a self-student confused.

A program proves nothing at all so I am not sure what you can understand from it? Typically in programming you are presented with a piece of code, a statement that this piece of code solves a specific problem and then a proof of that it actually works. Those proofs are typically far from understandable or rigorous.

> Typically in programming you are presented with a piece of code, a statement that this piece of code solves a specific problem and then a proof of that it actually works.

Not really. It's quite unusual to see a serious formal approach.

I'm working through a book that uses C++. Every few pages I find undefined behaviour (struct punning, illegal use of memset to zero-out objects, etc) and needless usage of non-standard compiler-specific language features.

This is the well it works on my machine mindset used widely in programming, even in books written by highly qualified people.

> Those proofs are typically far from understandable or rigorous.

I'm not sure what kind of thing you're thinking of here. Can you give an example?


But programs are proofs!

At least in the light of the Curry–Howard correspondence. :)

Anyways, I do agree that in programming it’s easier to see what are introductions, assumptions, definitions, functions, values etc. You can’t just invent a notation and go with it. Everything needs to be defined from the ground up. It’s constructive and I like that, probably because I’m a programmer.


That’s true but in most languages the things you prove are relatively obvious propositions like “(A and A)implies A.”

>I think you aren't buying very good math books then.

Just to follow up. A comment[0] on the book `Discrete Mathematics and Its Applications by Kenneth H. Rosen`

`I'm convinced that math gurus are incapable of teaching math.

Whenever I encounter math textbooks like this one, I'm reminded of Underpants Gnomes on South Park. For the unwashed: Underpants Gnomes stole underwear from the residents of South Park hoping that they'd profit from the thefts, but although their business plan included a Phase 1 ("Steal underwear") and a Phase 3 ("Make a profit"), Phase 2—the connective tissue—was just "?".

Rosen and others like him fail to grasp how much "?" connective tissue they're leaving out when they "teach" math. He'll describe a math concept using almost-but-not-quite human language, then—oh, there's always a "then" with these books—the math gymnastics begin. Math teachers can't resist the gymnastics, can they? No, they really can't. At some point they lose their ability to see how much they assume other people know, and when that happens they cease being effective at teaching. This describes almost every math teacher I've had, and it definitely describes almost every math textbook that I've read. Rosen's book isn't the worst math textbook that I've read, but it's still pretty awful.

Seriously, corner cases and extreme mind-bending problems don't help people learn math; giving students time to grasp concepts before baffling them with bullshit does. This is why math schools like Khan Academy are amazing and college/university level math courses are not. This stuff can be taught, but not like this.`

I strongly agree with this argument. I had this book in my undergrad and I felt barely learning anything and just kept getting drowned in the way the content was written. I switched to a different text in a few weeks and it became an easy subject. The difference was more intuitive examples and gradual difficulty in exercises and less of the proof is left for the reader as an exercise.

[0] https://www.goodreads.com/book/show/1800803.Discrete_Mathema...


May I ask what the "different text" was that you switched to? I have Rosen, and I'd like something more approachable and clear.

FWIW, I found Susanna Epp's book on Discrete Math to be more accessible than Rosen's. YMMV.

>the thing about math books I have read is that they overemphasize rigor at the expense of intuition

I think these are not contradictory notions! Bad maths books lack rigour. Many books are like you say: rigorous but difficult to make sense of. But truly great books both explain the concepts in the simplest, most lucid and succinct way possible AND maintain the rigour necessary to do mathematics.

I like this quote from Michael Spivak: "In addition to developing the students’ intuition [...], it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions."

I couldn't agree more. Rigour is not a "masturbatory" end in itself to feel very smart, but it is not also an obstacle to understanding.


Agree, additionally I've found that the key to thw very best books is explaining how the intuition and rigour link. I have found that many books (and teachers) exaplin both seperately, but fail to adequately connect the two.

Sentences like "It similarly follows that..." , "It is left as an exercise to the reader to show that..." , "Clearly, ..." always leave me dumbfounded.

You may enjoy this blog post, "I no longer understand my PhD dissertation".

A sample quote:

> This was not the casual read I had in mind. The notation was alien. I even had to scour the examiner’s report to direct me to the key results. And while I could have sworn this was a well-written thesis, I repeatedly found myself bamboozled by my own prompts. “The result now follows easily…” may have made sense back when, but now the author-turned-confused reader can profess that it most certainly does not follow easily, at least in his own mind.


Those phrases can definitely be abused. When used correctly I think they invite the reader to actively engage with the text and learn to think for themselves instead of passively have everything given for them. The author needs to make sure that they’ve actually given enough information up to that point that the reader, with maybe a few minutes of thought, can see why it’s “clear” or work out the similar case in their notes.

I'm reading Calculus, Multivariable from William L. Briggs (amazon link: shorturl.at/ltGVY) ; it has the answers at the back of the book ; I'm quiet happy about it for someone who always sucked in maths. If anybody has books to share, i'm especialy interested in linear algebra and signal processing

I like Gilbert Strang’s Introduction to Linear Algebra (there are videos too: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...). The book has answers for some, but not all, of the problems.

I think Strang does a good job balancing intuition and rigor, without too much mindless grinding. Other books I’ve used either leap right into “Let F be a field....” or have you learn a bunch of matrix manipulations but don’t really explain why these are interesting or useful.


How do programming books prove correctness of the algorithms taught? Hand waving. It is very frustrating to read for me, mathematical texts are a lot better at proving things.

In mathematics as well as in programming, there are rigorous and less rigorous texts. Compare “Linear Algebra for Dummies” with “Principia Mathematica”, for example. I haven’t read the first or more than a page or so of the second, but I think you’ll find the latter more rigorous.

Similarly, in programming, “Teach Yourself PHP in 24 Hours” is wildly different from “the Art of Computer Programming” (a series of books that you may like reading)


Advanced math topics are based on layers upon layers of abstraction (theorems, corollary etc.), like an enterprise software. I also dislike the hand-waviness, but I do understand that I have to read up on the basics in order to grasp the advanced concepts.

I would pay for the solutions as a book on its own. Solutions are essential for self learning, specially for tricky parts in mathematics i.e writing proofs, probability problems. I think the book “how to prove it” has good examples

I read once, the key to mastery is perceptual exposure and deliberate practice.

Don't know how the first part would work with Math.


Observing someones approaches at solutions and proofs would be perceptual exposure to mathematical analysis.

A good math book has examples and proofs of theorems in the text that demonstrate what a solution should look like. In a sense, these ARE worked solutions to some problems. If you ever get stuck on an exercise, it helps to go back and look at those. Nothing beats having an expert to look at your solutions and provide feedback, though.

There seems to be a github repo [0]. It mentions code solutions to examples.

  [0] https://github.com/pim-book/programmers-introduction-to-mathematics

I have a genuine question which might sound dumb but I really do wonder.

How do you actually read math, physics and programming books?

Reading them the same way as you'd read a novel doesn't seem right. I try to go chapter after chapter and make notes but I often get bored because I don't see the usage in my real life coding. Maybe I'm not working on problems that are challenging enough? Also after few chapters it often turns into a "job" of finishing the book. I don't have the pleasure of learning new stuff anymore.

Do you really finish such books? What am I doing wrong?


Here's Alain Connes, Fields medalist, on how a mathematician works and should read a book [0]:

"To understand any subject, above all, a mathematician SHOULD NOT pick up a book and read it.

It is the worst error!

No, a mathematician needs to look in a book, and to read it backwards. Then, he sees the statement of a theorem. And, well, he goes for a walk. And, above all, he does not look at the book.

He says, "How the hell could I prove this?"

He goes for his walk, he takes two hours ... He comes back and he has thought about how he would have proved it. He looks at the book. The proof is 10 pages long. 99% of the proof, pff, doesn't matter.

Tak!, here's the idea!

But this idea, on paper, it looks the same as everything else that is written. But there is a place, where this little thing is written, that will immediately translate in his brain through a complete change of mental image that will make the proof.

So, this is how we operate. Well, at least some of us. Math is not learned in a book, it cannot be read from a book. There is something active about it, tremendously active.

[...]

It's a personal, individual work."

[0] https://www.youtube.com/watch?v=9qlqVEUgdgo


This approach seems very similar to working backwards when trying figure out the behavior of a function that calls other functions or libraries and when trying to a debug an issue and following the stack trace to determine the root cause of the issue. Not a perfect analogy but I find it helpful to think about it that way.

The right way to read such a book really depends on your learning style and personal circumstances.

Personally, I tend to skim; I find most texts to be far too verbose. But then if I find a part that I'm interested in or want to explore at length, then I'd read it more in-depth.

I think classical books suffer from a lack of features like hyperlinks. This is, authors either have to say something or not say it; they can't just say something concisely with links for readers who want to read more (Note below). As a result, textbooks tend to be cluttered.

Skimming helps you to cut through the clutter to get what you need. But since you'll almost certainly miss points while skimming, it'll tend to involve bouncing back-and-forth rather than a straight read-through.

---

Note: Of course, footnotes like this, along with appendices and references, are partial substitutes for active content. They're far more limited (Meta-note below) and have a much higher overhead on both reader and writer, but they're still often better than cramming everything into the body of a text.

---

Meta-note: Because, seriously, who wants to trace recursively nested footnotes? For more discussion on this topic, please see "_References: Why they're inconvenient_", p.156.


