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(At the risk of being vapulated and downvoted, I'll ask this here.)

Suppose you work in a field that has nothing to do with the foundations of mathematics, but thanks to MO, you are becoming more and more interested in topics like axiomatic set theory, the different logical systems (intuitionistic, classical, finitist), category theory, type theory, etc. Thus, you want to understand all of this, and you can spend some time learning from books and articles, without any hurry. But you don't want to actually do research is this field. This is the case for me.

Question: Is there any organized way to achive this? Any book recomendatons to achive this goal? At my university there is no one working in this area, hence I have no one to ask this question directly. My background is naive set theory, naive category theory and some basic logic.

Let me explain with an example where I want to get. Suppose you saw the recent Zizek/Peterson debate, and you have read some of their books, you understood their ideas and you have an opinion. But you can't, and don't want, to sit in front of public and debate with either of them. (Of course, you would like to sit and chat with Zizek for hours :))

Thank you very much.

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    $\begingroup$ This Žižek–Peterson debate? What does that have to do with the foundations of mathematics? $\endgroup$
    – LSpice
    Dec 24, 2019 at 14:44
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    $\begingroup$ @LSpice That's why I begun with the phrase "Let me explain with an example where I want to get." You may extrapolate this. $\endgroup$
    – efs
    Dec 24, 2019 at 15:13
  • $\begingroup$ @MattF. I just flipped through that book, and is exactly the kind of book I'm searching for. Thank you very much. $\endgroup$
    – efs
    Dec 24, 2019 at 15:47
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    $\begingroup$ @MattF. : I encourage you to turn your comment into an answer. $\endgroup$ Dec 24, 2019 at 19:35
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    $\begingroup$ "Of course, you would like to sit and chat with Zizek for hours": no I wouldn't. Notice that for many people he's a charlatan. $\endgroup$ Dec 25, 2019 at 23:36

8 Answers 8

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One feature of the foundations of mathematics that poses a special challenge (compared to other branches of mathematics) is that it is very easy to get confused about certain distinctions—truth versus provability, theory versus meta-theory, formal versus informal, syntax versus arithmetic, etc. One book that I think is helpful in this regard is Torkel Franzen's Inexhaustibility: A Non-Exhaustive Treatment.

Beyond that, what I would recommend depends a lot on what you're specifically interested in. A topic that comes up quite frequently on MO is reverse mathematics, and for that, I'd recommend John Stillwell's book, Reverse Mathematics: Proofs From the Inside Out. For type theory, I think that Martin-Lof's original writings are still an excellent place to start.

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    $\begingroup$ <3 torkel, a nice guy i knew online for many years $\endgroup$
    – nomen
    Dec 26, 2019 at 21:01
  • $\begingroup$ What is/are the prerequisite(s) for Martin-Lof's writing's on type theory (especially the one you linked)? $\endgroup$
    – user57432
    Dec 27, 2019 at 4:31
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    $\begingroup$ @user170039: As I recall, there are not a lot of formal prerequisites for reading Martin-Lof's paper, but it will be very helpful to have some prior experience with some kind of sequent calculus for a formal language. Some prior exposure to axiomatic set theory will also help, not so much because he quotes any theorems, but because the motivation for various novel constructions will be clearer if you know how things are traditionally done. $\endgroup$ Dec 27, 2019 at 15:58
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    $\begingroup$ I recently learned about the book Type Theory and Formal Proof by Rob Nederpelt and Herman Geuvers. Besides being much more comprehensive than Martin-Lof's paper mentioned above, it is also beginner-friendly, and useful if you eventually want to move on to learning how to use a modern proof assistant such as Coq or Lean. $\endgroup$ Apr 11, 2022 at 12:40
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I recommend Varieties of Constructive Mathematics, a 1987 book by Douglas Bridges and Fred Richman, which covers constructive vs recursive vs intuitionist vs classical approaches, and has a brief introduction to topos theory too, all in 160 pages.

