...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

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"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Tuesday, June 18, 2013

Waxing Platonic…


The Platonic divide in math....


Ramanujan
The book I've highlighted recently, "The New York Times Book of Mathematics," ends with a chapter of readings on various notable mathematicians… Erdos, Ramanujan, Conway, Gödel, Wiles, etc. I suspect most (if not all) of the brilliant figures profiled were/are Platonists (mathematics is discovered, not merely created). Yet many other recent math figures (Reuben Hersh, William Byers, Keith Devlin, Jim Holt, and more) have forcefully argued that mathematics is indeed a mental creation that might even differ considerably in a different Universe than ours -- indeed some almost seem to find the notion of mathematical Platonism so wrong-headed as to be silly (while Martin Gardner found the non-Platonist view almost silly). And occasionally such writers cause me to sway toward their non-Platonist stance though I always seem to float back toward Platonism.

One thing that so many of the greatest, most productive mathematicians seem to share is an uncanny, almost inexplicable ability to tap into a realm of intuition or mental landscape not readily accessible to most of us. Ramanujan is certainly the unparalleled, most inexplicable, example of this; producing amazing mathematical results that are still today being explored and proven. Reading James Gleick's portrait of Ramanujan in the Times volume it really hit me… was Ramanujan, who routinely produced such results/theorems without ever showing the steps that led to the outcome, in direct access of the "Platonic realm?" He himself claimed his insights came in dreams and trances directly from the Indian Goddess Namagiri... Who are we to argue (and where did she reside)?!!

In many ways, Ramanujan's extraordinary talents are reminiscent of the incredible abilities of various mathematical savants and prodigies who usually can't explain how they do what they do. Their brains seem clearly to operate, or even be wired, differently from those of 'ordinary' people.

My point in all this is simply that such rare, yet nonetheless real, individuals DO give an appearance of tapping into a realm… call it perhaps the Platonist realm… that the rest of us lack ready access to, where numbers and math really DO exist apart from our day-to-day world.  Naysaying non-Platonists will simply argue that however Ramanujan and the rest gain their special knowledge, it ultimately still arises via the firing of neurons within a physical human brain situated between two ears… i.e. it is still a human creation. I can't prove that reductive view wrong, but the notion that there are worlds out there that only some of us can easily tap into, and only some of the time, through means we don't even comprehend… is so much more appealing! As Shakespeare put it long long ago, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.

I think Martin Gardner might well relate to this idea too… For all his empirical skepticism, Gardner also described himself as a "Mysterian," a philosophical view which holds that ultimately consciousness cannot be explained by any human brain. In the famous words of computer scientist Emerson Pugh, "If the human brain were so simple that we could understand it, than we would be so simple that we couldn't."  Is it possible that humans are able to draw upon a Platonic world, and can recognize 'consciousness,' yet perhaps never, with our limited minds, fully grasp either? Does the 'Platonic world' exist, but like the Continuum Hypothesis, fall into a nether land of things that simply can't be proved true or false by human logic?
Speaking of certain mathematical proofs, Paul Erdos would famously say, "This one is from The Book!" I'm not so sure he was speaking in metaphor... perhaps The Book, in some (Platonic) manifestation, exists. Is the alluring beauty of math only in our heads, or is it an integral part of all creation? MIT physicist Max Tegmark has argued for some time now that the entire physical universe, as we perceive it, is nothing more than mathematics, or a mathematical structure (called the MUH, or "mathematical universe hypothesis").

Anyway, read Gleick's beautiful 1987 portrait of Ramanujan and just imagine the Indian mystic-mathematician dreaming and dipping into a realm where numbers are as 'real' as rocks and chairs are to most of us:

http://www.nytimes.com/1987/07/14/science/an-isolated-genius-is-given-his-due.html?

a couple of brief lines from therein:
" 'When he [Ramanujan] pulled extraordinary objects out of the air, they weren't just curiosities but they were the right things,' said Jonathan M. Borwein of Dalhousie University in Halifax, Nova Scotia...
" 'He seems to have functioned in a way unlike anybody else we know of,' Dr. Borwein said. 'He had such a feel for things that they just flowed out of his brain. Perhaps he didn't see them in any way that's translatable.' "
It's probably also worth noting that the very first entry in the entire NY Times anthology is a 1998 George Johnson piece also addressing the subject of Platonism:

"Useful Invention or Absolute Truth: What Is Math?" by George Johnson

At the end of the piece, Johnson cites a 1995 book, "Conversations on Mind, Matter and Mathematics" that covered a debate between French mathematician Alain Connes and French neurobiologist Jean-Pierre Changeux over the subject of math Platonism. An interesting and rich review of that book here (even makes brief reference to Ramanujan):
 
http://www.timeshighereducation.co.uk/161513.article 

Connes and Changeux didn't resolve the debate... and we won't here... but still, nourishing food-for-thought.



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