Michael Spivak: Calculus (1967)

Edition: Addison-Wesley (World Student Series)
Review number: 1506

Calculus was the very first textbook I read for my university degree. As well as being a fine description of the basics of analysis (mostly real, with a toe in the deep water of complex functions), it is an excelent book to ease the transition from mathematics as taught at school level to the rigours of university mathematics.

Unlike many writers of textbooks in mathematics, Spivak makes a big effort to give more than a dry exposition: theorem - proof - next theorem etc. Considerable attention is paid to motivating the discussion, showing why each result is important (though mainly in the pure mathematics context, applications of calculus being mainly found in the problems at the end of each chapter). Of especial use to the budding mathematician are the points where Spivak discusses potential proof strategies for the theorems, often explaining the pitfalls that student taking a naive approach could fall into. There are even occasional jokes, both in the text and the index.

For students with an interest in how analysis can be used in apparently unrelated parts of mateematics, a number of advanced sections give proofs of such topics as the transcendence of the number e, and a construction of the real numbers from set theoretic principles.

Calculus was not just the first university textbook I read, but one of the best.

My rating: 10/10.

Arthur C. Clarke & Frederik Pohl: The Last Theorem (2008)

This is the novel with which Clarke rounded off his lengthy and prolific career. Like much of his later work (later in this case basically meaning novels published after Clarke was eighty), The Last Theorem is a collaboration. While most genre collaborations are between established authors and newcomers, this is different, in that Frederik Pohl is one of the very few authors who could be considered one of Clarke's near equals for prestige in science fiction.

The Last Theorem is a novel about an alien invasion of Earth, a theme of science fiction which goes all the way back to The War of the Worlds. Concerns today are not those which prompted Wells to produce a novel which is about colonial warfare, however; the motive for the invasion here is not a search for resources, but pest control. Immensely powerful aliens have detected the explosion of the first nuclear bomb on Earth in 1945 and applied their inflexible rule: eradicate the dangerous vermin who act so aggressively. This is surely not a very original scenario (even though I cannot immediately think of exact parallels), and it is indeed not the most interesting part of the novel.

For while the aliens are travelling to Earth (making use of some "loopholes" in the laws of relativity, but still slow enough to allow the plot to unfold), human beings are continuing their usual lives. The authors focus on one man, a Sri Lankan mathematics student at the beginning of The Last Theorem, who goes on to prove Fermat's Last Theorem. (This requires a certain amount of explanation, as Andrew Wiles was already famous for this feat before the novel was written. But Wiles' proof is far too lengthy to be the one Fermat was unable to write in the margin for lack of space, and that is the proof that Ranjit Subramanian finds. In addition, the authors feel - as indicated in their postscript comments - that a proof which relies on computer checking is not really as convincing as one in which the details can be grasped in their entirety by a human mind. So Ranjit's fictional five page proof is the "real" one.) The proof brings him international celebrity and a role in the alien encounter to come (though his daughter coincidentally has an even more important part to play).

At the start of The Last Theorem, the narrative voice is jocular and quite informal; and irritating. But one of the most impressive aspects of the novel depends on this. Once something unpleasant happens to Ranjit (the bridge between being a carefree student and an international celebrity), the narrative voice changes, and becomes more grown up.

The Last Theorem, while readable, is not the best work of either Clarke or Pohl by a long way. As well as the sloppy plotting of the coincidence already mentioned, there are other incidents in the story which don't really ring true. There is nothing new in the basic ideas in the novel. The mathematical components are well done, if you're interested in that sort of thing, and no prior knowledge is needed. But perhaps that is not really enough from two of the greatest writers of the science fiction genre - 4/10.




Edition: HarperVoyager, 2009
Review number: 1426

Carl E. Linderholm: Mathematics Made Difficult (1971)

Edition: Mosby-Wolfe, 1971 (Buy from Amazon)
Review number: 1083

When I was a mathematics graduate student, this book was passed around the department, delighting those of us working in pure mathematics. Basically, it takes apart the sort of mathematical ideas generally taken for granted, and shows that they are much more complicated than it first seems if you want to make them rigorous. (There is some cheating when ideas from category theory are introduced and make the explanations even more abstract than they need to be.)

There is, of course, a subject for a serious book in this; I can think of two without any effort (Rudy Rucker's Infinity and the Mind and the far older Bertrand Russell's Introduction to Mathematical Philosophy). Mathematics Made Difficult is not in any sense a book which aims to educate and inform its readers. Much of the mathematics is presented in a way which would probably not make a great deal of sense to anyone not already familiar with it (a course in the foundations of number theory is really the minimum needed to understand most of it, and one in category theory for the detail). What is enjoyable about Mathematics Made Difficult is that it is very funny, full of parodies of school textbook problems and bad puns. Mathematics is not easy to turn into humour, and this book is one of the very small number of consistently successful examples.