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I think at least one of the following books will be useful to you. I hope your university gives you access to the books available on Springer Link.

Next are two books on logic.

I recommend you read the book "Philosophical and Mathematical Logic", written by Harrie de Swart. It's very good and requires no prerequisites.

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    $\begingroup$ Why is the material on philosophical logic helpful as an introduction to foundations of mathematics? E.g. when we discuss foundations of math here on MathOverflow, we almost never talk about modal logics or the like. $\endgroup$
    – user44143
    Dec 24, 2019 at 23:48
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    $\begingroup$ @MattF. I think it is important to study philosophical logic because mathematics assumes several axioms of logic. Since there is not only one type of logical system, it is necessary to understand why certain branches of mathematics accept one axiom and reject another (for example, the principle of excluded middle in the case of Constructive Mathematics) . Also, mathematics assumes some rules of inference, so it is worth understanding a little about these things to have a better understanding of the foundation of mathematics. $\endgroup$
    – rfloc
    Dec 25, 2019 at 0:06
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One of my students liked Peter Smith's An Introduction to Gödel's Theorem because it takes the time to explain what's going on.

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I was really interested in this stuff as an undergrad, and spent a lot of my free time learning about it. So here is a roadmap, from a well-informed non-researcher.

  1. Enderton's (or Mendelson's) Mathematical Logic. You need the vocabulary, and to start seeing the "standard machines" asap -- e.g., a logic is a language definition, plus semantics, typically expressed using the vocabulary of model theory. In other words, you need to see the interplay between the language and its interpretation.

  2. Get a good advanced model theory book. You're going to need forcing. (But you're also going to need to understand things like the ordinals and transfinite induction, so some solid set theory or even descriptive set theory should come first)

  3. There are a lot of "philosophical" reasons to be interested in constructive logic. They may or may not be interesting to you. If not, at least read up on Brouwer and his intuitionism on a lazy Sunday afternoon.

  4. Boolos's Logic, Logic, and Logic has some interesting essays on hairs that shouldn't have been split, but also some good essays on interesting ideas (like the "intuitive" equivalence between a strong set theory and a second order logic).

  5. Definitely, learn a programming language like Haskell or ML. "The Gentle Introduction to Haskell" basically runs through the standard machine to teach you the language. Read that, and implement the "standard prelude" from the types. Ultimately, there isn't a big difference between intuitionism and computationalism, so it helps to understand the limits of computation -- i.e. Godel, Rice, etc.

  6. Category theory is a big topic. Topoi are natural objects that sort of implement the logical notion of a model, in category theoretic terms. If you build up topoi, you can do the "standard machine" in category theory. But there is a lot more to category theory than just this! It's basically "just" a powerful modelling language, so it's used in a lot of different applications.

  7. Don't forget lattices, like Davey and Priestley's Lattice Theory. Denotational semantics are another run through the standard machine, with particular relevance to computational languages.

  8. Depending on the direction you want to go, there's plenty of foundational work you can do in "descriptive set theory" i.e., what kinds of sets can you define measures for, if you do it generating from "simpler" by induction on the ordinals? (This is already the beginnings of a simple stratified type theory, specifically suited for real analysis or game theory, etc.)

  9. The newest "powerful" type theory to have textbooks about it is called "Homotopy Type Theory". Again, another run through the standard machine, this time with ordinal/induction constructions too.

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    $\begingroup$ This is a nice list, but some of these items strike me as unnecessarily advanced for someone who doesn't intend to do research in foundations. For example, I don't think that understanding forcing is really necessary if you're not going to do research. $\endgroup$ Dec 27, 2019 at 18:20
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Donald Monk’s book on set theory was an excellent starting point for me; in particular, he does a good job of developing enough ‘intuitive logic’ to present the axioms of set theory without getting lost in the details. He also does a fantastic job of introducing the theory of ordinals/cardinals; nothing hi-tech or fancy, but enough to get a sense for ‘what they are’ and how they behave.

