A question about the routine of usage of quantifiers and other logic symbols in Algebra courses

Question

I am nowing taking some 4th year algebra courses, for example Field Theory and Ring & Modules, and the instructors of both courses $\textbf{forbid}$ students from using quantifiers and other logic symbols like

$$\forall, \exists, \lor (\text{or}), \land (\text{and})$$

I once asked the instructor of Rings & Modules why he didn't allow students to use them, and he answered me that, "Have you ever seen any modern algebra textbook which uses quantifiers and other logic symbols in any theorem or exercise?" I checked the most famous textbooks, like Dummit & Foote, and they really didn't use these symbols.

But for courses of analysis, like real analysis, functional analysis, measure theory, the instructors of these courses never forbid students from using these symbols.

Therefore, my question is, does there exist an academic routine or requirement in the area of Algebra s.t. everyone should not use logic symbols when necessary? Hope someone who are expert in this can help explain a little. Thanks!

Answer 1

It sounds more like a dictum from an essay titled "How to Write Mathematics Well", the type of thing Halmos or Krantz might have written at some point, than anything to do with algebra per se. The overall point of the dictum would be that jargon and symbolism should generally be avoided if they do not aid in comprehension. Because then you have to spend a minute being distracted by and parsing the logical symbolism, while instead you could have spent that time considering what the problem is about.

(It could also be partly a pedagogical thing, e.g., some students may come to feel that they are being more formal and precise if they use the logical symbols, and get attached to them, and the instructors are possibly fighting delusions that students may have about how necessary they are in really understanding the mathematics. Gauss did very well before the time of Frege, after all.)

About all I can come up with, why you might see more of a concern with the quantifiers in analysis courses, is that in studying some concepts like uniform continuity, you have a certain alternation between three or more quantifiers $\exists, \forall$ where it becomes critically important to get the order straight, so you resort to the logical symbolism to get everyone on the same page. Cursory inspection of my memory is that the strings of quantifier alternations that crop up in algebra courses are generally shorter, along the lines "show that for all blah, there exists blah such that".

Answer 2

I can't speak to your professors intent but, generally it is best practice when introduced to a subject to either write completely algebraic sentences or complete prose. This is because it is often not entirely obvious when to use which one, and so switching between them can create awkward sentences that are repetitive or hard to read.

In analysis courses, this comes much more naturally to students, since you tend to deal with things you on some level already understand and have dealt with for a while. For example, a student is almost never going to write:

• $\forall \epsilon > 0$, there exists $N(\epsilon)$ such that for any $n > N(\epsilon)$ where $n$ is a natural number, that $|a_n - a| < \epsilon$

This is a harder to read sentence at the end, and its because I mixed prose and algebraic writing in a way that is purposefully hard to read. Also, you should note here that analysis without quantifiers tends to be a lot more writing (in my opinion). If you had to write "For every $\epsilon > 0$ there exists an $N(\epsilon)$ such that for all natural numbers $n > N(\epsilon)$ the inequality $|a_n - a| < \epsilon$ holds" your hand would fall off in a week.

However, in algebra you deal with generally foreign material. For example, if I wanted you to write "For any subgroup of a group, the order of the subgroup divides the order of the group itself", how would you do it if you wanted to mix quantifiers? Would it flow properly? Is it easier or harder to read?

• Let $G$ be a group and then $\forall H \leq G, \operatorname{ord}(H)| \operatorname{ord}(G)$; makes sense but how they fit in then subject is not inherently clear to all students.

There are other questions that appear, how would use $\forall$ when describe for all fields, rings, or groups? Do students know the difference reliably enough so that they can separate $\forall$ and "let" when appropriate? Or will it just give the grader a headache? There is also a pedagogical question as to whether or not students actually understand what they are writing, and whether or not using algebraic quantifiers would help or hinder them.

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In general, many textbooks at both the undergraduate and graduate levels don't use algebraic quantifiers, and instead opt to use prose, because it is more clear and there is less room for misunderstanding. Your professors likely agree.