(Part 2) Reddit mentions: The best mathematical logic books

I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.

On the CS side:

• Basic automata/formal languages/Turing machines; Sipser is recommended here.
  
• Basic programming language theory; I used University of Washington CSE P505 online video lectures and materials and can recommend it.
  
• Formal semantics; Semantics with Applications is good.
  
• Compilers. You'll need several resources for this; my personal favorites for an introductory text are Appel's ML book or Programming Language Pragmatics, and Muchnick is mandatory for an advanced understanding. All of the graph theory that you need for this type of work should be covered in books such as these.
  
• Algorithms. I used several books; for a beginner's treatment I recommend Dasgupta, Papadimitriou, and Vazirani; for an intermediate treatment I recommend MIT's 6.046J on Open CourseWare; for an advanced treatment, I liked Algorithmics for Hard Problems.
  
  On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
  
  
• You run into abstract algebra a lot in program analysis as well as in cryptography, so it's best to begin with a solid foundation along those lines. There's a lot of debate as to what the best text is. If you're never touched the subject before, Gallian is very approachable, if not as deep and rigorous as something like Dummit and Foote.
  
• Order theory is everywhere in program analysis. Introduction to Lattices and Order is the standard (read at least the first two chapters; the more you read, the better), but I recently picked up Lattices and Ordered Algebraic Structures and am enjoying it.
  
• Complexity theory. Arora and Barak is recommended.
  
• Formal logic is also everywhere. For this, I recommend the first few chapters in The Calculus of Computation (this is an excellent book; read the whole thing).
  
• Computability, undecidability, etc. Not entirely separate from previous entries, but read something that treats e.g. Goedel's theorems, for instance The Undecidable.
  
• Decision procedures. Read Decision Procedures.
  
• Program analysis, the "accessible" variety. Read the BitBlaze publications starting from the beginning, followed by the BAP publications. Start with these two: TaintCheck and All You Ever Wanted to Know About Dynamic Taint Analysis and Forward Symbolic Execution. (BitBlaze and BAP are available in source code form, too -- in OCaml though, so you'll want to learn that as well.) David Brumley's Ph.D. thesis is an excellent read, as is David Molnar's and Sean Heelan's. This paper is a nice introduction to software model checking. After that, look through the archives of the RE reddit for papers on the "more applied" side of things.
  
• Program analysis, the "serious" variety. Principles of Program Analysis is an excellent book, but you'll find it very difficult even if you understand all of the above. Similarly, Cousot's MIT lecture course is great but largely unapproachable to the beginner. I highly recommend Value-Range Analysis of C Programs, which is a rare and thorough glimpse into the development of an extremely sophisticated static analyzer. Although this book is heavily mathematical, it's substantially less insane than Principles of Program Analysis. I also found Gogul Balakrishnan's Ph.D. thesis, Johannes Kinder's Ph.D. thesis, Mila Dalla Preda's Ph.D. thesis, Antoine Mine's Ph.D. thesis, and Davidson Rodrigo Boccardo's Ph.D. thesis useful.
  
• If you've gotten to this point, you'll probably begin to develop a very selective taste for program analysis literature: in particular, if it does not have a lot of mathematics (actual math, not just simple concepts formalized), you might decide that it is unlikely to contain a lasting and valuable contribution. At this point, read papers from CAV, SAS, and VMCAI. Some of my favorite researchers are the Z3 team, Mila Dalla Preda, Joerg Brauer, Andy King, Axel Simon, Roberto Giacobazzi, and Patrick Cousot. Although I've tried to lay out a reasonable course of study hereinbefore regarding the mathematics you need to understand this kind of material, around this point in the course you'll find that the creature we're dealing with here is an octopus whose tentacles spread in every direction. In particular, you can expect to encounter topology, category theory, tropical geometry, numerical mathematics, and many other disciplines. Program analysis is multi-disciplinary and has a hard time keeping itself shoehorned in one or two corners of mathematics.
  
• After several years of wading through program analysis, you start to understand that there must be some connection between theorem-prover based methods and abstract interpretation, since after all, they both can be applied statically and can potentially produce similar information. But what is the connection? Recent publications by Vijay D'Silva et al (1, 2, 3, 4, 5) and a few others (1 2 3 4) have begun to plough this territory.
  
• I'm not an expert at cryptography, so my advice is basically worthless on the subject. However, I've been enjoying the Stanford online cryptography class, and I liked Understanding Cryptography too. Handbook of Applied Cryptography is often recommended by people who are smarter than I am, and I recently picked up Introduction to Modern Cryptography but haven't yet read it.
  
  Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.

Physicist here so don't pretend I don't know what science is. (Though like the ancient Pythagoreans I'm sure as soon as I discuss something that has been proven that goes against a purely scientific worldview out comes the pitchforks.) And though I love science, unlike some people here I am willing to admit to the limits of science. Science can lead to all truth in the same way that rational numbers define all numbers: it can't! and Godel proved it.

The real problem with science is that it has been mathematically proven by Godel that there are more things that are true then are provable and thus you can't ever have a scientific theory that can determine the truth or falsity of all things. As soon as you write down that theory, assuming it allows for arithmetic, Godel's incompleteness theorem immediately shows if the theory is true there will be true statements about reality that are beyond provability. Read Godel Esher Bach or Incompleteness or work through it yourself in this textbook as I have.

So like I said above, science is great in it's sphere (and in that sphere let me emphasize it is awesome!) but leads to all truth in the same way that rational numbers leads to all numbers. (And the analogy is precise since Godel used the famous diagonizational argument in his proof.) Russell and Whitehead set out to show in the early 1900s that if we could determine the axioms of reality then through logic work out everything that was true and Godel spoiled the party.

It it would be one thing if these truths were trivial things, but they are not. Some examples of true or false statements that may fall into this category of being unprovable are:

• Goldbach's conjecture and an uncountable number of mathematical theorems (by the diagonalization argument) for that matter.. (Search the pdf for Goldbach)
  
  
• Issues related to the halting problem in computer science.
  
  
• Issues related to recursive logic and artificial intelligence.
  
  
• And again, this list goes on uncountably.
  
  Now, at this point critics almost always tell me: but Joe, Godel's incompleteness theorem is only relative to your set of logic. (Ie... we can prove Goldbach by just adding axioms needed to do so.) Fine. But two things: (first) adding axioms to prove what you want willy nilly is not good science. (Two) You now have a new set of axioms and by Godel's theorem there is now a new uncountable set of things that are true (and non-trivial things like I listed) that are beyond proof.
  
  Now usually comes the second critique: But Joe, this doesn't prove God exists. And this is true. But at least it has been proven God gives you a chance. It has been proven that an oracle machine is free from the problems that hold science and logic back from proving the truth of all things. At least something like God gives you a chance (whereas science falls short).
  
  Or, like Elder Maxwell says so well: it may only be by the "lens of faith" that we can ever know the truth of all things. He maybe be right, and hence the importance to learn by study, and also by faith...

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)

​

(Link to the proof). After doing the proof, I also did a proof analysis to show where the different decisions in the proof came from. If you think something is unclear, don't hesitate to ask down below. I'm not a Math major but if some of the Math majors here have some suggestions, they're welcome.

​

A detour:

Don't be discouraged! While you can use templates for the different proof techniques, doing mathematical proofs involves a lot of insight and that insight can only be gained trough practice, practice and more practice so that different definitions become second nature to you. I currently taking my first proof-based course (mainly because I failed my Discrete Math class last semester) and if you've told me before "Let p be an odd number", I would've thought nothing of it. Nowadays the first thing that comes to mind is that I could write it as "p = 2n + 1 for some integer n". Although it might not be always the case that you need it in that form, it helps you to extrapolate and relate things. For example, another fact that stems from "p being odd" is that "p^(2) is also odd". My point is that it takes practice and if you're willing to put some effort, you can do it. Remember that it took Andrew Wiles seven years to prove Fermat's Last Theorem and even then his proof wasn't entirely correct which meant he had to go back for another year, find where he went wrong and correct it. Lastly, don't take much of the mathematics we know today for granted; it took centuries and many brilliant minds, who achieved more than what any of us could probably achieve, to develop it. So again, don't be discouraged.

I think one of the easiest way to get a gist of how proofs by induction work is by doing several problems involving some series of integer numbers where you can express the sequence in terms of an arbitrary number k and k+1 (or alternatively, in terms of k-1 and k). For instance, proving that 1 + 2 + 3 + ... + n equals n(n+1)/2.

​

Books that I recommend:

• The Book of Proofs by Richard Hammack (Free)
  
• A Gentle Introduction to the Art of Mathematics by John Fields (Free)
  
• Mathematical Proofs: A Transition to Advanced Mathematics. This one is quite expensive but I plan to get this one when I have some positive cash flow. You can read Chapter 0 for the 3rd edition in Amazon preview.
  
