Rational Mathematics

by G. P. Jelliss


By a sign we will mean any easily recognisable geometrical mark, such as the letters of the Roman alphabet (which may be printed in upright, bold, italic or script forms), or letters borrowed from other alphabets such as the Greek, or any other specially designed signs for particular purposes such as the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and geometrical signs such as +, –, =, |, ×, /, <, >, (, ), [, ] and so on, but we will try to keep our use of special signs to a minimum.

A sign used to represent something other than itself is called a symbol. All reasoning could, if we so chose, be conducted in normal language, but it would be extremely long-winded in many cases. The use of short-hand abbreviations for long phrases that recur can result in great savings in verbiage and probably be clearer too. The use of signs as symbols to clarify or systematise arguments is symbolism (or algebra in a very general sense of that term).

Since the number of signs available to us is limited, for example by the number of keys on our typewriter keypad, it is often necessary to use groups of signs placed close together to form a compound symbol. The most familiar compound symbols are the words of our languages. In these each sign also symbolises a vocal sound, thus enabling us to vocalise the word as well as write it, though the correspondence between sound and spelling is often far from reliable.

Another way of expanding the number of signs available to us is to print them in a different styles, such as italic or bold (though we will avoid this is far as we can). This can also be used to distinguish individual signs used as symbols from signs used in words or other compounds.

The ‘thing’ that a symbol represents, the meaning of the symbol, is not necessarily a physical object but can be a mental concept or an imaginary entity of any kind. An explanation of the meaning of a symbol is called a definition. A definition may be symbolic, that is expressed in terms of other symbols, or it may be demonstrative, that is by showing or demonstrating what the symbol represents without using symbols of equivalent meaning.

One sign we cannot do without is the equals sign (=). If symbols r and s are used to represent the same concept we say that r equals s, symbolised r = s. It is an important convention that we allow the possibility that different symbols may represent the same concept. It may be possible to write the same statement in different ways symbolically, in which case each particular representation is an expression of the statement. Any proposition of the form r = s, where the symbols r and s may represent more complex expressions, is an equation.

The meanings of symbols may depend on the context; that is the general assumptions, often unstated, on which an argument is based. If the context changes then the meanings of some of the symbols may change. For instance the meaning of the word ‘mate’ in the context of chess is very different from its meanings in the contexts of biology or of seamanship. If we are not very careful, it may be possible for the context to change without us being aware of the change. Changes of context are the basis of much humour, such as the pun (words of different meaning that sound alike), the malapropism (confusing similar words with very different meanings) or the double entendre (using words with one apparent meaning but another meaning to those in the know). There are also many logical paradoxes that result from carelessness on this point.

The context need not necessarily be one of reality, it may be one of fantasy. For example in the real world of history there was no such thing as ‘Julius Caesar's collection of American Indian totem poles’, but we could imagine a fictitious ‘parallel universe’ in which Caesar extended his conquests improbably to the Americas. Similarly the flying horse ‘Pegasus’ does not exist in the real world, but it does exist in the world of Greek Myth as told in the story of Perseus. There are things that probably cannot exist in any world, such as a ‘square triangle’ unless we give the terms ‘square’ and ‘triangle’ new meanings. These are meaningless concepts in the real world.

A symbol whose meaning changes in the course of an argument is a variable. The meanings that a variable may take are called its values. Many writers tend to use the later letters of the alphabet for variables and the earlier letters for symbols that take only one value, which are constants. Variables are useful since they make it possible to make many similar propositions at one go. To avoid confusion however, it is necessary to be clear what values a variable may take. In the course of an argument we often let constants vary or make variables fixed, so the distinction is not always as firm as it sounds.