I try and write a tutorial about a subject-chunk - the kind of tutorial that i think would have made sense to me if I'd read it before writing it. you have to suspend the obvious fact that by writing the tutorial you've learnt more and probably wouldn't have understood it at the beginning anyway but let that go for a while.

part of writing the tutorial is exercises - i like a stripped down flash card system. i try and do random-10 as regularly as possible. at the beginning, there's lots of jargon with scripted answers (ie, what is a blah?) but after a while you can move those to the back and you're left with some multi-step questions - over time (because i'm not trying to memorise the answer pattern) i find that i internalise a kind of narrative that uses the concepts/jargon to grope my way to the answer ie, hmmm looks like an aqueous equilibria question, probably need the equilibrium expression (jot that down) and HCl is a strong acid, so i can ignore those two terms and oh look... there's the answer if I just do this and that. Over time, it's the 'groping for an answer' that sticks as a habit of thought and seems to be quite robust. having said all that, i'm craphouse at maths but i'm hoping to get into one day.


"...study actively. Don't just read it: fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?...it is not a good idea to open a book on page 1 and read it, working all the problems in order, till you come to the last page. It's a bad idea. The material is arranged in the book so that its linear reading is logically defensible, to be sure, but we readers are human, all different from one another and from the author, and each of us is likely to find something difficult that is easy for someone else. My advice is to read till you come to a definition new to you, and then stop and try to think of examples and non-examples, or till you come to a theorem new to you, and then stop and try to understand it and prove it for yourself --- and, most important, when you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not."

(P.R. Halmos)


There's a classical book titled "How to Read a Book" by Mortimer Adler. It's about reading texts for information, arguments, etc., but not pleasure. The heart of it is to not read the book linearly from cover to cover. Ideally you'd do multiple passes, often non-linearly, at different speeds, skipping different parts, concentrating on different parts. You'd start with parts that are known to have most information, like introductions and conclusions, the table of contents. Then introducing and concluding paragraphs, then introducing and concluding sentences. All the meanwhile, you'll be getting a top-down view of the material and you'll be naturally honing in on the parts that are pertaining to you.

Reading mathematics has a whole series of additional challenges. In my limited experience with reading those: nurturing comfort in incomprehension is important; it's often better to skip a part than get stuck on it; look for multiple accounts of the same concept.


Holding my copy of this excellent book, I'm reminded of Mortimer & Van Doren's comment on 'How to Read Science and Mathematics'

249.2 "Until approximately the end of the nineteenth century, the major scientific books were written for a lay audience. Their authors--men like Galileo, and Newton, and Darwin--were not averse to being read by specialists in their fields; indeed, they wanted to reach such readers. But there was as yet no institutionalized specialization in those days, days which Albert Einstein called 'the happy childhood of science'. Intelligent and well-read persons were expected to read scientific books as well as history and philosophy; there were no hard and fast distinctions, no boundaries that could not be crossed. There was also none of the disregard for the general or lay reader that is manifest in contemporary scientific writing. Most modern scientists do not care what lay readers think, and so they do not even try to reach them.

Today, science tends to be written by experts for experts. A serious communication on a scientific subject assumes so much specialized knowledge on the part of the reader that it usually cannot be read at all by anyone not learned in the field. There are obvious advantages to this approach, not least that it serves to advance science more quickly. Experts talking to each other about their expertise can arrive very quickly at the frontiers of it--they can see the problems at once and begin to try to solve them. But the cost is equally obvious. You--the ordinary intelligent reader whom we are addressing in this book--are left quite out of the picture."

The authors apologize for not being able to offer specific advise to guide the reader of technical books, as those texts are on subjects of which they do not possess the necessary expertise. They suggest the reader fall back on 'scientific popularizations'.

I've found this to be a very useful strategy. Beginning with a readable book, well researched book. Then systematically read the book's bibliography, recursively.

The one problem I ran into, specifically in mathematics, was it seems some notations are not universal. Furthermore, some authors take no pains to define their symbols at any point in their texts. (I quite understand a paper would assume a specific audience, but for a published book I think it's just rude.)

For this I needed to rely on communities such as math.stackexchange.com.

Good luck. The journey is part of the reward.


I am going to quote Manuel Blum's advice for graduate students [1].

"Books are random access -- a great innovation over scrolls. "

[1] https://www.cs.cmu.edu/~mblum/research/pdf/grad.html


How do you actually read math, physics and programming books?

Reading them the same way as you'd read a novel doesn't seem right.

Your intuition is correct. When people read a novel, they often "forget" that they're reading words and picture the action. The optimal reading speed for that is 200 to 250 WPM. You can't read mathematical papers, where everything has precise definitions, at that speed and expect a high rate of comprehension if everything's new to you. It takes a lot of time.

Do you really finish such books?

Yes, if I'm interested in the subject matter. It takes a lot of time, though.


Unless you are working on a course of study under guidance, you probably won't go chapter by chapter. Indeed, you won't even read it linearly.

Once I finished my coursework, I tend to have a problem and go searching for material that may get me some traction on it. So a lot of keyword searches, a lot of reading prologues (prologues are where the author tends to give his conceptual overview of what the material in the book is for...I've learned a lot from reading random prologues in a math library).

Once I find something that looks interesting, I'm usually down to a few theorems or definitions that I think will give me traction. Then I start poking around the book or any other source I need/can find to figure out what I need to understand those definitions/theorems.

Usually at some point it becomes clear if they're a) irrelevant, or b) partially relevant. If (a), I drop the entire line right there and go back to searching. If (b), I start running counterexamples from the problem I'm working on against them, looking for where it breaks and seeing what I can salvage.

Prologues usually read like a novel, but occasionally stopping to look up words, so more like reading a novel in a foreign language. Once you're into the meat, your rate of progress is typically measured in hours per paragraph or days per page. But you probably only end up using ten or twenty pages out of a two hundred page book in this method.


Depends how they're written. Some are written to be read cover-to-cover, whereas others are references that you dip into as you need.

If you're getting bored because you're reading something that's not relevant to what you're doing, skim over it instead and make a mental note of what the content covers so you'll know what you have available to you should the topic come up later.


Relevant discussion :

Ask HN: How do you read programming books? (https://news.ycombinator.com/item?id=15733745)


I haven't read any math textbooks cover to cover or anything but I did try to speedrun my way through some concepts needed for ML and from what I can tell the most important thing to learn is the insight.

Doing exercises or taking notes is only necessary if you feel the need for it, otherwise try to understand what the thing is and try to find where it is applied. Even though you probably won't ever need it in the future if you do picking it up will be much easier and this way you get the quick endorphins from figuring stuff out without the drudgery of actual work.

just my 2c.


> What am I doing wrong?

I don't think you are doing anything wrong. Depending on how the book is written, some might be really boring and unchallenging, pick another one! If you are looking for brain-wrecking adventure and some good story telling, pick something that starts with number theory. If you are looking for understanding the bread and butter for most recent big discoveries and theory crafting, pick something that focus on Complex Analaysis, High dimensional calculus or partial differential equations. If you are planning to stick in the field of computer science for good, pick something that focus on abstract algebra _and_ probabilty theory.

> I often get bored because I don't see the usage in my real life coding.

Then you don't need mathmatics. Depending what you do in coding, there is really rare type of work that need mathematics. Sciences and Mathematics are so far from practical use they are there purely for the sake of knowledge. This isn't just you, it's the case for everyone. Unless some day you find the interest to battle that boredom, the probability of you finding some usage from mathematics might be very unlikely.


> Reading them the same way as you'd read a novel doesn't seem right.

I think most technical books end up being organised like a reference manual for some reason - not necessarily as dry - but in a way that appears to facilitate some some externally driven learning activity by augmenting it.

> Do you really finish such books? What am I doing wrong?

Not personally, i'm terrible at penetrating any such book until I have a problem at hand that requires it. However I have found a few books that are technical while being enjoyable and engaging at the same time, in much the same way as a novel: "The Feynman Lectures" (Physics), "Michael Abrash's Graphics Programming Black Book", "Thinking Forth" (Programming). These books are well known, I think their popularity is in no small part due to how accessible they are without diminishing the quality of their technical content - I would like to find more books that have this quality.


Prof. Manfred Lehn has some very good advice on his web-site (in German):

http://www.alt.mathematik.uni-mainz.de/Members/lehn/le/semin...

Here is the Google Translate transcript:

"""

How do you read mathematical texts?

If you are ready to give a seminar lecture in your studies, you have already studied one or two semesters and read one or the other book and know what is important: If you read mathematical texts, there are two modes in which You can proceed: From the bird's eye view: What are the rough lines? What is the subject of the present text? What are the central concepts and definitions, what are the central statements and sentences? What are the rough evidence structures? Why do you do it all? From a frog's perspective: how is it done in detail? How does a proof work? Why do you need the prerequisites in the sentence? What happens if you leave them out? You often have to switch between these modes. First, one has to get an overview of where one is actually going, otherwise one bites oneself in the first technical lemma and gets stuck. At the first reading one can skip all the evidence and focus on the statements of the sentences. At some point, however, comes the point where one no longer understands the sentences, because one has developed no feeling for the introduced concepts. Then it's time to take a closer look at the evidence as well. If you have understood more technical details, you should step back a bit and ask yourself again what the overall context is, etc. In an adapted form, this also applies to the way you approach individual sentences or examples. If you are confronted with a new sentence, you may ask questions of the following kind before, after, or even while studying your proof: What are simple examples of the sentence (such as special cases)? What are simple counterexamples where certain conditions are not met? Does the sentence, or the term used or the proof, refer to already known things? Is there a characteristic example of observing all the essential phenomena? Work in circles in the literature to your presentation (and his position in the seminar).

