1.8 What are mathematical proofs
The beginning student in mathematical reasoning might find it surprising that there is no agreement as to what exactly constitutes a mathematical proof.15 Despite this lack of agreement, hundreds if not thousands of mathematical proofs are being published every day. For the working mathematician, a proof is a sequence of statements that starts with the premise and ends with the conclusion such that each statement in the proof follows logically from earlier statements. You have already seen some examples of proofs in this chapter. And in this day and age of computers, formal proofs in which every logical step can be verified against a predefined set of rules of inference are also possible.16 It can be argued that formal proofs are the ideal because they leave little room for doubt but formal proofs are at the moment very tedious to construct and most mathematicians do not construct such proofs for their theorems — all they need to do is to convince their peers that their results are correct in what is known as a peer-reviewed process for publication in an academic journal.
In the following chapters, you will learn various techniques for writing convincing arguments for establishing the truth of mathematical statements. You will also recognize the need for rigour. As an illustration, consider the following derivation: \[\begin{align*} -2 & = -2 \\ 1-3 & = 4-6 \\ 1-3+\frac{9}{4} & = 4-6+\frac{9}{4} \\ 1^2-2\cdot 1\cdot \frac{3}{2}+\left(\frac{3}{2}\right)^2 & = 2^2-2\cdot 2 \cdot \frac{3}{2}+\left(\frac{3}{2}\right)^2 \\ \left(1-\frac{3}{2}\right)^2 & = \left(2- \frac{3}{2}\right)^2 \\ 1-\frac{3}{2} & = 2- \frac{3}{2} \\ 1 & = 2. \end{align*}\]
The derivation starts with the obviously true statement “\(-2 = -2\)” and ends with the absurdity “\(1 = 2\).” Unless you truly believe that \(1\) and \(2\) are the same number, you should feel that there must be an error somewhere. Do you spot the error?
Every step in the derivation is valid up to \[1^2-2\cdot 1\cdot \frac{3}{2}+\left(\frac{3}{2}\right)^2 = 2^2-2\cdot 2 \cdot \frac{3}{2}+\left(\frac{3}{2}\right)^2.\] However, moving from this to \[1-\frac{3}{2} = 2- \frac{3}{2}\] is where the trouble occurs. Remember, if \(a\) and \(b\) are numbers satisfying \(a^2 = b^2,\) it is not necessarily true that \(a = b\); \(a = -b\) is another possibility which is precisely the case here!
Many people find the exactness required in doing mathematics difficult or even scary — a little blunder can cause the whole enterprise to collapse. But this exactness is also what makes mathematics easy in some sense — once a statement is correctly proved, it stays true and is immune to the fads of fashion…well, provided that the logical foundation itself is not being called into question.
Exercise 1.5 What is wrong with the following derivation? \[\begin{align*} 1 & = \sqrt{1} \\ & = \sqrt{(-1)(-1)} \\ & = \sqrt{(-1)}\sqrt{(-1)} \\ & = (\sqrt{(-1)})^2 \\ & = -1. \end{align*}\]
An introductory article on formal proofs by Thomas Hales can be found at https://www.ams.org/notices/200811/tx081101370p.pdf.↩