Rational Mathematics

by G. P. Jelliss


How do we know what we claim to know? We wish to begin here by assuming as little as possible. However, we cannot begin by assuming nothing. As soon as a writer sets down his first word he is already assuming a great deal, whether he acknowledges it or not. The ability to communicate through language, written or spoken, is a very complex achievement.

So how does language work? An argument proceeds by a series of statements, which are expressed in sentences made up of words, which may be in spoken or written form.

The use of language in a manner seeking to gain agreement or at least tolerance of our point of view, without the use of force or shouting down the opposition or any more subtle forms of compulsion, is reason. The way we do this is by reasoned (or rational) argument in which we set out our arguments in simple stages, seeking to get agreement that each step is acceptable to our audience. If all the steps are accepted, then the whole argument should be accepted, or else our opponent is being unreasonable (or irrational).

By truth we mean suitable to be accepted by our audience. The nature of truth may thus depend on our audience, some audiences will be prepared to accept our assumptions, or presumptions, for their beauty, ingenuity, common sense, self-evidence or other qualities, while others may simply be prepared to suspend judgment, to see what our assumptions lead to, and some may refuse to accept our assumptions, or may wish to subject our assumptions to tests of some sort designed to assess their ‘correspondence with reality’. Statements which have passed the particularly strict tests of ‘scientific method’ may be termed scientific truth.

Any audience usually has an accepted body of truths, constituting their collective knowledge (or belief) against which the truth or falseness of new propositions is judged. Historical studies show that what is accepted as true by the scientific community, or by a religious community, evolves over time. A collective world-view, against which new propositions are assessed, is known as a paradigm. Major new ideas may require deep changes in the paradigm, that may amount to paradigm-shifts (or revolutions) in ideas, often accompanied by revolutions in social organisation.

The attempt to make reasoning more reliable by paying close attention to the way statements are formulated and the steps by which arguments are constructed is logic. The most basic kind of logic deals only in propositions, that is precisely defined statements, that are either true or false. This is two-valued logic. There have been attempts to develop a three-valued logic, but the two-valued form is so much simpler that it in effect chooses itself for study.

A proposition has a truth value, that is it is either true (1) or false (0). We use these number-symbols for truth values (rather than say the letters T and F which are often used) since we will eventually develop a more comprehensive system in which intermediate values can be assigned that measure the probability (or degree of truth or certainty) that may be assigned to a proposition. It is perhaps surprising that by using two-valued logic we can develop this more complex system that can cope with multiple or fuzzy valuations.

Reasoned arguments, at least when tidied up, consist of sequences of propositions whose truth we assert. An argument to gain acceptance of one proposition from the accepted truth of others is a proof, and a proposition together with a proof of it is a theorem. (A minor theorem that is used in the proof of a more important one is sometimes called a lemma, while a minor theorem whose truth follows in a simple way from that of a major theorem is called a corollary or rider.) The propositions with which we start a proof are termed assumptions (or premises) and those we finish with are conclusions. The conclusions are said to be inferred (or deduced) from the assumptions, and the process is one of inference (or deduction). A conclusion of one argument may be an assumption of another.

So what are the simplest steps by which rational arguments proceed? An inference from one statement to another is an implication and takes the form ‘If X then Y’. It seems apparent that Y must either say the same as X or say less than X, which doesn't seem to be progress, but in fact the implications of an assumption may not be obvious and can often be overlooked, particularly special cases. An inference from two statements to one is a syllogism and takes the form ‘If X and Y then Z’. If Z follows by implication from X or Y alone then this is not a genuine syllogism, since Y or X is not needed. So it is clear that Z must combine part or all of X with part or all of Y. A series of syllogisms is called a sorites (pronounced sore-eye-tease). Most arguments however are far more complex than these simple forms.

A long series of connected arguments constitutes a theory. An axiomatic development of a subject proceeds by linear or branching chains of arguments that can be traced back to certain initial assumptions, which are called axioms (or postulates). There may however be many different ways of developing a subject in this manner, and there may not be any one way that is best or right. A non-axiomatic development of a subject, in which concepts may be circularly related or form a complex network, is also possible and can be equally logical and correct, so long as the circularity is recognised.

A theory whose axioms contain undefined terms, or whose terms can be interpreted in different ways, is an abstract (or mathematical) theory, while one whose terms get their meaning from outside the system itself, i.e. in the ‘real world’ is a concrete (or scientific) theory. As a matter of history, the first abstract theories were developed from concrete examples, but as mathematics developed, it has often happened that abstract theories have been developed before any applications of them have been found.

In applying an abstract theory to concrete uses, or in other words using a theory to model a real situation, our success or failure depends on the accuracy with which the terms of the theory correspond in their theoretical behaviour to the observed phenomena. One can say all sorts of things in theory, but whether they work in practice depends on the fit. The fitting of theories more and more closely and economically to the facts is what science is about. Abstract theories can often be applied to other abstract theories, or indeed to themselves. In fact this process is the principal way in which we develop complex abstract theories from initially simple beginnings.