Mathematics for Computer Science [pdf]

https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

by lainon • 9 years ago

1059 153 9 years ago

courses.csail.mit.edu

vayarajesh2• 9 years ago

Is this a good start for Math required for Machine Learning ?

jonnybgood• 9 years ago

Yes, discrete math is a good start for anything CS, but you'll definitely need more than what's presented here.

mwambua• 9 years ago

Not really. You're better off looking at introductory calculus and statistics. This book places more emphasis on discrete math.

One good way to go about it is to audit an online course [like Andrew Ng's] and figure out what gaps you need to fill in your knowledge to understand the material.

jpau• 9 years ago

From what starting position are you asking from; what is your current math background?

vayarajesh2• 9 years ago

I am familiar with Linear algebra and at most average understanding of graph theory. I know there is lot more math to cover for ML but I find it overwhelming to start

__john• 9 years ago

Not really. The paper deals with "conventional" CS math. Here are some good resources for ML/DL math.

https://hn.algolia.com/?query=machine%20learning%20math&sort...

_qhtn• 9 years ago

Needs more:

Linear Algebra: http://joshua.smcvt.edu/linearalgebra/

Calculus: https://cnx.org/contents/i4nRcikn@2.48:H2TLb2-S@2/Introducti...

https://cnx.org/contents/HTmjSAcf@2.40:rrzms6rP@2/Introducti...

https://cnx.org/contents/oxzXkyFi@2.49:72YaCFgv@2/Introducti...

austenallred3• 9 years ago

I can't thank you enough for this comment

sn93• 9 years ago

MIT OCW Scholar has those subjects and more with HW, exams, and lecture notes and is intended for autodidacts.

gumby• 9 years ago

No, this is for a general introduction to the mathematics of computer science. This looks like a basic (and good!) text any MIT freshman should be able to master. Perhaps it's for what 6.001 has morphed into?

If you understand this stuff, you really need linear algebra for today's "deep learning", which perhaps is 18.03 (I can no longer remember).

vecter4• 9 years ago

This class is 6.042 and is not required for CS majors. 6.001 morphed into 6.01, which is something to do with python.

18.06 is linear algebra. 18.03 is differential equations, which you don't really need for machine learning.

beisner• 9 years ago

This book is a really accessible primer on the basics. Much more readable than a lot of other textbooks, which kind of go all-out in terms of rigor rather than present things intuitively. This was one of the textbooks we used in Princeton's introductory CS theory class (COS 340: Reasoning About Computation). The probability section is especially good.

raoulj• 9 years ago

lolhi

dang2• 9 years ago

I almost banned you before realizing you were probably talking to your instructor. It's good to build up a substantive comment history before posting like this!

whitenoice• 9 years ago

OCW Video lectures for the course - https://ocw.mit.edu/courses/electrical-engineering-and-compu...

dmlittle2• 9 years ago

It'd be nice if they are able to record Leighton's 6.042 lectures. Meyer teaches the class 3 out of every 4 semesters in an inverted classroom setting (you read and learn topics on your own before lecture and lecture time is spent in smaller groups with TAs helping you solve problems) while Leighton teaches the class lecture style (which is better for OCW imo).

jmstfv• 9 years ago

You might want to check the older version of this course https://ocw.mit.edu/courses/electrical-engineering-and-compu...

sjroot2• 9 years ago

I printed off the 2015 version (yes, the whole thing) and it has served as a very valuable reference. I wish there was a way to purchase a nice printed edition to support the authors. If they read this - thanks! Also, is a 2017 version of the course coming to OCW? I imagine that's why they'd release the updated PDF.

moinnadeem• 9 years ago

Freshman at MIT here taking this class -- the lectures are actually taught in a flipped classroom format, so I wouldn't imagine they would release the course considering there are no lectures to follow. I could see them releasing problem sets, however.

sjroot2• 9 years ago

Interesting! Who is your instructor? Tell them to release a hard copy!

bad_hairpiece• 9 years ago

Well, shiiiiit. I came to Hacker News to avoid doing my discrete math homework. While this is number 1, guess I'll plug Fleck's textbook from UofI http://mfleck.cs.illinois.edu/building-blocks/

impendia2• 9 years ago

This fall I will be teaching the required "Discrete Math for CS course" to about fifty students at the University of South Carolina. Previously I used Epp's book [1] which in my opinion is an outstanding book but regrettably is $280.44. Many of our students are working minimum wage jobs to make ends meet, and I don't want to make them pay so much if I can at all help it.

