Saturday, February 9, 2013

Plato in a Non-Euclidean World: Review on Berlinski's King of Infinite Space

Book Cover of Euclid with clouds by David Berlinski

There are two known sins that one can commit in relation to books: a) to prematurely judge a book by its cover; and b) to believe in the tempting correlation that a short book can be read in a short amount of time.

I have managed to stay clear from the first bookish sin since the name of the author has more value to me than its graphic representations, but in the case of David Berlinski's The King of Infinite Space: Euclid and his Elements I made the error of thinking that its slimness, the relatively and comparatively few number of pages meant that I could finish it in a jiffy. Not so. I learned my (mathematical) lesson the cumbersome way because a lot of complex information can be packed in small parcels of space.

Entering the world of Euclid, as imagined by Berlinski, is like sitting in a wide-angled out-of-focus high school classroom where the bespectacled math teacher is hardly audible and the clock keeps receding instead of advancing in time. Or so was my own general feeling when I attempted to read and re-read this book.

I must make a clear confession before I go on: I lost my math ability and most enjoyment thereof after age 17. It happened overnight and there is, in fact, no reasonable explanation for it. Math simply vanished out of my mind, and I carefully circumvented and kept out of its shadows and surroundings whenever I could.

Fortunately, math and literature are not the closest of friends, so I have been safe for most of my life. But in philosophy you will stumble upon math every once in a while, while the fascinating and overwhelming world of quantum mechanics will take even seasoned mathematicians for a hearty spin.

I cannot deny the fact that I envy mathematicians out there: they have access to a world that I will never understand. The Greek double Pythagoras and Plato were great mathematician-philosophers, so were their French double counterparts Descartes and Pascal. And each and everyone had powerful things to say about human nature using math as a their guiding light.

Reading about Euclid's theorems and axioms and postulates re-vibrated my Platonic sensibilities. I realized that both Plato and Euclid either lived in an elaborate world of fantasy or that they correctly imagined the existence of a world outside of this world, one that defied imagination and general conventions. They indeed spoke and were inspired by the same abstract and grammatically precise language of mathematics.

As I was walking home one day immersed in Euclidean thoughts I could not help feeling slightly overwhelmed by the world of geometry myself. In fact, everything is geometrical! Buildings construct a right angle to the ground and stretch out in an imaginary infinite line through the sky; streets are straight lines that are parallel to each other; shop signs and ladders on the street form triangles. You cannot escape shapes; even my face and pot belly are round-shaped and my hanging arms are straight lines.

And yet, it turns out that Euclid needed to be modified when it comes to spheres and the physical world because the shortest distance between two points is not always a straight line. Curves and geodesics within spheres, gravity and elliptical forces make it not so what on paper looks perfectly sound and fine. And the sum of the angles of triangles is not always 180 degrees in real life.

And then it came to me. This is what Plato must have meant by the heavenly forms! Yes, there must be a perfect triangle out there, one that fully represents what is drawn on two-dimensional space. Whatever shape we encounter may then be an imperfect replication of its heavenly and otherworldly ideal. We live in a non-Euclidean world because it is the shadow of the infinite space that was imagined by Plato and reformulated by Euclid.

Oh, my God! And it had seemed that this book by mathematician David Berlinski was a bore or drudgery! Not so if math is your cup of tea of course, but to me all the abstract logical talk had given me a slight headache. Until the information somehow seeped in into my subconscious, I suppose.

Euclid spent a lot of time to prove his theorems with his various definitions and postulations. Why did he take such pains, I asked myself? And yet, it was this strategy of taking pains to try to convince others through logic that has shaped the Western consciousness. Socrates and Plato had used reason to debate points and to persuade and enlighten others. Euclid used geometry. Descartes applied a similar mathematical approach to the methodology of philosophy itself.

And Euclid's ideas had withstood time over more than two thousand years (compare that with the nimbly two hundred years or so that Newton's mechanical outlook lasted!) until 19th century mathematicians turned it upside down, and Einstein put the proverbial dagger through the hearts of Euclid and Newton respectively.

And yes, the very foundations of the Earth had begun to shake! Berlinski quotes how Bertrand Russell and others were shocked at the opening cracks in a once stable foundation of mathematical precision. Suddenly nothing was certain anymore and everything became relative. And the problem with relativity is that there is no absolute truth or theorems, no clear formulas.

Plato will say I told you so! He will insist on his version of another idealized Euclidean plane of existence where triangles are as they should be, namely perfect. Life can become more predictable again; we can calculate shapes and movements without elements interfering with our studies. Put differently, it is not the triangle that is wrong; it is our faulty perception of it.

Of course, had I more knowledge about Euclid or the subject matter I would have gotten more out of this book. As an afterword under his Teacher's Note, Berlinski talks about Euclid's book, something that could be easily applied to his own work: “The book demands both effort and concentration. The proofs do not come easy.”

Although there is humor to help us along, it comes as dry as unsalted crackers and the language is quite sparse. Where I would expect to be taking off on flights of fancy, there is but a cracked line. It is all as geometrically efficient as it gets.

At first sight, the book seems to have absolutely nothing to do with one's own life being as abstract as it can be, but it is not unlike Euclid's work itself: Dig in there for a while, and you will find something worth your time and effort.


Vincent said...

You make some sweeping high-level assertions here. How can you say Euclid was wrong? The existence of non-Euclidean geometry alongside doesn't make Euclid wrong, just as Newtonian mechanics is not made wrong by Einstein's theorems etc which apply to other scales of magnitude, limiting cases and so on.

And I well remember when I was first taught geometry, probably from a translation of Euclid, for we had old-fashioned books. We were taught straight away that a point has no dimension at all, so cannot physically exist, and a line has length but no width, so cannot be physically drawn, etc.

One thing was very easy to understand: that the shortest difference between two points is (provably) a straight line. We were also told about the curvature of the earth, the inadequacies of the Mercator's Projection (used for our wall map of the world) and that the most direct aeroplane journeys are segments of Great Circles (circles whose centre is the centre of the earth).

With respect, Plato never came into it! But when we did find out about him, his ideas seemed rather Christian. (Doubtless because his philosophy did influence Christianity.)

Arashmania said...

Well, I have given the guy credit since he has been right for two thousand years! That's a really long time.

And I do not imply that he is "wrong" in an Earth-is-flat-kind-of-way, a binary either with us or against us outlook.

Newton was the absolute norm until we realized that it was not a perfect encompassing model. So we had to readjust his theory. The same happened to Euclid. Neither is wrong necessarily; they are not completely right either.

And this could lead to a more philosophical discussion of what we mean as right or wrong in the first place and where what applies.

Unfortunately my math background will not be able to suffer such stretches. And most likely I have myself solely to blame for this because I am sure that my high school math teachers were rather competent.

Arashmania said...

Dear Vincent, I stand corrected and my sincere apologies to you, to all high school math teachers and particularly to Sir Euclid himself.

After an enlightening and instructive chat with a respected mathematician I decided to change a few lines here and there in order to soften the margin of error. It turns out that Euclid was not necessarily wrong as I had myself erroneously and somewhat prematurely claimed.

But I still stand by the claim that Newton was not fully right as his theory was not as absolute as it was thought to be in a pre-Einsteinian world.

Thanks again for your comment and input!