Amir Aczel: Probability 1: Why There Must be Intelligent Life in the Universe (1998)

Edition: Little, Brown & Co, 1999
Review number: 795

The question of the existence of extra-terrestrial life is one which has fascinated the human race since classical times. In recent years, various attempts have been made to estimate the likelihood that such life exists, prompted in some part by the popularity of science fiction. This sort of speculation begins with what is known as the Fermi paradox, which is basically that we should already know if there is intelligent life more advanced than we are because they should have contacted us already.

This way of looking at things actually suggests other questions, which are rather more interesting that whether life exists at all; it would be hard to get excited by a universe populated only by micro-organisms other than on our own planet. Basically, these other questions amount to speculation about what form putative extraterrestrial life might take - could there be advanced civilizations? How could we know? They actually lead back to close study of life on earth, to try to see how things could have developed differently.

There is an equation, the Drake equation, which predicts the likelihood of contact by an advanced alien race; this is put together by assigning probabilities and expected values to various events, most of which is guess work - the probable lifetime of a civilization, for example. Aczel's book, after a discussion of some of the issues raised in the previous paragraph, makes an estimate for the first few values in the equation, those which refer to the existence of life itself rather than levels of technology, and infers from this that the probability of life existing somewhere else in the universe is essentially indistinguishable from 1, certainty.

This is done through some elementary probability theory, which essentially amounts to saying that the universe is so big that, no matter how unlikely, life must exist somewhere. This is saved for the end, but much more interesting is Aczel's explanation of why he thinks humanity might well be the most mature civilization in the universe (as a result of the inspection paradox, unfortunately not as clearly explained as most of the mathematics in the book).

The book as a whole is not as successful as Aczel's earlier explanation of Fermat's Last Theorem, at least as far as I am concerned. A lot of it is over-simplified - I ended up skipping a lot of the early part of the book. It picks up in the middle, when nuggets of interesting information start to be included, but unfortunately becomes less interesting again as it starts to concentrate on the existence of life in general rather than intelligent life.

Ian Stewart: From Here to Infinity (1996)

Edition: Oxford University Press, 1996
Review number: 315

Ian Stewart has written several versions of a survey of the current state of mathematics, dating back as far as 1975 with Concepts of Modern Mathematics. The more snappily titled From Here to Infinity, published in 1998, is the latest version of his 1987 book, The Problems of Mathematics. As mathematics progresses, the important and interesting areas change rapidly, as new breakthroughs come and new applications and connections spark renewed interest in hitherto obscure areas. Thus a considerable revision has been made with each new incarnation.

The original book has had a considerable influence on my own life. It helped confirm my desire, as an A-level student, to study mathematics at university. From Here to Infinity is as inspiring as this suggests, and made me want to return to some of the books I have not opened for years.

Stewart tends to focus on those areas which are of interest to him personally, and his enthusiasm helps to make his account more accessible. There is one part of the book which reads as though it was included because he felt he had to rather than because he wanted to, and that is the section on Fermat's Last Theorem. However, his writing is never opaque, and should be comprehensible to the (more or less) general reader. He gives a picture of the mindset behind modern mathematics, something very difficult to obtain in the English and American school systems, where little more recent in date than 1900 is taught even under the "new maths" banner. The change between school and undergraduate mathematics is marked, and at a very fundamental level, as proof rather than correct calculation becomes the important skill.

Amir D. Aczel: Fermat's Last Theorem (1996)

Edition: Viking, 1997
Review number: 102


This book is a popular history of Fermat's Last Theorem, from its original conjecture by Fermat to its solution by Andrew Wiles. As a history, it works quite well, with occasional infelicities (mainly to do with forced, false sounding connections between unrelated parts of the narrative, such as linking mathematicians because they were both interested in some fairly large division of mathematics). From a mathematical point of view, I felt it was perhaps a little less successful. It managed to be over-simplified for the mathematically trained and at the same time potentially confusing for the non-mathematician. I suppose you can assume a minimal level of interest in mathematics to be held by anyone wishing to read a book on this subject, but I'm not sure the book always reached that level. The order used to detail the contributions - and contributors to the theories involved include just about every single really famous mathematician - is neither strictly chronological nor really by mathematical subject; the book could have done with perhaps some appendices to help orient the reader who wanted to use it as a reference.

These criticisms apart - and they are fairly severe - I enjoyed the book; I wanted to go on and read more on the subject.

Alan Baker: Transcendental Number Theory (1975)

Edition: Cambridge University Press, 1998
Review number: 75

This book has been the standard survey of the theory of transcendental numbers for some time now - the first edition was published in the mid-seventies. The author is a prominent researcher in the field, and several chapters draw heavily on his own work.

My own mathematics training is in areas other than number theory, and I found large parts of the book difficult to follow. I have a copy of the basic number theory textbook written by Baker, A Concise Introduction to Number Theory, and I would have expected some discussion of concepts used in Transcendental Number Theory but absent from the more basic book.

The other problem is with Baker's writing style, and it is shared with An Introduction to Number Theory. His mathematics is extremely condensed in style, and it is often difficult to work out what is going on. In a survey this is less of a problem than in a textbook, but I still felt I would have found it easier to follow if I were attending a course of lectures based around the book at the same time as reading it.