Discrete Thoughts: Essays on Mathematics, Science, and Philosophy by Mark Kac, Gian-Carlo Rota and Jacob T. Schwartz (Birkhauser Boston: $34.95)
For reasons unknown, people have different ways of apprehending the world. So, it is said, if you find truth, you shouldn't say, "I know the truth," but rather, "I know a truth."
Beauty, like truth, comes in many guises. Some people find it in art, music or dance; others see it in literature; still others find it in mathematics, the 2,500-year enterprise that is the most abstract expression of the human mind.
Mathematics is the science of organized thought. Or, as Jacob T. Schwartz of New York University writes in this book, "To find the simple in the complex, the finite in the infinite--that is not a bad description of the aim and essence of mathematics."
World of Specialists
I am not a mathematician.
On a purely personal level, I would rather spend time with mathematicians than with any other group. In a world of specialists and people who know a few things, mathematicians' breadth of knowledge and interests has no match. They are living proof, in the words of Mark Kac, that "everything is connected to everything else, and it is impossible to separate completely any intellectual endeavor from any other."
Kac (who died in 1984) is one of three eminent mathematicians whose writings are excerpted in this anthology, which is part of the "Scientists of Our Time" series being published by Birkhauser Boston. This is not an elementary book. Though much of it is accessible to interested readers, some of it requires more background.
I couldn't follow all of the details of the arguments, but I nonetheless appreciated the clarity and precision of the thought. Though I didn't grasp all of the objects, their shadows made me smile. Besides, it is possible to like a book without being able to pass a final exam on it.
Interplay of Cultures
And what is the subject addressed herein? There are many--reflecting the varied interests of the authors--including the current state and future of mathematics, science and philosophy; education (mathematics and otherwise); computers and artificial intelligence; statistics; game theory; economics, and the interplay of the two cultures--the sciences and the humanities.
Not only do the authors know a lot, they are wise, which is a much rarer attribute. So their essays are studded with insights that could stand alone as "The Sayings of Mathematicians":
- "The computer is just an instrument for doing faster what we already know how to do slower."--Gian-Carlo Rota of MIT
- "Mathematics lives on an interplay of ideas. The progress of mathematics and its vigor have always depended on the abstract helping the concrete and the concrete feeding the abstract."--Kac (who spent his last years at USC).
- "Mathematics is a cruel profession. Solving a mathematical problem is for most mathematicians an arduous and lengthy process which may take years, even a lifetime. The final conquest of the truth comes, if ever, inevitably tinged with disillusion, soured by the realization of the ultimate irrelevance of all intellectual endeavor."--Rota
- "The main purpose of professional education is development of skills ; the main purpose of education in subjects like mathematics, physics or philosophy is developing of attitudes."--Kac
This is a book that will stretch you and will reward your effort. The authors are so smart that the rest of us can never catch up, but they share their wisdom as best they can. And though that wisdom has its basis in the rarefied field of mathematics, it has applications throughout human experience. As Kac writes, "There is now hardly a corner of human activity which has not for better or for worse been affected by mathematics."
All of that said, some complaints should be noted. Each of the 26 essays is a reprint or excerpt from something else, but there is no clue as to where or what they come from. All that is provided is a list of acknowledgements for the right to reprint without any indication of the source of each excerpt.
In addition, the reader has to look back to the table of contents to find out who wrote each essay. And the date it was originally published appears at the end of each piece. So you have to look in one place to find the author, in another place to find who published it and in yet a third place to find when it was written. All of these problems could have been solved with a brief paragraph at the start of each of the essays.
Overtaken by Events
Further, though almost all of the points and ideas remain valid today, some of them have been overtaken by events, or at least could have benefited from an updating sentence or two. Rota's discussion of the Four-Color Theorem, for example, was written in 1969 and does not mention that it was proved in 1976.
Allow me another quibble. In several places, the authors quote a sentence in a foreign language, typically French or German. But they do not translate these occasional sentences.
So why, given all of that, did I enjoy reading this book? Perhaps it is just an idiosyncrasy, but when mathematicians speak, I tend to listen. Their playful minds come up with things that I would otherwise never know, and while I cannot share the doing, I can share their delight in knowing.