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> But I think it would be helpful for many people to first learn logic and deduction "the precise way", and then do actual mathematical proof where you can jump over more obvious parts.

Absolutely. And I can not imagine a mathematics degree which doesn't place formal logic front and center in the first semester. When you are starting out you absolutely need to suffer through the rigourous formal arguments.

>Ironically, the one subject which often teaches formal logic as an early introductory class for undergraduates isn't mathematics, computer science, or physics, it's philosophy.

I can not believe that this is the case. Even engineering undergrads have to learn formal logic in their first semester.

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> Even engineering undergrads have to learn formal logic in their first semester.

I got my engineering B.S. from a top 5 college in the US, and have known many people who have gone to similar schools. None of us have had to take a class that goes into first order logic or proof writing. I don't know what college you go to where that is a thing, but it would be exceedingly rare.

>I don't know what college you go to where that is a thing, but it would be exceedingly rare.

German technical university. I was a tutor there. AFAIK this is normal.

As part of the first semester you take linear algebra and analysis, starting out with the basics of formal logic. Of course the courses are less focused on proof writing than the mathematics "major" courses.

I should also point out that German universities have very loose entry standards (except when places are very rare compared to applicants) and use the first two semesters to filter out students. These courses are often designed to have around a 50% failure rate.

Your student filtration system sounds like a long and expensive waste of everyone's time.

Not really. First year courses usually have hundreds of students in large halls. There is some more effort as you need more tutors, but that is basically it. (Students usually do not live on campus)

The enormous upside is that all students are judged equally on their ability to academically succeed in their chosen field. I think US university admissions are ridicolous for many reasons.

> As part of the first semester you take linear algebra and analysis, starting out with the basics of formal logic

There is not enough time to learn the basics of formal logic and linear algebra and/or analysis in a single class, but I think what you’re referring to is an introduction to proof techniques like induction, modus tollens, quantifiers, etc.

Every math and computer science department in the US that I’ve ever heard of teaches these topics, but I wouldn’t call it a formal logic class.

For me basic formal logics means learning the symbols (conjunction, disjunction, implication, equivalency, not, etc.) and the rules of inference to maniuplate these symbols and using these rules to prove new things.

How can you teach analysis without that anyway. It is absolutely essential for set theory and how would you e.g. define the reals (in a "proper" math course, not engineering) without a good understanding of set theory?

If you don't believe me, here is a link to the contents of a first semester engineering math course from some german technical university: https://page.math.tu-berlin.de/~joswig/teaching/notes/Joswig...

The symbols should be enough to tell you what the contents are.

> For me basic formal logics means learning the symbols (conjunction, disjunction, implication, equivalency, not, etc.) and the rules of inference to maniuplate these symbols and using these rules to prove new things.

That's really only baby logic. Which, probably, is what will be sufficient for most mathematician most of the time.

A real first introduction to formal logic would introduce an actual formal proof system and go at least as far as proving completeness of first order logic.

>That's really only baby logic.

Instead of (literally) infantalizing the name, you could also call it basic formal logic.

I don't want to quibble about names (and I wasn't the person to come up with the term "baby logic"), but the point is that just introducing a couple of connectives and proof strategies doesn't even constitute the basics of what mathematical logic really is about. Which btw I'm perfectly fine with, most people don't need more than that.

If you do want to study logic formally, the basics start with well-formed formulas, signatures, etc.

I guess what you call it doesn't matter a lot, but the discussion seems to have started with the assertion that most students, even in mathematics, never really learn formal logic, and I would agree with that (under my definition of "formal logic"), while also agreeing with you that you can't pursue a degree in maths without knowing how induction works or what a bijection is. But still, most people don't need to know exactly how to formalise induction and that it's actually (in its full form) a second-order axiom.

That level of logic taught to every mathematics and computer science student, and it’s really not what I was thought others in this thread were talking about.

Formal logic is taught in Mathematical Logic class (intermediate, optional) or Discrete math (intro/intermediate, often not taken)

Geometry class in high school sometimes teaches some of it.

Computer science Binary logic teaches some of (De Morgan's laws)

Outside of New Math of the 1970s, it is a glaring omission from the curriculum.

Even enriched classes like Art of Problem Solving that put heavy emphasis on proofs, do not teach formal logic.

To be fair, I think some people just quickly and intuitively "get" what's expected from them when doing proofs, without any formal introduction. But for others (like me) it is very much helpful to at least list the basic natural deduction inference rules, and to do a bunch of exercises where they have to use these rules explicitly. Otherwise they are floating in thin air, with only a hazy idea of what a "valid logical step" even is.

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