"""

(You have to swap the word "sentence" with (mathematical) "proposition" or "theorem" at some places for this to make more sense. In german those are the same word ("Satz").)


Start a software project that uses advanced math, and learn as you go in order to build the project. Learning on the job is the best way to go since you only learn the fundamentals of what’s necessary to accomplish the goal, you immediately see the utility of what you’re learning and you get a useful product at the end.

I learned substantial amounts of algebraic topology by building an open source topological data analysis library, for example.


I read programming books just like normal books, and take time to examine the example code. I am currently reading a book about Swift design patterns. I dont try to learn how to implement it but instead on how to use a certain design pattern what it does etc. By the time I need it I look it up and implement it for my problem.

I'd add to that, how do you read programming language library documentation? Often I'll be reading along in the documentation for a function in say a python library and not really understand what the documentation is trying to say.

I was just as lost with the K & R C book.


For each section, start from the exercises, then work backwards towards the material necessary to understand and solve the exercises.

If there are no exercises, make some up (do the proofs yourself, try to come up with unsolved scenarios and solve them, etc)

For me it helped immensely when I had a fun side project that demanded new knowledge. For example, my chat bot side project was a fun way to learn natural language processing:

* its an "unsolved" problem (for various definitions of "solved"). I didn't expect to "solve" it, but every time I'd try something new there was the excitement that maybe, just maybe, this might be "it" (i.e. an immense improvement)

* the responses it produces can be really funny. If its online, others can have fun with it too.

* you can try lots of strategies and see the immediate effect. Not only do you learn about the math and technique by implementing it, but you get an immediate "feel" about the kind of effects it has.

Trying to over-generalize, it helps to have some "element" that you consider valuable, fun or rewarding to drive the whole process.


I have heard that they way Einstein read scientific books is by reading until he doesn't understand, stop there, start again from the begining and repeat until you reach the end.

My first computer science course was a life-changing moment for me. I was two years deep into general-ed, I was trying to manage my mother's alcoholism that I had just started to understand, and my girlfriend and best friend wanted me to move away to state University with her. I read that Java book start to finish, every chapter, and eventually realized the potential that computers can bring the world.

Anecdote aside, there's a difference between reading something and understanding something. My friend who is really in tune with the nature of the subject, is also spending an incredible amount of time surrounding it.


> How do you actually read math, physics and programming books? > Reading them the same way as you'd read a novel doesn't seem right.

Indeed it doesn’t.

As the author of the book in this post points out on his blog [1] in math you focus on learning the syntax first, just like in programming.

Off the top of my head I can recommend pre-reading a chapter before reading it: go through theorems, bold/italic text, graphs, tables, conclusions, exercises. Also, try proving theorems yourself before reading proofs in a textbook.

My general advice for you is to check out a book “How to Read a Book” by Mortimer J. Adler and Charles Van Doren. It’s a great manual for reading, which contains both general advice (in the form of 15 rules) applied to any reading material, and specific advices for math, philosophy, history, social science, etc.

> I try to go chapter after chapter and make notes

Notes are good, but only if they “active” notes. If you are able to formulate something in your own words it means you’ve done the work towards understanding, if you’re only able to express a thing in author’s words — you have not, and if you leave it at that the chances are you’ve wasted your time.

In math you also have to do problems. Your ability to solve problems w/o the help of the textbook is the ultimate test of whether you’ve learned what you’ve read.

Interleave: in your practice session work simultaneously on problems from different (usually sequential) chapters. IOW: don’t practice one topic during several study sessions until you’ve exhausted all exercises for that topic in a book — go read the next chapter (or chapters), and include its exerices in your next practice.

Interleaving is harder, but more efficient. The reasons being that (1) our brain is bi-modal processing system, it needs time to process things both consciously and unconsciously, it does the latter while your attention is busy with other things; (2) it also needs time to consolidate memories, grow and strengthen new synaptic connections (which it primarily does in sleep) — so spreading a topic/exercises over time is generally a good idea.

> but I often get bored because I don't see the usage in my real life coding > Also after few chapters it often turns into a "job" of finishing the book. I don't have the pleasure of learning new stuff anymore.

This is about psychology of study.

Don’t turn it into a job, it’s not a competition, there are no bosses, no deadlines. Don’t push too hard, allow yourself time to integrate new knowledge by keeping yourself busy with other things and studying several subjects/books simultaneously — when you are tired of one thing, switch to another. It’s more efficient way of learning (as explained above), despite increase in absolute time for mastering a single subject.

And work on your motivation. You can start by read what Paul Graham has to say on why study math [2]. Remember, math is hard; our brains are not wired for it.

[1] http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...

[2] http://www.paulgraham.com/college.html


While I like most of your points, I've got to stand up for our brains (or mathematics, depending on who you think is getting shorthanded there :)).

I would say the exact opposite of "our brains are not wired for [math]": mathematics is the simplest expression of our ultimate thought ("brain") processes.

It is only hard because it is so abstract and builds on a large corpus of what has already been constructed, but it was "designed" exactly after how our brains operate.

For a layperson, proving even the basic arithmetic might seem like magic, but once you go through the simple exercise of defining everything in terms of "One" and "Successor(One)" and compare that to how we teach our kids numbers, it's clear mathematics matches exactly how human brains operate.

Then again, I concede that not everybody's brain is "wired the same way," but I am sure there are plenty of us for whom mathematics is just natural: the only problem being the fact that majority of people have not been exposed to true mathematics.


As someone who works as a programmer but wasn’t super interested in mathematics, I’ve found this blog to be a fantastic read time and time again. I’d highly recommend going through Jeremy’s previous posts if this is your first time seeing his site on here.

If this topic piques your interest I would also recommend Mathematics for Computer Science: https://courses.csail.mit.edu/6.042/spring17/mcs.pdf


To the author:

You might want to extend the preview PDF to include a few pages from later chapters. The issue's that the [current preview](https://pimbook.org/pdf/pim_first_pages.pdf) only gets into polynomials over its 45 pages. But since polynomials are typically taught to students during early childhood, it seems like most readers are liable to just skim that content, being more interested in the topics discussed later. For example, the start of Chapter 14 (on optimization) would be neat to see.

That said, I like the parts that translate between analytical expressions and programming code. Such mappings seem like high-value content to readers; the language barrier can keep people from understanding mathematical writing, while a few helpful translations can help to tear down those language barriers.


The Amazon preview has more pages, in case you're still curious.

> But since polynomials are typically taught to students during early childhood, it seems like most readers are liable to just skim that content, ...

My view was that the initial chapter was more about how to learn mathematics generally using polynomials as a source of examples, rather than being on the subject of polynomials like you'd get in grade school.

Agreed though—I would like to see some of the later material, too :)

Edit: actually you can skip around and see more on the Amazon preview.


I think the choice to showcase the polynomial chapter is deliberate. The topic is likely to be familiar to many, but he's using it as an example to explain why mathematicians do things they way they do. I've often expressed the same frustrations that he describes when trying to explain to programmers why mathematics can't be replaced with a rigid programming language.

I really wish this book would include probability and statistic sections. My guess is that a lot of people, like me, will want to read it because they're getting started with ML and need help getting used to the math, and probability/stats is an important part of it that's missing.

Any chance you could add it in the future?


Well, two chapters have singular value decomposition and neural networks as the applications. So it does have a lot of ML :)

But yes, I unfortunately had to cut a probability chapter. I think someone who reads this book would have a much easier time learning probability after, and a better foundation.


> I think someone who reads this book would have a much easier time learning probability after, and a better foundation.

So will you be following up with a prob & stats book? ;)


Thank you for this book. There is a huge need for this.

It saddens me that schools may be handing out CS degrees without having first required students to at least have taken linear algebra, multivariable calculus, and discrete math that covers basic counting, sets, graphs and groups. How can this be?

Probability and statistics should also be a required part of every CS program.


/shrug misaligned incentives probably. Schools are also handing out CS degrees without students being all that good at writing programs either.

Have just ordered the e-book. Didn't expect to find a chapter on Hyperbolic tessellations. Nice!

Unlike the other subjects contained in this book, I do not think that any useful understanding of Probability and Statistics could result from only one or two chapters of coverage.

I would recommend a book I found that covers Statistics using Python:

Statistical Methods for Machine Learning https://machinelearningmastery.com/products/

Personally, I much prefer that book to the free "Think Stats" one available.

You can also find a book there on Linear Algebra and others related to Deep Learning, all containing real implementations with Python.

I have no affiliation with the site; these books have proven so valuable to me that I need to share them.

Anyway, the book mentioned in the OP looks like a great foundation and I picked up a copy as a refresher. I would have also liked to have seen coverage of number theory in this book, with RSA being an application, but that too would have required lengthy explanation for any useful insights to be learned.



That's rather like saying that a Java book should include a section on processor architecture so you can design your own chips. Plus, what's in this book isn't really enough to do probability and statistics at any but the most basic level.

The path to a deep understanding of inference that makes most modern ML seem straightforward goes through measure theory, Lebesgue integration, some functional analysis, a smattering of topology, and an intuition around dynamical systems. Then you go do probability and random processes, detour into game theory for a bit, learn decision theory, and at that point it all goes from mystical to bleedin' obvious.


It's good to have something that lowers the bar for programmers so they could learn themselves some math without much fear. Knowing math is very important if you are a coder - and not just linear algebra: knowing a formula, for example, might let you do certain things in constant rather than linear time or, perhaps, reduce the cost of the iteration. Unfortunately, too many of those who can call themselves programmers by trade know very little math (you'd be lucky if they remember what they learned in high school).

