vol 4: Glossary
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... to restore theology to the mainstream of science
Mathematics
There are many ways to define mathematics:
Mathematics is the science of quantitative relations and spatial forms in the real world. Hazewinkel, Mathematics.
... a group of related subjects, including Algebra, Geometry, Trigonometry and Calculus, concerned with the study of number, quantity, shape and space, and their inter-relationships, applications, generalisations and abstractions. Borowski, Mathematics.
Somebody once said that philosophy is the misuse of terminology which was invented for just that purpose. In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented for just this purpose. The principal emphasis is on the invention of concepts. ... The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. Wigner
According to the formalist point of view, mathematics should be regarded as a purely formal game played with marks on paper, and the only requirement this game need fulfill is that it does not lead to an inconsistency. Cohen, 3.
This project takes the formalist point of view. The principal constraint on mathematics is consistency. Within this constraint, mathematicians, as characterised by Wigner, are free to dream up anything they like.
We do not want to outlaw inconsistency completely, however, since it seems to be part of reality, and mathematically specified machines must be capable of dealing with it. "For example, an aircraft aloft might be receiving contradictory data from its sensors but be unable to take the luxury of shutting down before solving the problem of what is the nature of the physical environment." Mortensen, 9.
To get attention, of course, mathematics must be interesting and/or useful. In this sense it is no different from any other art. Casti In the real world, things must work, that is be consistent within themselves and with the constraints imposed by physics, aesthetics and all the other relevant elements of reality.
Further reading
Books
| Borowski, Ephraim J, & Johnathan M Borwein, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Casti, John L, Five Golden Rules Great: Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). Amazon back |
| Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cumings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Goedel's proof of the consistency of the continuum hypothesis. ..' (i) Amazon back |
| Frege, Gottlob, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Nature of Number, Northwestern UP 1980 Jacket: 'The book represents the first philosophically sound discussion of the concept of number in Western civilisation. It influenced profoundly developments in the philosophy of mathematics, general ontology and mathematics.' Amazon back |
| Gellert, Walter, The VNR Concise Encyclopedia of Mathematics , Van Nostrand Reinhold 1994 Preface: '... there is a wide demand for a survey of the results of mathematics ... Our task was to describe mathematical interrelations as briefly and precisely as possible. ... Colours are used extensively to help the reader. ... Ample examples help to make general statements understandable. ... A systematic subdivision of the material, many brief section headings, and tables are meant to provide the reader with quick and reliable orientation. The detailed index to the book gives easy access to specific questions. ...' The Editors and Publishers Amazon back |
| Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Hofstadter, Douglas R, Goedel Escher Bach: An Eternal Golden Braid, Basic/Harvester 1979 An illustrated essay on the philosophy of mathematics. Formal systems, recursion, self reference and meaning explored with a dazzling array of examples in music, dialogue, text and graphics. Amazon back |
| Kneebone, G T , Mathematical Logic and the Foundations of Mathematics, van Nostrand 1975 Preface: 'The present book ... is designed to serve in the first instance, when supplemented by reference to original sources, as a comprehensive introduction to the earlier phases of the historical development of the philosophy of mathematics. p vi.back |
| Kramer, Edna E , The Nature and Growth of Modern Mathematics, Princeton UP 1982 Preface: '... traces the development of the most important mathematical concepts from their inception to their present formulation. ... It provides a guide to what is still important in classical mathematics, as well as an introduction to many significant recent developments. (vii) Amazon back |
| Monk, Ray, Bertrand Russel: The Spirit of Solitude, Vintage 1997 '"the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a deep ocean of anguish, reaching to the very verge of despair." ... The aim of this book is to chart this course ... To understand the course his life took is to understand the power of each of these great passions and the tensions that existed between them, forcing him, on occasion, to abandon, in turn, philosophy for love, politics for philosophy, love for politics, and so on.' pp xviii-xix. Amazon back |
| Mortensen, Chris, Inconsistent Mathematics, Kluwer Academic 1995 'The argument from pure mathematics for studying inconsistency is the best of reasons: because it is there. ... It is always dangerous to think that a physical use will never be found for a given piece of mathematics. Nor is present-day mathematical physics anomaly free: witness the singularities at the beginning of time or in black holes, delta functions in elementary quantum theory, or renormalisation in quantum field theory.' p 8-9. Amazon back |
| Stewart, Ian, Life's Other Secret: The new mathematics of the living world, Allen Lane 1998 Preface: 'There is more to life than genes. ... Life operates within the rich texture of the physical universe and its deep laws, patterns, forms, structures, processes and systems. ... Genes nudge the physical universe in specific directions ... . The mathematical control of the growing organism is the other secret ... . Without it we will never solve the deeper mysteries of the living world - for life is a partnership between genes and mathematics, and we must take proper account of the role of both partners.' (xi) Amazon back |
| Tymoczko, Thomas, New Directions in the Philosophy of Mathematics: An Anthology, Princeton University Press 1998 Jacket: 'The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. ... By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, TT reveals an evolving effort to account for the nature of mathematics in relation to other hman activities.' Amazon back |
Links
| Eugene Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back |
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