Is it true that, nowadays, logic isn't held in high esteem by the mathematical community? if so, why?

Question

A couple of years ago, before I enrolled in my math undergrad. I took out of enjoyment a course on logic. The teacher realized that I liked math, and also logic, to which he said it was strange since logic is nowadays -kind of- looked down by mathematicians. It was, at the moment, a bit strange to hear that, since math and logic are really closely related, and they were kept in my mind as "kinda' the same thing". However, I didn't pay much attention.

Nonetheless, those words have been resonating in my mind lately. I'm undergoing my second semester in math, and lately I have noticed a kind of -academical not personal- "disdain", "weirded out attitude", "uncomfortableness" regarding my really formal style when writing theorems, proofs, definitions, axioms, etc. As an example, I will include how I write the definition of a vector space

$\{$ $<K,+,·>$ $\text{ is a field}$, $V$ $\text{ is a set}$, $V≠∅$, $⊕ : V×V ↦ V$, $∘ : K×V ↦ V$ $\}$ $⊨$ $[$$<V,⊕,∘>$ $\text{ is a K-vector space}$ $≡$

$<V,⊕>$ $\text{is an abelian group}$

$∧$ $∀α,β|α,β∈K,∀w|w∈V$ $:$ $(α+β)∘w = α∘w⊕β∘w$

$∧$ $∀u,w|u,w∈V,∀α|α∈k$ $:$ $α∘(w⊕v)=α∘w⊕α∘v$

$∧$ $∀α,β|α,β∈K,∀w|w∈V$ $:$ $(α·β)∘W = α∘(β∘W)$

$∧$ $∀u|u∈V$ $:$ $1∘V=V$ $]$

I like this writing style because it's not ambiguous (for example, most definitions I see in text books are written in a "if... then" fashion, when it should be "...if and only if..."), it actually emphasizes quantifiers as a special element of language ("for all", "there exists" when written in normal language don't seem to do that), it expresses that operators are different ( field's $+$ is a whole different thing than vector space's $⊕$ ), among other advantages.

The most insistent teacher in this regard has been my undergraduate program's director. We get along pretty well, and generally when I have a question he is happy to answer, but as soon as I ask him things regarding foundations of mathematics (for example why the properties of exponentiation hold with exponents that are not natural numbers, even though in real, complex, or rational numbers multiplication is not defined as repeated addition), type theory, mathematical logic, etc, he tells me those are really old mathematics, that I shouldn't even bother with that. He's even told me that this semester his mission will be to "set me free" (in his words) from "that math". It has been kind of the same with every teacher, except with the one that is designated for set theory class. However, they have never explicitly stated why is wrong that I'm really formal in my mathematical writing. This has led me to think that what my old logic teacher said is actually true.

So, if it is true that logic is, nowadays, not well regarded in the mathematical community, why is that the case?

Thanks in advance.

Answer 1

We need to distinguish between two very different things:

• Writing style: heavy on notation, or more prose?
• Mathematical logic: a branch of mathematics.

Some of your examples bring up yet a third issue:

• Writing style: rigorous or hand-wavy?

One last throat-clearing point: you have a very small sample size (a few professors at one undergraduate institution). Posting your question to math.SE will give you a wider sample, but it's still not statistically valid.

OK. My degree is in math logic, so I very much don't want to believe the field is looked down on today, nor have I seen any evidence to that effect. (Of course, your mileage may vary at any one institution.) On the other hand, I cringe a little at notation-heavy definitions like your vector space example. A prose-heavy style is in no way incompatible with precision and rigor.

The "if" vs. "iff" issue is an old one; I've even seen a math writing guide somewhere that said, roughly, use 'if' in definitions and 'if and only if' in theorems. I agree it's a potential source of confusion. Before Halmos (?) introduced the "iff" abbreviation, people favoring a more succinct style might have figured that definitions "by definition" state equivalences. ("Omit needless words!"---the first sentence of Strunk & White's style guide, celebrated by some.)

Personally, I agree about quantifiers: unless authors are careful, this can lead to serious confusion. For a historical instance, read Jeremy Gray's The Real and the Complex, where he shows how sloppiness with quantifiers lead to (among other examples) Cauchy's famous "proof" that the limit of a sequence of continuous functions is always continuous. (This was nearly a century before the birth of modern logic, so we should cut Cauchy some slack.) But you can be careful with quantifiers without using the $\forall$ and $\exists$ symbols. (I like to use them, but an analyst and an algebraist of my acquaintance find them irritating, with no apparent detrimental effects.)

On the other hand, I see no reason to distinguish notationally between the field and vector space addition operators. I would find it distracting. I claim it's always clear which is meant in any given context. I also think you want to reserve the $\oplus$ symbol for when it really matters, like with direct sums: $V+W$ and $V\oplus W$ ($V$ and $W$ being subspaces) mean different things, and you want to have a concise way to indicate which is meant.

In short, lots of factors go into good writing style, and you won't find universal agreement---maybe even not broad consensus---on many aspects.

I would hope that any mathematician, pure or applied, would not cavalierly dismiss your question about the rules of exponentiation. Pedagogically, most calculus textbooks regard these issues as best left to a later course. (Spivak's Calculus is a famous exception.) Is it possible that's what he meant? An aside: Gray's book talks about the almost mystical faith 18th century mathematicians had in the persistence of formulas from the real to the complex domain.

Finally, mathematicians are human, and have their likes and dislikes for ice cream, music, and math. Some people hate category theory. Some look down on combinatorics. Personally, I've never warmed up to measure theory, despite the ministrations of a good friend who's an analyst. De gustibus.

[Added later:] I've now read the intriguing article by Lolli you found. Lolli spends almost all his ink refuting Halmos; it could almost be titled, "Why Halmos is wrong about logic". (The stuff about Descartes, while interesting, is irrelevant to modern logic and more broadly to modern math. Some of the footnotes need more historical context, e.g., the one about Poincaré.)

I side with Lolli on most of his points, but the bottom line is this quote from Halmos, included by Lolli:

I didn't say all of this to convince anyone of anything---I said it in an attempt to describe the reaction, as I see it, of many mathematicians, myself included, to formal logic. It is not a question of right or wrong. It is a matter of native competence, professional upbringing, and mathematical taste...

No accounting for taste.

Answer 2

In regard to your last paragraph, about logic being "old" - I won't consider myself as an expert. I also got really hooked on logic, and my master thesis is about intersection between logic and probability.

I think that at begining of the $20$'th century, logic had given a solution to a lot of interesting questions in mathemathic. Computer science had started to evolved and the possibilities had seemed endless. Why can't we have a computer going over all proofs written in a formal language and find a solution to some hundred years unsolved questions?

However, we found out those tools also have their own limitations (See Gödel's incompleteness theorems). Also, there wasn't a lot of progress in regard to open problems in the field. In my mathemathical logic class, our proffesor applied tools from mathemathical logic in his group theory research. But sadly, I guess that if mathemathical logic extremly useful, you will see people using it more often, but sadly, it isn't the case (right now!).

It is also important to remember trends are a reall thing in mathemathics, and they tend to come and go.