Keith Devlin, mathematician, has written another fine book: The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time (Basic Books, 2002). It's a popularization that's not likely to be very "popular", alas. Devlin is above all honest. He remarks in his preface:
... I knew from the start that no matter how hard I tried, I could not make this book an easy read. The Millennium Problems are the hardest and most important unsolved mathematics problems in the world; they have resisted numerous attempts at solution, over many years, by the best mathematical minds around. Even achieving a layperson's appreciation of what they are about takes considerable effort. ...
In 2000 the Clay Mathematics Institute put up seven prizes of $1 million each for solutions of these problems; see its web site (http://www.claymath.org) for detailed descriptions. The names of the challenges are, in Devlin's choice of ordering toward increasing abstraction:
But as Lewis Carroll observed in Alice in Wonderland, simply naming something is far from knowing what it is.
Throughout his book Keith Devlin paints an impressionistic image of these conundrums, but as he nears the end he almost runs out of pigment. He throws up his hands, however, in a most charming and forthright fashion. For example:
Thus, if you're feeling pleased with yourself for having got this far (even if you had to bail out halfway through Chapter 6), and within a page or so from now you have a sudden sinking feeling that you just aren't getting it, please don't despair. In fact --- and this is not something I say often --- if you find the going too hard, then the wise strategy might be to give up. The Hodge Conjecture ... is a highly technical question, buried deep in a forest of highly abstract advanced mathematics known to few professional mathematicians. It deals with objects that are so far removed from the intuitions of even the experts that not only is there no "smart money" on whether the conjecture will turn out to be true or false, there isn't even a consensus as to what it really says.
Whew! ... I get out of breath just reading that. But after his disclaimer Devlin goes manfully to work, beginning with a section modestly titled "The Hard Stuff, Made as Easy as I Can". First, though, he offers a lovely statement on abstraction, the essence of deep thought:
The Hodge Conjecture illustrates perhaps most clearly of all the Millennium Problems the point I raised in Chapter 0, that the nature of modern mathematics makes much of it all but impossible for the layperson to appreciate. For a century now, mathematicians have built new abstractions on top of old ones, every new step taking them further from the world of everyday experience on which, ultimately, we must base all our understanding. As I have observed before, it is not so much that the mathematician does new things; rather, the objects considered become more abstract --- abstractions from abstractions, and abstractions from abstractions from abstractions. In the case of the Hodge Conjecture, the operations of calculus play a major role (differentiation, integration, etc.). But the calculus is not done on the real numbers, as many high school students learn it, or even on the complex numbers. It's calculus done in a much more general, more abstract setting.
To the layperson, the very inaccessibility of the problem is perhaps its most interesting feature. A hundred years ago, any problem in mathematics could be explained to an interested layperson. Today, some problems cannot be explained even to most professional mathematicians.
The human brain has to work hard to achieve a new level of abstraction. Only when one new level has been mastered is it possible to abstract from that level to yet another level. ...
Heady stuff, this exploration of the high country of the mind ....
TopicScience - Datetag20021205