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We come finally, however, to the relation of the ideal theory to real world, or "real" probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: "If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician." In practice he is apt to say: "Try this; if it works that will justify it." But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
Loci: Convergence
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Prime Numbers: The Most Mysterious Figures in Math
Prime Numbers, The Most Mysterious Figures in Math, David Wells, 2005. 272+xv pp., $24.95, ISBN 0-471-46234-9. John Wiley & Sons, Inc., Hoboken, NJ, http://www.wiley.com/
In Prime Numbers, The Most Mysterious Figures in Math, David Wells has compiled a comprehensive A-to-Z guide containing a vast amount of information about prime numbers. Every reader of this book is bound to find many new and interesting facts about the primes. Mathematics teachers in particular will find much material here to enrich their classes. Indeed, the subject of prime numbers can provide teachers with many examples that are accessible to students and illustrate important points about what mathematics is and what mathematicians do.
The book contains many entertaining anecdotes and fun curiosities about prime numbers. There is a wide variety of topics, including:
· Famous conjectures about prime numbers, such as: Brocard's conjecture; Fortune's conjecture; Giuga's conjecture; Goldbach's conjecture; the Hardy-Littlewood conjectures; Polignac's conjecture and Shank's conjecture.
· Famous constants, such as: Brun's constant; Champernowne's constant; Euler's constant; Linnik's constant and Merten's constant.
· Famous functions, such as: d(n) (the number of divisors of n); σA(n), (the sum of the divisors of n), and the Euler phi-function, (the number of positive integers less than n and relative prime to n).
· Famous sequences, such as: the Bernoulli numbers; the binomial coefficients; the Fibonacci numbers and the Fortunate numbers.
· Famous mathematicians, such as: Euclid, Euler, Erdös, Gauss, Hardy, and Lucas.
The book includes an index, a glossary, and an extensive bibliography, all very useful for the reader wishing to delve more deeply into a particular topic. I recommend this book highly as an excellent reference book and a fun read.
Gabriela Sanchis, Professor of Mathematics, Elizabethtown College,PA