A fascinating introduction to some higher mathematics for those who are afraid of it.

Ach, I really wanted to give this five stars. Byers does a great job of showing how ambiguity and paradox are at the core of what mathematics is about. Of course it is also a paradox that mathematics is paradoxical, because mathematics is the prime example of a discipline where paradox has been banished or at least securely caged. Byers discusses briefly how this paradoxical nature of mathematics is important for science and culture at large.

But in the end his conclusion falls a bit flat. He sees computers and software and algorithms as being stuck on one pole of the paradox and therefore essentially trivial. I must say, he triggered one of my pet peeves. On page 383 he says:

"The theory of chaos arises from the study of nonlinearity. Complexity is fundamentally nonlinear. If mathematics is non-linear, then its essence cannot be captured by algorithmic procedures or by the linear strings of reasoning that characterize both mathematical proofs and deductive systems."

This is really disappointing. Through most of the book he has been quite careful to be clear and avoid confusing concepts. But clearly the term "linear" above is used in two very different senses. Deductive systems don't look like vector spaces very much at all!

The crazy thing is, the whole business of complexity theory and chaos, this arose because of computers. It is just too much work to try to simulate those differential equations, to compute solutions for a variety of parameter values.

Here is a huge question that Byers just avoided. It is very nice to say that the human mind is not a deductive system. Sure, there is a school of cognitive science that would like to model minds as computers. I'm not sure whether very many folks in that field work from that premise any more.

But, it seems pretty clear that the world of physics can be modeled quite nicely with differential equations. It seems quite reasonable, in principle, to simulate a human being, i.e. all the atoms, the chemical bonds, etc. OK, the computer would probably require cosmic-scale resources to pull this off. But there was recently some huge simulation of a decent sized chunk of a cat's brain, and it did simulate some interesting behavior. This is not modeling the human mind as a deductive system, this is modeling brain behavior as a biochemical system.

This whole area is vast and deep. I think Byers is making a valuable contribution to the philosophy of mathematics. But when he discusses computer science and cognitive science, he falls short. Both of these research areas are much more fertile than he seems to imagine. What would be much more fun is to extend his notions of the fundamental roles of ambiguity and paradox to those disciplines, to show how the internal conflicts in those disciplines are actually fertile, rather than flaws.

For example, in computer science, contrast the view of Edsgar Dijkstra, that it is a mistake for students of computer science to run their programs on computers, with the common practice of agile development. Maybe computer science should be totally separated from software development? That is really a beautiful paradox!

Of course the whole mind-body distinction is an ancient paradox. Byers seems to be landing on one pole, mind is not body. Even life is not body. He seems to be proposing some kind of vital essence or soul. Wait a minute, Byers acknowledges, on p. 17, his practice of Zen. Buddhism is practically founded on, hmmm, not exactly the non-existence of the soul, but the paradoxical nature of that issue.

What if the paradoxical nature of the mind actually points to a paradoxical nature of the body? Does that mean that, after all, we can't really simulate the physical world? Playing with Byers's idea of objective subjectivity... one problem with simulating the physical world is that causality is necessarily tied up with a free choice among possible stimuli.

The idea that depth is associated with paradox, this is really nice. I was just disappointed that he couldn't maintain that depth but ended up driven to resolution and hence landing in the shallows, just where the real fun could have begun. ( )

Uses more words than necessary to explain his ideas. I kind of understand what he's trying to say but not really. I'm sure there is a more eloquent way to convey his ideas. ( )

Here we have one of those important books we should approach with some care and about which I am somewhat divided (or, should I say, ambiguous...). The main argument of the book, as the subtitle clearly points out, is how do people (and, in particular, mathematicians) use "ambiguity, contradiction, and paradox to create mathematics". Written by an active research mathematician (hence, by someone how knowns what is he talking about!) every research mathematician will certainly recognize the truth of atributing to those non-logical elements a central role in the production of new mathematics, that is: is the creative aspects of the field, both when producing new results and when trying to understand some body of existing mathematics. The stress of the argument is, thus, in these non-logical components, and, although the author repeatedly points out the importance of the logical component for the overall mathematical enterprise, I am affraid the point will be lost by most readers without a solid mathematical education, since most of the main examples are rather sophisticated (about the level of first year undergraduates). This can have as a consequence that the reading of this book by people from the humanities, with no mathematical training but with a propensity for post-modernist thinking, can result in a misrepresentation of what is mathematics and how mathematicians work that could be more off the mark then, say, their (ab)use of Gödel's incompleteness theorem. In short: an excellent book that should be required reading for someone with an undergraduate mathematical education, but that should require the same mathematical education as a prerequisite to be read: a necessary and sufficient condition... ( )

A lengthy and meaty argument for attaching much more importance to those aspects of mathematical work *other* than the formal structure (definitions, theorems, proofs, etc) that emanates from it. Includes a good discussion about the infinity concept's role.