Anatomy of a Proof - Forms of Argument

Mathematical Writing - Vivaldi Franco 2014

Anatomy of a Proof
Forms of Argument

Franco Vivaldi1

(1)

School of Mathematical Sciences, Queen Mary, University of London, London, UK

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

A mathematical theory begins with a collection of axioms or postulates. These are statements that are assumed to be true—no verification or justification is required. The axioms are the building blocks of a theory: from them we deduce other true statements called theorems, and a proof is the process of deduction that establishes a theorem. As more and more theorems are proved, the theory is enriched by a growing list of true statements.

When we develop a mathematical argument, we need to organise sentences so that every sentence is either an axiom or a true statement derived from axioms or earlier statements. In practice only significant statements deserve to be called theorems. In the proof of a theorem many true statements may be derived, but these fragments are not usually assigned any formal label. Sometimes a proof rests on statements which, if substantial, are not of independent significance: they are called lemmas. Finally, a proposition is a statement which deserves attention, but which is not sufficiently general or significant to be called a theorem.

In this chapter we survey some methods to give shape to a mathematical argument. Our survey will continue in Chap. 8, where we consider induction techniques.

7.1 Anatomy of a Proof

A first analysis course begins with a list of axioms defining the real number system. Then one may be asked to prove that

$$\begin{aligned} \forall x\in \mathbb {R},\,\,-(-x)=x. \end{aligned}$$

(7.1)

But isn’t this statement obvious? In recognising the truth of this assertion we make implicit use of knowledge derived from experience, which is not part of the axioms. A proof from the axioms requires erasing all previous knowledge.

We now introduce some axioms, state a theorem equivalent to (7.1), and then prove it from the axioms. We put the proof under X-ray. Our purpose is to dissect a mathematical argument articulating every step; we use the language mechanically, to facilitate the identification of the logical elements of the proof. To avoid making implicit assumptions, we represent familiar objects with an unfamiliar notation.

We are given a set $$\varOmega $$ and a binary operator ’$$\odot $$’ on $$\varOmega $$, with the following properties:

G1:

G2:

G3:

G4:

(These axioms define a general object: a group.) Setting $$\varOmega =\mathbb {R}$$, and ’$$\odot {\text {'}}=`+$$’, one recognises that $$\diamondsuit $$ represents $$0$$ and $$x^\prime $$ represents $$-x$$. We are ready to state and prove our theorem.

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Items 1, 3, 11 are logical tags—expressions or symbols that say things about the text. They announce the statement of the theorem, and the beginning and the end of the proof, respectively. The symbol ’$$\Box $$’ in item 11 may be replaced by the acronym Q.E.D.1

Items $$5,8,9$$ begin with the logical tag ’then’, used here to mean that what comes after is deduced from the axioms. Formally, item 5 is our first theorem, being a true statement deduced from the axioms. Items 7 and 10 are deductions of a slightly different nature, and we have flagged them with the tag ’hence’. These items use not only axioms but also facts assembled from previous statements.

Items 4 and 6 are not statements but instructions. Item 4 instructs us to assume that $$x$$ is an arbitrary element of a set. This is a standard opening sentence which mirrors the expression $$\forall x\in \varOmega $$ in the theorem. Item 6 instructs us to define two new quantities. Many different instructions may be found in proofs, such as the instruction to draw a picture.

In item 10 we use implicitly the transitivity of the equality operator (if $$x=y$$ and $$y=z$$, then $$x=z$$), which depends on the fact that equality is an equivalence relation (see Sect. 4.6).

Proofs from axioms are necessarily rare. Our next proof—the irrationality of $$\sqrt{2}$$—is quite removed from the axioms of arithmetic, as it relies on several definitions and facts without justification. In this respect this proof is more typical than the previous one, even though the style and conventions are the same.

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A fair amount of mathematics is assumed in the proof:

the definition of the rational numbers and of $$\sqrt{2}$$ (items 2, 4, 5);

the definition of co-primality (item 6);

some properties of equations (items 8, 9);

the definition of an even integer (items 12, 17);

a theorem of arithmetic, referred to as Theorem A (items 11, 16).

The argument contains novel elements. The core of the proof is item 4, the assumption that $$\sqrt{2}$$ is rational. This is puzzling: how can we assume that $$\sqrt{2}$$ is rational when we know it isn’t? How can we let $$\sqrt{2} = m/n$$ when we haven’t been told what numbers $$m$$ and $$n$$ are? This is not so strange; assumptions belong very much to common reasoning.

Suppose that I meet Einstein on top of Mount Everest.

Suppose that people walk laterally.

We are clearly free to explore the logical consequences of these assumptions. In mathematics, assumptions are handled formally via the implication operator $$\Rightarrow $$ developed in Chap. 4.

The assumption that $$\sqrt{2}$$ is rational eventually leads to a contradiction: items 6 and 17 are conflicting statements. From this fact we deduce that the assumption in item 4 is false (item 18). Here we abandon the assumption 4. But if the statement $$\sqrt{2}\in \mathbb {Q}$$ is false, then its negation is true. This is item 19, which is what we wanted to prove.

We now abandon the robotic, over-detailed proof style adopted in this section, and analyse various methods of proof in more typical settings.