Ahhh, The Calculus

Sunday reflection....

"The calculus is humanity's great meditation on the theme of continuity, its first and most audacious attempt to represent the world, or to create it, by means of symbolic forms that in their power go beyond the usual hopelessly limited descriptions that we habitually employ. There is more to the calculus than the fundamental theorem and more to mathematics than the calculus. And yet the calculus has a singular power to command the attention of educated men and women. It carries with it the innocence of an abstract pursuit successfully accomplished. It is a great and powerful theory arising at the very moment human beings contemplated the infinite for the first time: sequences without end, infinite additions, limits flickering in the far distance. There is nothing in our experience that suggests that mathematics such as this should work, so that the successes of the calculus in unifying aspects of experience are tantalizing but incomplete evidence that of the doors of perception, some at least may open and some at least may lead to someplace beyond."

-- David Berlinski (from "A Tour of The Calculus")


[p.s., over at MathTango this morning I recommend two recent books.]

Complex Numbers (Sunday Reflection)

"…there is never a need to go beyond the complex numbers. No further numbers are needed. They suffice and so they bring to completion the very long effort at construction that over thousands of years yielded first the natural numbers, then the fractions, then zero and the negative numbers, and after that the real numbers. The complex numbers complete the arch.
"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love.
"I once mentioned the beauty of complex analysis to my great friend, the mathematician M.P. Schutzenberger. We were riding in a decrepit taxi, bouncing over the streets of Paris.
" 'Perhaps too beautiful,' he said at last.
"When I mentioned Schutzenberger's remarks to Rene' Thom, he shrugged his peasant shoulders sympathetically.
"This is one of the charms of the theory of complex numbers and their functions. It has broken men's hearts."

-- David Berlinski from "Infinite Ascent"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Proofs As Artifacts...

Sunday Meditation (to wrap your brain around)…

"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.
"Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.
"These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."
-- David Berlinski (from "The King of Infinite Space")

Brief Berlinski Book Blurb….

As simple as one, two, three....

Of the math books I've been perusing/reading lately, the only one grabbing and holding my attention much (and oddly a volume I wasn't expecting to like), is David Berlinski's 2011 volume, "One, Two, Three." Indeed, I've enjoyed the book quite a bit, but must throw in my frequent precaution that it won't suit everyone's taste… or, to put it more bluntly, I suspect MOST readers (even math readers) may NOT relish it.
Leonard Mlodinow, in reviewing the volume, called it, "candy for the intellectually curious"… but I'd feel more comfortable deeming it 'candy for the epistemologically curious' -- some 'intellectuals' (and mathematicians for that matter) don't have the patience for philosophy that this volume requires. The book deals with matters at such an elementary level, where ideas tend to be ingrained and taken-for-granted, that many readers simply may fail to appreciate the significance of what is expounded.

Berlinski calls his subject matter "absolutely elementary mathematics" (which is also the sub-title of the book) or AEM for short -- it's not clear to me if that is his own concocted term or comes from somewhere else (and one Web commenter mistakenly bought the volume thinking it would be a nice introduction to math for her youngster... this it definitely is NOT!). At any rate, the subject matter includes the very most rudimentary foundations of math, logic, arithmetic. You either like this sort of thing… or, you don't. The author romps through a playground of ideas regarding numbers, algebra, axioms, sets, integers, fractions, rings, polynomials, proof, and clearly enjoys himself while doing so (even if it isn't always such a merry-go-round for the reader!).  Parts of it will read quite pedantically to many readers, but this kind of elementary and obvious-sounding subject matter can't help but be pedantic at times… and it is FAR less so, in Berlinski's deft hands/voice, than it would be within some other textbook relating the same notions.

Berlinski's style is terse and economic, and yet with such careful, and even interesting, word choice that I'd be tempted to call it breezy; at times surprisingly engaging; even witty and humorous (though still with interspersed and inevitable dryness at points). His rhetoric can often be fun, even when it is challenging.
There are also interesting brief sidebars about various historical mathematicians throughout the volume, though it wasn't always clear to me just why some of these were included… yet they did make a nice interlude between some of the more tedious sections. The penetrating focus though is always on pulling back the curtain on the fundamental logical and axiomatic thought processes/assumptions that underlie mathematics. I do recommend the book, but not to the mathematically-nervous nor philosophically-naive, nor to readers who read for pleasure and not to be challenged. Whereas a Keith Devlin or Ian Stewart book might connect with a large audience of the mathematically curious, this volume may only resonate with a narrow slice of the popular-math crowd.

I'll admit that I find it hard to objectively read Berlinski's math writings because I find some of his other philosophical/science/political writings rather distasteful (…and yet having said that, there are elements of his "radical skepticism," as some call it, that I'm in sympathy with). He remains a highly controversial figure. Worth noting though, I did also enjoy one of his earlier math works, "Infinite Ascent," a nice historical look at major ideas in mathematics.

The Wikipedia entry on Berlinski is here:

http://en.wikipedia.org/wiki/David_Berlinski

and here are a few more Web entries that lend some sense of the controversy surrounding the man:

http://tinyurl.com/7sdcuj9

http://www.observer.com/1998/06/is-the-big-bang-just-a-big-hoax-david-berlinski-challenges-everyone/

http://recursed.blogspot.com/2008/04/david-berlinski-king-of-poseurs.html

(…of course googling his name will lead you to many more references on him, including clips on YouTube)