Seconded for this. Going back and taking an actual structured math course at community college after working professionally as a programmer for years was a real eye opening experience for me. So many things that seemed so abstract and meaningless as a 12 year old were instantly apparently useful in an unimaginable number of ways.

What courses did you take?

>What courses did you take?

Nothing advanced, just Trig, Geometry (proofs based), Algebra and Precalculus. Basically what a sharp high school senior in college prep classes would be familiar with. Geometry was probably the best. If you've never encountered formal proofs before, it's probably the best training for pure logic I can imagine. They start out incredibly frustrating but become addictive once you figure out how to crack them. Precalc was insanely useful too. Learning how to build up a function and shape it with various operators, and work with polynomials is absolutely essential for things like graphics shaders and timing functions and all kinds of other stuff that comes up in UI development (my work). All I can say is that it will give you a much greater appreciation for studying the purely abstract rather than always focusing on practical matters that come up in engineering, as well as making apparent more elegant solutions you may have brute forced in the past.


Where are you that high school only covers up to precalculus? In Australia (depending on the level) you do multivariate calculus among other things. Perhaps other subjects aren't covered as deeply.

>Where are you that high school only covers up to precalculus? In Australia (depending on the level) you do multivariate calculus among other things. Perhaps other subjects aren't covered as deeply.

I'm from the US. The state of mathematics education here is absolutely embarrassing to pretty much any other developed country. Taking precalculus is actually considered advanced, most students graduate with nothing more than an "Algebra" class that covers basic algebraic notation, quadratic equations, linear equations, and not much beyond that. Calculus 1 (single variable) is considered an advanced math class that is only taken by STEM majors for freshman university students here.


That's really me to a "tee". I just never had a teacher in high school who really understood more than basic math, and I went to work instead of college afterwards.

I expect I'll have to work at it to follow this book but that's why I need something like this. And I need to spend some time with Python too so I'm looking forward to digging into it.


I started on the Project Euler problems years ago, and like most naive coders I took the brute force route. After discovering the problems have a deeper mathematical basis I decided to start again and learn some math. Wikipedia illustrates with the first problem, the difference in elegance and efficiency knowing some math can make. https://en.wikipedia.org/wiki/Project_Euler

I'm sorry if this isn't your intent but the way you structured your comment comes off as very condescending.

Aside from that, I'm also skeptical that the frequency in which these maths apply to practical, commercial programming is really that high.

Im not anti math or something. I just think the practical value gets way over sold by some people. And sometimes it feels like it's because of the dislike of "those who can call themselves programmers by trade."


It is an incredibly important foundation for analyzing any kind of data. That is a need that crosses many different fields, be it sales forecasting, quantitative finance, econometrics, deep learning, signal processing, any sort of scientific computing etc.

I would be more interested in hearing an argument about why math knowledge is not useful or lucrative.


If anything, I think the practical value of mathematics is generally undersold. It seems OK if some people tend to oversell it.

I am basically trying to understand the formulas in machine learning papers, will this book help achieve improvements in speed?

I just realize I fear math because the educational system I grew up in was violent (like beating kids for getting a quiz wrong wtf).

It was only through psilocybin mushrooms did I discover math and calculus again.


Plenty of cliffhangers in this comment

Sure, I grew up in South Korea much of my childhood, corporal punishments was the norm.

So I have this phobia of calculus and math. Anytime I'm faced with a formula I get this panic attack. Some may call it PTSD. But it explains why I had such problem with calculus and it really made me feel inferior.

it's still a cliff hanger, searching for my arc.


I was teaching a college class for high schoolers last year and thought it would be great to record my lectures for them. Then I put it up as an entire youtube channel for everyone:

https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA

It basically builds up mathematics rigorously from basic definitions, while trying to stay very accessible.

If anyone has the time, or desire to learn math this way, let me know what you think, and if I should make more in this series!


I am a huge fan of Jeremy's blog. Found his primers on a multitude of topics very useful - https://jeremykun.com/primers/

As a Math guy who got into the world of programming relatively recently, I am on the opposite side of the spectrum I suppose but I'm gonna order this nonetheless to support him.


Would you read "A Mathematician's Introduction to Programming"?

> The problem is that the culture of mathematics and the culture of mathematics education--elementary through lower-level college courses--are completely different ... I've had many conversations with such students [...] who by their third year decided they didn't really enjoy math. The story often goes like this: a student who was good at math in high school (perhaps because of its rigid structure) reaches the point of a math major at which they must read and write proofs in earnest. It requires an earnest, open-ended exploration they don't enjoy.

I found this interesting because I too discovered this difference in approach but had the complete opposite reaction. I absolutely hated math in middle and high school. It wasn't until I took a discrete math course for my CS program that I got exposed to dealing with real proofs, which I found required a level of creative thinking, and I totally loved it. This admittedly wasn't an "advanced" university math class, but the difference from high school math was still quite stark.


That's exactly how I felt. I didn't really discover math until college.

Hmm, the "first few pages" end just before the "meat" begins.

From the table of contents, there seem to be short prose sections inteleaved with the teaching sections. I hoped to see an example of the teaching section, not the prose.


Yeah, I should update that to have the full first chapter.

First chapter is up now at https://pimbook.org/pdf/pim_first_pages.pdf

Also note that the Amazon "Look Inside" lets you see basically any page. Some readers have told me the first chapters were too slow, and so I think more advanced readers will want to breeze through that (though the applications in the first two technical chapters have a coolness to them that is hard to beat!).


Would love to see a little more of the math - am hooked on the idea though!

Amazon lets you browse more of the book and you can get a better feel for it. The E-Book you have to suggest a price, which in theory seems nice to pay what you what, but now I don't know what I'd pay. :)

Jeremy, been reading your blog for years. Just wanted to say thanks for the wealth of readable intros to interesting mathematics.

I'd like to tag onto this: I've been reading your blog for nearly a decade and can say that you were one of my inspirations for studying math and CS.

Nice.

I tried to start something similar which tried to explain all weird maths symbol via code. It went nowhere.

But feel free to check https://github.com/mkagenius/mathsymbol2code


What's provided currently at [1] doesn't provide the reader with any impression of the teaching style or quality/quantity of visual aids.

I'm very likely to buy a hard copy of such a book, but not unless I can do the equivalent of flipping through it like I would in a book store.

Consider changing the preview instead to a scattered sampling of some of your proudest pages.

[1] https://pimbook.org/pdf/pim_first_pages.pdf


On a related note, I'm curious if anyone has taken the Mathematics for Machine Learning (https://www.coursera.org/specializations/mathematics-machine...) courses on Coursera, and whether it really covers enough to be comfortable with ML. The course bills itself as enough math knowledge for folks who barely remember high school math.

Looking at the courses, it doesn't cover calculus and probability. That's two topics that you should already know and that you might not have mastered in High School. Otherwise, you are good to go.

I have completed all three courses in the series. It was a good supplement to other resources, especially 3blue1brown's Linear Algebra course on youtube[0] (mind-blowing, do check it out) but I wouldn't recommend it as a first course. The first two courses weren't rigorous enough for my taste (I am yet to find a rigorous course on Coursera), but the third was pretty good. You should take up books if you are serious.

MIT OCW Scholar(independent study) course on Linear Algebra by Prof. Strang[1] is really good and is designed for self-study. If you have the time, you could look up Coding the matrix[2] too. I read probability from Mathematics for Computer Science-MIT[3] and also referred Khan Academy[4] and PennState STAT 414/415 [5] for statistics and probability. StatQuest channel[6] on Youtube has handwavy but easy to understand videos on statistics for ML too. The Deep learning book[7] by Ian Goodfellow et al. has a couple of chapters at the beginning that gives you a fairly good idea of the mathematics required to get into Deep learning. Communities like r/AskStatistics and r/statistics on Reddit were really helpful when I got stuck.

I also chanced upon Mathematics for Machine Learning[8] book recently and it seems to be good. It has a chapter on optimization that is left out in most books but skips statistics.

[0] - https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

[1] - https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb...

[2] - http://codingthematrix.com/

[3] - https://courses.csail.mit.edu/6.042/spring18/mcs.pdf

[4] - https://www.khanacademy.org/math/statistics-probability

[5] - https://onlinecourses.science.psu.edu/stat414/

[6] - https://www.youtube.com/user/joshstarmer/videos

[7] - https://www.deeplearningbook.org

[8] - https://mml-book.com


These are great insights! Thanks so much. Do you think it's worth going through pre-calc/calc deeply? I assumed I should do that first, but it would take quite a while (I haven't taken calc in ~8 years and barely remember more than the basics).

Essence of calculus[0] by 3blue1brown for the basics and the second course in the Coursera Mathematics for Machine Learning would let you get started. You would rarely need calculus more advanced than that covered in the above, and if need be you will be in a position to look it up quickly. If you can sustain your interest in ML over a long period of time and are in no hurry, I would recommend going through all the math mentioned. If you are a top-down learner, the fast.ai course on ML and deep learning for coders will get you started head-first. All the best!

[0] - https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...


I have. It’s great because it’s developed using geometric intuition (a la 3b1b). Just missing probability/stats.

Can we get that GitHub solutions repo going? I feel like that's super important for this book. Bought it anyway.

Let's get it started, and initial thoughts, questions, or suggestions can be in Github issues for now.

https://github.com/pim-book/exercises


I guess it's best if Jeremy starts the repo himself, and we contribute to it. But I agree; cannot understate the importance of solutions to make the best use of this book.

