Chapter 3 Mathematics as a Language
Figure 3.1: A whiteboard with mathematical symbols.
The Language of Mathematics
Did you realise that Mathematics is a language? It has rules, structure and grammar. For example in the sentence “\(x+1=2\)”.
the word “equals” is the verb, (it is denoted by the symbol “\(=\)”),
“\(x+1\)” is the subject,
“\(2\)” is the predicate.
Symbols, such as “\(=\)”, are a shorthand for complex ideas (concepts) that have precise meanings. Symbols in themselves are abstractions, but their power lies in the fact that they can be manipulated to calculate solutions to problems.
Getting to grips with these ideas is certainly not easy but it will be
necessary if you are to be successful - reading and writing mathematics is as important as the calculations themselves. Being able to construct a proof that others can follow is a vital skill to demonstrate your understanding and convey your ideas.
One way to get to grips with it is to read mathematics; starting with your lecture notes and suggested textbooks, you can ‘graduate’ to journal articles as you progress. If you feel yourself getting bogged down or stuck, slow down, try to identify the source of your
confusion and if you still get nowhere ask a friend, tutor or lecturer.
There are many excellent books that can help you learn this ‘language’ as well as the ‘mathematics’ it describes. We have chosen a few references at the end.
Notations and conventions
Get to know the meanings of any special symbols (such as \(\forall\) \(\exists\), \(\mbox{iff}\), \(\sum\) and \(\mathbb{R}\) ) including the Greek alphabet and how to say them. Try making your own list, or find a good reference sheet. Try to find out the different meanings of words such as ‘theorem’, ‘axiom’, ‘definition’, ‘proof’, ‘example’, ‘whenever’, ‘unless’, ‘if’, ‘only if’, ‘necessary’, ‘sufficient’, ‘for some’, ‘for any’, ‘for each’. The distinction between a general proof and a particular example is commonly misunderstood. Be aware that many words like ‘bound’, ‘limit’ and ‘infinity’ have technical meanings superficially similar to but actually quite different from common usage.
Some of these symbols and their meanings are given in the Appendix.