Philosophy of Mathematics An Introduction to the World of Proofs and Pictures Author: James Robert Brown Publisher: Routledge ISBN: 0-415-12275-9 --------------------------------------------------------------------- Rambling Preamble: Now, gentle reader, I know you're interested in the Philosophy of Mathematics. How do I know this? Have I been rifling through your things? Maybe. Have I been stealing your underwear whilst rifling through your things? Quite possibly, and, if you're interested, it's rather a snug fit. But that is not how I know you're interested in what this book has to offer. I know because you're interested in Mathematics, otherwise you wouldn't be looking in this section. If you're interested in Mathematics, you are undoubtably interested in the philosophy of Mathematics. I stress that there is no possibility here of me being wrong, no matter what you might think. If, against all that is good and decent in the world, you are not interested in this then you are not interested in Mathematics at all! You're merely a calculator, interested in calculation! Interested in little but the execution of routine algorythms and the manipulation of data! We have no need for you, we have computers, with flashing lights and everything, to do that for us! Get out, get out I say, nay, demand! Leave the sanctity of this place, head into the woods and offer yourself to the wolves. I might add that you had better pray that God has mercy upon your soul, for you shall find none here. Anyway, now that we have gotten the pleasantries out of the way, I shall continue. Mathematics and Philosophy are inseparable. When we wish to prove something, it is necessary to first ask "What constitutes 'proof'?", this, in turn, entails that we have some concept of 'certainty'. Unlike in the physical sciences, in Mathematics there are no 'approximations' to the 'truth' as there are in Physics (i.e. approximating the magnitude of actual forces acting between planets, using Newton's Law of Gravitation). A Mathematical statement is either correct, or incorrect. How we determine whether or not such a statement is correct, and how we can be certain in either case are questions that have had continued relevence in Mathematics since the Classical era (a contemporary issue being what role computers have in the derivation of proofs). It is this rigour of definition, and the concept of a 'Mathematical Reality' (such a reality is the only place where a circle can be found, such an object does not exist either in the 'mind's eye' or in physical reality, it exists only as a mathematical defintion of an object in geometry) that sets Mathematics apart from other subjects and prevents it from being little but a useful tool for the Sciences. Introduction: This book is, I would say, a 'first text' introduction and sourcebook to and for the study of the Philosphy of Mathematics. It describes and comments upon various matters, including: the meaning of a 'definition' in Mathematics; Computation and Proof, different approaches (constructivism etc.); the role of 'picture proofs', the connection with 'Applied Mathematics'; as well as discussing the contributions made by Philosophers such as Wittgenstein, Frege, Kant and Plato to the debate surrounding proof, certainty and Mathematical reality. Pros: A succinct, yet comprehensive introduction to the major issues in the Philosophy of Mathematics. A 'conversational' and entertaining prose that occassionally discusses practical applications of Mathematics and their implications (such as the sometimes objectionable use of Mathematical concepts in the social sciences). Such is the variety of subject matter and the skill with which it is written, there is rarely a dull moment in my humble, and quite possibly dull opinion. Happily, a great many prominant philosophers are mentioned, often with quotes, allowing many a gleefull hour of name dropping to be had with your contempories. Cons: Whilst the passion with which the author writes helps to carry the reader through subjects that he/she may otherwise find dry and inaccessible, without ever being overbearing, this really does represent one point of view. Although in many areas this may be of little consequence, in others, such as the discussion of Platonism and the potential of picture-proofs, it may be beneficial to suspend judgement until a wider collection of views have been read (that's not to say that I disagree with Mr Brown's conclusions). Cover: Yes, this issue now has its very own section. The cover is, quite frankly, amazing. This book has the equivalent of high art, cinematic sex with the 1812 overture as its soundtrack wrapped around its pages. Compared with the banal, middle aged, self loathing marital romp that can be symbolised by most Springer publication covers, and the grubby, pedestrian sub soft-porn fumble of most Mathematical texts' covers (oh look! a roller coaster with superimposed, relevant equation, bravo.), this cover truly is a wonder. Want to know why? Because Routledge have it on the money as far as covers are concerned. A bookcase of Routledge is as beautiful to the eye as the content of the books themselves is beautiful to the mind. Conclusion: Philosophy of Mathematics, an Introduction to the World of Proofs and Pictures, is an excellent book with which to become acquainted with the subject. Regardless of whether or not it acts as a catalyst to a growing interest in the Philosophy of Mathematics, or represents the reader's first and last encounter with such ideas as are contained within, reading it can only prove a beneficial and, thanks to the style adopted throughout, an enjoyable experience to those who are interested in Mathematics beyond the mundane matter of plugging in numbers and rearranging ever more complex equations. Having said that, it is not, and does not pretend to be the definitive word on the matter, but, thanks to a comprehensive bibliography, could easily be used as a springboard for those who wish to further their interest beyond the book's limitations (a personal recommendation is 'Philosophy of Mathematics, an Anthology', edited by Dale Jacquette and published by Blackwell Philosophy Anthologies).