# Rational Mathematics

## Quotations

### (a) In support of the thesis that mathematics is the study of structure

The Hyperbolic Geometry was the forerunner of numerous non-Euclidean Geometries, differing more or less from the Euclidean. In fact, almost any body of doctrine, based on Axioms, which resembles in some way the Euclidean Geometry, is now styled by courtesy a Geometry. [H. G. Forder, The Foundations of Euclidean Geometry, Cambridge University Press, 1927, p.2.]

The most striking characteristic of modern algebra is the deduction of the theoretical properties of such formal systems as groups, rings, fields and vector spaces. ... Modern algebra also enables one to reinterpret the results of classical algebra, giving them far greater unity and generality. Therefore, instead of omitting these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra. [G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Macmillan, 1941, p.v.]

Mathematics, as conceived today, is fundamentally the study of structure. Thus, although arithmetic is ostensibly about numbers and geometry about points and lines, the real objects of study in these branches of mathematics are the relations which exist between numbers and between geometrical entities. As mathematics develops, so it becomes more abstract, until at last it is seen to be concerned with networks of formal relations only, and not with any particular sets of entities between which the relations hold. [J. G. Semple and G. T. Kneebone, Algebraic Projective Geometry, Oxford University Press, 1952, p.3.]

The elements of systems need not be numbers: they can be transformations, substitutions, polynomials, ... What we are interested in is the structure of the system rather than the nature of the elements; examples will show systems with different kinds of element but the same structure. [J. W. Archbold, Algebra, Pitman, 1958, p.237.]

### (b) Concerning infinity and large numbers

For centuries, mathematicians and philosophers have been intrigued by the concept of infinity. Is there a number greater than all [other] numbers? If so, how large is such a "number"? Can we calculate with it as we do with ordinary numbers? And on the small scale of things, can we divide a quantity — say a number or a line segment — again and again into smaller quantities, or will we eventually reach an indivisible part, a mathematical atom that cannot be further split? Questions such as these troubled the philosophers of ancient Greece more than two thousand years ago, and they still trouble us today — ///. (p.30) Archimedes, however, was careful to formulate his solution in terms of finite sums only; the word infinity never appeared in his argument, and for a good reason: the Greeks banned infinity from their discussions and refused to incorporate it into their mathematical system. (p.43) [Eli Maor, e: The Story of a Number, Princeton University Press, 1994.]

We may talk about immense numbers with gay abandon, but their size is so great as to make the imagination boggle at any attempt to understand their full significance. (p.9) [W. J. Reichmann, The Fascination of Numbers, Methuen & Co Ltd, London, 1957 (reprint 1963).]

An algorithm must always terminate after a finite number of steps. /// Note however that the number of steps can become arbitrarily large /// A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a "computational method". (pp. 4–5) We should remark that the "finiteness" restriction is really not strong enough for practical use; a useful algorithm should require not only a finite number of steps, but a very finite number, a reasonable number. For example there is an algorithm which determines whether or not the game of chess is a forced victory for the White pieces /// ; here is an algorithm which can solve a problem of intense interest to thousands of people, yet it is a safe bet that we will never in our lifetimes know the answer to this problem, because the algorithm requires fantastically large amounts of time for its execution, even though it is "finite". (p. 6) [Donald E. Knuth, The Art of Computer Programming, Vol.1 Fundamental Algorithms, 2nd edition, Addison-Wesley Publishing Company 1973]

### (c) Concerning real numbers

We use real numbers in physical theory out of convenience, tradition, and habit. For physical purposes we could start and end with finite, discrete models. Physical measurements are discrete, and finite in size and accuracy. To compute with them, we have discretized, finitized models physically indistinguishable from the real number model. The mesh size (increment size) must be small enough, the upper bound (maximum admitted number) must be big enough, and our computing algorithm must be stable. Real numbers make calculus convenient. Mathematics is smoother and more pleasant in the garden of real numbers. But they aren't essential for theoretical physics, and they aren't used for real calculations. (p.175) [Reuben Hersh, What is Mathematics, Really? Vintage 1998]

The square root of 2, for instance, can only be calculated to the nearest significant figure, according to the degree of accuracy required; it can never be calculated exactly. In the same way, there is no exact value for p, the symbol used to show, amongst other things, the relationship between the diameter of a circle and its circumference. For practical purposes, this is not a serious complication since the approximate values satisfy most requirements. (p.15) [W. J. Reichmann, The Fascination of Numbers, Methuen & Co Ltd, London, 1957 (reprint 1963).]

