Of Dinosaurs and Mathematics... and Pathos

Anyone up for some philosophical reflection on a lazy Sunday morning...?

(image via Danny Cicchetti/WikimediaCommons)

Vocal math Platonist Martin Gardner famously wrote that if 2 dinosaurs met 2 other dinosaurs in a clearing, there would then be 4 dinosaurs present… whether or not there was any human mind around to appreciate the fact, and whether or not there existed any such words as "two" or "four," the relationship would still exist completely apart from human recognition of it. In fact, Gardner (who was actually more a philosopher than a mathematician) largely scoffed at the few professional mathematicians in his day who claimed that mathematics was merely some sort of human/cognitive creation. The case against Platonism has grown since then, with fewer, though still many adherents.

Non-Platonism comes in several different varieties and degrees, but Platonists tend to more uniformly feel that mathematics is a real, ubiquitous component of the Universe (or, in the case of Max Tegmark, they believe math is ALL there is -- it is ultimately the only component or structure of the Universe; p.s., Tegmark's new book "Our Mathematical Universe" will soon be out).

Much of this debate continues to seem semantic, hinging on what one means by words like "real," "existence," "component," and other words that simply can't be defined in terribly rigorous/consistent ways (even "mathematics" is not that easy to define). Is mathematics 'out there' in the Universe, apart from us and our existence, or is it only inside our heads, generated from neurochemical processes?

I bring all this up because recently Jason Rosenhouse broached the Platonist/Non-Platonist divide in this post where he addresses interview comments from Platonist/recent-author Ed Frenkel:

http://scienceblogs.com/evolutionblog/2013/12/13/in-what-sense-do-mathematical-objects-exist/

I tweeted a link to the above post when it first came out, but deliberately held off posting about it here right away, because I suspected it might lead to some engaging discussion in Jason's comment section… and, it has. Eventually, Ed Frenkel himself responded (twice), and all the discussion has been quite interesting... unless that is, you're in the camp that finds such philosophical debate sleep-inducing! ;-)

It certainly seems to me that if "science" has any reality in the Universe (and we're not just living in a simulation imposed by highly-advanced aliens) then there are elements of mathematics that must be real and integral to the Universe's operation -- if math ISN'T "real" in some sense, then "science" (which is based upon it) must also not be real, which implies that the Universe, rather than having 'order,' 'laws,' and causation as we perceive, is a rather hopelessly random/chaotic place (...which in turn assumes that "randomness" can truly exist!???)... but whether all of mathematics exists in some Platonic sense or only elements of it is more of a leap, and perhaps, as I say, more an argument over words and meaning, than mathematics itself... In any event, read Jason's take and the conversation that follows it.

....I wrote the above words yesterday for posting today, then this morning woke up to a tweet from Alexander Bogomolny linking to this quote on a discussion forum, from "Thelonious Mac," which somehow seems worth closing with:

"I just saw a television program in which a mathematician lost his daughter. Unable to express himself in the language of platitudes that most people use at such a time, he created a series of equations to represent her life, a work of art, expression, in math. Yes math is beautiful. There is absolutely no aspect of our lives for which you will not find math at its foundation. If I have a glass of clean water today, it is because of the math behind the engineering that brought it to me.
Math is the mother of all science. Without it, our lives would be incomprehensibly pathetic."

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.