Lucky I saw this!!

I do have one reservation though -- many of our students come in with a weaker mathematical background than MIT students; for example we spent several weeks doing proofs by induction (and no other kinds of proofs) and this text doesn't seem to feature a couple of weeks worth of examples.

I think I'll probably go with this and supplement as needed. Really it looks quite wonderful. (And hell, the book seems to be open source which would mean that I could potentially write supplmentary material directly into the book and make my version available publicly as well.)

This thread seems like a particularly good place to solicit advice: experiences with this book or others, what you wished you'd learned in your own undergraduate course on this subject, etc. I've taught this course once before -- I feel I did quite well but I still have room to improve. Thanks!

[1] https://www.amazon.com/Discrete-Mathematics-Applications-Sus...

llzes2• 9 years ago

You're the type of professor everyone wishes they had. That's so phenomenal you're looking out for your students like this. That and if you end up writing supplementary material for the book.

Sean1708• 9 years ago

Is it an American thing to force your students to buy textbooks? In Britain (at least at the universities that I know about) there tends to be 2 or 3 recommended texts, of which the library will have several copies, and they're meant for reference when you get stuck rather than learning the entire course.

sn93• 9 years ago

A Schaum's book can be a cheap supplement with lots of solved example problems.

For proofs, you can find the free Book of Proof online [0]. It includes solutions to the odd exercises.

Velleman's How to Prove It is also a great text, though not free.

Regarding math pedagogy, I think this post might prove useful [1].

[0] http://www.people.vcu.edu/~rhammack/BookOfProof/

[1] https://bentilly.blogspot.com/2009/09/teaching-linear-algebr...

pasbesoin2• 9 years ago

$280??!!

I knew prices were "high", but good grief! I guess I really am out of touch.

Glad I held onto my vintage copy. Don't imagine the intervening 20-ish years have added circa $200 (at a guess) of value to the latest edition.

chrisper3• 9 years ago

Yes. But why people buy books is what I don't understand. At least for college and if it's not something you want to own for the future.

Throughout most of my semesters I just rented the books. Much cheaper than buying.

mljoe• 9 years ago

Textbook prices and college costs in general are amongst the most notable products outpacing inflation.

nathanyo• 9 years ago

I took a course that used that textbook last semester and I got the international edition for around $30, and it was the exact same book with less problems.

dizzystar2• 9 years ago

I'm not very mathematical, and I was much less mathematical when I worked that book (calc 1, not calc 2 yet, etc).

I found it highly accessible and fun to work through. The concepts are all well-explained and I don't recall anything that would be out of reach for someone with some probability and foundations in classical logic. If they are at discrete math, this book is well withing their reach.

It doesn't teach hard math per se, but how to think in a mathematical way, showing how to break apparently diabolical problems into manageable pieces, and how to reconsider the perspective of the problem, if that makes sense. The book is among the most mind-expanding books I've ever read.

I'm fully autodidact and far from MIT pedigree.

kayoone• 9 years ago

I am interested in which book you are referring to, the Book on Amazon Op lined too or the MIT material linked in this thread ?

x0x0• 9 years ago

I waffle back and forth on my opinion of scihub and genlib, but ffs, $280 plus tax is simply theft. Don't forget the $90 student solutions manual!

mattmanser5• 9 years ago

Honest question, why would you spend weeks doing proof by induction?

Looking it up, I think I did that when I was 16, but I've never used it while programming. I can't see a single benefit to knowing it for programming. Been programming professionally for 12 years now. What am I missing?

Why do you think it's important?

popcorncolonel• 9 years ago

It's helpful for truly understanding recursion.

tarsinge• 9 years ago

They are the basis of recursive algorithms proofs, sure you may not need them for everyday programming but that's different for computer science

naasking• 9 years ago

Looking it up, I think I did that when I was 16, but I've never used it while programming.

