Concepts Of Modern Mathematics - Ian Stewart

[VinnieK]

For those looking for a popular science treatment of mathematics, Concepts of Modern Mathematics by Ian Stewart is without a doubt the best available. Although there are other popular accounts (those by Simon Singh and Marcus du Sautoy being amongst the most prominant), these all too often say very little in a very attractive way, replete with appealing allusions and imagery that actually replace rather tham complement meaning, and the reader comes away from them knowing little more than when they began, apart from a few new terms and some fuzzy analogies.

 

Rather than rely on one particular historical problem to grab the reader's interest (unlike Singh with Fermat's Last Theorem, or Marcus du Sautoy with Riemann's zeros), Stewart's attempts, and with considerable success, to explain what modern mathematics actually is, what mathematicians do, and to introduce the reader to the central topics and concepts in mathematics. What's more, he does so in a manner that's enjoyable and interesting, without once patronizing his audience or resorting to awkward similies - he does his reader the courtesy of assuming that they're already interested in the subject, resisting the urge to use grand literary flourishes or appeals to "relevance" to try and bribe them into reading further.

 

Originally intended as a guide for parents to the ill fated "new math" that in some countries' schools replaced basic maths with cursory treatments of fundamental mathematic concepts, Concepts of Modern Mathematics transcends it's original purpose and is, if not the most well known, perhaps the most sophisticated and lucid accounts of maths the general reader could hope to find.

[Chinahand]

There are quite a few maths books I'd like to read at the moment:

 

Dr Riemann's Zeros - as mentioned above

 

The Nothing That Is: A Natural History of Zero by Robert Kaplan

 

Zero: The Biography of a Dangerous Idea by Charles Seife

 

Brief History of Infinity: The Quest to Think the Unthinkable by Brian Clegg

 

An Imaginary Tale The Story of the Square Root: The Story of "I" (the Square Root of Minus One) by P Nahin

 

Who knows when I'll get round to reading them. Beyond just numeracy I think mathematics is important - there's an aesthetic to it - a beauty of pure thought - and beyond that there is almost no thing we currently use which doesn't need applied mathmatics (and a bit of engineering!) to create it.

 

I also thought I'd pass on one of the most unusual books I've ever read: Godel, Escher and Bach by Douglas R. Hofstadter . It is a truly stunning book - brimming full of word play, the amazing art of Escher and other equally thought provoking graphics, mathmatical puzzles and ideas.

 

The book sets out to explain Godel's famous mathematical proofs, but really it is an attempt to show how it is possible for a computer/machine to be conscious and a refutation of people who use Godel's theories to say artificial intelligence is impossible (a famous and equally complex book covering similar ground to say the complete opposite is Penrose's The Emporer's New Mind).

 

I've always had a fascination about whether computers will ever be able to trully think - Hofstadter firmly believes its possible and is one of the major people in the field. Godel Escher Bach is definitely one of the most unique books ever published - its a real challenge - it took me something like 6 months to get through it, but was a very worthwhile experience!

[VinnieK]

There are quite a few maths books I'd like to read at the moment:

 

Dr Riemann's Zeros - as mentioned above

 

The Nothing That Is: A Natural History of Zero by Robert Kaplan

 

Zero: The Biography of a Dangerous Idea by Charles Seife

 

Brief History of Infinity: The Quest to Think the Unthinkable by Brian Clegg

 

An Imaginary Tale The Story of the Square Root: The Story of "I" (the Square Root of Minus One) by P Nahin

 

Who knows when I'll get round to reading them. Beyond just numeracy I think mathematics is important - there's an aesthetic to it - a beauty of pure thought

 

Those are very good as histories of a single idea (such as the history of the complex numbers, or the history of Riemann's hypothesis), but the problem I have with them is that you don't actually get that great an insight into the actual subject of the book or mathematics in general, a problem that plagues much of popular mathematics. They're a bit like reading a book on Monet without any illustrations - you'll get a great idea of who he was and how he painted, but unless you've already seen all his major works you'll be none the wiser as to what he actually did. It's telling that amongst the reviews of those books, those from people with a mathematical background tend to raise the complaint that often they're actually unclear and sometimes outright misleading, which is the experience I've had with most popular mathematics texts - in an effort to spruce up the subject matter or make it "exciting" (which to a certain degree is patronizing to the reader), they often distort or obscure it to the point where reading it would be pointless. That's why I recommend Concepts: it actually takes the reader inside mathematics and guides them through. Plus it's written by a working and well respected mathematician. Most popular mathematics texts tend to be written by scientists or scientific journalists (strangely they tend to be physicists), and whilst there's a temptation to lump them all together, a large majority of scientists have at best a limited knowledge of mathematics.

[Chinahand]

For those looking for a popular science treatment of mathematics, Concepts of Modern Mathematics by Ian Stewart is without a doubt the best available.

Well its only taken me 3 and a half years to get round it, but I have at last finished Professor Stewart's book!

 

I did enjoy it, but to be frank I found it quite difficult! I did think it was a helpful introduction to the various branches of mathematics and do think it gave me an insight into them, but I was too often left behind as Prof Stewart ploughed on.