I’ve found myself unable to do even elementary maths recently just because I’m sorely out of practice. Hasn’t really affected my performance as a programmer. Wondering, what kind of math I could learn that would benefit me in my job?

What kind of work do you do? For some types of development, basic knowledge of mathematics will take you reasonably far. Knowing more math opens up possibilities. For example, I recently found myself wanting to add features to some flight control software for drones. I wanted a return-to-home feature, which involved implementing a PID controller for gps navigation. I studied control theory in college decades ago, and hadn't used it for anything professionally until this year. Also, autonomous navigation requires taking vectorial inputs from sensors that must be rotated to the frame of reference of the drone (e.g. the accelerometer). I could not have participated in this project if I didn't have enough knowledge of calculus and algebra.

> Wondering, what kind of math I could learn that would benefit me in my job

It depends on what you want to do. I’ll get to your question in a bit, but I think this passage from the book on page (i) describes why you might want to learn mathematics.

> So why would someone like you want to engage with mathematics? Many software engineers, especially the sort who like to push the limits of what can be done with programs, eventually come to realize a deep truth: mathematics unlocks a lot of cool new programs. These are truly novel programs. They would simply be impossible to write (if not inconceivable!) without mathematics. That includes programs in this book about cryptography, data science, and art, but also to many revolutionary technologies in industry, such as signal processing, compression, ranking, optimization, and artificial intelligence. As importantly, a wealth of opportunity makes programming more fun! To quote Randall Munroe in his XKCD comic Forgot Algebra [0], “The only things you HAVE to know are how to make enough of a living to stay alive and how to get your taxes done. All the fun parts of life are optional.” If you want your career to grow beyond shuffling data around to meet arbitrary business goals, you should learn the tools that enable you to write programs that captivate and delight you. Mathematics is one of those tools.

You should never feel like a lesser programmer, because you don’t know mathematics it’s just something that might make things a little more interesting and fun while adding some value. Of course it’s not always fun for everyone, so let yourself decide if it’s for you. Mathematics can create value for a programmer in a lot of different ways like as stated above, but yet again people should not feel belittled not having this knowledge because there are many other ways you can add value to your job as a programmer without mathematics.

With that said, I’ll give you a list of programming topics below that are enabled by mathematics and what kind of mathematics can help with those topics. If you’re not needing to study any of these topics or needing to use them in your daily job I think one of the most useful and fun types of mathematics is discrete math for everyday programming. This can build foundations of logic and reasoning needed for programming. Someone else already mentioned the mathematics for computer science course from MIT [9] in this thread and that’s a great intro to discrete though it can be pretty challenging at times especially if you’re new to the topic.

Programming Topic - Math Topics to Study

cryptography - number theory, abstract algebra, probability, basic combinatorics, information theory and asymptotic analysis of algorithms [1]

data science - Linear algebra, Regression techniques, Probability theory, Numerical analysis [2]

art/computer graphics - geometry, linear-algebra, physics-based calculus, topology and numerical methods [3]

signal processing: Fourier transforms, Laplace transforms, differential equations, statistics, linear algebra, and complex analysis [4]

Data structures and algorithms: mostly discrete math (set theory, logic, combinatorics, number theory, graph theory, formal proofs), asymptotic notation.

compression: Information theory, statistics, probability, linear algebra [5]

ranking: statistics, markov chains (think PageRank), probability, linear-algebra [6]

optimization: numerical analysis, computational geometry, discrete mathematics, probability, linear-algebra, calculus [7]

artificial intelligence: discrete math, statistics, analysis, linear-algebra [8]

[0] see the hover over: https://www.xkcd.com/1050/

[1]: https://crypto.stackexchange.com/a/10468

[2]: https://www.datascienceweekly.org/articles/how-much-math-sta...

[3]: https://math.stackexchange.com/questions/830856/how-is-math-...

[4]: https://www.reddit.com/r/ECE/comments/4i7jq9/what_math_will_...

[5]: https://en.wikipedia.org/wiki/Data_compression#Theory

[6]: https://en.wikipedia.org/wiki/Learning_to_rank

[7]: https://en.wikipedia.org/wiki/Mathematical_optimization

[8]: https://math.stackexchange.com/questions/791326/what-maths-a...

[9]: https://ocw.mit.edu/courses/electrical-engineering-and-compu...


It appears the ebook is in PDF format[1], does anyone know if an EPUB will become available?

[1]: https://gumroad.com/l/pim-book


I prefer EPUB for most ebooks, but maths books work far better in PDF because mathematical notation gets turned into image files in an EPUB and don't render nearly as nicely.

I have been searching for something like this for a while. Just bought the book, I will get back to you. I hope the book will give me exactly what it promises.

I would have loved to have the printed version of this book. Unfortunately not available in India.

I've been wanting to peek into mathematics and an "introduction" is exactly what I need.

Thank you for sharing this.


This was the book that Richard Feynman used to teach himself calculus when he was 10 or 11 as I recall: https://www.amazon.com/Calculus-Practical-Man-J-Thompson-ebo...

This book looks great. Thank you for sharing!

Thank you!

Long-time programmer without a CS degree here. I've studied polynomial factoring, adding, subtracting, graphing them, etc. Sites like Khan Academy break things down in little bits but the underlying theory seldom emerges. But after working through the preview pages of this book I feel like I finally have a feel for some of the underlying theory ideas behind polynomials. This really emerged during the proofs section. The bits of code and analogies to programming really help. It was like a lightbulb going off in my brain. As a result I have ordered the book from Amazon and can't wait for it to arrive. Thank you for this book.

I would be interested to see someone post their experiences after working through at least half of the book. I am completely outside the target audience, and it would be really useful to know what works to teach mathematics to programmers and what doesn't.

I recently started reviewing mathematics on Khan Academy to brush up on my math skills and learn more Calculus so I can better understand ML. Really looking forward to reading this!

This looks really nice, from the preview content---I really like the approach of explaining the background assumptions of reading mathematical definitions and such. Ordered!

Some comments/corrections on the first chapter, if you don't mind:

1. Theorem 2.4 is stated incorrectly. Given the context, I feel like this is worth correcting. Specifically, it says "degree n" rather than "degree at most n". Part of the proof purports to prove that the degree is indeed n but of course it doesn't because that needn't be true.

There are other cases where you say "degree n" for "degree at most n". Again usually this would be a minor error not worth pointing out, but in this context it seems worth getting right.

2. At one point you introduce a convention that deg(0)=-1. Later, in the exercises, you ask, is this really such a good convention? (The answer being, of course, no.) IMO you should anticipate this. Indeed I don't think you should state, as you do, "By convention the zero polynomial is defined to have degree -1", because that suggests it's some standard universal convention, which is definitely correct, and it's neither of those. Rather you should say something like "We'll use the convention that the zero polynomial is defined to have degree -1". But anyway, the point I made is that, if you're going to question its correctness later, you should anticipate that here, maybe saying something like "(Think about whether this convention makes sense.)" Or maybe not, and just getting rid of the absolutism of your current wording is sufficient. Either way, getting rid of that absolutism and certainty is good; you want to encourage to people about this sort of thing immediately, not encourage them not to think about it until later.

3. You say that when you see a definition you should write down examples. I would add, "and non-examples". Ideally non-examples that come as close as possible but don't quite make it. You touch on this a little with your polynomial examples, but it's worth stating explicitly.

(In some cases non-examples are unnecessary, but in the generic case one should look for them.)

4. Regarding your polynomial examples, you don't justify that they are, in fact, not polynomials. Now of course you don't, that would be too hard to do here and take up lots of space you want to use for other things. That's fine. But if you're not going to do it, you should call out that you're skipping over it, like you do with other things. After all, all sorts of nonobvious things can be polynomials -- such as (x-1)(x+6)^2, as you pointed out earlier, but included no similar examples here. (Yes that's obvious to anyone who knows anything about polynomials, but my point is that it's not in the correct syntactic form.) Like, x^e - x^e is a polynomial, you know? Because it's 0. So without some more knowledge, you can't immediately conclude that your example x + x^2 - x^pi + x^e is in fact not a polynomial! You should make a note of that, as I said.

5. I feel like it's likely worth noting somewhere in this chapter that actually in general in math it's the "syntactic" definition of polynomial that turns out to be the right one (you don't want to define polynomials to be functions if you're working over a finite field, say!). Maybe not and that would just be confusing, I dunno.

6. This is just nitpicking, but I'd suggest rewriting Theorem 2.3 in a clearer, more standard way. "A nonzero polynomial of degree n has at most n distinct roots." What you wrote down is equivalent, of course, but (IMO) harder to read.

Otherwise, this is pretty nice. I remember being distinctly confused by stuff like "the product over j not equal to i" when I was a kid. I imagine it'll be quite helpful to a number of people that you're laying things out like that explicitly.

Actually, sorry, on that note, one further comment:

7. You comment on how sigma and pi notation are special cases of fold, but you might want to make a further note about how (unlike general folds) these are folds where the order doesn't matter, and that the fact that the order doesn't matter is one of the things that allows notation like "product over j not equal to i".


Of course you're right about degree n vs. at most n, with many easy counterexamples (the list (1,1), (2,1), (3,1) being a simple one). If you'd like, please submit an erratum using the link at pimbook.org, and other readers can see it, and I will include the fix in the next version of the book and credit you.

