What does "because" mean, in the context of an answer to a mathematical problem?

Question

Many times in a math textbook, we see the phrase "$X$ is true because of $Y$", or "$X$ is false because of $Y$". For example, consider the relation on $\mathbb{R}^2$ which is the set-theoretic union of the two functions defined by the equations $f(x)=x$ and $f(x)=-x$, respectively. Suppose some high school textbook asks a multiple choice question about whether that relation is a function, and one of the choices is "It is not a function because there are multiple $x$-values for some $y$-values", and another choice is "It is not a function because there are multiple $y$-values for some $x$-values". Intuitively, the teacher would mark the first answer wrong and the second answer right. However, since that relation is in fact not a function, any statement of the form "If $P$, then that relation is not a function" is true no matter what the statement $P$ is, since any conditional with a true consequent is automatically true.

To give another example, suppose I write the two statements, "The derivative of $x^2 + sin(x)$ is $2x + cos(x)$ because of the derivative sum rule" and "The derivative of $x^2 + sin(x)$ is $2x + cos(x)$ because Fermat's Last Theorem is true". Intuitively, the first statements is true and the second false, and in fact if, in a calculus class, a student writes the latter instead of the former, he would probably be marked wrong. However, again, both implications are true because the consequent is true. So, these and many other examples show that "because" is not the same as "implies". My question is, has there been, in some text, a formal and rigorous analysis of "because", at least as applied to mathematical statements? Perhaps that text also gives a definition of "because". Or, is this, in fact, not a notion amenable to rigorization and formalization, and just one of those "You know it when you see it" concepts that are not precisely defined?

Answer 1

I agree with univalence's comment: "$X$ because $Y$" means, typically, and at least in your examples, that $Y$ is a sketch of a proof of $X$, or perhaps the key step or key theorem that needs to be used in a proof of $X$. It does not refer to the material implication, and you're pointing towards some of the paradoxes of material implication in your question, that material implication just does not capture a lot of what we mean by implication, even in a mathematical context.

Your example is quite nice: nobody would say that

"The derivative of $x^2 + sin(x)$ is $2x + cos(x)$ because Fermat's Last Theorem is true"

because if you were writing down a proof that the derivative of $x^2 + \sin x$ is $2x + \cos x$, you would not mention Fermat's last theorem, because it is not relevant. Note that this notion of relevance is actually not captured by the strict logical notion of a valid proof: you could write down a valid proof of the calculation of the derivative and then, at any point in that proof, insert the proof of Fermat's last theorem, and it would still be a valid proof. But that FLT segment would be totally irrelevant, and including it violates one of Grice's maxims, the maxim of relevance.

As far as I know, no mathematical formalization of "relevance" exists. Possibly it can't exist. This has a similar flavor to metamathematical questions like "when can two theorems meaningfully be said to be equivalent" (given that all true statements imply all other true statements, in the material sense; there is an interesting old MO discussion of this question which I can't find) which also do not have a mathematically formal answer as far as I know.

Answer 2

A true story (just for fun).

Once upon a long time ago, the logician Geoffrey Hunter (the author of that excellent old book Metalogic) told me that he used to give a low-level logic course. Near the very beginning of the course he handed out a list of not-entirely-straightforward informal arguments, and asked the students to sort them into valid and invalid. At the end of the course, he handed out the same list as another test. Typically, students did worse at the end of the course than at the beginning!

Why? Because they had learnt some shiny new tools -- the material conditional! the existential quantifier!! -- and they applied them without sufficient thought.

So, as it might be, they took an argument using "implies" or "because". They thought "Ah hah, that looks kind-of-"if"-related", so they hit the argument with the material conditional .... disaster!!

In this case, e.g., both implications are true because the consequent is true is not something anyone would dream of saying before corrupted by half-understood logic ....