*A History of Mathematics*by Carl Boyer, 2nd edition revised by Uta Merzbach. It is a 600+ page tome that traces mathematical ideas back to considering the origins of the very "concept of number," all the way to the mid-twentieth century's obsession with logic and computing. It does so in a way that is informative to both the mathematician and layperson, because it illustrates the development of mathematical ideas with actual examples and theorems used by the mathematicians grappling with them at the time, all the while eschewing technical complexities for the sake of covering the contributions of every major mathematician in history.

In short, the book is a masterpiece. I bought it as a senior in college for a variety of reasons, one of which was because I, as a math and economics major in college, was writing my honors thesis on art, infinity, and Godel's incompleteness theorem, and I needed the book in order to understand the historical context of formalism and the parallel postulate. I've come back to it time and again, mostly for a before-bed perusal when I don't want to think about anything related to my discipline. And in fact, I've used it on this blog before as well -- when I wrote a post on Keynes' methodological revolution (it's still one of my favorite posts I ever wrote, here it is). I also plan to use it when I write some blog posts on Godel's incompleteness theorem and methodology in economics.

At any rate, I just now noticed that Isaac Asimov wrote the forward for the second edition of Boyer's book. His comments aren't much, but they are strikingly inspiring, so I thought I'd summarize them here with some quotes:

Asimov's premise is that "nearly every field of human endeavor is marked by changes which can be considered as correction and/or extension" (vii). And "only among the sciences is there progress", in the sense that art, for example, has "continuously and chaotically changed." But in science, he refines the progress as "one of both correction and extension" -- for example, "[e]ven Newton, the greatest of all scientists, was wrong in his view of the nature of light, of the achromaticity of lenses, and missed the existence of spectral lines. His masterpiece, the laws of motion and the theory of universal gravitation, had to be modified by Einstein in 1916" (vii).

"Only in mathematics," he writes, "is there no significant correction -- only extension.

...Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever.

"Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like

*A History of Mathematics*, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago.Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak" (vii-viii).

Who wants to go do some abstract algebra?

I would argue that in all disciplines there is only extension...no significant correction.

ReplyDelete"extension" may reshape the truth claim of any theory but does that not make it a new theory instead of a "corrected" version of the old one?

(A wonderful example of the incompatibility of modernism (math) and post-structuralism.)

Hey Daniel, can I borrow the book in the summer? In exchange, you can borrow one of my books of Jean Van Heijenort. (Not the famous "From Frege to Godel" but a less well known).

ReplyDeleteBests. Leopoldo

Blog(fermaton.over-blog.com), No-30. - THÉORÈME DE L'IGNORANCE. - Gödel ou Asimov ?

ReplyDelete