Thanks for reading! ^_^


OK, will do! Like I said, I think a lot of people will find this book quite helpful. :)

Thanks for the effort. I purchased the e-book and will start working through it immediately. I'll let you know what I think.

Just ordered from Amazon! I used to read his blog a few years ago and I loved the articles and the breadth of topics. Thank you!

From the ToC, is the any reason there is no chapter covering probability/statistics? It has chapters for single/multivariable calculus and linear algebra, and all CS programs I know have all three, especially when there are some nice connections between them, not to mention how useful they are in other CS/engineering subjects.

Can you create an EPUB version? I'd like to buy it but I need EPUB for my reader.

Please tell me you wrote it in LaTeX. It would be another reason to buy it.

Of course!

I tried to buy it but for some reason Paypal refuses to use my existing balance and instead asks for my card.

Is this something you can control on your end?


Ebook a tad too expensive for me.

minor suggestion: make it easier to see the table of contents. A survey book leaves uncertain exactly what is included.

Does it compare to Don Knuth, et al, "Concrete Mathematics: A foundation for Computer Science"?

It might have some crossover, but my guess is it's probably much more introductory than that book. I didn't know this until fairly recently, but Concrete Mathematics was used in a Graduate level course at Stanford as the textbook (with the course name following the book's, Concrete Mathematics). Kind of threw me off when I first found out, because the book says it's a foundation for computer science, so I thought it would be an undergraduate course. So, I don't think you need to be a graduate student or in particular a Stanford level computer science graduate student to read Jeremy Kun's book.

A section on quaternions would be nice.

I think the author is confused. His book is not for programmers his book is for "programmers lacking computer science education" as computer science is a branch of applied match. If somebody says they have a computer science degree and they don't know math, I would doubt their degree.

The description got me excited, but looking at the table of contents, the level is ultra basic - appears to roughly correspond to first year of a cs degree.

The title is "introduction to mathematics." I think you are probably looking for a different book.

Not all software developers have been through a CS curriculum. Not all CS curriculums teach rigorous math.

Certainly more than just the first year, and I don't think the majority of CS degrees require multivariable calc or any group theory.

I do wish there were more of a preview than just the TOC to see how novel the examples are and how much it helps with intuition for these mathematical concepts beyond what you would learn in a plain CS sequence. That would be my reason for buying the book and I wouldn't write it off just because the list of topics covers the first two years of college math.


My uni did this weird thing where they put all the math courses into the first year (except for one stat course in the second year). The first semester had highschool basics like calc and trig, followed up by another two courses the following semesters that covered linear algebra and ... something else. I don't remember, I because I was too busy with girl problems.

Looking back, that first year was brutal with each successive year getting way easier and way more fun.


Still potentially useful for people who got into programming by some path that doesn't include a CS degree.

I see a lot of users are learning maths for machine learning. I did the same and here is what I found:

I started with 3blue1brown's Youtube course[0] on Linear Algebra and loved it. I had already done a college course on LA, but this made me truly understand what I was doing.

MIT OCW Scholar(independent study) course on Linear Algebra by Prof. Strang[1] is really good and is designed for self-study. If you have the time, you could look up Coding the matrix[2] too. I read probability from Mathematics for Computer Science-MIT[3] and also referred Khan Academy[4] and PennState STAT 414/415 [5] for statistics and probability. StatQuest channel[6] on Youtube has handwavy but easy to understand videos on statistics for ML too. The Deep learning book[7] by Ian Goodfellow et al. has a couple of chapters at the beginning that gives you a fairly good idea of the mathematics required to get into Deep learning. Communities like r/AskStatistics and r/statistics on Reddit were really helpful when I got stuck.

I also chanced upon Mathematics for Machine Learning[8] book recently and it seems to be good. It has a chapter on optimization that is left out in most books but it skips statistics.

[0] - https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x....

[1] - https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb....

[2] - http://codingthematrix.com/

[3] - https://courses.csail.mit.edu/6.042/spring18/mcs.pdf

[4] - https://www.khanacademy.org/math/statistics-probability

[5] - https://onlinecourses.science.psu.edu/stat414/

[6] - https://www.youtube.com/user/joshstarmer/videos

[7] - https://www.deeplearningbook.org

[8] - https://mml-book.com

Copied from my comment here: https://news.ycombinator.com/item?id=18582022


I'm a little skeptical of CreateSpace.

Why did you decide to self-publish?


I actually used to work for CreateSpace! I think they do a splendid job on the printing, and the royalties are much better than a publisher. I think I will write a longer blog post with more details.

I had a CreateSpace textbook once and the mathematical notation was so grainy that it was unpleasant to read.

why is it not working

I guess this is for the nontraditional programmers since computer science is a mathematical field and programming is simply applied mathematics in some sense. I don't see how you could get a CS degree without being competent in mathematics to some degree since CS is a mathematical field.

From the bit I've read from https://pimbook.org/pdf/pim_first_pages.pdf it seems to be very very poor.

* 19 pages of droning before you start with something concrete. Much talk talk talk about your experiences before you get to the point. I can't put into words how much it frustrates me when I'm expecting to read something interesting and the author takes 3 paragraphs talking about nothing (usually with lots of overexcited exclamation marks).

[sorry if I am being blunt, but it's how I feel]

* Imprecise definitions. This defeats the purpose of learning mathematics. Like Leslie Lamport says, rigour in mathematics is not a hurdle or a chore one must endure, it's the whole point of learning the damn thing. You give imprecise definitions, and then obscure it even further with neverending paragraphs of confusing explanations. This to me kills the whole pedagogical value the book might have. Here is a rule of thumb that in my experience applies well to almost everything in mathematics: the simpler your explanation is, the better. Your goal is to explain a concept as succintly and beautifully as possible. This exposes the idea behind it. A long and meandering explanation only serves to obscure the idea behind. Less is more.

* Attempting to shoe-horn programming "lingo" into mathematics. Sometimes, the best way to explain something, even to programmers themselves, is not to force an awkward analogy with Java programming. EDIT: 5 pages later: "The best way to think about this is like testing software." oh boy...

* The graph in e.g. page 8 (20 of the pdf) is terribly typeset. The axes text is way too small to read and in a font that doesn't match the rest of the content.


Regarding the first point, if you are the type of person who just wants it to get to the equations with a minimum of talk, you're probably not the intended audience of this book.

The culture of mathematics (including its lingo) has always been a big barrier for me. Thus far, I've only skimmed it, but that part looks promising to me.

It seems reasonable to guess that if someone has learned a good bit about programming but managed to avoid learning about math, reasons like culture, terminology, accessibility, and motivation might be a big part of why. So including that sort of material in this book makes sense to me.


Regarding the first point, if you are the type of person who just wants it to get to the equations with a minimum of talk

I don't want "no talk" and "just equations, I want the "minimum of talk possible" that communicates the ideas and nothing more. I find the more stuff you write the more there is to confuse people. The best way is to strip everything that is superfluous. You are left with the essential, in the clearest form there can be.


Math books already exist for that audience.

Indeed, most of the canonical texts for undergrads would likely fit the bill.


There is a common category of misguided book review that the parent comment falls under: the reviewer has some particular things they are looking for from a book and fails to see that there are more uses for it than their own; so they generalize, reading the book's failures to meet their personal criteria as a failure intrinsic in the book's content, and mistakenly lambaste the work.

> 19 pages of droning before you start with something concrete.

One person's "droning" may be another person's "rare, illuminating presentation"; we are not all interested in the same things. I can offer another perspective that what the author wrote on polynomials within the first 19 pages there is in fact pretty unique and interesting.

> Imprecise definitions. This defeats the purpose of learning mathematics.

The commenter is again taking a minority viewpoint and baselessly extending it. There is a special kind of paranoia that often manifests in discussions of mathematical pedagogy that any departure from perfect rigor will weaken the minds of students. This is a blatantly shallow and one-sided view of things: the fact that rigor has value does not imply that the most effective method of teaching—which is a matter of both mathematics and human psychology—is to offer nothing but the most rigorous presentation possible.

This particular book is explicitly targeted at helping to transition programmers into mathematics. Given that goal, do you really think the pedagogically superior approach would be to offer a more rigorous definition of polynomial than:

"A single variable polynomial with real coefficients is a function f that takes a real number as input, produces a real number as output, and has the form: f(x) = a0 + a1x + a2x^2 + ... + anx^n"

—in the introductory chapter of the book?

I would urge potential readers to take the parent comment with a grain of salt. And also, if you're among this book's target audience and trying to teach yourself mathematics, be aware of such personalities preaching this particular dogma of mathematical instruction: it's fairly common on the internet. But it basically represents the same corner of the mathematics world as that of the programming world where folks insist on using nothing but VI/Emacs on Linux with C++ and/or Haskell, and are all too ready to belittle any alternatives perceived as softer/weaker. These aren't necessarily the most impressive programmers—they just project the most intimidating auras. Don't let their counterparts in mathematics scare you off.


>One person's "droning" may be another person's "rare, illuminating presentation"; we are not all interested in the same things.

You are absolutely correct in that regard. I've talked to many people who hated the textbooks I most liked, and preferred ones that I found unreadable. Just goes to show that what works for you may not work for me.

You are very wrong on the other point: none of what I said is borne out of a view that mathematics should be difficult and "hard" an impenetrable (rather than soft and approachable). Indeed I view this sort of overlong prose as impenetrable and confusing, and the succinct style of e.g. Landau textbooks much clearer (see my other comment).

>The commenter is again taking a minority viewpoint and baselessly extending it.

