This blog is, in part, an effort to connect dots --- particularly among the books that have come off my Bookshelf. A number of dots connected in my mind as I read Leonard Mlodinow's Euclid's Window.
Euclid's Window opens in Greek antiquity in the port of Miletus on the west coast of what is now Turkey. Mlodinow asserts that a "revolution in human thought, a mutiny against superstition and sloppy thinking," occurred here in the 7th century BCE. Around 620 B.C., Thales of Miletus, who Mlodinow describes as humanity's "first scientist or mathematician," lived here and is purportedly responsible for the systematization of geometry, a methodology that would later be incorporated in Euclid's Elements. This blog first mentioned the scientific contributions of the Milesians in a prior post (see March 28, 2010 post), noting historian David Lindberg's comment, "[I]n the answers offered by these Milesians we find no personification or deification of nature; a conceptual chasm [that] separates their worldview from the mythological world of Homer and Hesiod. The Milesians left the gods out of the story. What they may have thought about the Olympian gods we do not (in most cases) know; but they did not invoke the gods to explain the origin, nature or cause of things." (See March 24, 2010 post). The inference from Lindberg's observation is that when the human mind frees itself from myth, religion, and superstition --- the types of "beliefs" that Michael Shermer wrote about in The Believing Brain (see June 12, 2011 post) --- scientific progress is unshackled.
While Mlodinow does not make the same observation about the Milesians, he does seem to fall into a trap that Lindberg encourages historians to avoid: blaming Christianity entirely for Europe's failure to maintain the scientific progress that the Greek's initiated before the first millennium A.D. (see March 24, 2010 post). Apparently relying on Edward Gibbon, Mlodinow cites the Christians for burning down the greatest library of its era at Alexandria, Egypt and all the scientific and philosophical works that were part of that library. This claim may not be true or entirely true, but the fact the Mlodinow seems to harbor this belief, as revealed in this reference and other statements he makes, strains his credibility as a writer of science history (at least about science and mathematics in the era of the Dark and Middle Ages). It is true that once the institutions of the Dark and Middle Ages lost touch with Greek scientific inquiry and knowledge, institutional biases developed that made it very difficult for that knowledge to be rediscovered, and among those institutions was the Catholic Church. Yet, as Lindberg documents, those institutions also had a small role in the rediscovery of Greek science and thought.
As Mlodinow moves from his portraits of the geometers and Euclid to Descartes to Carl Gauss (and Reimann), the reader senses the impending merger of geometry and physics (or perhaps the impending takeover of geometry by physics) with the development of non-Euclidean geometry. Since the times following Euclid, Euclid's Fifth Postulate (the parallel postulate) had proven troublesome. Euclid stated a proposition that would determine whether two co-planar lines were parallel, converging, or diverging: take two lines and cross them with a third line; if the sum of the two inner angles on the same side of the crossing line is less than two right angles (180 degrees), then the two lines are converging (on that side of the crossing line). The postulate seems intuitively correct. The problem is that the fifth postulate could not be proven as a theorem would be proven. It was assumed as a fact, until non-Euclidean geometry began to address surfaces that are curved and the parallel postulate failed. Geometry began not only to take a hard look at spherical surfaces and topography --- the earth, but it began to turn its attention to space.
Enter Albert Einstein, relativity, and the influence of gravity on the shape of space. Even Mlodinow's brief discussions of Euclid, Descartes, and Gauss made me recall Rita Carter's discussion of the posthumous study of Einstein's brain in Mapping the Mind (see November 6, 2011 post). Carter reports that researchers at McMaster University in Canada found that Einstein's brain "was different from most in several ways, the most notable being that two sulci (infolds) in the parietal cortex had merged during development, creating a single enlarged patch of tissue where usually there would be a division. In normal people, one of these areas is primarily involved in spatial awareness, while the other does (among other things) mathematical calculation. The merging of these two areas in Einstein's brain," Carter speculates, "may well account for his unique ability to translate his 'vision' of space-time into the most famous mathematical equation of all time, e=mc2." Here Carter has been discussing synaethesia, the phenomenon where, because of the close proximity of two parts of the brain, there is a merging of sensory phenomena: e.g., hearing the sound of a certain word or number takes is associated with a certain color. Have the brains of certain mathematicians who can develop mathematical theories or even practical algorithms that describe physical phenomena or physical space developed in a way that facilitates their mathematical skill and insights, in contrast to the brains of most humans? Mlodinow's historical survey of the history of geometry certainly makes one wonder about that.
Reading Euclid's Window also reminded me of a quote from Michael Shermer that I mentioned in the June 12, 2011 post, "We are not equipped to perceive atoms and germs, on the one end of the scale, or galaxies and expanding universes, on the other end." Yet clearly, as Mlodinow's portrait of Einstein and later Edward Witten moves from relativity and quantum mechanics to the "standard model" and ultimately to string theory, it is clear that some minds are clearly capable of not only envisioning atoms, but even smaller particles, and some minds (sometimes the same minds) are capable of envisioning galaxies and expanding universes. Without this capability, Mlodinow would never have had a story to tell. Mlodinow sums this up as follows: "Through Euclid's window we have discovered many gifts, but he could not have imagined where they would take us. To know the stars, to imagine the atom, and to begin to understand how these pieces of the puzzle fit into the cosmic plan is for our species a special pleasure, perhaps the highest. Today, our knowledge of the universe embraces distances so vast we will never travel them and distances so tiny we will never see them. We contemplate times no clock can measure, dimensions no instrument can detect, and forces no person can feel. We have found that in variety and even in apparent chaos, there is simplicity and order."
This is not a deep book. It is written for the general public who has an interest in mathematics and the history of science. I began by criticizing Mlodinow for his knowledge of history in the Dark and Middle Ages, but by the end of the book and the discussion of string theory, I came to conclude that I wished I had read this book before embarking on other deeper books about string theory.