Any reasoning about loops or recursion requires an implicit understanding of proof by induction. Making it formally explicit seems like a perfectly fine idea.

gjm113• 9 years ago

In some sorts of code, you can get a lot of mileage from "loop invariants" and "loop variants". Understanding these is more or less the same as understanding proof by induction.

Whether you are missing anything depends on whether you ever write the sort of code that benefits from this. (Even if you do, you still might not be missing anything. You might be comfortable with loop invariants but not have connected them with proof by induction. Or you might be good at reasoning about these things in other ways that don't involve invariants.)

(You can stop reading here if you're already very familiar with loop variants and invariants.)

Here's the sort of thing I mean by loop variants/invariants. Suppose you're writing a binary search. (Don't write a binary search. Use someone else's that's already had the bugs taken out.) This is basically pretty simple but surprisingly easy to get subtly wrong, which means it's the sort of code it may be useful to write and document in such a way that you have an informal proof of its correctness. Like this:

  function find(a, x):
    # find x in sorted array a; return index i such that a[i]==x
    # or -1 if no such index exists. Takes time at most
    # proportional to log(#elements).
    lo,hi = a.index_bounds()
    while lo<hi:
      # INVARIANT: min index <= lo < hi <= max index+1
      # INVARIANT: if x is in the array then lo <= its index < hi
      # VARIANT: D := hi-lo reduces to at most D/2 next iteration
      mid = lo + floor((hi-lo)/2) # may equal lo, can't equal hi
      if a[mid] == x: return mid # done!
      if a[mid] < x:
        lo = mid+1
      else:
        hi = mid
      # INVARIANT: if x is in the array then lo <= its index < hi
    # now lo >= hi which means there's nowhere for x to live
    return -1

The above is pseudocode in no particular language. Is it correct? Those variants and invariants (1) divide that question in two and (2) provide a strategy for answering the second half of the question. First sub-question: if the (in)variants always hold, does that imply that the code is correct? Second sub-question: do they always hold? (Strategy for second sub-question: proof by induction on array size and/or number of iterations.)

The first invariant implies that since mid is always between lo (inclusive) and hi (exclusive) it's safe to access a[mid]. The second, given that it holds at both start and end of the loop body, implies that if we exit the loop then indeed x isn't in the array. The only ways to leave the function are via the return in the middle, which only happens when we have explicitly found x, and via the return at the end, which only happens when x is known to be absent; therefore, if we return anything, we return the right thing. And the variant says that the quantity D, which starts by equalling the number of elements in the array, is reduced by a factor of at least 2 at each step; as it's always a (strictly) positive integer, this implies that the number of steps is at most log_2(n), so the function does return.

(Note that that last bit is more or less a proof by induction on array size that the function always returns.)

So if the invariants hold then the function does the right thing in the right amount of time.

Proving that the invariants hold is the induction-like bit. The first invariant holds at the start (since lo,hi start out being exactly min index and max index + 1). Each time around the loop we replace either lo or hi with something in between the two, so we still have min <= lo <= hi < max+1; and we will exit the loop unless in fact lo<hi. So the first invariant always holds.

The second invariant holds at the start (since lo,hi are bounds on the entire array). Because the array is sorted, our adjustments to lo/hi make this invariant true again for the next time around the loop. So the second invariant holds at the end of the loop -- and of course therefore at the start of the next iteration.

(Note that those were both proofs by induction, though I wasn't super-explicit about the fact.)

The variant works because if we write D = hi-lo and E = floor(D/2) then the new value of D is, depending on which branch of that if we take, either E or D-E-1; note that D-E-1<=E; so new-D <= E <= D/2 as claimed.

It's probably pretty clear what sort of code this is useful for: highly "algorithmic" code that's basically doing something fairly simple but that's tricky to think about and easy to get wrong. If you're writing a language's standard library, or implementing some sort of iterative mathematical algorithm, then you're quite likely to find it useful. The less like that your code is, the less often this technique will be useful to you. (Fairly extreme example: if you are writing a CRUD webapp then it is unlikely that anything you do will benefit from this.)

jimhefferon• 9 years ago

You need it for a follow-on course, Theory of Computation.

dave1629• 9 years ago

I used this book for my course at the University of Virginia last semester, https://uvacs2102.github.io/.