This is not a minority viewpoint. I'd daresay is the majority opinion among mathematicians. It is also the opinion of prominent computer scientists: Dijkstra, Leslie Lamport, Donald Knuth, to name a few off the top of my head.

It is also plainly true: mathematics is about rigour. It is not an obstacle, nor is it an end in itself, but it is a fundamental part of mathematical study and reasoning. If you aren't being rigorous, you're not doing mathematics, it's just a waste of time. I'll let Michael Spivak speak for me:

"In addition to developing the students’ intuition [...], it is important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions."

>be aware of such personalities preaching this particular dogma of mathematical instruction: it's fairly common on the internet. But it basically represents the same corner of the mathematics world as that of the programming world where folks insist on using nothing but VI/Emacs on Linux with C++ and/or Haskell

Completely nonsensical comparison (C++ and Haskell?? two languages who could not be further apart), and again, not in the least bit what I mean with my criticism.


> I'd daresay is the majority opinion among mathematicians. It is also the opinion of prominent computer scientists: Dijkstra, Leslie Lamport, Donald Knuth, to name a few off the top of my head.

There are reasons partly historical and partly due to the subject matter of CS that mathematicians closer to it tend to place relatively higher value on formality.

> It is also plainly true: mathematics is about rigour.

This is taking taking it too far and you're placing yourself in the formalist camp, which is a minority viewpoint. You gave a nice quote from Spivak where he characterizes rigor, "...the natural medium in which to formulate and think about mathematical questions."

Labelling it a 'medium' is very informative. Computer programs likewise depend totally on the formality of the medium; it's in intrinsic part of how they are able to operate. Same with the magic of formality + inference in mathematics.

That said, the psychological questions of pedagogy involve much more than the intrinsic features of some subject. In learning to write completely formal computer programs, one benefits immensely from instruction which departs from total formality. I'm assuming this would be obvious to the readership here so I won't go into more detail.

> Completely nonsensical comparison (C++ and Haskell?? two languages who could not be further apart)

I was referring to the social standing of the languages (which is all that matters for the points I was making), not their intrinsic features.


I'm sure the author would have liked for you to enjoy his book, but if you've already read Spivak and Landau, then you are most definitely not his intended audience.

Your criticism is like that of a professional mathematician complaining of the elementary results presented in some high school math book.


wtf is wrong with hn.

>[sorry if I am being blunt, but it's how I feel]

you should learn to keep your feelings to yourself when the only function they serve is to denigrate others and derive cruel satisfaction for yourself.

>Here is a rule of thumb that in my experience applies well to almost everything in mathematics

this is aspirational pretension - everyone claims to appreciate formal purity /after/ they've learned something but when you're /learning/ none of that matters because you're just trying to develop intuition. to be one of those people that understands after their own stumblings/ruminations and then begrudge the next person the same is despicable. shame on you and i hope you're never in a position where someone depends on you to teach them absolutely anything.

>Imprecise definitions. This defeats the purpose of learning mathematics. Like Leslie Lamport says, rigour in mathematics is not a hurdle or a chore one must endure,

but that's just like your opinion man (or leslie lamport's). there are shelves and shelves of books for people like you - go read bourbaki or rudin or mochizuki or whomever you'd like. this book is not for /you/ - it's stated purpose is to excite and entice people that don't have formal mathematical training to learn mathematics and those sorts of people decidedly don't enjoy austere definitions and succinct theorems and terse proofs.

hence the only purpose your comment serves is to hurt the author's feelings, an author whom i might add has done infinitely more for the math community than you have with your pedantry and vitriol by maintaining a blog https://jeremykun.com/ with literally reams of interesting mathematical content that is simultaneously exciting /and/ rigorous. and furthermore iirc jeremy was originally a math ed phd student so i trust his opinion of the right way to teach math infinitely more than i do yours mr random internet physics guy.

next time think twice before posting this kind of lowbrow mean shit.


This was close to being a great, meaty comment, but personalizing your criticism kind of ruined it. Which is too bad! I’m inclined to agree with the substance of what you’re saying.

https://news.ycombinator.com/newsguidelines.html ("Guidelines" link at bottom of every page)

> In Comments

> Be civil. Don't say things you wouldn't say face-to-face. Don't be snarky. Comments should get more civil and substantive, not less, as a topic gets more divisive.

> When disagreeing, please reply to the argument instead of calling names. "That is idiotic; 1 + 1 is 2, not 3" can be shortened to "1 + 1 is 2, not 3."


I'm grateful for OP opinion and your opinion. I agree with you except in "next time think twice...", I think that opinion trigger your response which was great!

Enjoy your weekend, both of you!


It's funny, because I discuss Lamport's view in the book as well in a later chapter. I think some folks are annoyed that the first chapter is slow, but I promise it ramps up :)
krick 13 days ago [flagged]

As far as I can tell, you are what's wrong with HN. His post is a well-justified opinion, with reasoning provided, whether you chose to agree or disagree with that. Somebody may (or may not) take it into consideration. Yours are simply "how dare you!"-post, adding absolutely no value to the discussion. Also, learn capitals, it's hard to read you like that.

there is zero justification in the post I responded to outside argument from authority and prescription so i have no idea what you're talking about
andrepd 13 days ago [flagged]

I have not one single time invoked any kind of authority to make the points I'm making. It was only below, after you've (ironically) invoked the writer's credentials and insultingly proclaimed I never should be allowed to ever teach anything, that I let you know my qualifications.

Erm...

> Like Leslie Lamport says

Pretty clear appeal to authority, IMO.


20 lines of reasoning and one quote that I think conveys what I mean. I think if you can't honestly engage with that besides accusing me of "appeal to authority" there is nothing to be gained from continuing a discussion.

Btw it's actually Lamport quoting Spivak.


"wtf is wrong with hn"

One thing that's wrong with HN is that perceived "negativity" often gets condemned in exactly the way you have done here.

It seems as if a significant number of HN readers have never really participated in a spirited discussion with arguments made from multiple different perspectives. Maybe any kind of apparent conflict scares them, maybe they project their own aggression onto a comment that seems to go against the grain of the discussion. It will never change.


There are many easy ways to rephrase OP's comment into one that isn't so direct and denigrating. The only thing "apparent" here is that OP has trouble with empathy.

What's the problem, too much empathy? I mean, if the opposite of empathy is appathy, then I don't see that in the OP.

The opposite of empathy is antipathy.

The opposite of antipathy is sympathy.
andrepd 13 days ago [flagged]

"Direct", yes. "Denigrating", how?? I'm genuinely asking so I can fix that in the future, unless you think criticising is offending.

>The only thing "apparent" here is that OP has trouble with empathy.

Again, I can do without the online pretend-therapy. Amazing how perceptive some people are that they deduce the most profound things from a dozen lines of text!


I wanted to say I found your original comment critical, but not offensive. But I found the reply made to you offensive, because personal and aggressive. Which I think is how you see it too.

However, it seems several people took the side of the replier.

So I reviewed your original comment, and I think I've found the problem: it exaggerated and labelled, e.g. droning, talk talk talk, talking about nothing, neverending paragraphs.

Many of these aren't literally true ("nothing", "neverending"). Others are emotionally loaded ("drone"). It's probably almost always better to speak directly, without exaggeration or emotion... but this is particularly important when criticizing.

I didn't notice these at first because I tend to filter out decoration, and just hear the content (i.e the literal meaning) - though this is much easier to do when I'm not personally involved!

I think, "to be blunt", to speak plainly, to get to the point, really mean to be factual and accurate - without emotional language, exaggeration or labeling.

Anyway, I notice dang asked to not continue this thread, but I was troubled by it, and reviewing it helped me - maybe it will help you too.


Maybe you should take a hint huh? If many people are telling you something about yourself (a subject you're inexorably biased on) you think maybe you should reconsider your position?

[flagged]

andrepd 13 days ago [flagged]

I have not one single time invoked any kind of authority to make the points I'm making. It was only below, after you've (ironically) invoked the writer's credentials and insultingly proclaimed I never should be allowed to ever teach anything, that I let you know my qualifications.

I do justify why simplicity and succinctness are better than meandering. More superfluous information obscures the "nugget" or the essence of we really want to talk about. Strip away the accessory, write the simplest, most lucid explanation that captures the idea, and you will see the concepts and connections in their elegance, more clearly than when that is obscured in the middle of pages and pages of talk. In summary: "idea" is more understandable than "idea+cruft".

What do you mean no reason why? Are you expecting a scientific trial? I'm sorry to disappoint you: these are but my opinions.


Please stop now. Regardless of who's right, and you probably both are, this is the kind of tedious spat we don't need.

You're absolutely right. It feels frustrating for your comment to met with ill-faith attacks and insults but that's no excuse to lower the level in turn. Apologies for that.

The comment of yours that I responded to only has an allusion to a quotation and nothing else substantiating any of your points. Yes if you're going to proclaim and critize on something material (like the best way to teach someone something) I expect you to have scientific trials.

Please stop now. Regardless of who's right (which you probably both are) this is the kind of tedious spat we don't need.
andrepd 13 days ago [flagged]

>wtf is wrong with hn.

I believe you must have mistaken me for someone else. My name is andrepd. Is this "hn" a friend of yours?

>you should learn to keep your feelings to yourself when the only function they serve is to denigrate others and derive cruel satisfaction for yourself.

It never ceases to amaze me how some people are so perceptive that they can confidently deduce the intentions and personality of a person half a world away by reading a short text online. Are you available for therapy sessions? I could use your valuable insights.

