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Comment by threwythrw | Hacker News Reader

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?

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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-...

They have a repo.

https://github.com/pim-book/programmers-introduction-to-math...

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.

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.

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.

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

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