Common logical pitfalls when learning to write mathematical proofs

Mathematics is needed in many scientific disciplines, even though it’s used in somewhat different ways in them. A theoretical physicist usually has to study not only applied math, but also some pure mathematics which involves logical proofs of statements. A mechanical engineering student can usually do without anything containing formal logic and such, but has to learn basic differential equations, volume integrals and similar methods of applied mathematics. Chemistry and biology students also benefit from mathematical knowledge (at least one has to be able to solve a quadratic equation before one can calculate reactant and product concentrations from an equilibrium constant of a chemical reaction).

Rigorous logical thinking is not something that a human being has a natural instinct for (just as the concept of inertia and the Newton’s 1st and 2nd laws are not obvious from everyday earth-bound experience). Let me list some of the most usual logical errors that a beginner does when trying to learn to prove mathematical statements.

1. Inappropriate reversal of an implication

It is easy to see that if x = 3, then also x^2 = 9. But the reversal is not true – if x^2 = 9, we don’t necessarily need to have x = 3, as it could also be -3. Even the statement x^3 = 27 does not mean that x = 3, if we allow for the possibility that x┬áis a complex number (can you find the points on complex plane that solve this equation?).

A more obvious way to demonstrate this is to say that the statement x = 3 implies x \in \mathbf{N} but not every natural number is 3.

To avoid doing this kind of logical errors, make sure that you never start proving a statement by assuming that it is true. In fact, any logical statement can be “proved” if you assume things that contradict each other – it is logically correct to say that “If the Moon is made of cheese and it’s not made of cheese, then also cows are able to fly.”

2. Playing with something that does not exist

Sometimes it seems possible to prove that an object X has properties A, B and C, but after some more thinking you find out that X doesn’t even exist at all. You may have seen some Youtube videos where a math teacher proves something like 1 + 2 + 3 + 4 + \dots = -\frac{1}{12}, which is an absurd statement as the sum on the left side of the equality does not converge, but it is possible to make it look like true if you take a liberty to divide the integer terms into many pieces (with different signs) and rearrange them as you want.

Actually, the terms of a sum with an infinite number of terms can’t be arbitrarily rearranged even in all cases where the sum converges.

3. Assuming without proof that something is unique

This is what we already did in case 1, by assuming that x^2 = 9 implies x=3. There are also some more obviously crazy results that can be made by doing this mistake. One example is to use De Moivre’s formula

e^{iz} = \cos z + i\sin z,

to show that e^{0} = e^{2\pi i}, and then take the natural logarithm of both sides of this equation to “show” that 0 = 2\pi i. The problem with this logic is that the complex logarithm function is not single valued – there is an infinite number of complex numbers that can be called logarithm of a given complex number.

4. Assuming that an infinite set contains everything

I remember seeing in some book a historical claim that the ancient Greeks tried to prove that there is only a finite number of types of an atom in the following way: “If there were an infinite number of different kinds of atoms, there would exist atoms of all possible sizes, and then some of them would have to be large enough to be visible which is not true as we have never seen an individual atom”. This, of course, is poppycock, as we can also say that there is an infinite number of integers that are divisible by 2, but that set still does not contain the number 5.

There are many other ways to do logical errors by making false assumptions about infinity. For one thing, the above mentioned set of even integers is “just as infinite” as the set of all integers, even though it may seem to contain half as many numbers. The set of real numbers is “more infinite” than the set of natural numbers, which you will learn to prove if you study the concept of cardinality and injective/surjective/bijective mappings.