Philosophy of Mathematics: An introduction to the world of proofs and pictures by James Robert Brown, Routledge, 拢16.99,ISBN 0415122759
Conversations with a Mathematician: Math, art, science and the limits of reason by Gregory Chaitin, Springer, 鈧�24.95, ISBN 1852335491
Mathematics Elsewhere: An exploration of ideas across cultures by Marcia Ascher, Princeton University Press, 拢17.95, ISBN 0691070202
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Algebraic Number Theory and Fermat鈥檚 Last Theorem (3rd edition) by Ian Stewart and David Tall, A. K. Peters, 拢28, ISBN 1568811195
MANY mathematics undergraduates, used to a diet of theorems and proofs, will hold a comforting belief that all mathematical statements can be proved to be true or false, even if some of these proofs have not yet been discovered. There is, however, a remarkable and unsettling truth lurking in the foundations of mathematics: G枚del鈥檚 first incompleteness theorem.
Consider the statement 鈥淭his sentence is false鈥�. Is this statement true or false? If it鈥檚 true, then it follows from the statement that it鈥檚 false, whereas if it鈥檚 false, then it similarly follows that it鈥檚 true. The statement is in fact a paradox, neither true nor false.
Your initial reaction might be to dismiss this as a mere linguistic trick, but G枚del showed that statements like this exist in arithmetic too. There are basic assumptions, or axioms, about the familiar 鈥渘atural鈥� numbers 1, 2, 3 鈥� that mathematicians usually assume to be true, and arithmetic is built on those axioms. G枚del showed that, whatever axioms you choose, there exist mathematical statements about the natural numbers that are neither true nor false. It is impossible to find a proof that such a statement is true, or that it is false. This area of metamathematics is mentioned only briefly, if at all, in many mathematics degree courses, often being seen as philosophy rather than mathematics.
Students who would like to learn more about this and other philosophical issues not covered in their degree course will find James Robert Brown鈥檚 Philosophy of Mathematics a good starting point.
G枚del鈥檚 incompleteness theorem and some related ideas are also discussed, in greater detail, in Gregory Chaitin鈥檚 enjoyable and thought-provoking Conversations with a Mathematician (reviewed 12 January, p 44). The book consists of chatty lectures and interviews given by Chaitin. Many of these cover similar ground, but the repetition of topics mostly serves to illuminate them further. For a mathematician attempting to discover proofs of statements, G枚del鈥檚 incompleteness theorem can be very disconcerting.
Chaitin explains that unprovable mathematical statements are far from rare, so it is conceivable that some mathematicians out there are trying to justify statements that have no proof. So Chaitin makes a case for an experimental approach to mathematics. He argues that more mathematicians should consider working in a similar manner to other scientists 鈥� assume that mathematical statements are true if there is evidence for them, and work on using them to obtain further useful results.
The mathematical theory to which G枚del鈥檚 theorem applies is constructed by combining some key undefined terms together with a set of axioms about these terms. This approach originates from early Greek mathematics and is at the core of modern Eurocentric mathematics. Books such as Marcia Ascher鈥檚 Mathematics Elsewhere provide a reminder that there are other approaches to the subject and that the view of mathematics varies across different cultures. Ascher illustrates that non-Western cultures have developed sophisticated mathematical ideas often without having any formal concept of mathematics. This stimulating book deserves a wide audience, especially among those involved in teaching the subject.
Despite G枚del鈥檚 findings and the possible limitations of our approach to mathematics, mathematicians continue to discover wonderful new theorems as well as cracking seemingly impossible problems such as Fermat鈥檚 Last Theorem.
Ian Stewart and David Tall鈥檚 Algebraic Number Theory and Fermat鈥檚 Last Theorem gives a good account of how this old chestnut finally succumbed after more than 300 years of effort. It is no coffee-table book, being intended for advanced undergraduate students.