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?

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    Many proofs are simple enough to be understood without a course of formal logic. For example a proof of the form $A\Longrightarrow C \Longrightarrow D\Longrightarrow B$ to prove that $A\Longrightarrow B$ is trivial for most students. – BPP Sep 12 at 12:11
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    @BPP, what do you mean by "simple enough"? Before studying logic I always got stuck when I was asked to prove something as I didn't even know what it means to prove something. Even that "simple enough" statements like showing if a is even, then $a^2$ is also even. So, from such situations I could be satisfied by any explanation or "proof" in a reasonable fashion which might be not even a proof because I didn't have enough information to justify correct sequence of statements of the proof and validate how a proof should look like. – Turkhan Badalov Sep 12 at 12:20
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    $a$ is an even integer if it's the double of an integer. $a$ integer $\Longrightarrow$ there exist an integer $k$ whose double is $a$ $\Longrightarrow$ $a=2k$ $\Longrightarrow$ $a^2=4k^2$ which is the double of the integer $2k^2$. Where's the hard logic part in this proof? Students don't need a course in logic to understand this proof (often taught in middle school). – BPP Sep 12 at 12:31
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    I never had a logic course and I still got big benefit from proofs. I suspect I am the norm. – guest Sep 12 at 20:49
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    I think that there are probably many mathematicians who have never had a course in mathematical logic, and they do just fine. You do not need to know about truth tables to be a professional mathematician. – Steven Gubkin Sep 13 at 0:43
up vote 4 down vote accepted

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.

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.

  • The question was not about coming up with a proof, it was about understanding what a proof is, and whether a certain assertion is a proof according to presently accepted logical rules. – Rusty Core Sep 12 at 21:22
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    Exactly, and I tried to make a point that the knowledge about logic *to understand * a proof is rather limited and one can therefore use proofs before all students have taken a course in logic. – Jasper Sep 12 at 21:48
  • The point is not whether students will understand it. The point is that adding "redundant" or even funny statements in between still will be accepted as something okay because "professor knows what he does". He could have said after second statement something sounding clever: "let's now divide $a = 2k$ by $2$ to explicitly see it is divisible. We get $a = k$, so it is divisible as we don't get fractions. $a^2=k^2$ but remember that reduced $2$? Putting it back again gives us $a^2=4k$", etc. Did he say false statenents? No. But they are useless while most of the students will just accept it – Turkhan Badalov Sep 13 at 5:22
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    Of course there is a false statement because a/2=k. And studens should be able to point this out. – Jasper Sep 13 at 5:27
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    @Jasper you are demonstrating really clever and ideal students actually that can notice every mistake. Why do you think students make mistakes, leave gaps in their proofs? Let's put aside that concrete simple proof that I just gave as a simple example. – Turkhan Badalov Sep 13 at 5:43

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