Part 2

New skills

Formulating Maths
Introduction
Hypotheses &c
Implications
And & or
For all & there exists
Dependence

Proof
Introduction
Counter examples
Constructing proofs
Understanding
Experimentation
Precision
Examples
Assumptions
Contradiction
Induction
The End

 

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In everyday language, many statements do not mean literally what they say, either by design or by oversight on the part of the speaker or writer. For example, ``I'll be with you in a second'' never means that the speaker will be available in exactly one second; after all, it takes more than a second to say it. However, the situation in mathematics is very different-mathematical statements have very precise meanings, and there is no room for metaphor, hyperbole, or other figures of speech. Moreover, some of the terms used will have specific mathematical meanings, which should have been defined (for example, ``vector'', ``group'', ``continuous function''), while others will express the same logical concepts as in ordinary English language, but be used in a very precise way (for example ``if'', ``for all'', ``or''). The statements are often quite complicated in structure, with several dependent clauses; a small change in the syntax of a statement often produces a large change in the mathematical meaning, which may change a true statement into a fallacy. At first, undergraduates often have difficulty grasping the precise meaning of mathematical statements made in their courses, but it is extremely important to learn to master the formulation of mathematical statements, both in order to understand the meaning of statements made in books and lectures, and also in order to write down mathematics accurately and unambiguously. In this chapter, we discuss various points concerning the way mathematical statements are commonly formulated, in particular the precise use of everyday words to express logical ideas.

There are two points of detail which we should make about the material in this chapter. Firstly, the phrase ``mathematical statement'' or ``statement'' is not intended to encompass everything that is written in a textbook or said in a lecture or tutorial. It refers only to formal mathematical statements, such as the statement of a theorem, proposition, lemma, or corollary, or part of a formal proof, or a formal definition, or certain other statements. Books and lectures will include other material whose role is to motivate the theory, or to explain it informally; this material may include some imprecise statements. Tutorials, and other forms of discussion of mathematics, may well include many very imprecise statements!

Secondly, we shall sometimes refer to two statements, A and B, as having the same meaning, or being synonyms. This indicates that the statements are equivalent for purely formal logical reasons, involving only use of language, notation, and definitions; it has nothing to do with any mathematical content of the statement. In other words, A and B are different ways of expressing the same logical concept. This concept may or may not be true. However, if A is true then B must be true (because they have the same meaning); if A is false, then B is false. Any statement can be expressed in many different ways in English (i.e., there are many statements which have the same meaning); in our examples, we shall only be able to mention a few.

To assist the reader, we shall label synonymous statements in the form S1a, S1b, S1c, ... whereas statements with different meanings will have different numbers (e.g. S1a and S2).