A good followup is Steve Awodey’s book on category theory; it is geared towards logicians more than some other category theory books, but I found it very friendly to read after Monk.

Rounding things out, Bell and Machover’s book on logic is very helpful —- I haven’t read it cover to cover (and couldn’t find a free pdf online), but it is on my shelf and has served very well as a reference on the logical notions pervading the foundations of mathematics.

Any one of these books could probably be used as a starting point; which works best will depend on how you intuitively view mathematics. A good text once you’ve got a feel for the topic is Kunen’s book on foundations; it would be a bit much to dive in cold, but it serves as a great ‘bringing things all together’ text. (Note that the pdf linked above is missing a chapter titled ‘Philosophy of Mathematics’ present in the printed version, which is one of my favorite parts of the book for getting a ‘birds eye view’ on foundations. If your institutions library has a copy, I would highly recommend grabbing it to peruse at your leisure.)

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  • $\begingroup$ I think it’s misleading to praise Kunen’s book for “bringing it all together”: he basically omits all the topics in the top-voted answers here (by Timothy Chow and me). Instead, he starts from the premise that “we study set theory first because it is the foundation of everything” (a premise which I find obviously false given the definite article); on that basis he covers a range of topics in logic with a set-theoretic perspective. $\endgroup$
    – user44143
    Apr 12, 2022 at 5:41
  • $\begingroup$ @MattF. Yes, when I say ‘bringing it all together’ I don’t mean that he literally uses set theory, category theory and logic simultaneously or provides a foundational roadmap between them. I just mean that the tone of the book is more casual than the others in this list, which Kunen leverages to discuss what we mean by ‘a foundation of mathematics’ like he’s talking to someone who’s already familiar with the subject. It is similar to the Feynman lectures on physics imo; not a great first text and certainly not comprehensive, but a master of the craft intuitively explaining their understanding. $\endgroup$
    – Alec Rhea
    Apr 12, 2022 at 8:37
  • $\begingroup$ @MattF. In the version on my shelf, section 0.4 is titled ‘The Foundation of Mathematics’; this appears to be the case in the draft linked above as well. That being said, on further review the book is very set theoretical in flavor; the main part I felt was a nice summary is the ‘Philosophy of Mathematics’ section absent from the above draft. I stand by my assertion that it is a great overview of the mindset behind foundational considerations; I’ll see if I can find a complete free version. $\endgroup$
    – Alec Rhea
    Apr 12, 2022 at 8:58
  • $\begingroup$ Indeed — ‘the foundation’ is the problem, ‘a foundation’ would be the start of something reasonable! $\endgroup$
    – user44143
    Apr 12, 2022 at 9:02
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Strongly recommend Category Theory for the Sciences (David Spivak) if you are on a budget, you can access the book from the author's page, only difference is that solutions to the problems are not in this version.

I am not a mathematician, but find category theory fascinating, the more I study it, the more I see it as one of the fundamental blocks of mathematics.

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    $\begingroup$ This is such a bad reccomendation. $\endgroup$ Apr 12, 2022 at 6:45
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    $\begingroup$ @IvanDiLiberti I'm interested to know why this is a bad recommendation, in your opinion. $\endgroup$
    – efs
    May 18, 2022 at 5:33
  • $\begingroup$ @efs While the book has to mention some topics related to foundations, the book is not about foundations at all and is not written by an expert in foundations of mathematics. It's as if the question was "Learning roadmap for Darwin Evolution theory" and someone suggests a very generic introduction to biology for engeers written by an expert in gardening. It's fine for the book to exists, it just does not belong to the list. $\endgroup$ May 18, 2022 at 6:32
  • $\begingroup$ @IvanDiLiberti I see. Thanks. $\endgroup$
    – efs
    May 18, 2022 at 16:23
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Paul Taylor's book should definitely be on this big list! It's not the book I would start with, but the sooner you open it, the better.

Paul Taylor, Practical Foundations Of Mathematics.

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