Posted by: Alexandre Borovik | October 7, 2018

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, $$2\leqslant 3$$, 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.

Responses

1. Your second problem is mentioned on the first page of a Soviet textbook that my American teacher had us use in high school (Elementary Mathematics, by Dorofeev, Potapov, and Rozov):

“The student usually finds no difficulty when using [the signs ≤ and ≥] in formal transformations, but examinations have shown that many students do not fully comprehend their meaning

“To illustrate, a frequent answer to, ‘Is the inequality 2 ≤ 3 true?’ is ‘No, since the number 2 is less than 3’…”

2. Interesting; I did not remember that. Actually, Dorofeev’s textbooks were for for me a paradigm of proper exposition of inequalities: in particular, need for treating not only systems of simultaneous of inequalities, but also совокупности, aggregates of inequalities, that is, disjunctions of inequalities, and for a very good reazon: an inequality x^2 >1 is equivalent to disjunction of inequalities x < -1 and 1 < x. More on that in http://bit.ly/1FZDj0P .

3. Because of the continuum hypothesis, I have studied foundational mathematics as an amateur for the last thirty years. I have no doubt that your observations are correct, but your students should be given more credit than you give them. The law of identity can fail relative to a semantic conception of truth. Logicians will disagree on this matter. But where you find argument in support of failure, it is because the term is non-denoting.

At issue here is the reliance of logical systems on well-formedness presuppositions. The law of identity is an artifact of metaphysical reasoning that is encoded into first-order logic relative to an analytical conception of truth. The use of the negated sign of equality in Frege’s definition of zero is correctly understood as an instance of the indiscernibility of non-existents from negative free logic. That is, no existent can satisfy the predicate.

Formal systems and the instrumentalist mindset they engender obfuscate these issues because logical calculi must interpret the sign of equality as justification for substitutability. It would be more correct to develop theories that explicitly implement warrants for substitutivity. This is, in fact, what the identity of indiscernibles provides (the axiom of extensionality is an instance). Logicians have rejected this principle as a logical principle.

My point here is that students arrive with natural language experience. There is no reason to believe that formal system presuppositions are faithful to natural language reasoning. Indeed, the literature suggests otherwise.

Aristotle speaks of demonstrative arguments as a subclass of deductions. His example does not carry existential import. So, his works have not been fully debated with respect to mathematical contexts. It reappears in Russell’s work. Of course, Goedel and Skolem addressed a miniscule part of what Russell did and havr received all of the glory.

Give more credit to your students and less to fallacies of authority which abound in mathematical logic.

4. I was struck by the phrase “presence of clear rational argumentation in everyday conversations within family” which struck a chord with me. “Clear rational argumentation” is an abiding memory of my earliest years, which gave me the confidence to believe that I could decide every question, no matter how difficult, for myself.