Tuesday, April 07, 2015

Staking a claim

Came across a quote from René Decartes in an article about his mathematics:

"..having determined as I did [in La Géométrie] all that could be achieved in each type of problem and shown the way to do it, I claim that people should not only believe that I have accomplished more than my predecessors but should also be convinced that posterity will never discover anything in this subject which I could not have discovered just as well if I had bothered to look for it (To Mersenne, end of December 1637; AT 1, 478; CSMK, 78–79)."

He seems to me to be staking a claim not only on the innovations that are immediately apparent, but also on all other future discoveries on the subject, that had he only "bothered to look for", he would have made those as well... talk about self-confidence!

Monday, April 06, 2015

Opening the Eye of the Soul

I was talking to someone who noted that though they had done AP calculus in high school, all that math seemed to have been of no use or meaning whatsoever in their life.  Indeed, this person noted, setting aside its role in the narrow fields of science and engineering, is it not true that there is no point to teaching all the math we do in schools, even geometry, to every student given only arithmetic is practically encountered on a daily basis and even that could mostly be done on a calculator?

This is an age old question. A dismissive response to it was said to have been offered by the great Euclid, who, when a student asked what good he would derive from learning Geometry, ordered his assistant to "pay him three obols, for he must profit from what he learns!"

But while Euclid, in this story, treats with contempt the questioner for seeking a material use for Geometry, he doesn't express in clear and positive terms what other good it might bring.

I came across the following quote, from a T.Taylor, in his "Dissertation on the True End of Geometry" (1792), wherein he addresses the most important reason to study geometry more explicitly, rather poetically:

".. if geometry is a speculative science (I mean the geometry of the ancients), it is both desirable for its own sake, and for still higher contemplations, the visions of intellect, to which it is ultimately subservient. For, when studied with this view, it opens the eye of the soul to spectacles of perfect reality and purifies it from the darkness of material oblivion. Away then, ye sordid vulgar, who are perpetually demanding the utility of abstract speculations, and who are impatient to bring down and debase the noblest energies, to the most groveling purposes..."

As it happens, I have been going through a few propositions from the first book of Euclid's Elements with my son in recent weeks. I think I understood why I was doing it only dimly till now...


For anyone that may be interested, here are a few excellent links to study the Elements:

The Elements are quite accessible to be explained to a middle-schooler. There is something really inspiring and meaningful about learning Geometry directly from the Ancient Greeks, from a book first written circa 300 BC. It is still current and makes for excellent training in rigorous thinking and deductive reasoning. It makes it possible to gauge for oneself how brilliant and sophisticated they were, how far along they got with abstract thinking, and helps one see the unbroken threads that connect philosophical and intellectual investigations and the growth of knowledge through the ages.