"the wonders of the world of algebra"

Sir Roger... and algebra:

Currently lacking time for longer posts, so just another quotation today, this time from the eminent Roger Penrose remarking on his introduction to algebra as a youngster (taken from his Foreward to Mircea Pitici's "The Best Writing on Mathematics 2013"):

"My earliest encounter with algebra came about also at an early age, when, having long been intrigued by the identity 2 + 2 = 2 X 2, I had hit upon 1.5 + 3 = 1.5 X 3. Wondering whether there might be other examples, and using some geometrical consideration concerning squares and rectangles, or something -- I had never done any algebra -- I hit upon some rather too-elaborate formula for what I had guessed might be a general expression for the solution to this problem. Upon my showing this to my older brother Oliver, he immediately showed me how my formula could be reduced to 1/a + 1/b = 1, and he explained to me how this formula indeed provided the general solution to a + b = a X b.  I was amazed by this power of simple algebra to transform and simplify  expressions, and this basic demonstration opened my eyes to the wonders of the world of algebra."

Quote... Unquote

Just a quotation today, from P.R. Halmos, from his 1968 essay "Mathematics as a Creative Art":

"…a brief and perhaps apocryphal story about David Hilbert… When he was preparing a public address, Hilbert was asked to include a reference to the conflict between pure and applied mathematics, in the hope that if anyone could take a step toward resolving it, he could. Obediently, he is said to have begun his address by saying 'I was asked to speak about the conflict between pure and applied mathematics. I am glad to do so, because it is, indeed, a lot of nonsense -- there should be no conflict -- there is no conflict -- in fact the two have nothing whatsoever to do with one another!' "

 ;-)

A Few Quotes

Just a few quotations for your reflection today, taken from Simon Singh's "The Simpsons and Their Mathematical Secrets":

From famous British mathematician G.H. Hardy: "Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly  word, but probably a mathematician has the best chance of whatever it may mean."

From Simpsons writer (and Harvard physics grad) David Cohen, on the satisfaction derived from slipping mathematics into Simpsons' episodes: "I feel great about it. It's very easy working in television to not  feel good about what you do on the grounds that you're causing the collapse of society. So, when we get the opportunity to raise the level of discussion -- particularly to glorify mathematics -- it cancels out those days when I've been writing those bodily function jokes."

And from author Singh: "It would be easy for non-nerds to dismiss the mathematical shenanigans that appear on The Simpsons and Futurama as superficial and frivolous, but that would be an insult to the wit and dedication of the two most mathematically gifted writing teams in the history of television. They have never shied away from championing everything from Fermat's last theorem to their very own Futurama theorem.
"As a society we rightly adore our great musicians and novelists, yet we seldom hear any mention of the humble mathematician. It is clear that mathematics is not considered part of our culture. Instead, mathematics is generally feared and mathematicians are often mocked. Despite this, the writers of The Simpsons and Futurama have been smuggling complex mathematical ideas onto prime-time television for almost a quarter of a century."

Quote... Unquote

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that "laws of nature" exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

-- Eugene P. Wigner

"I believe that scientific knowledge has fractal properties, that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think, is the secret of the Universe."

-- Isaac Asimov

Quote... Unquote

 First, once again, thanks to RJ Lipton for another update on Deolalikar's P ≠ NP proof, in which Neil Immerman expresses major concerns (2 major, fatal?, flaws):

http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/

This will be well beyond the comprehension of most of us... which simply makes it all the more fascinating that there are those who do grasp the reasoning communicated!

On to other things... Just a few miscellaneous quotes to ponder for today, as food for thought!:

 "[Alain] Connes thinks that expert mathematicians are endowed with a clairvoyance, a flair, a special instinct comparable to the musician's fine-tuned ear or to the wine taster's experienced palate that enables them to directly perceive mathematical objects: 'The evolution of our perception of mathematical reality causes a new sense to develop, which gives us access to a reality that is neither visual, nor auditory, but something else altogether'." --- from "The Number Sense" by Stanislas Dehaene

"The sciences have developed in an order the reverse of what might have been expected. What was most remote from ourselves was first brought under the domain of law, and then, gradually, what was nearer: first the heavens, next the earth, then animal and vegetable life, then the human body, and last of all (as yet very imperfectly) the human mind." --- Bertrand Russell

"Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings."  --- Alfred North Whitehead

"It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class."  --- Author Unknown

"All generalizations are false, including this one." --- Mark Twain