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  • Are all people who do mathematics applying (whether they know it or not) formal logic?
  • Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm not asking whether it can be translated into mathematical logic, but whether at its core it already is formal.)

Most of the time, when I hear my professors speaking about mathematics, they use natural language - with some precise terms thrown in. For example, one professor might say,

"All positive real numbers have $2$ square roots."

If we were to write down this statement formally, on the other hand, it might go something like this:

$\forall x(x\in\mathbb R^+ \implies \exists y \exists z(y\in\mathbb R \land z\in\mathbb R \land (y\neq z) \land (y^2 = x) \land (z^2 = x))$.

If I were to ask my professor if they really meant the second statement, they might squint at it for a second, tell me I'm using the exterior product incorrectly, then (after clarification) remark, "Well, I suppose you could write it that way, but you know what I meant." (The dialogue here is fictional, but reflects some of the responses I've gotten from professors when I tried to draw out extremely formal statements from them.)

What this brings me to is the question of, when they said the first statement, were they

  1. expressing in natural language a formal statement in mathematical logic (such as the second statement, for example), or
  2. simply stating a truth about the positive real numbers?

I feel like most of my math professors would assert that they were merely stating a truth about the positive real numbers, not trying to express in natural language some convoluted formal statement of mathematical logic. It appears to me that not all of my professors have training in, or even care about, formal logic. This is incomprehensible to me, especially when I read about the foundations of math being entirely based on formal logic.

If mathematics is as its core formal, then how could one do any math without formal logic?

In an answer elsewhere on the site, user Yuval Filmus stated that "From the point of view of the acting mathematician, foundations are only needed if they're your specialty." He goes on to request of the asker, "please don't take this formal stuff too seriously. While entertaining and fruitful, it is only one way to look at math." But my response to this is:

Why? If they're the foundations for everything we do, shouldn't everyone care about them?

I realize this question was rather long-winded, but I hope I successfully expressed some of my confusion and frustration, and I would appreciate any advice in making sense of these topics.

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justin
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    One question here that isn't really addressed by the answers on the duplicate question is that the 'formal sentence' that you gave in your example is just an instance of a particular flavour of first-order logic. There is nothing preventing us from defining the syntax differently, so that "For every $x \in \mathbb{R}^+$, there are $y,z \in \mathbb{R}$ such that ( $y \ne z$ and $y^2 = x$ and $z^2 = x$ )." is a syntactically valid sentence in the formal system! If you try asking your same professors in the kind of format I give in this comment, I'm sure they won't have trouble understanding. – user21820 Jul 29 '16 at 02:28
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    Historically, some logicians used "$\supset$" to denote "implies"! Now it's largely used in philosophy rather than mathematics. But imagine the confusion if you asked your professors using this (now) super-set notation for implication! – user21820 Jul 29 '16 at 02:30
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    Anecdotally, I can confirm that the majority of mathematics professors in my institution have only a cursory knowledge of formal logic as might be described in a modern logic textbook, but that does not mean that they cannot understand and manipulate intricate logical deductions. (Some can't, but most can.) The reason is that they are mostly using intuitive deductions. In fact, if their intuitive grasp of logic includes all the rules of natural deduction, they would be able to perform all valid logical deductions without knowing modern symbolic syntax for logic. – user21820 Jul 29 '16 at 02:36

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