The Darkness of Axioms

A little Sunday reflection from Bernhard Riemann:

“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.” 

Happy Birthday Bernhard!

It's Bernhard Riemann's birthday today... or would be if he were around to celebrate his 187th.
Died at just 39, in his "prime," and never having a chance to prove the hypothesis that bears his name... and earn a million dollars! ;-)

https://en.wikipedia.org/wiki/Bernhard_Riemann


(image via "Dark Meadow"/Wikimedia)

One Heckuva Mathematician…

  …but not my grandparent :-(

Bernhard Riemann is likely my favorite historical mathematician (...and thus my adoption of his name for this blog); he is easily one of the greatest mathematicians who ever lived… no telling how much more he would've accomplished had he not died at the all-too-young age of 40.

 A biography of him here:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html 

And someone has put together this nice (10-min.) video tribute to
him (the treatment of the zeta-function/"Riemann Hypothesis" beginning at the 4:30 min. mark is especially entertaining):





ADDENDUM: This is WAAAAY beyond my neuronal capacity, but Dick Lipton/Ken Regan have just put up a seemingly fascinating new post concerning Riemann's zeta function for those able to follow it here:

http://rjlipton.wordpress.com/2012/12/04/the-amazing-zeta-code/

If I Were Stranded on a Desert Island...

...and could only have one book to read I think I know what it would be. . . .

Yes, I'll squeeze in one more bit about Martin Gardner before his day of honor tomorrow (to any who are tired of hearing about Martin Gardner by now, I apologize, but HEY! it's MY blog so deal with it ;-))
When Gardner's "The Colossal Book of Mathematics" came out in 2001, I looked at its size, price, and the many chapters covering topics I wasn't particularly interested in, and ignored it (I already had plenty of Gardner books on my shelf).  It was only years later that I checked it out from a public library and discovered not only how many chapters were of great interest to me, but also how many of the topics I wouldn't normally have found interesting were made so by Gardner's deft and insightful writing.
So again I highly recommend this volume to anyone lacking it on their shelves. If you're ever stranded on a desert island it would offer you weeks/months of mental entertainment (...and, as an alternative, just in case I got tired of math, I might take along Gardner's essay-anthology, "The Night Is Large," as well!).

Meanwhile, 'Mathematics Rising' blog has just covered the 3rd of my 'Fab Four,' Bernhard Riemann, here:

http://mathrising.com/?p=270

Finally, a couple of recent math blog carnivals here for your enjoyment and delectation:

http://tinyurl.com/2bhva6c

http://tinyurl.com/2bcdscb

And I'll end today with a factoid lifted off an old Twitter posting:

A 10,000 number gap in prime numbers: 9973! + 2 through 9973! + 10006 are all composite (non-prime).

...someone please double-check those for me over the lunch hour and get back to me to confirm.