The book is terrific, and wonderful that it is available under a CC license. Most students found it accessible. We covered about Chapters 1-8 (and some things not in the book). (This is only the first part of the book, so a much slower pace than the MIT course. Some of this is to have more time for additional background and to go in depth in things, but also that our courses are scheduled with about 2/3 the amount of time per class as an MIT course is.

chrisweekly• 9 years ago

https://betterexplained.com/ is hands-down the most transformative free resource for understanding math I've ever encountered.

primodemus• 9 years ago

Buy older or used my friend:

http://marginalrevolution.com/marginalrevolution/2004/09/the...

http://marginalrevolution.com/marginalrevolution/2004/12/the...

https://www.amazon.com/gp/offer-listing/0534359450/ref=dp_ol...

saycheese• 9 years ago

> "the book appears open source"

Yes, these are the terms of the license:

https://creativecommons.org/licenses/by-sa/3.0/legalcode

Webpage for the course is here: https://learning-modules.mit.edu/class/index.html?uuid=/cour...

That page includes an email contact for the course; (6042-gradesmaster@mit.edu). If you're planning to release an edit, I'd reach out as soon as possible to give them time to get back to you on if they'd be interested in working with you and the best way to do so, since you posting it randomly somewhere on the net would likely do less good in my opinion, but would be a huge help if done with them.

bogomipz6• 9 years ago

I appreciate your practical comments. This particular caught my eye:

"Previously I used Epp's book [1] which in my opinion is an outstanding book but regrettably is $280.44."

This is extortionate. What is the justification for a book being priced at nearly $300? This is one book of how many a student will need that semester? Are school's getting a piece of this as well? Is this why such a racket is permitted?

Sorry I know its tangential to your original comment but I would be curios to hear anyone's feedback, especially someone from Europe or elsewhere. I wonder if this pricing is specific to the U.S.

dboreham• 9 years ago

Perhaps those of us who seek to make money from entrepreneurial activities should instead wonder at the business skills of people who are able to extract $300 from their customers for a book?

jcranmer2• 9 years ago

This is extortionate. What is the justification for a book being priced at nearly $300? This is one book of how many a student will need that semester? Are school's getting a piece of this as well? Is this why such a racket is permitted?

That is a bit high for textbooks, most textbooks are usually "only" in the realm of $80-160. That said, math textbooks are usually far more expensive than others.

The excuse given for high prices is that there is a strong used book market for college textbooks, as most students sell off their textbooks at the end of the semester, so from the point of view of the publisher, they see only one sale in 5-20. Of course, publishers discovered that the "cure" for this is to release new editions of textbooks with minor updates (most importantly, reorder the questions in the question banks!) every few years to kill off the used textbook market and haven't lowered their exorbitant prices to compensate.

As an undergrad, I did have one semester where my total book costs were about $1000. Most of the semesters, they were more like $500.

Oh, and the prices usually go to lining the publishers' pockets. Not the authors', not the schools'.

dragonwriter4• 9 years ago

What is the justification for a book being priced at nearly $300?

It's a product of a capitalist, for-profit producer. The justift for its price is that the market will bear it.

feyman_r• 9 years ago

I heard an interesting take on this on the Planet Money podcast. While I still think books are over-priced here, it allowed me to think of different 'forces' that shape book prices: http://www.npr.org/sections/money/2014/10/03/353300404/episo...

jly• 9 years ago

The textbook prices are ridiculous. I buy a lot of textbooks for personal library/study, and I have found for 'classic' texts that have many editions, going back 2-3 revisions can be super cheap. I have purchased many used from Amazon in great condition for $5-10 that cost $100-200+ new in the latest edition. This might not work for a class, but it's been indispensable for my personal use.

mindcrime3• 9 years ago

There are a number of freely available online resources for proofs & mathematical reasoning. Besides the already mentioned "Book of Proof", try:

https://sites.google.com/site/mathematicalreasoning3ed/

http://www.math.vt.edu/people/day/ProofsBook/

http://giam.southernct.edu/GIAM/

For anybody looking for freely available maths related texts online, three great resources are:

http://aimath.org/textbooks/approved-textbooks/

http://web.archive.org/web/20170126161813/http://people.math...