### (d) On the nature of mathematics

... No one ever learned mathematics by reading a book; the learning comes about through doing – through practising the techniques which have been described. So make sure you do the exercises as you go along, and that if you disagree with one of the answers provided, you study the solution and ensure that you understand where you went astray before carrying on to the next section. [I have learned mathematics mainly from reading books, but it is true that you should practice calculations and problem solving.] Mathematics is to a large extent a linear subject – that is, one topic builds on another, so if you are confused about arithmetical operations you have little chance of getting to grips with algebra, and if algebra poses problems your grasp of calculus will probably be hazy. Thus the first five chapters of the book need to be read in sequence. (p.xiii) [I would definitely put algebra before arithmetic.] Many people believe that mathematics is a rather mysterious and elusive subject, composed of abstractions, and accessible only to a minority of the population possessed of extraordinary minds. While it is true that there are areas of advanced mathematics to which this description would apply, it is also the case that elementary mathematics is extremely practical stuff, rooted in common sense, and useful for dealing with everyday problems in an efficient manner. [Clare Morris & Emmanuel Thanassoulis, Essential Mathematics, A Refresher Course for Business and Social Studies, Macmillan 1994.]

### (e) Misuses of infinity (hyperbole)

As far as I have been able to learn, from both written or verbal accounts, this complaint [hepatitis], so frequent on the Coromandel coast, ought to be treated with large bleedings, and copious active purgatives, in the very first attack. The quantity of blood taken, I think, ought not to be regulated by rules laid down by systematic writers for the inflammatory fevers of this country; if it is done at the proper period, it ought to be infinitely larger. If these evacuations have been neglected, or sparingly employed, suppuration, torpor, or infarction of the liver, to a certainty succeed, which render the use of mercury indispensible. [Thomas Trotter, A View of the Nervous Temperament, Longman and partners, London, 1807, p.99.]

E-mail offers the most obvious benefits. It is infinitely cheaper than either post or telephone and the fastest way of delivering copy. [Jane Dormer, 'Writers and the Internet', in Writers' and Artists' Yearbook 2000, A&C Black, London, 2000, p.549.]

## Bibliography of Sources, and Recommended Reading

### Logic and Set Theory

John G. Kemeny, Hazleton Mirkil, J. Laurie Snell and Gerald P. Thompson, Finite Mathematical Structures, Prentice-Hall, 1959.

Patrick Suppes, Axiomatic Set Theory, D. Van Nostrand Company, 1960.

C. L. Liu, Elements of Discrete Mathematics, (first published 1985), second edition, McGraw-Hill Book Company, 1987.

### Algebra

G. Chrystal, Algebra, (first edition 1886), seventh edition two volumes, Chelsea Publishing Company, 1964.

Clement V. Durrell, Advanced Algebra, volume 1, (first published 1932), G. Bell and Sons, 1968.

Garrett Birkhoff and Saunders MacLane, A Survey of Modern Algebra, (first published 1941), The Macmillan Company, 1946.

J. W. Archbold, Algebra, Isaac Pitman & Sons, 1958.

Kenneth S. Miller, Elements of Modern Abstract Algebra, Harper & Brothers, 1958.

Neal H. McCoy, The Theory of Rings, (first published 1964), The Macmillan Company, 1970.

D. E. Rutherford, Introduction to Lattice Theory, Oliver and Boyd, 1965.

### Numbers

Robert D. Carmichael, The Theory of Numbers and Diophantine Analysis, (first published 1914 and 1915), both books bound as one, Dover Publications 1959.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, (first edition 1938), fifth edition, Oxford Clarendon Press, 1979.

H. Davenport, The Higher Arithmetic, Hutchinson's University Library, 1952.

W. J. Reichmann, The Fascination of Numbers, (first published 1957), Methuen & Co, 1963.

Albert H. Beiler, Recreations in the Theory of Numbers, (first published 1964), second edition, Dover Publications 1966.

Waclaw Sierpinski, Elementary Theory of Numbers, (first English edition 1964), second English edition, revised and enlarged by A. Schinzel, PWN - Polish Scientific Publishers, 1987.

C. Berge, Principles of Combinatorics, (first published in French 1968), Academic Press, 1971.

L. F. Taylor, Numbers, Faber and Faber, 1970.

Charles R. Wall, Selected Topics in Elementary Number Theory, University of South Carolina Press, 1974.

Ian Anderson, A First Course in Combinatorial Mathematics, (first published 1974), Oxford Clarendon Press, 1979.

Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, (first published 1986), revised reprint, Penguin Books 1987.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1989.

Malcolm E. Lines, Think of a Number [Thanks to Rod Ball, email 02/09/2008, for this information, since I mislaid the reference.]