This mostly started because I was trying to learn stochastic differential equations and to a lesser extent topology. I became unsatisfied with my understanding of set theory (not sure how to answer questions like “when I construct a set, what am I iterating over?”), and to a lesser extent measure theory. When I went to get the foundations of set theory, I realized I wasn’t even very familiar with first order logic, and I continued down the rabbit hole.
At the moment I am not especially interested in questions like “is this theory consistent”. I am primarily interested in how one does the fundamental theories of math in a way that bottoms out, meaning I can see and enumerate the notions or procedures I am just taking for granted or defining. If propositional logic was just constructing a specific context free grammer and saying statements constructed in this manner are called ‘proofs’ for this grammar I think that would satisfy me (though it doesn’t look like this is all logic involves). I could easily be using the phrase “foundations of math” incorrectly; please tell me if I am.
Then foundations texts are not what you’re looking for. If I understand you correctly, you seem to be confused about the way sets and other basic constructs are used in normal mathematical prose, and you’d like to learn formal logic and formal proof systems, and then use this knowledge to tackle your problem.
Unfortunately, that’s not a feasible way to go, because to learn metamathematics, you first have to be proficient in regular mathematics—and even when you learn it, it won’t help you in understanding standard human-friendly math texts, except insofar as the experience improves your general math skills. Moreover, formal set theory is about esoteric questions that are very rarely relevant for non-foundational areas like differential equations, in which informal naive set theory is nearly always adequate. (In topology you might run into foundational issues, depending on what exactly you’re after.)
So, what you really need is an introductory text about classical mathematical reasoning. I’m not familiar with any such books in English, but the book nhamman recommended (How to Prove It) seems to be exactly what you’re looking for, judging by the Google preview.
I’m not sure what you mean by “necessarily metamathematical.”
Propositional logic isn’t powerful enough to be of that much use in metamathematics. Its main applications are technical. Most notably, it’s the fundamental basis for digital systems, but it’s also used in various methods for optimization, formal verification, etc. Consequently, it also has huge importance in theoretical computer science.
First-order logic, on the other hand, is principally a tool of metamathematics. Sometimes it’s used in a semi-formal way as a convenient shorthand for long and cumbersome natural-language sentences. But its principal applications are metamathematical, and its significance stems from the fact that it’s powerful enough to formalize “normal” mathematics, which then enables you to reason about that formalism mathematically, and thus examine the foundations of math using mathematical reasoning. (Hence the “meta.”)
This mostly started because I was trying to learn stochastic differential equations and to a lesser extent topology. I became unsatisfied with my understanding of set theory (not sure how to answer questions like “when I construct a set, what am I iterating over?”), and to a lesser extent measure theory. When I went to get the foundations of set theory, I realized I wasn’t even very familiar with first order logic, and I continued down the rabbit hole.
At the moment I am not especially interested in questions like “is this theory consistent”. I am primarily interested in how one does the fundamental theories of math in a way that bottoms out, meaning I can see and enumerate the notions or procedures I am just taking for granted or defining. If propositional logic was just constructing a specific context free grammer and saying statements constructed in this manner are called ‘proofs’ for this grammar I think that would satisfy me (though it doesn’t look like this is all logic involves). I could easily be using the phrase “foundations of math” incorrectly; please tell me if I am.
Then foundations texts are not what you’re looking for. If I understand you correctly, you seem to be confused about the way sets and other basic constructs are used in normal mathematical prose, and you’d like to learn formal logic and formal proof systems, and then use this knowledge to tackle your problem.
Unfortunately, that’s not a feasible way to go, because to learn metamathematics, you first have to be proficient in regular mathematics—and even when you learn it, it won’t help you in understanding standard human-friendly math texts, except insofar as the experience improves your general math skills. Moreover, formal set theory is about esoteric questions that are very rarely relevant for non-foundational areas like differential equations, in which informal naive set theory is nearly always adequate. (In topology you might run into foundational issues, depending on what exactly you’re after.)
So, what you really need is an introductory text about classical mathematical reasoning. I’m not familiar with any such books in English, but the book nhamman recommended (How to Prove It) seems to be exactly what you’re looking for, judging by the Google preview.
Thanks for your help, I think you’ve clarified a lot for me.
Would you classify propositional logic/first-order-logic as necessarily metamathematical?
I’m not sure what you mean by “necessarily metamathematical.”
Propositional logic isn’t powerful enough to be of that much use in metamathematics. Its main applications are technical. Most notably, it’s the fundamental basis for digital systems, but it’s also used in various methods for optimization, formal verification, etc. Consequently, it also has huge importance in theoretical computer science.
First-order logic, on the other hand, is principally a tool of metamathematics. Sometimes it’s used in a semi-formal way as a convenient shorthand for long and cumbersome natural-language sentences. But its principal applications are metamathematical, and its significance stems from the fact that it’s powerful enough to formalize “normal” mathematics, which then enables you to reason about that formalism mathematically, and thus examine the foundations of math using mathematical reasoning. (Hence the “meta.”)
Thanks, that clears up quite a bit.