Although Peterson paints a detailed picture of Galileo's world and influences, several names come up repeatedly. Perhaps the most important is Pythagoras (although it must be admitted that so little is definitively known about Pythagoras that we are probably giving deference more to the Pythagoreans than the man himself). By the time Galileo was born, Pythagoras was like a mythic god whose teachings, always consciously mysterious, had been garbled to the point that most of what remained wasn't knowledge but lore.
Incredibly much of that lore wasn't just accepted but untested. We see this particularly in music theory. The ratio of two strings that were an octave apart was known to be 2:1 thanks to the Pythagoreans. This was true, but for almost 2000 years it had been accepted that this meant simply doubling the tension on one of the strings. Experiments performed by Galileo's father Vincenzo revealed that the tension needed to be increased by four- the square of two. Similar experiments showed the same results for the accepted ratios for other intervals. Pythagoreans had realized centuries before that the relationship was exponential- geometric, if you will- but somehow that result had been lost and needed to be re-derived.
I thought this was the most fascinating chapter, and not only because I am not an expert in music theory. The problem with tuning different instruments to each other that arose (in part) as a result of the misunderstood mathematics had been noticed, of course, by anyone who could hear. However, because mathematics was so intertwined with philosophy, music teachers continued to insist that instruments be tuned by theory, even if the resulting sound was awful.
(As an aside, I note that the math around tuning seems dependent on a seven major note arrangement- CDEFGAB, for example. I'd love to see how cultures that used a different system managed.)
Another name that comes up frequently: Dante, as in the author of The Divine Comedy. Two short chapters will convince even the most skeptical that the poet not only knew classical mathematics but understood it well enough to be moved by it. Peterson gives a close to literal translation of some of the last lines of the Paradiso that, on its face, seem fairly unintelligible ("I wanted to see how the human image/conforms itself to the circle, and finds its place there"). However, after Peterson explains some of Archimedes' (another important influence) work, the reader not only believes but understands that what Dante is talking about is the measurement of the circle. Peterson's interpretation of Dante's description of Heaven and the physical and spiritual worlds is no less impressive; it is entirely possible that Dante anticipates the field of topology when he describes what we now call the hypersphere to describe the space of the philosophized universe.
|Dante's universe, or the Hypersphere|
It is easier, perhaps, to "see" the math in painting than poetry, and here we come to another important figure: Euclid. His work on optics is an explicit influence on the perspective technique used by many painters. While painting may seem (at first) to be a strange place to apply mathematics, Peterson talks about a number of painters fascinated with the subject, Piero della Francesca being among the most important but not, oddly, Leonardo da Vinci.
If painting is an unexpected place to find math, it is more surprising to find how much mathematics wasn't used in architecture (and engineering). Brunelleschi's Santa Maria del Fiore in Florence, built a century before Galileo, is an example of a structure that pushes the physical limits of area versus weight. This issue was instrumental in helping Galileo make a name for himself at the beginning of his career, then later threatened to undue him. His jumping off point wasn't Santa Maria del Fiore but, rather, the dimensions of Dante's Hell. (Although he took great pains to stress that this issue was theoretical and not physical, it is still a little horrifying to the modern mind that the subject was taken so seriously.) He asserts as a young man in the Inferno Lectures that the roof of Hell, which is shaped like a dome, must be a certain dimension and is scalable. It is only a few years later that he realizes scale has its limits; while the area will only be squared, the volume of the space will be cubed, or increased by another factor. This probably contributed to the collapse of the cathedral at Beauvais. According to Peterson, Galileo probably spent many years terrified that someone else would realize this too and spent a significant amount of time preparing his response. (Surprisingly, the challenge never came.)
|Santa Maria del Fiore|
Peterson does more than just make his case for why a learned young man of the arts would see the beauty of math and science more than someone trained in the science of the time. (And it should be noted that while Galileo apparently spared little respect for Kepler and, shockingly, Copernicus, there were of course other scientists at the time who were also making amazing leaps in scientific understanding.) He also helps the reader understand the intellectual history of the Western world and explain how mathematics and to some extent science came to be understood more as philosophical examinations of perfection and the role Galileo played in reviving them as concrete methods that could be used as tools. Even something as basic as measurement was understood differently before Galileo's famous experiments.
It must be said that while much of the historical background is exciting, at a certain point the rigidity imposed by authorities- and not simply those of the Church- becomes stifling. By the time Peterson starts mentioning the Arabs and Indians, they feel like a breath of fresh air.
Excellent book- recommended for the math and science enthusiast, as well as those who already the see the poetry contained in both.
"... the beauty of physics, the wonder of mathematics..."