Not quite the question you asked, but "Classic Set Theory for Guided Independent Study" by Derek Goldrei is a great self-study intro to ZFC, which is the formal foundation of any other math you'll read about. I think the early chapters are simple enough that they'd make good practice for reading and writing proofs (though I didn't encounter it until I already had some experience doing that. I still think it's an exceptionally good book for self-study, though)
With proofs, I also think there are three layers:
1. How do I draw logical conclusions from premises. This is the most straightforward part.
2. What are some of the clever tricks mathematicians use for doing this (e.g. constructing non-intuitive counterexamples, finding equivalences between two seemingly incompatible things, etc). This requires reading other proofs, and is slower, but can be very fun if you like math and find clever proofs beautiful.
3. Finding the right English words and phrases to capture the logic you have in mind (the language used in proofs is not normal English, and it has its own idiosyncrasies and conventions. Like other mathematical notation, it's often specific to particular fields of math and sometimes to a specific author). This also requires reading proofs, I think, and is also where one benefits the most from formal instruction ("how do I say X in my proof?") but I think you can get there on your own with some persistence. It's not Klingon either—proofs are supposed to be readable—but it's a bit like code, maybe, or legalese. If you just try to write nice prose, other mathematicians may find it confusing or non-rigorous.
It can be good to separate the three. Specifically, when learning a new field of math, the proofs sometimes don't feel rigorous to me right away, but once I get used to the the basics and the linguistic conventions, I'm more able to fill in the holes in my head
> Specifically, when learning a new field of math, the proofs sometimes don't feel rigorous to me right away, but once I get used to the the basics and the linguistic conventions, I'm more able to fill in the holes in my head
the most complicated proofs i ever dabbled in were things like proofs of convergence for algorithms, but i encountered various types of proofs in several intro courses: computability and theory of computation, discrete math, introduction to higher math and then some upper div cs courses in ai/ml.
what you say here rings absolutely true to me. as someone coming in with a long time background in coding and computers, i always found myself wanting to apply the same unambiguousness and precision that one uses to express computations in a programming language in mathematical proofs. this was a huge stumbling block for me! proofs are written in shorthand (!) by humans for other humans who have the same base knowledge, obvious things are omitted. once you have that knowledge it makes sense, but before that it seems like giant leaps are being taken without a rigorous line of reasoning between them.
to answer op's question: "an introduction to mathematical reasoning" by peter eccles was helpful for me. it's basically an expanded version of what you'll find at the start of many intro cs/math courses. another option, if sets, number theory and such are confusing to you is the computability angle (cs theory, computation). personally i found this material a lot easier to reason about which then imputed confidence, which is really the magic ingredient for good proofs.
I think that if you want to go away from "proofs written in shorthand by humans for other humans", then you are talking about formal verification of proofs. So, you are talking about something like making your proofs understandable to Coq or Isabelle, or maybe reading how others did it: you can start from http://us.metamath.org/
With proofs, I also think there are three layers:
1. How do I draw logical conclusions from premises. This is the most straightforward part.
2. What are some of the clever tricks mathematicians use for doing this (e.g. constructing non-intuitive counterexamples, finding equivalences between two seemingly incompatible things, etc). This requires reading other proofs, and is slower, but can be very fun if you like math and find clever proofs beautiful.
3. Finding the right English words and phrases to capture the logic you have in mind (the language used in proofs is not normal English, and it has its own idiosyncrasies and conventions. Like other mathematical notation, it's often specific to particular fields of math and sometimes to a specific author). This also requires reading proofs, I think, and is also where one benefits the most from formal instruction ("how do I say X in my proof?") but I think you can get there on your own with some persistence. It's not Klingon either—proofs are supposed to be readable—but it's a bit like code, maybe, or legalese. If you just try to write nice prose, other mathematicians may find it confusing or non-rigorous.
It can be good to separate the three. Specifically, when learning a new field of math, the proofs sometimes don't feel rigorous to me right away, but once I get used to the the basics and the linguistic conventions, I'm more able to fill in the holes in my head