Few works on mathematics define the meaning of the word. The original meaning of the Greek root from which it derives was simply ‘learning’. The viewpoint taken here is that mathematics is essentially a method that can be applied to other branches of learning, rather than a subject in itself. The method is that of the formalisation and systematic use of language, through the use of abbreviatory symbols.
An important aspect of mathematics is its generality. The methods of mathematics should be developed so as to be readily applicable to as wide a range of subjects as possible without need for modification. Some subjects, like geometry, mechanics and physics are highly mathematical and indeed are sometimes regarded as themselves being branches of mathematics.
In order to achieve this wide applicability, mathematics seeks to develop forms of language that will describe the simplest possible structures, such as sets and relations, from which more complex systems, such as algebras and geometries, can be constructed. An alternative name for this subject, in its general aspects, is morphology, from Greek roots meaning ‘shape-study’. The advantage of this morphological approach is that it guides us to develop the ideas in a philosophically sound way, instead of as a mosaic of apparently disconnected tricks.
This text is intended to be an introduction to mathematics as used in common scientific and recreational applications, presented in a manner suitable for those new to the subject or wishing to refresh their knowledge, and forming a complete course, not needing reference to any other works. My aim is to clarify these ideas for myself, and to hope that this will also prove helpful to others. The work is as yet far from complete.
The main unorthodoxy in my approach to the subject is that I have undertaken to develop as much of mathematics as possible without the use of infinite concepts. My view is that most of mathematics, certainly any that has any practical application, can be expounded purely in finite terms.
One of the guiding principles followed is Ockham's Razor: that concepts should not be introduced unnecessarily. Applied to terminology and notations this implies that they should be kept clear and concise. The use of several different notations for the same operation, or several alternative terms for the same concept, is likely to confuse the beginner in any subject. Terms used throughout are printed bold when defined on their first occurrence. Within brackets we mention alternatives in italics that the reader is likely to encounter elsewhere, but otherwise we use only one scheme consistently throughout. Some thought has been given to using clear notation and terminology, even if it is contrary to traditional forms.
A related principle that I have tried to follow is not to introduce unmotivated definitions. The sudden introduction of a new idea with a complex definition simply at the dictat of the expositor that this is an important idea and that subsequent results will show its significance is something at which I have always bridled. The aim therefore is to introduce ideas only when they arise naturally, and preferably when they are needed and can show immediate results.