http://www.openculture.com/free-math-textbooks

xenihn• 9 years ago

I'm a big fan of these books:

https://minireference.com/

bloodbeard• 9 years ago

impendia,

I'm currently taking a discrete mathematics class that Applied Discrete Structures by Alan Doerr and Kenneth Levasseur (http://faculty.uml.edu/klevasseur/ads2/). It's freely available, though I recommend steering away from the PDF on their site as it's got quite a few errors that are corrected in the source on Github.

trentmb5• 9 years ago

Margaret M. Fleck at UIUC has one for their Discrete Math course:

http://mfleck.cs.illinois.edu/building-blocks/

devnonymous• 9 years ago

many of our students come in with a weaker mathematical background than MIT students;

Maybe this would be a better first couple of weeks introduction:

http://bookboon.com/en/gentle-introduction-to-mathematics-fo...

More information about BookBoon:

  http://bookboon.com/blog/en/bookboon-com/our-mission-and-concept/
  http://bookboon.com/blog/en/bookboon-com/about-the-concept-and-books/

Note, I have no affiliation with BookBoon, I've just found their textbooks/books very useful.

40acres• 9 years ago

Looks like an excellent resource. What would the prerequisite math skills be before diving into this work?

kevindeasis• 9 years ago

It depends on the person, but what is the median time in hours, for a person to finish this book?

Curious42• 9 years ago

I don't think you should be starting with a goal to finish the book cover to cover. It's an unrealistic, and ineffective goal. Although, if you think that works for you, go ahead.

voycey2• 9 years ago

I failed so hard at this in my CS degree :(

canadiancreed• 9 years ago

You're not the only one. Only course that I took that I crashed and burned on.

cs02rm0• 9 years ago

Yeah. I had what was termed a "tolerated fail" on what I think was the equivalent on my (UK) course - "Foundations of Computer Science". Tolerated as in because I scored high in all other modules I didn't have to resit, fortunately, as I'm not convinced I'd have passed.

I'm not convinced either that it's negatively affected my career. I do occasionally wonder if it was just not taught in a way I could get, whether I was just bored to tears because the theory was so separate from the application, or if it's actually just not that important a prerequisite.

voycey2• 9 years ago

Yeah I had the same, I have been developing now at various levels for 15 years since university and I can honestly say I have never once come across anything that has needed it! Then again I would say seeing as I am primarily web based it's not likely I would vs someone coding realtime systems

trueSlav• 9 years ago

This is so stupid, the "primer on the basics" should be "Algorithmization for Computer Science", not a 900+ page math textbook (that doesn't even explore graph theory, calculus and differential calculus).

Smaug123• 9 years ago

So if I understand correctly, this book is both too long and doesn't cover enough? (I'm confused about the book to which you're referring. If it's "Mathematics for Computer Science", then it does cover graph theory from p445.)

sn93• 9 years ago

Calculus is a prereq for taking the course in which this book is used. I'm pretty sure Differential Equations is also a required course for CS majors at MIT, too.

qntty2• 9 years ago

Read the TOC again, there's plenty of graph theory. Calculus isn't important for CS.

mindcrime3• 9 years ago

Calculus isn't important for CS.

It can be, depending on what you're doing. Look at back-propagation in neural networks for example, where the whole thing is based on the chain rule from Calculus.

dang2• 9 years ago

This breaks the HN guidelines against calling names in comments: https://news.ycombinator.com/newsguidelines.html. Please don't do that here.

We detached this subthread from https://news.ycombinator.com/item?id=13800495 and marked it off-topic.

ode• 9 years ago

On this subject, has anyone taken this (https://www.coursera.org/learn/discrete-mathematics) MOOC on the same subject?

Thoughts?

nvarsj• 9 years ago

Any opinions on how this compares to the Rosen and Epp books? I'm looking for something to prep for a graduate level CS algo course.

lucb1e• 9 years ago

This document is over 900 pages. How long are you supposed to take to read and understand all of this? And is this really all necessary?