  Youtube series:
  
  
• Discrete Math 1 by TheTrevTutor
  
• Arsdigita- Discrete Mathematics
  
  **
  
  Now I'm off to learn how to prove statements involving sets. Good luck!

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

I would advise you not to start with category theory, but abstract algebra. Mac Lane and Birkhoff's book Algebra is excellent and well worth the money in hardback. It covers things like monoids, groups, rings, modules and vector spaces, all of which are -- not coincidentally -- typical examples of structures that form categories. Saunders Mac Lane invented category theory along with Samuel Eilenberg, and Birkhoff basically founded universal algebra, so you cannot find a more authoritative text.

Edit: The other thing that will really help you is a basic understanding of preorders and posets. I don't have a book that deals exclusively with this topic, but any introduction to lattice theory, logical semantics or denotational semantics of programming languages will treat it. I would recommend Paul Taylor's Practical Foundations of Mathematics, though the price on Amazon is very steep. You can look through it here: http://paultaylor.eu/~pt/prafm/

There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)

Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!

The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!

The Stanford Encyclopedia of Philosophy is a gem. It contains articles on any topic of relevance to philosophers, typically with a great deal of attention paid to the history.

You will likely want to look at modal logic. Apparently this is a good introduction.

As for history, this book and this book will be very, very good.

There are a number of excellent (scholarly) survey articles on certain subjects within the philosophy and history of mathematics, but I would need more specific guidelines on what you'd like to learn, and what you know.

Finally, if you dig through the archives of the n-category cafe you can find some interesting posts and discussion from working mathematicians and philosophers of math... You will have to do some digging, as most of what's there is pure math or mathematical physics, but the more philosophical posts have wonderful discussions.

The specific example I had in mind is the one I mentioned in my response to esthers, but I had quite a few more experiences like that when studying A Logical Journey and the Logical Investigations. Both books are impressively obtuse, but also have a very high idea density and reward thorough study. (You might not want to start out with them, though, since they're the kind of books that make you stop and think for half an hour on every page just to keep up with what they're saying.) Another book that's easy to read and unreasonably effective for problem definition is The Back of the Napkin. It doesn't look like much, but it presents a fully-developed methodology for problem definition using visual cognition, based on results from current neuroscience. Out of all the books I've mentioned in this thread, the Back of the Napkin is the one that I use all the time as a desk reference, so you may find it worth your while to pick up.

I can't say in advance what sorts of books/ideas will have the same impact on you, so I encourage you to do a sampling of the field, find out which authors grab your attention the best, learn everything you can about them, and then work your way outwards into more divergent schools of thought.

Hey cool, thanks a ton!

I have been looking at logic and set theory a lot lately so this is all pretty helpful.

If you were interested, I could add to the list a textbook I'd recommend - http://www.amazon.com/gp/product/0716730502/ref=oh_aui_detailpage_o03_s00?ie=UTF8&psc=1

EDIT: the book really goes through logic in a way i can understand, it may be too elementary for you though.

Either way, I hope that is a useful weapon to add into the ol' arsenal!

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

Just to add a little more philosophy to what was said above:

Platonism (the theory of mathematics we are taught for most of our life) treats mathematics as something we discover. It asssumes that all proofs already exist `somewhere' and we just uncover them. Essentially all proofs are, a priori, true or false.

Intuitionism takes the constructive approach mentioned above. In the intuitionistic foundations of mathematics we don't have a distinct notion of a priori truth as we do not assume that everything is either true of false from the outset. Mathematics is treated as a human construction and our notion of x is true'' in intuitionism is actuallywe have a proof of x''. This means that we can avoid dubious ideas such as a completed infinity and universal excluded middle.

If you are interested in learning about the theory of intuitionism this dialogue is really handy : https://www.amazon.co.uk/Intuitionism-Introduction-Heyting/dp/B000JNON1U

If you are more interested in the mathematics then I can reccomend this book : https://www.amazon.co.uk/Constructivism-Mathematics-Vol-Introduction-Foundations/dp/0444702660/ref=sr_1_2?s=books&ie=UTF8&qid=1463047729&sr=1-2&keywords=Introduction+to+constructivism

For an old, but very servicable text, on the subject that discusses a lot of crucial ideas I can reccomend this book (which has been like my bible during my PhD thesis) : https://www.amazon.co.uk/Elements-Intuitionism-Oxford-Logic-Guides/dp/0198505248/ref=sr_1_1?s=books&ie=UTF8&qid=1463047805&sr=1-1&keywords=Elements+of+intuitionism

Finally, if you would enjoy something a bit more lighthearted on the subject of constructive mathematics (written by a solid intuitionist) I can reccomend : https://www.amazon.co.uk/Truth-Proof-Infinity-Constructive-Reasoning/dp/9048151058/ref=sr_1_2?s=books&ie=UTF8&qid=1463047950&sr=1-2&keywords=Truth+proof+and+infinity

Ah, cool.

