Importance of the role of First Order Logic (FOL) in understanding higher level mathematics

In this case FOL is essentially referring to proof techniques
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I would guess that the pedagogical value of learning something like first order logic before undergrad algebra/topology/analysis is less about the specific content and more about getting used to and getting practice in formal expectations in math (specifically, advanced undergrad/early grad).

The expectation is that you need to take a semi-formal sentence formulated within a semi-formal language, set of axioms, and key theorems and generate a semi-formal proof about it. The motivation and context in which these sentences originally found value are missing, and the only mode of work that is available is this semi-formal setting. Therefore, you need to be ready to work without context and motivation while still making some kind of progress in understanding - which is what this semi-formal setting allows.

But group theory is unintuitive, and so people can have a hard time if they have to learn this formality AND group theory at the same time. On the other hand, logic (at this level) is pretty intuitive. We can reason about how things "should" be and so we're grounded in something familiar and building up the formal on top of it. And since you're already familiar with the structure of a theorem  or how to use symbols to make a proof, you'll be able to focus on the group theory part of a group theory class.

So First Order Logic is a good way to train these skills. A different question would be whether or not we actually need Abstract Algebra to be done in this formalized setting? It's not like "real" math is just people solving dry textbook problems that have no meaning or motivation. A more pedagogically friendly context for Abstract Algebra could also fix the problem of people hitting a wall in the course, but it would take work from professors to try and think about different ways of teaching which is the last thing many of them want to do.
As others mentioned, what the tweet author calls first order logic is actually just mathematical rigor.

Mathematical rigor is indeed a prerequisite for abstract algebra, or indeed for any serious mathematics. It is indeed true that there is no book that will teach you mathematical rigor. It's like trying to learn how to swim by reading a book. The book is of no use to you unless you actually get in the water and practice.

There are some books that cover this material, such as Velleman's How To Prove It, but the book is not the main requirement. You need to practice what's in the book.
When I started university in Italy, I came from a poor math background, and had absolutely no knowledge in formal languages or even propositional logic. First semester was a pain; every course took for granted a basic understanding of first order language/logic and logic (propositional logic, quantifiers, proofs by contradiction, and so on). On the second semester I took a mathematical logic course, essentially first order logic. After that, everything went painlessly: you get know what's going on, how to follow a proof and tell if everything's going well or if there are some mistakes. It's just knowing the "rules" of the language, making it easier to express yourself and understand others.
First order logic is rather important as it is the language in which set theory is formulated. There's a graduate level book by Jean Gallier that's easy to read and explains what it means to conduct proofs in fol.
Some day proof technique just clicked and now i transport every theorem i see in the form of first order logic.

I mean, almost averything can be expressed as an implication and sometimes a double implication (iff).

Then the proofs techniques just follow...i mean, the hypothesis and what you have to proof is clear, you just have to connect the dots.
Reading the rest of the thread, it sounds like the argument that the most important skill building a house is laying a foundation, since without that no other skill comes to play. Similar, if you can't write a basic proof, then you're limited to high school level math. Which is true, but mostly shows that we need to constrain better what level of mathematics we are talking about.
I really liked "how to prove it". I did it on the side and didn't have too much trouble.
What level of first order logic? Elementary discrete math course or Godel incompleteness theorem?
90% of the time advice that begins with "I disagree with everyone" is not going to be super helpful advice. As many other people have said this person may just be referring to the basic structures and technique of modern mathematics as is usually taught in "Intro to Proofs" courses or better Discrete Mathematics courses.

My undergraduate program send you through an advanced calculus with proofs (basic real analysis) AND an advanced version of discrete math with proofs before you did any level of abstract algebra. That was pretty good preparation, but neither taught "first order logic" more than superficially.

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