The books I love best are the ones that connect me to a larger story. In the case of Galileo’s Muse, the story Mark Peterson, Professor of Physics and Mathematics on the Alumnae Foundation at Mount Holyoke College, connected me to was about the essential role the arts played in the preservation and rediscovery of mathematics as we now understand it. I am honored that Doctor Peterson took time out of his busy schedule to chat with me about why the differences between idealized and “applied” mathematics, “Pythagoras’ method of philosophizing” and what lessons we can take from Galileo’s journey.
What was your first hint that Galileo's inspiration was the arts and not contemporary science?
The relationship between the Renaissance arts and Galileo's scientific work came into focus for me only gradually. I began to notice that Galileo frequently implies that his thought is formed by the arts. You don't see this if you only read secondary sources, like standard biographies in English, though. I learned Italian in order to immerse myself in Galileo's world and to understand the ideas that would have been most natural and familiar to him. A key moment, a kind of confirmation, came when I read his first biographers, authors who knew him personally. They all emphasized his love of the arts and the importance of the arts for him.
In the Renaissance people seemed to look at mathematics as part of a philosophy of perfection and not "real" in the way we do now. How did Galileo change that, and what was his motivation to?
That was my original question! To paraphrase your observation, I noticed something odd, namely that Galileo's mathematics was different from that of his contemporaries, even though it was also, strictly speaking, the same mathematics, just geometry, etc. So what was different? I gradually realized that Renaissance assumptions about mathematics idealized it and made it useless for much of what we do with mathematics now. Mathematics was on a kind of philosophical pedestal, revered, but not applied in any very interesting way (astronomy always being the exception). Galileo's mathematics (which was almost never about astronomy) was different, I found, in being explicitly part of a new earthly philosophy. Since modern science didn't yet exist, he couldn't point to scientific examples, but he could point to the arts. Mathematics in the arts was surprisingly effective, hinting at perhaps more surprises to come. A philosophical shift like this doesn't come quickly or easily, and I believe Galileo was in his sixties before he was really clear about it: the evidence I would point to is in the last chapter of my book, a discovery that finally pulled everything together for me, the Oration in Praise of Mathematics by his closest student [Niccolò Aggiunti], a work that finally articulates a new philosophy of what mathematics could and should be. (This Oration has been completely ignored by Galileo scholarship, by the way.)
I was heartbroken to read how little DaVinci, possibly the ultimate Renaissance Man, knew about mathematics. And yet he was able to make impressive advances in the arts and sciences himself. Which begs the question: how much math do you need for science?
I don't think we should feel sorry for Leonardo, whose well deserved reputation for brilliance is secure for all time! If 19th century romantic notions of his genius might lead us to have impossibly high expectations for him, well, maybe it is right to realize that he was also a man of his time, and not ahead of his time, certainly, in mathematics. In fact, he illustrates very well the mysterious and almost magical quality that mathematics seemed to have for many of its enthusiasts in the late 15th century, an interesting phenomenon in itself. Both he and we are fortunate that his greatest achievements did not rely on his understanding higher mathematics.
But your question returns to the role of mathematics in science, and here it does seem to me that for some purposes mathematics is essential. In the 17th century new science and new mathematics developed together, especially in the work of Newton and afterwards. And there is additional evidence for the importance of mathematics in all the sciences, even biology, as they have become increasingly mathematical over time. Why is this?
One possibility: mathematics, in abstracting away from the details of a situation, makes it possible to transfer insights gained in one place to another place; it facilitates non-obvious analogies, the sort of thing that only has to be pointed out once to effectuate rapid innovation. When Newton cited Galileo and Kepler, it was their mathematical results that he quoted. They had encapsulated in mathematics insights that Newton could generalize, also in mathematics. In this way the revolution could build on itself.
My favorite chapters were about Dante (and I didn't start out with a great affection for The Divine Comedy), particularly the one about measuring the circle. How did you come to see Archimedes in Dante?
I hope you are feeling more affectionate toward The Divine Comedy now! This loveliest of epic poems seems to me, almost by the way, to be a significant part of the scientific revolution, although I may be alone in thinking so. Galileo's involvement with Dante (in the Inferno Lectures) is a matter of record, although the significance that I give to this incident is controversial. And my contention that the climax of Paradiso paraphrases a proof of Archimedes is also controversial. So many controversies!
I see Archimedes in Dante because he is palpably there. I would say it is obvious. I gave a talk about this in the Italian Department at Yale, and the audience was not convinced until I showed them the 14th century commentary that very clearly says the same thing. Lucky for me that this commentary exists. But I would say it is obvious anyway (while acknowledging that it is controversial).
The other chapter about Dante and his conception of the topology of the universe blew me away. How is it that Dante, pretty firmly rooted in the Middle Ages, came to see all of this advanced mathematics?
This was the start of my involvement with Renaissance mathematics, when I noticed the 3-sphere in Dante's Paradiso, over 40 years ago (and I was not the first to see it). The idea has become respectable, if not universal, among Dante scholars. But how is it possible? I try to suggest how Dante did it in one chapter of Galileo's Muse, and Dante himself tries to show us how he did it.
What seems like advanced mathematics to us was not so advanced for Dante, I suggest. He could imagine it precisely because he was firmly rooted in the Middle Ages. The idea is hard for us because we have to overcome certain common sense ideas, always a hard thing to do, ideas like infinite space. Dante did not have those ideas. In fact, common sense for him was finite space. His fondness for the circle in geometry, and the notion of a reflected space, as he first sees heaven reflected in Beatrice's eyes -- these things lead quite directly to this most beautiful conception.