 

Often this was a bit frustrating - in the first few chapters I felt I understood his examples and could see when he was making general statements or ones simply concerning the particular example, but as the book advanced this distinction was lost and I often was left wondering why an example was being presented in a particular way, and how it could be different but still fit the general rules.

 

I'm not sure what to make of my confusions - I suspect I was over confident of my mathematical skill and simply got lost - but I also think the book was limited by its ambition. There simply wasn’t enough space for Prof Stewart to explain in more detail and give more examples in a book trying to cover all the major fields of the subject.

 

I do think of myself as being reasonably numerate – I was good at engineering mathematics, though I am sure VinnieK will roll his eyes at that as I probably wouldn’t be able to spot a proof at 10 feet! But I used to be able to follow reasonably high level mathematical concepts such as control engineering and aerodynamics with relative ease – warning to any budding Concorde designers out there – aerodynamics is at least a year of advanced maths!

 

I fully admit this was all quite a long time ago now, but given that back ground I thought a general introduction to the “New Mathematics” taught in schools wouldn’t be too hard. I was wrong!

 

I do recommend the book, but be aware Prof Stewart is covering a lot of ground in quite a short book – he goes at quite a pace and expects you to keep up. Often I would go back to earlier examples and then try to work back again and again trying to understand the leaps Prof Stewart was making with only partial comprehension.

 

Maybe I’m just getting old and slow!

 

Maths is such an important skill to be able to rationally examine the world – numeracy has a major correlation with an individual’s or economy’s success. Keeping people engaged with it is a real challenge and so to give an overview of mathematics’ depth is very worthwhile. I’ve not read any other book even approaching Prof Stewart’s commitment to explain the reality of maths rather than a journalistic gloss. The trouble is, with such a vast subject matter to cover, and a society sadly loosing touch with it, I worry Prof Stewart has over-reached and only maths nerds like me would attempt to read it – and I found it hard going.

 

I fear that many would become lost by about the fourth chapter – and that would be a shame as the more advanced material is still fascinating.

[VinnieK]

I'm not sure what to make of my confusions - I suspect I was over confident of my mathematical skill and simply got lost - but I also think the book was limited by its ambition. There simply wasn't enough space for Prof Stewart to explain in more detail and give more examples in a book trying to cover all the major fields of the subject.

 

I do think of myself as being reasonably numerate – I was good at engineering mathematics, though I am sure VinnieK will roll his eyes at that as I probably wouldn't be able to spot a proof at 10 feet! But I used to be able to follow reasonably high level mathematical concepts such as control engineering and aerodynamics with relative ease – warning to any budding Concorde designers out there – aerodynamics is at least a year of advanced maths!

 

I fully admit this was all quite a long time ago now, but given that back ground I thought a general introduction to the "New Mathematics" taught in schools wouldn't be too hard. I was wrong!

 

Hoi! I'm not that much of a prick, though I concede that I may often seem it

 

Glad you enjoyed it overall. I do agree with you that the project Stewart embarked on was very ambitious (although no more ambitious than was the decision then to teach all this stuff formally to school children in the New Math syllabus), and how well he meets the challenge is very much up for discussion, but that's kind of why I enjoyed it: to my knowledge it's one of very few popular maths book around that actually tries to explain maths to the layman, rather than describing it.

 

I wouldn't feel to crestfallen at finding parts of it difficult, despite having a mathematical/scientific background. In my experience the difficulty when first meeting a decent chunk of pure mathematics is that it's asking you to think about things in very different ways: there's a subtlety and conceptual tricksiness in there which is actually quite disorientating. In fact, 'disorientation' is probably a better word that 'difficulty', and it's this which trips up and even alienates a large number (possibly the majority) of undergraduates when they first meet these subjects, something that's compounded by the fact that pure reason and logic can be a very unforgiving and inhospitable environment to work in!

 

Where Stewart fell down is perhaps in not providing more motivation. For instance, the chapter on functions seems a bit arbitrary and, well, 'meh' , but functions and the cardinality of sets can be presented in a way that doesn't make them seem quite so much an intellectual parlour game (namely by regarding a function as a formalization and generalization of counting and labelling objects), the chapter on Linear algebra isn't too great either, unfortunately. But this in part is a quirk of the time in which Stewart was writing: back in the 70's it was pretty unfashionable in mathematics to focus on motivation, application, or even to include examples!

 

Overall, I think your assessment is fair, and that the way to approach Stewart's book is not so much as a popular science book, but more in the vein of an introduction to a branch of philosophy. That sounds a bit pompous, but I think it's a fair distinction: rather than providing a narrative, it provides the basic objects, concepts, and arguments, but expects the reader to do a fair bit of the work.

 

Often I would go back to earlier examples and then try to work back again and again trying to understand the leaps Prof Stewart was making with only partial comprehension.

 

That is actually pretty much the way everyone reads maths, including experienced researchers! Mathmo's in their typically condescending way, describe it as a matter of gaining 'mathematical maturity': an understanding or comprehension that only comes with time, and the development of which is utterly mysterious and follows little rhyme or reason. As awful as the terminology is, there is something to this idea. Were you to go back to Stewart's book in say, six months, I can almost guarantee that you will understand more of it and better, even if you have nothing to do with it in the interim!