Probably? But the number of people who study formal logic to the required degree is dwarfed by the number of people who need this skill.
Also, mathematical logic, studied properly, is hard. It forces you to conceptualize a clean-cut break between syntax and semantics, and then to learn to handle them separately and jointly. That’s a skill many mathematicians don’t have (to be fair, not because they couldn’t acquire it, they absolutely could, but because they never found it useful).
I have a personal story. Growing up I was a math whiz, I loved popular math books, and in particular logical puzzles of all kinds. I learned about Godel’s incompleteness from Smullyan’s books of logical riddles, for example. I was also fascinated by popular accounts of set theory and incompleteness of the Continuum Hypothesis. In my first year at college, I figured it was time to learn this stuff rigorously. So, independent of any courses, I just went to the math library and checked out the book by Paul Cohen where he sets out his proof of CH incompleteness from scratch, including first-order logic and axiomatic set theory from first principles.
I failed hard. It felt so weird. I just couldn’t get through. Cohen begins with setting up rigorous definitions of what logical formulas and sentences are, I remember he used the term “w.f.f.-s” (well-formed formulas), which are defined by structural induction and so on. I could understand every word, but it was as if my mind went into overload after a few paragraphs. I couldn’t process all these things together and understand what they mean.
Roll forward maybe a year or 1.5 years, I don’t remember. I’m past standard courses in linear algebra, analysis, abstract algebra, a few more math-oriented CS courses (my major was CS). I have a course in logic coming up. Out of curiosity, I pick up the same book in the library and I am blown away—I can’t understand what it was that stopped me before. Things just make sense; I read a chapter or two leisurely until it gets hard again, but different kind of hard, deep inside set theory.
After that, whenever I opened a math textbook and saw in the preface something like “we assume hardly any prior knowledge at all, and our Chapter 0 recaps the very basics from scratch, but you will need some mathematical maturity to read this”, I understood what they meant. Mathematical maturity—that thing I didn’t have when I tried to read a math logic book that ostensibly developed everything from scratch.
I think this notion of “mathematical maturity” is hard to grasp for a beginning student.
I had a very similar experience. Introduction to (the Russian edition of) Fomenko & Fuchs “Homotopic topology” said that “later chapters require higher level of mathematical culture”. I thought that this was just a weasel-y way to say “they are not self-contained”, and disliked this way of putting it as deceptive. Now, a few years later I know fairly well what they meant (although, alas, I still have not read those “later chapters”).
I wonder if there is a way to explain this phenomenon to those who have not experienced it themselves.
Interesting off-topic fact about Fomenko—I’d read his book on symplectic geometry, and then discovered he’s a massive crackpot). That was a depressing day.
He is a massive crackpot in “pseudohistory”, but he is also a decent mathematician. His book in symplectic geometry is probably fine, so unless you are generally depressed by the fact that mathematicians can be crackpots in other fields, I don’t think you should be too depressed.
Probably? But the number of people who study formal logic to the required degree is dwarfed by the number of people who need this skill.
Also, mathematical logic, studied properly, is hard. It forces you to conceptualize a clean-cut break between syntax and semantics, and then to learn to handle them separately and jointly. That’s a skill many mathematicians don’t have (to be fair, not because they couldn’t acquire it, they absolutely could, but because they never found it useful).
I have a personal story. Growing up I was a math whiz, I loved popular math books, and in particular logical puzzles of all kinds. I learned about Godel’s incompleteness from Smullyan’s books of logical riddles, for example. I was also fascinated by popular accounts of set theory and incompleteness of the Continuum Hypothesis. In my first year at college, I figured it was time to learn this stuff rigorously. So, independent of any courses, I just went to the math library and checked out the book by Paul Cohen where he sets out his proof of CH incompleteness from scratch, including first-order logic and axiomatic set theory from first principles.
I failed hard. It felt so weird. I just couldn’t get through. Cohen begins with setting up rigorous definitions of what logical formulas and sentences are, I remember he used the term “w.f.f.-s” (well-formed formulas), which are defined by structural induction and so on. I could understand every word, but it was as if my mind went into overload after a few paragraphs. I couldn’t process all these things together and understand what they mean.
Roll forward maybe a year or 1.5 years, I don’t remember. I’m past standard courses in linear algebra, analysis, abstract algebra, a few more math-oriented CS courses (my major was CS). I have a course in logic coming up. Out of curiosity, I pick up the same book in the library and I am blown away—I can’t understand what it was that stopped me before. Things just make sense; I read a chapter or two leisurely until it gets hard again, but different kind of hard, deep inside set theory.
After that, whenever I opened a math textbook and saw in the preface something like “we assume hardly any prior knowledge at all, and our Chapter 0 recaps the very basics from scratch, but you will need some mathematical maturity to read this”, I understood what they meant. Mathematical maturity—that thing I didn’t have when I tried to read a math logic book that ostensibly developed everything from scratch.
I think this notion of “mathematical maturity” is hard to grasp for a beginning student.
I had a very similar experience. Introduction to (the Russian edition of) Fomenko & Fuchs “Homotopic topology” said that “later chapters require higher level of mathematical culture”. I thought that this was just a weasel-y way to say “they are not self-contained”, and disliked this way of putting it as deceptive. Now, a few years later I know fairly well what they meant (although, alas, I still have not read those “later chapters”).
I wonder if there is a way to explain this phenomenon to those who have not experienced it themselves.
Interesting off-topic fact about Fomenko—I’d read his book on symplectic geometry, and then discovered he’s a massive crackpot). That was a depressing day.
He is a massive crackpot in “pseudohistory”, but he is also a decent mathematician. His book in symplectic geometry is probably fine, so unless you are generally depressed by the fact that mathematicians can be crackpots in other fields, I don’t think you should be too depressed.
Yes.