I was shocked that Pythagoras’ ratios for musical intervals were never tested during the Renaissance- or the Middle Ages- before Galileo’s father. Is it fair to say that measurement and testing- empiricism itself, perhaps- was Galileo's most enduring contribution?
I agree with you, this is most surprising, and Galileo's father (and Galileo himself) must have been truly shocked when they discovered by experiment the real meaning of this ancient story about Pythagoras. It is perhaps the first discovery of modern mathematical experimental physics, so it is very significant. It seems to have directed young Galileo's attention to what he came to call Pythagoras' method of philosophizing, a combination of experiment and mathematics. Many difficult ideas have to come together to make this method work, especially a feeling for the randomness in measurement, but that is something that practicing artists are comfortable and familiar with as they use measurement and testing in the context of real things. So I think I would say yes to your question, but with the proviso that there are philosophical dimensions to this idea that are also essential, and not captured in mere "measurement and testing."
Here is a rather sad story to make that point. Fourteen years after the death of Galileo the Medici princes, wishing to build on Galileo's work and further the new science, founded one of the world's first scientific societies, the Academia del Cimento, the "Academy of Experiment." Because they collectively doubted their abilities in mathematics, they confined themselves to doing many, many careful experiments, aiming just to see, very precisely, how nature actually behaves, drawing on all the impressive technological resources of their rich court. Their beautiful surviving apparatus can be seen in the Museo Galileo in Florence, but alas, that is all that remains. In all their empirical observations there is virtually nothing of lasting value, because it was simply experimental, without any mathematical theory to make sense of it.
You talk about Euclid's and Archimedes' influence on Galileo and his contemporaries, but eventually it all comes back to Pythagoras (or the Pythagoreans). He- they- didn't make it easy for people to figure out what they were doing or saying, and as you note it doesn't get any easier when math is removed from its context. Dante maybe gets a glimpse of it and Galileo maybe most fully grasps it. Did any of Galileo’s contemporaries "get it”?
I think Galileo uses the name Pythagoras and "Pythagoras' method of philosophizing" in a way that was peculiar to him. Meanwhile, Pythagoreanism had another meaning for most of his contemporaries, including most significantly Kepler, a kind of number mysticism that Galileo regarded with disdain. Galileo thought of Euclid and Archimedes as continuing and furthering his idea of the Pythagorean tradition, and that is the tradition that he imagined himself to be a part of. The clearest way to address your question is to ask how Galileo's contemporaries regarded Archimedes, whose significance in this tradition, unlike that of Pythagoras, is not ambiguous. Kepler, for example, was curiously ambivalent about Archimedes. Guidobaldo del Monte, Galileo's first significant patron, was a great enthusiast of Archimedes. It is a revealing distinction, and perhaps this is a question that researchers could do more with.
In light of your thesis, it’s ironic if not tragic that arts in the public schools have been threatened for at least three decades. And yet we've spent a long time bemoaning the lack of creativity in our students- and the lack of qualified scientists and mathematicians that we’re producing (which may or may not be true). Have you thought about mailing your book to Secretary of Education Arne Duncan?
The issue of arts in the schools was one of my motivations in writing Galileo's Muse, and not just because arts education is useful -- it enriches life in any case. But I especially wanted professional scientists to see how much they owe to the arts. Since I teach mathematics and physics at the college level, I get to see the results of various kinds of school preparation. It is startling to see, for example, that the most strictly trained students in mathematics may have had the creativity literally beaten out of them. They are quick and accurate, but consumed with anxiety when they make a mistake, and they can no longer think of something on their own initiative. They themselves are frustrated when they realize their limitations.
An arts education, with its attitudes of experimentation, creativity, engagement with tangible things, and the willingness to innovate, would have served these students better. On the other hand, it is even more depressing to see students who lack the elementary mathematical skills that all students should have. An appreciation that the arts are, among other things, mathematical, should do away with the fictitious and imagined divide that supposedly separates arts from supposedly more useful subjects.
But I don't think Arne Duncan would be interested in my book. People in his position must have their minds already made up, for better or worse. My book is for the rest of us, who might make more local, spontaneous, and imaginative decisions.
Well, I haven't heard such calls, but equally lunatic suggestions are made, I know, in other scientific subjects. The corpus of school mathematics, geometry and algebra, is small, easily sampled, and it connects us directly to the Hellenistic Greeks, to the Indian arithmeticians, and to so many others, whose discoveries are as perfect now as when they were first made. It is a uniquely beautiful and useful thread in our common intellectual heritage, completely uncontroversial, I would have said. At least these calls do not, I imagine, dispute that mathematics is true!
What's the essential point you want your readers to understand when they're done with your book?
I want my readers to savor the pleasure of intellectual surprise. I want them to see, for example, that the story of Galileo, which most people already have an opinion about, can be read in a completely different way. This was Galileo's own way of thinking: an almost instinctive and reflexive readiness to imagine the contrary. I'm not trying to convince my readers of my thesis, even if I am very fond of it. The more essential point is the possibility of imagination subverting the obvious. That is what was at stake in the 17th century scientific revolution, the ostensible subject of my book, but on a less grandiose scale it is always at stake. There are always other possibilities: that is the point that I want to make, both in the substance of my story and in the way that I tell it. This is not how most people think of science and mathematics: I want to surprise them, and that is why I don't end with some kind of scientific triumphalism and a fait accompli, but with the Muse, and the beginnings of things only glimpsed.
