How is it correct for a lecturer to prove and "explain" a proof while explicitly knowing students are not familiar with logic itself?

Question

I often see a situation when professors use words "logic", "mathematical proof" and even prove logically while actually knowing that students are not even familiar with logic itself, i.e. no formal understanding of equivalence, implications, inference rules, etc.

How are students supposed to understand such "proofs" because they can actually accept as a proof any "intuitively" explained reason why a theorem is true and never even suspect they were deceived unless they know the exact definition of an argument and true (sound) argument? Is knowing logic always taken as granted like a prerequisite? Even so, shouldn't the lecturer at least designate some time for explaining basic logic?

Answer 1

In practice, one does have to explain (even to university engineering students) that although A implies B, it need not be the case that B implies A. However, there is a confounding factor, which is that mathematical logic is not what is used in scientific reasoning. As is explained by Polya in one of his books (the one on plausible reasoning) and by V. I. Arnold in some of his essays (he tells a story about his paper being rejected), in scientific contexts, when it is known that A implies B, the truth of B provides evidence for the plausibility of A (this is precisely what Polya calls plausible reasoning). It is therefore incumbent on the teacher of mathematics to distinguish mathematical logic from ordinary reasoning (dare I say common sense reasoning?) and it is generally helpful to do this explicitly rather than implicitly.

In general, one does not need to teach logic formally, but the basic consequences of negations of implications should be explained if they are used, because they are not obvious to many, and they are less so still in contexts in which reasoning is not strictly mathematical (an error often made by mathematicians is to assume that there are no such contexts). This is particularly so if one proves a statement by proving its contrapositive, or by contradiction. The validity of either line of argument is not clear to a novice, and the strategy underlying such an argument needs to be explained to such a student.

Answer 2

A mathematical proof has (among others) the purpose to convince someone of some fact, given some already established facts.

Whether or not a proof is valid does not depend on who presents it. That is one of the key features of math - it does not matter at all if the "professor knows what he does" or if she sounds clever. If I can't help but agree that the arguments clearly show that the new fact follows from known facts, I have to agree to the new fact as well.

There are however several levels of difficulty, for example:

1. Direct proof: showing $A \Rightarrow B$ by showing $A \Rightarrow C_1 \Rightarrow C_2 \Rightarrow \dots \Rightarrow B$
2. Proof by contradiction.
3. Proof by (complete) induction.

I agree that point 3 is somewhat hard to employ without any prerequisite in logic, but the first two points are doable. I will give examples for each.

There is a huge difference between explaining (or understanding) a proof and finding and writing it down. Since the question is asking about a lecturer explaining a proof, I'll not bother with the latter.

Using the $a \text{ even} \Rightarrow a^2 \text{ even}$ example:

1. The lecturer can explain the bigger picture of the proof: "We take any even number and show that its square must also be even".
2. The lecturer can remind the students that for even $a$, there exists $k \in \mathbb N$ so that $a = 2k$. Give examples if neccessary. This should be working knowledge already. If not, this has to be considered a gap in mathematical basics and not in logic.
3. The manipulation to express $a^2$ in terms of $k$ can be executed by either students or lecturer, leading to $a^2 = 4k^2$ if done correctly -- regardless of whoever did it.
4. The last step, showing that the number $4k^2$ is also even because $4k^2 = 2 \cdot k'$ with $k' := 2k^2$ can also be done by the lecturer. This decomposition always works and again does not depend on who does it.

None of these steps requires "deep" understanding" of logic except "simple" implications. But in the end, the students should be able to understand that for any even $a$, also $a^2$ must be even.

It is indeed a completely different story for the students to come up with a similar proof.

Regarding proofs by contradiction: it is important for the students to understand that apart from the cleverly chosen starting fact, each conclusion is right and again does not depend on who presents the reasoning. If one then arrives at a contradiction, the only remaining option is that the premise was wrong.

Let's look at Euler's proof of the fact that there are infinite prime numbers.

1. Assume that there are finitely many prime numbers. We don't know yet if this is true or false, but we can assume either.
2. If the students have mathematical background about divisors, they must agree that the product of all primes, increased by one, is prime or has a prime divisor that is not in the list we started with. This argument is somewhat complex, but uses only known facts about divisibility.
3. The reasoning in point 2 is sound. The only way to fix the error that we did not start with "all primes" is to assume that there is no such thing.

To summarize: Understanding proofs does not require formal knowledge of logic. Common sense is enough, together with the attitude that arguments are not valued by authority.