On the surface this looks like an insurmountable task with questionable benefits. Don't get me wrong, but in the past 6 years of casual and professional programming, I've needed only a basic understanding of math, the most difficult thing being collision detection in games, and that includes doing RSA cryptography by hand. This paper starts off with proofs, something I've never had in school and which always seemed to me as if they only belong in scientific math papers, not practical life of someone doing computer science.

I don't mean to criticize the document or math in general, I would genuinely like to understand what 900 pages of this is going to bring me, especially when it starts with something I've never needed or seen outside of theoretical math discussions.

Synaesthesia• 9 years ago

It's hardly insurmountable, but to answer your question, maybe people will read it because you're just curious, want to learn. Also you don't have to learn all, or any of it -maybe you could just use it as a reference book.

bboreham• 9 years ago

I have found the mechanics of mathematical proof invaluable when programming, because they show you how to be very clear about what you do know and what you don't know. Basically "Is this thing just a random example, or something that is true in all cases?"

When I work on complex algorithms or concurrent systems I really feel the benefit of proof, because there's no way I could just test if the code is correct.

In the book, explaining what a proof is, how proofs are typically constructed and how to write a good one takes them ten pages, all written in very down-to-earth language. Then there are eight pages of problems you could do if you wanted to test your understanding.

I bet if you read those ten pages you'd feel there was nothing special there, just obvious ways to reason about things.

crphang• 9 years ago

In professional programming, most of the time, system design is most crucial and would use less of these mathematics. These 900 pages is not insurmountable. A discrete mathematics course for year 1 computer science students would have covered a good 70% of it. Such a course takes only 3 months and is 1/5 of a student's workload.

The benefits of these is hardly questionable. Its use is apparent when you take a Design and Analysis of algorithms course like https://www.youtube.com/watch?v=JPyuH4qXLZ0. A good example of using knowledge of the pdf is analysing expected runtime of a hashtable. Which turns out to be theta(1) average case. Good analysis of algorithms inspire better design of it in general.

Data structure and software engineering courses would probably be sufficient for many software engineering jobs out there. Databases, networks, OS and security are good to have knowledge. However, if an engineer is building cutting edge stuff, Mathematics will be his/her best friend.

One good property of Mathematics is that it provide guarantees in the form of equality, inequalities or equivalences. Such guarantees can help you ensure that your system holds quantitatively. It is thus your job to reduce your computer science/engineering problem to a mathematics problem.

I find this remark by Terence Tao particularly good: "If you don’t have mathematical background, the classes you take will help you train to analyze existing systems and build things that haven’t been built before. If you want to design something really new, at some point you’ll have to model what you’re doing, which might be different from previous models, and you have to do some mathematics somewhere." This remark is made in this interview: https://docs.google.com/document/d/1rinL25rC8LnMTzZcGjg1axT-...

data4science2• 9 years ago

Yes, mathematics has been, is, and will be at the cutting edge of software engineering.

For historical examples: consider Turing, Knuth, and Page & Brin.

For current examples: consider High Performance Computing (HPC), Data Science, and Data Engineering.

For future examples: wait and see (or study math in the present).

Also, consider etymology: computers are machines that preform computations; software is code that directs computations and their results on computers; software engineers create code that directs computations on computers; computational scientists define the limits of computing on computers that have given properties, and create novel computational solutions to problems; mathematicians discover the theory that computational scientists (and, more generally, all scientists) utilize.

Note_0: There is a field at the cutting edge of computation, computational science, that requires the most mathematical knowledge of any discipline that includes the compute stem.

Note_1: Math goes beyond providing guarantees of equality, inequality, and equivalence: it is the quantitative basis of computer science (and all science); it provides Boolean logic; it guarantees boundaries/possibilities; and it provides scope.

snmartin• 9 years ago

This is, in fact, a textbook for a discrete mathematics course for year 1 computer science students :)

agentultra• 9 years ago

It depends on what you want to achieve. Strong fundamentals in maths gives you the tools to solve big problems. The more ambitious the problem you want to help solve, like say a traffic-control system for a high-speed train network, then the more maths you will need.

crazygringo• 9 years ago

It depends on what you mean by "read and understand".