Also, I was under the impression that I was making constructive criticism. I made concrete points and showed examples of books that in my opinion did those things right (see my other comment). I regret the "very very poor" phrase at the start though. I've changed it to just "poor".

>everyone claims to appreciate formal purity /after/ they've learned something but when you're /learning/ none of that matters because you're just trying to develop intuition

Oh, I see you know more about how I felt learning stuff than I do myself! Lovely.

I can tell you that I found it was almost always reading the most succinct treatments that I truly understood the greatest insights. Although from your tone I suspect that none of this will make a difference about how you feel. I won't repeat myself. You can (re)read my other comments if you're interested (and if you find I didn't explain myself well I'd be happy to discuss it more), and also check the examples I've mentioned.

>shame on you and i hope you're never in a position where someone depends on you to teach them absolutely anything

>an author whom i might add has done infinitely more for the math community than you have with your pedantry and vitriol

>jeremy was originally a math ed phd student so i trust his opinion of the right way to teach math infinitely more than i do yours mr random internet physics guy.

I am doing a maths PhD and (woe is you) I teach undergraduate classes AND I've been tutoring students of physics, maths and computer science for over 4 years now, online and in person. I can say with no false modesty that I've been repeatedly complimented by students on my ability to teach and explain concepts.

What Nobel prizes have you got, since that seems to be so important to you?


Is there any way we can keep this somewhat civil?

You're totally right. Apologies for my part. I should know better than to engage with

"next time think twice before posting this kind of lowbrow mean shit".

It was meant to attack and irritate and I fell for it.


on the internet everyone is a dog and has read all of the volumes and TAOCP and all of the volumes of landau lifshitz (and also peskin and shroeder and read baby rudin at 5). no one cares about /you/ man - it's a public forum so the point is to give advice that is generally relatable.

*>on the internet everyone is a dog and has read all of the volumes and TAOCP and all of the volumes of landau lifshitz (and also peskin and shroeder and read baby rudin at 5). no one cares about /you/ man - it's a public forum so the point is to give advice that is generally relatable.

I am again very sorry to disappoint I've not read TAOCP or Landau or even Peskin from cover to cover. I'm however curious if you can let me know where you got that idea from, cause I scoured my comments and can't for the life of me find anywhere I indicated so.

Seriously though, I have no idea what your point is supposed to be. I shouldn't give my opinion because... it might not be relevant to someone else, is that it? Someone might not agree? If it interests you I've met many people in the past few years who share my view (and also many personalities, again, Lamport, Dijkstra, Knuth, in the field of computer science). I've also met many who don't agree! And that's fine, what works for some might not work for others. Clearly this concept does not sit well with you.

This is being an uninteresting conversation which is only serving to irritate me (and you, it seems). I will stop replying now.


I personally think this is exactly wrong. Most people struggle with Maths because they don’t have the implicit understanding that books like you’re talking about rely on. It’s like telling a novice programmer to read the documentation. I’m not sure how successful this book is but it seems a worthy attempt.

That's not at all the point I was trying to make. Indeed, if you check my other comment, both the books I give as an example have precisely the feature of being self-contained. Landau's Mechanics starts with the basic concept of a particle. Cohen-Tannoudji's Quantum Mechanics starts with the simplest possible description of the double-slit experiment. Again, they assume no prior knowledge about what they are trying to teach (wouldn't be much of a textbook about the subject if it did require previous knowledge about the subject, now would it? :))

The counterparts of these books in the mathematical domain may be self-contained with respect to the subject matter, but you fail to consider the fact that most of them require a high degree of mathematical sophistication and are really targeted at people who have already acquired the required level of intuition. The "self-contained" nature of these books is deceptive - they might start with some innocent looking stuff about counting, integers etc and very soon drill deep into abstract stuff through complex chains of reasoning which a beginner will find extremely dry, bewildering and unfathomable. When you are a total beginner and trying to learn a subject through self-study, you are more likely to appreciate a book in which the author tries to recreate the experience of sitting in a classroom and listening to a professor holding your hand and showing you stuff from various angles.

Attempting to shoe-horn programming "lingo" into mathematics.

This statement is particularly amusing and highlights to me the fact that mathematics are a natural language, and not a programming language. Sussman had a really great presentation on this point; anyone who’d argue this please watch!

https://www.infoq.com/presentations/Expression-of-Ideas


Do you mind summarizing Sussman's presentation? I can't view it on the link you provided without making an account. I can't tell from your comment whether you agree or disagree with the parent's point.

Here's a YouTube link of the presentation...

https://youtu.be/iqlxsSrd-M4

NB: The slides aren't visible in the video, but here's the corresponding AI Memo by Gerald Jay Sussman and Jack Wisdom from November 2012...

AI Memo 2002-018: The Role of Programming in the Formulation of Ideas (November 2012) [pdf] https://dspace.mit.edu/bitstream/handle/1721.1/6707/AIM-2002...


That was very interesting! Certainly a different perspective.

I'd recommend reading Tao's blog post "There’s more to mathematics than rigour and proofs":

https://terrytao.wordpress.com/career-advice/theres-more-to-...


I enjoyed reading this opinion, do you have any recommendations for math textbooks that reach the standards you've described?

Not exactly math, but physics, since that was my background. Perhaps my favourite textbooks are Landau and Lifschitz[1]. They pretty much exemplify what I mean. They are concise, lucid, entirely self-contained. All the relevant information is there and not one bit more. The material is reduced to its key ideas, therefore making their exposition as clear as it can possibly be. Many people I've spoken to, however, don't like it precisely for being "too terse", so there you have it as well: ymmv.

Another book that fits this is Cohen-Tannoudji's Quantum Mechanics[2]. The first two chapters explain quantum mechanics from absolute scratch. It starts as all physics does: with an experiment. Then everything else follows from looking carefully at the consequences of that observation, and the concepts are truly explained for what they are, because they are presented in the simplest way possible. "Idea" is more understandable than "Idea+Cruft". This is remarkable to me: much fuss is made about how quantum mechanics is strange and confusing and difficult to grasp, and it is indeed so. In that first chapter, however, I've found the most illuminating explanation of quantum mechanics I've ever read.

[1] https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics

[2] https://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji#Selecte...

EDIT: I just opened Landau's Mechanics Vol I, and here's what his colleague had to say about his writing and teaching style: "[These are] all the features of his characteristic scientific style: clarity and lucidity of physical statement of problems, the shortest and most elegant path towards their solution, no superfluities.". Indeed.


I find this a bit funny. You are comparing books that are used by advanced undergraduates or graduate students to a book written essentially for the subset of programmers with almost no math background. There are beautiful mathematics books by Springer (Undergraduate Texts in Mathematics), Carus Mathematical Monographs and Dover that provide both intuition and rigor at an introductory level. The main problem for a beginner though is knowing where to start and getting a survey of the arena they are entering. They might think linear algebra is adding vectors together with a vector being just a list of numbers.

This book hopefully fills that gap. There are other introductory books like Mathematics: Its Content, Methods & Meaning by Aleksandrov and Kolmogorov but that requires a serious investment of time.

Lastly, I apologize but I am going to rant about this physicist phenomenon. I am a physicist myself (high-energy theory) that entered quantitative finance after my PhD (a heavily disliked field on HN). I found a lot of physicists outside academia who tend to be aggressive and condescending and often haven't actually gone through the rigors of a PhD program themselves. Maybe it makes one feel smarter to tell other people that they are reading easy books or can't jump straight to Landau or Bourbaki (those two couldn't be more different but you get the point) and there are no other physicists around to correct the notion that there are multiple ways to gain knowledge and build intuition.

It doesn't matter who says what about which book. Sample some books, find what you like as long as its not junk science/mathematics, do plenty of exercises, and try to derive things yourself and don't fool yourself that you understand something deeply by studying it for a year.


Landau-Lifschitz as a reference is also very polarizing. There are people that really like the style (myself included), but I have also met many people that really dislike it, think it is too hard, too brief, too dry, too little description of intuition... . Similarly, the Feynman lectures are loved by some and disliked by others.

I am curious how many people the author has taught mathematics to.

It doesn't seem a good idea to jump into writing a textbook teaching mathematics unless one has experience of teaching mathematics.

But the author makes the very point about early failures of programming, due to lack of experience, so perhaps he can supply information about what his pedagogical experience consists of.

Teaching other people is a craft, similar to programming or mathematics, with its own necessities.


It's a good question, and sort of hard to balance. I don't have any formal training as a teacher, or really a _pedagogy_ per se. I did teach five years of discussion sections among calculus, python programming, and differential equations. I do guest lectures for high school math classes, volunteered with groups like Black Girls Code and Hour of Code. I also did years of tutoring-center style tutoring, which meant I worked with students from the entire math curriculum. One time I even did an impromptu linear algebra course for a group of co-interns when I worked at MIT Lincoln Lab.

So in terms of number of people face to face, somewhere in the low thousands seems right. In terms of writing, my blog has on the order of millions of all-time page views.

I think it's a stretch to call the book a textbook. I think of it as an O'Reilly-type general technical book, but for math. If someone uses it to teach a course, that would be pretty wild, and I'd feel honored.

Hope you enjoy it :)


Apparently he was a TA as a math PhD student at UIC. Now is a programmer working at Google. He wrote a bunch of posts on his website over the years, https://jeremykun.com/main-content/

His blog has appeared here before and I've found it well written.

It’s a shame you don’t get to read any of the actual maths in the sample pages. Otherwise I might have been able to evaluate whether I want to buy this book. Intros don’t really tell us much



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