I used Understanding Symbolic Logic in my logic course. I was not a huge fan of it, but it is not a bad book. There were just a few things that Klenk does that even the professor said were 'meh' but weren't wrong. She said she was going to change textbooks but that was 3 years ago, and it's still the text they use.

It's got tons of books listed below that are also probably pretty good companions. But yeah, it really is just a ton of working the proofs and truth tables and everything.

> Never read it, will google them after this reply.

It's so fucking cool it's unreal. Not up to date with recent developments but wanna check it out again properly soon.

>Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.

That'd explain why it was so dense lol - I dived from no mathematical logic (apart from like basic predicate calculus) and using first order symbols sparingly.

>There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommendthat you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626
>
>Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.

Thanks :D I'll check it out. Given your breadth of knowledge on it I imagine your background is pure mathematics?

Sorry. The sentence is not circular. It only appears to be circular. The ideas are clearly explained in text books on computability theory. Or if you are smart and patient you can just read Turing's original paper, or if you are really smart and really patient Goedel's work as well. Take your time and use your own mind to form your own opinion.

https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

My introduction was in this book:
https://www.amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/0486614719

For more original sources:
https://www.amazon.com/Undecidable-Propositions-Unsolvable-Computable-Mathematics/dp/0486432289#reader_0486432289

Certainly recommend getting a copy of Rosenlicht if you're in this situation. This is what I'm currently reading in preparation for an honors-level analysis class. Should be pretty easy to follow even without much of a strong proof-based background.

As far as proof mechanics and mathematical foundations go, I recommend "How to Read and Do Proofs". It's not a bad reference to have at your side, just in case.

> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.

The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.

If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.

So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.

> What's a good textbook for that field, anyway?

The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.

> Second and third semester calculus

Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.

In general Stewart's big fat calculus book is a nice thing to have for autodidacts.

Obviously what you describe might include analysis, which these books won't help with.

>Formal logic theory (Think Kurt Godel)

I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.

That's a great book. There's a good section about triangle numbers relating to squares as well.

Another good one with lots of visual proofs is QED, which features diagrams showing geometric formulas (such as the volume of a pyramid).

I don't know if there is any useful distinction between the two. Metamathematics may be thought of as strictly the reduction of math to a single, consistent formal system, whereas philosophy of mathematics can also be concerned with epistemology and existence (as well as metamathematics).

Well, anyways, good reading for this stuff that's also "pragmatic" (usefulness of the concept of pragmatic questionable in this context) starts with theoretical computer science.

No, wait, it starts with internet encyclopedias. I'm not going to link to Wiki, you can do that yourself. You however might not know about Stanford's encyclopedia of philosophy:

http://plato.stanford.edu/entries/philosophy-mathematics/

Okay, now for books. You should know logic first. I suggest Hinman, "Fundamentals of Mathematical Logic". It's a big part of fundamental mathematics, and touches a lot of its philosophy.

For computer science, you want to look at complexity theory I think. That should cover things like decidability and the Halting problem. You should be able to find a good book for free from the Prof. Arora:

http://www.cs.princeton.edu/~arora/

For more philosophy-oriented books, I don't know. Good luck!

http://www.amazon.com/Q-E-D-Beauty-Mathematical-Proof-Wooden/dp/0802714315/ref=cm_cr_pr_product_top

It's a nice little book, very small, notebook sized and only 60 or so pages, but it's quite enjoyable. :)

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

I guess you're looking for a book that focuses on Number Theory only, and goes really in depth. I don't have any recommendations there, unfortunately. But if you like "pure math" and need an introduction book on a variety of topics, I recommend A transition to Advanced Mathematics : A survey course

It has about 90 pages dedicated to number theory alone, and have chapters for abstract algebra/set theory, groups, formal logic, introduction to mathematical proofs, real analysis, complex analysis, combinatorics and graph theory. The chapter on number theory goes through what primes and primality is, inspecting polynomials and diophantine equations in combination with Fermat's last theorem() and show how to rigidly define rationals with integers, etc. The whole book (minus the answer section) is 600 pages, and it has a lot of exercises.