My approach with something like this is to breeze through it a first time, just to understand the main point of each section and how it all fits together. If you spend 30s/page on average (obviously there will be a lot of variation), that's 7.5 hrs -- two mornings, afternoons, or evenings.

And for most people, that will provide 90% of the value. And you know what to look up, where, when you need it in the future. And you also know whatever part you might be personally interested in more, and you can go back and read those sections in more detail.

This approach works well with all types of non-fiction, and it can actually be really helpful to first "skim" through every non-fiction book you read, before reading in detail -- knowing in advance where the author is going can often help remove a lot of confusion.

mindcrime3• 9 years ago

There's a lot you can do in programming without knowing a ton of math, just like these days you can drive a car quite well without having a degree in mechanical engineering. But just like with a car, the more you know about the deep underpinnings of things, the more you can do. A person with deep knowledge of how their car works may rebuild the engine and overbore the cylinders for more displacement, put in a camshaft with more lift, and replace the factory exhaust manifold with headers with better flow.

A computer programmer who understands the mathematical underpinnings of his work might replace the standard library implementation of HashMap with one that's better suited to his use case, or do his own derivations to figure out how to implement back-propagation in a neural network, etc.

No, it's not a perfect analogy, but the point stands I think - having the deep knowledge of the theory and foundations in any field enables you to push the boundaries beyond any "black box" that is provided to you.

csense• 9 years ago

It seems like about two semesters' worth of material in here, so 3 hours classroom time + 7 hours homework/self-study time, for 36 weeks, would give 360 hours. 900 pages divided by 360 hours is 2.5 pages per hour, a rather reasonable rate for study (this is not as dense as an academic paper).

This covers intro to proofs, mathematical logic, number theory, graph theory, combinatorics, and probability. In a typical undergrad math curriculum, these would be five or six different courses. So if you have a two-semester course or self-study covering this 900-page textbook, it's probably more efficient than learning the material in the math department by a factor of 2.5:1 or 3:1.

Yes, stuff's been cut. This is more of a survey of these fields with the emphasis on the most important points, and informing you of the most basic parts. So if you encounter one of the more specialized problems that might have been covered in a more specialized course, you'll have hints on where to look, what to Google for, and what sort of reasoning to use.

oerpli• 9 years ago

I just skimmed the whole book and the content were about 70% of what we did in Discrete Mathematics in one term.

A term has ~15 weeks with 4 hours per week.

The exercises are not included in this - so I think 100-150 hours should suffice.

ralmidani• 9 years ago

I'm still brushing up on Calc, but may have to learn discrete math on my own, since Harvard Extension School offers discrete and algorithms during the same semester, and I don't want to wait 2 years before I can take algorithms.

To that end, I bought a copy of the 1995 "C Edition" of Foundations of Computer Science by Aho and Ullman. It combines data structures and discrete math into one massive text.

Does anyone have an assessment of that book? Is the combined data structures and discrete math a good approach? Does the book hold up well against more "modern" ones?

treehau5• 9 years ago

Donald Knuth's "Concrete mathematics" is also a really good one, and can be found used on Amazon for roughly 30 dollars.

MrInvisible• 9 years ago

Was about to recommend this. Best discrete math book out there from my standpoint as an applied mathematician.

hyporthogon• 9 years ago

This is awesome, thanks.

Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.

aashu_dwivedi• 9 years ago

I had used the second edition to refresh my probability before taking a class in AI and found the material succinct and very useful. It's only today that I found out that one of the authors is the CEO of the company I work for.

I work for Akamai Technologies and the author I am talking about is Frank Thomson Leighton.

kylemccann• 9 years ago

Does anyone know where I could find out more about Semantics, Operational Semantics, Denotation semantics and Type Systems?

mattfrommars• 9 years ago

From someone with an engineering background (mechanical engineering undergraduate 2014), do I need to study this if I want to transition into computer science/software developer/engineer post?

GoodbyeEarl• 9 years ago

From a person with only pre-high school math knowledge, what books/pdfs/moocs would you guys recommend in order to prepare for a future computer science degree? I've read that I would need linear algebra, discrete mathematics and calculus. Is that right? Is there any resource that targets compsci subjects?

Mathematics for Computer Science [pdf]

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