What are some common gaps in logic that students make in mathematical proofs that lead to inaccurate results?

My answer to a question in Quora:

What are some common gaps in logic that students make in mathematical proofs that lead to inaccurate results?

I wish to comment on two specific flaws exhibited by students who encounter proofs first time in their lives.

The first one is

inability to accept the Identity Principle: “\(A\) is \(A\)”, and arguments related to it, as a valid ingredient of proofs.

For many students, a basic observation

For all sets \(A\), \(A \subseteq A\) (\(A\) is a subset of \(A\)) because every element of \(A\) is an element of \(A\)

is very hard to grasp because of the appearance of the same words about  the same set \(A\) twice in the sentence: “element of  \(A\)  is an element of  \(A\)”.  I have observed that many times and I think that students cannot overcome a mental block created by their

expectation that a proof should yield some new information about objects involved

– and this is the second fundamental flaw.

And, of course, reduction, removal of unnecessary information, is seen by many students as something deeply unnatural.

Every year, I hear from my Year 1 students the same objection:

How can we claim that 2 is less or equal than 3, that is, \( 2\leq 3\), if we already know that 2 is less than 3, \(2 < 3\)?

I think we encounter here a serious methodological (and perhaps philosophical) issue which I have never seen explicitly formulated in the literature on mathematics education:

a proof of a mathematical statement can illuminate and explain this statement, it may contain new knowledge about mathematics which goes far beyond the statement proved; but

  • elementary steps in proofs frequently do not produce any new information,
  • moreover, sometimes they remove unnecessary information from consideration.

A proof can be compared with a living organism built from molecules which can hardly be seen as living entities — and even worse, from atoms which are definitely not living objects.

This is closely related to another issue which many students find difficult to grasp: statements of propositional logic have no meaning, they have only logical values (or truth values, as they are frequently called) TRUE or FALSE. Any two true statements are logically equivalent to each other because they are both true; moreover, the statement

if London is a capital of England then tea is ready

makes perfect sense, and can be true or false, even if constitution of the country has no relation to the physical state of my teapot.

When my students express their unhappiness about logic which ignores meaning (and I provoke them to express their emotions), I provide an eye-opening analogy: numbers also have no meaning. The statement

The Jupiter has more moons than I have children

compares two numbers, and this arithmetic statement makes perfect sense (and is true) even if Jupiter has no, and cannot have any, connections whatsoever with my family life. Numbers have no meaning; they have only numerical values. Arithmetic, the most ordinary, junior school, sort of arithmetic is already a huge and deep abstraction. We did not notice that because we are conditioned that way.

Learning proofs also involves some degree of cultural conditioning. As a side remark, I suspect (but have no firm evidence) that the role of family — presence of clear rational argumentation in everyday conversations within family — could be important.