I'm not sure if this book would be useful for you or not. Partially because I haven't gone to school in a long time (A lot of it could all be really undergrad'ish material for all I know), and in part because you didn't mention your grade level. But it might be worth your time to give a peek.

() It only demonstrates Fermat's n=4 proof though; not Andrew Wile's more general proof that introduced elliptic curves.

There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.

• Bridger: Real Analysis - A Constructive Apporach
  
• Bell: A Primer of Infinitesimal Analysis - not constructive per se, but an exposition of non-classical mathematics.
  
• Vickers: Topology via Logic
  
• Lombardi: Commutative Algebra - the odd-one-out in that one does need a good working knowledge of classical commutative algebra to read the book, but it presents a whole lot of genuinely new material textbook-style.
  
• Taylor: Practical Foundations of Mathematics - Chapter 3 is an intro to constructive order theory.
  
• Troelstra: Choice Sequences - again, hardly constructive, but a good exposition of an important part of intuitionistic mathematics.
  
  While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.

Never read it, will google them after this reply.

Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.

There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommend that you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626

The price is outrageous, so get a pdf here if available.

Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.

Since mathematical logic has split after the Second World War in four or five branches, it is uncommon for more advanced texts to have a broad focus on logic in the general sense like beginner's books.

That being said, the most 'advanced' (and quite recent) logic book that attempts at doing justice to the main (though pretty introductory) results of all the major branches of logic is Hinman's Fundamentals of Mathematical Logic (it claims to be an update on Shoenfield's Mathematical Logic, which has some more info on proof theory).

In the spirit of the answers below, where 'advanced' is taken to mean most unreadable or with the most prerequisites, I would nominate another of Shelah's books, Classification Theory.

If you want to learn elementary set theory from the "inside", I recommend Axiomatic Set Theory. It's old, but everything in it is still valid and relevant to modern mathematics, and it's easy for self-study. You can learn "the" set-theoretical construction of the integers (i.e., not the Zermelo version), the reals, and so on, in a very rigorous context, yet it's not difficult for anyone who's had a basic course in logic.

This book does not treat set theory from the "outside", e.g., with modern techniques like forcing. That's a good thing, tho, 'cause the fundamental theorem of forcing is about 100x more difficult to fully understand than the fundamental theorem of algebra.

You need to develop an "intuition" for proofs, in a crude sense.

I would suggest these books to do that:

Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.

How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.

How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.

Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."

Have you tried the print shop at the university? they are used to binding dissertations for people and have some nice options (at least at my local one) .....in that general vein its a printing service you want, not a publisher , so try local print and copy shops ....ones with digital printing are better for short run things and one offs. Most of these places wont care what you are printing , they might ask you to sign a box saying you have the right to print, but that's as far as it goes.

Also seems like there are a few books available of his work:
http://www.turing.org.uk/sources/biblio.html

https://www.amazon.com/exec/obidos/ASIN/0198250800/alanturingwebsit

https://www.amazon.com/exec/obidos/ASIN/0486432289/alanturingwebsit

https://www.amazon.com/Alan-Turings-Systems-Logic-Princeton/dp/0691155747/alanturingwebsit

This is the book I used at university. I thought it was pretty good. Velleman's book is also popular. I've heard good things about this book, but I've not read it.

> Why do people push Velleman's "How to Prove It" so much?

Well the alternative book to this rather narrow subject (textbook speaking) is the classic (i.e. has been around for ever - the early eighties) is this book, "How to Read and Do Proofs". The advantage Velleman's book has are the included exercises which the former does not even have. Reading Amazon's reviews indicates that Velleman's book has a more favorable reception with reviewers liking the prose.

>I thought the book was very lacking, especially in the exercises.

I really haven't met many exercises/"math problems" that I really "liked"! Maybe that's why some people call them "problems".

I heartily recommend Solow's How to Read and Do Proofs which would serve as a good foundation for thinking about math.

I actually found Hughes & Cresswell's A New Introduction to Modal Logic to be a great text.

To add to gnomicarchitecture's suggestions, there's also Klenk's 'Understanding Symbolic Logic', it's a great book with lotsa exercises.

Frege's calculus / symbolic lexicon was most obscure... However, his essay 'On Sense and Reference' is a good read, pre-analytic / phil.of lang. stuff.

As far as the Tractatus (link to a very funky PDF) is concerned, I'd just go ahead and read it, really.

edit btw, regarding Klenk's book: there at least used to be used paperback versions of previous editions (at least on the British Amazon) that were dirt cheap; I've also seen ebook/pdf versions on ebook/torrent sites, I think I decided that I had to own a physical copy after skimming through my pirated version..

Q.E.D.: Beauty in Mathematical Proofs by Burkard Polster.

Read this book: Proof, Logic and Conjecture. I read and studied it for a class that was a bridge course going from basic calculus types of mathematics into formal proof and analysis. If you understand this material, there pretty much is nothing stopping you from understanding anything you want afterwards.

Isn't that true of any subject one likes?
Regardless, besides the Linear Algebra textbook, here are some books you should look at as well. These should give you a taste of what your introductory classes might be:

http://www.amazon.in/Transition-Advanced-Mathematics-Survey-Course/dp/0195310764

http://www.amazon.in/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025

PM me if you want pdfs.

Smith's Introduction to Godel's Theorems is very good. So is Franzen's Godel's Theorem

I liked David Marker's book Model Theory: An Introduction (Amazon, Google Books, SpringerLink, Errata)

You might also want to see the responses to this MathOverflow question.

I found this book immensely helpful.

Really? Nolt's Logics? Besides the numerous errors, it's telling that the book has not come out in a second edition.

I think Quine's Methods of Logic remains a fantastic text, if it is a bit dated and filled with Quinean quirks. A more recent text, Ted Siders' Logic for Philosophy is also very good, although the exercises are sometimes quite difficult. I would combine Sider's text with a book on metalogic, since he skips over some of that. Kleene's Mathematical Logic is a classic text by a real giant in the history of 20th century logic. Those should keep someone busy for a good year of study. If you want to branch out, Graham Priest's Introduction to Non-classical Logics will get you started in modal, tense, epistemic, paraconsistent and dialethic logics, also by a contemporary giant in the field.

After that, I would go on to set theory, and stop when I had a grasp of forcing.

I haven't used the set theory books myself so I can't comment on their quality, but anytime I hear someone looking for reasonably priced math books I immediately think of the Dover Books on Mathematics series.

https://www.amazon.com/Theory-Logic-Dover-Books-Mathematics/dp/0486638294

https://www.amazon.com/Axiomatic-Theory-Dover-Books-Mathematics/dp/0486616304/ref=pd_bxgy_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486616304&pd_rd_r=c84f7c07-350b-4ec3-8fc5-adf30ae9b20c&pd_rd_w=74PSR&pd_rd_wg=RG95z&pf_rd_p=a2006322-0bc0-4db9-a08e-d168c18ce6f0&pf_rd_r=RKXGP4020J1B5GVED0PH&psc=1&refRID=RKXGP4020J1B5GVED0PH

https://www.amazon.com/Book-Theory-Dover-Books-Mathematics/dp/0486497089/ref=pd_sbs_14_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486497089&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96

https://www.amazon.com/Naive-Theory-Dover-Books-Mathematics/dp/0486814874/ref=pd_sbs_14_1/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486814874&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96

edit: added more books

What is Mathematical Logic? by Crossley will give you the foundations to understand the theorem in full.

It depends what you're trying to get out of it.

There are literally hundreds of introductory texts for first-order logic. Other posters can cover them. There's so much variety here that I would feel a bit silly recommending one.

For formal tools for philosophy, I would say David Papineau's Philosophical Devices. There's also Ted Sider's Logic for Philosophy but something about his style when it comes to formalism rubs me the wrong way, personally.

For a more mathematical approach to first-order logic, Peter Hinman's Fundamentals of Mathematical Logic springs to mind.

For a semi-mathematical text that is intermediate rather than introductory, Boolos, Burgess, and Jeffrey's Computability and Logic is the gold standard.

Finally, if you want to see some different ways of doing things, check out Graham Priest's An Introduction to Non-Classical Logic.

>EDIT: Wink? ;)

Yup. (I was trying to be facetiously "cute"... so many Q.E.D. proofs end up as geometric drawings! )

But on the serious side, yes. Keynes liked to "play around" with math-like "quasi-proofs" (and to expropriate and abuse physics concepts) and succeeded in baffling many with what was basically a lot of literary "fancy footwork" -- but his assumptions are so many (and so questionable) and he piles on so many things as "givens" (when they in fact are nothing of the sort -- falsely dismissing the inevitable failure of his equations with "in the long run we're all dead" and similar faux-witty rhetoric) -- that nothing of his really qualifies as a Q.E.D. He simply ignores that anything really NEEDS to be demonstrated.