How not to write about Renaissance mathematics

This is a book review. It is a review of Mark A. Peterson’s Galileo’s Muse: Renaissance Mathematics and the Arts (Harvard University Press, 2011) that I have to admit I’m writing with some reluctance. Why? I’m writing this review with some reluctance because it is going to be an extremely negative review. Now regular readers of this blog are probably asking themselves, “is he ill?” “There’s nothing the Renaissance Mathematicus likes more than putting the boot in, so why not now?” These would of course be justified questions so before I review Mr Peterson’s book I want to take some time to explain why I am in this case reluctant to put the boot in.

Some time back my Internet histsci soul buddy, Darin Hayton of the PACHS Network wrote a really good piece on book reviewing. He compared different reviews of Robert Westman’s new Copernicus book and explained why, in his opinion, the review written by Renaissance historian David Wootten Wootton is relatively worthless. Darin then goes on to give a set of guidelines for what he considers to be the right way to review (history of science) books. The whole article is well worth a read and I think his guidelines are absolutely spot on. One of his recommendations reads as follows:

Do not accept for review a book you are predisposed to dislike, or committed by friendship to like. Do not imagine yourself a caretaker of any tradition, an enforcer of any party standards, a warrior in any ideological battle, a corrections officer of any kind. (The bold emphasis is Darin’s).

Now the publisher’s blurb for Peterson’s book had instantly predisposed me to dislike it, which I’ll explain later. This being the case I felt a certain reluctance to even read it let alone review it. However not every book is as bad as the publisher’s blurb makes it out to be and the author often has very little control in how the publisher markets his efforts. This being the case I ignored my feelings of doubt and read the book. It turned out to be worse than I had feared. Worried that given my reaction to the publisher’s blurb I had read the book with prejudicial eyes I put it aside for several weeks and then re-read it trying very had to view it objectively and not to let my, possibly, prejudices get in the way. This didn’t improve my opinion of Peterson’s tome, what to do?

I have had several exchanges with various history of science colleagues in particular with Becky Higgitt, who as well as being a very good historian of science is also the book review editor of an important history of science journal, on the subject of bad popular history of science books and their reviews. On the whole real working historians of science refuse to review popular books on their subject written by non-historians. The reviews get written instead by fiction authors, journalists, professors of Italian[1] and other non-experts who often praise the reviewed volumes for their literary and entertainment qualities completely ignoring the fact that they are historical rubbish. Said volumes go on to become best sellers and the professional historians of science moan about the fact that they are factually incorrect, misleading and so on and so forth. Weighing up both points I have in the end decided to write my highly negative review of Peterson’s Galileo’s Muse, because although I had a negative view of this book before I even read it, it is a book that in my opinion should not be offered to the general public without saying that it is very, very far from being good history of science.

Prejudice! Why was I put off this book by the publishers blurb? On the flyleaf one can read the following statement:

Mark Peterson makes an extraordinary claim in this fascinating book focused around the life and thought of Galileo: it was the mathematics of Renaissance arts, not Renaissance sciences that became modern science. Galileo’s Muse argues that painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day, steeped as they were in a medieval cosmos and its underlying philosophy.

 This is an example of what I call the “my God Newton was an alchemist” syndrome. Every few years a new book or newspaper or magazine article trumpets out “Shock, Horror, Outrage Isaac Newton the father of modern science was a secret alchemist!” That Newton was a practicing alchemist has been known since at least the middle of the 19^th century and that his alchemy even influenced his scientific work is also no longer new. It’s actually a rather sad comment on peoples’ knowledge of the history of science that such stories can be recycled about every five years.

That there was a vital and extensive interchange of knowledge between the artisan and scientific communities in the Renaissance has been the subject of extensive research for several decades and has produced a significant amount of literature. Peterson claim is not extraordinary or in anyway new but is the daily bread of a fairly large number of Renaissance historians of art, science, culture, literature and mathematics. Just to name a couple of the better-known products of these efforts we have Pamela H. Smith’s very impressive The Body of the Artisan: Art and Experience in the Scientific Revolution, Chicago University Press, 2004. The complete life’s work of art historian Samuel Edgerton who has written numerous books and articles on the discovery of linear perspective in the Renaissance and his, highly disputed, theory that this triggered the scientific revolution. We have Martin Kemp’s masterpiece The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, Yale University Press, 1990. Horst Bredekamp’s Galilei der Kunstler, Berlin 2007 and a whole lot more. Peterson is not doing anything new he is following an already well-worn path and doing it badly.

This article is already quite long and I haven’t even started on the review proper. If I were to go into detail on everything in Peterson’s book that annoys, worries or angers me then my review would be substantially longer than the book itself. Instead I shall make some remarks about various aspect of the book that are in my opinion are wrong but with the caveat that there is more of the same that I haven’t commented on.

The book opens with a biographical sketch of Galileo’s youth and his decision to become a mathematician. Strangely Peterson does not reference modern scholarly biographies of Galileo but uses as his sources the two contemporaneous partial accounts of Galileo’s life. What’s wrong with that you might ask? Very simply both accounts are known to be at best dubious and at worse false. So why does Peterson use them? He uses them to sustain the myth that Galileo came to mathematics by accident as an adult and against opposition to his choosing this career. Why will Peterson sustain this myth? Because he is in reality writing a hagiographical account of how Galileo the autodidact singlehandedly re-introduced ‘real’ mathematics into science. The first and only person to do so since the classical age of Greek mathematics! This is the real message of Peterson’s book and it is historical rubbish. To achieve this aim Peterson proceeds to twist, misrepresent and falsify the history of science and mathematics. He starts by explaining that Galileo’s apparent ignorance of mathematics until he was an adult and university drop out although implausible can be verified by the similar experience of Thomas Hobbes. Peterson relates Hobbes’ exhilaration on first viewing an open page of Euclid’s Elements, as an adult. Peterson argues so Hobbes, so Galileo. Unfortunately the comparison doesn’t work, which Peterson would know if he had really studied the history of Renaissance mathematics. Galileo grew up and went to university in Northern Italy where the universities had had specialist chairs for the mathematical sciences since the last quarter of the 15^th century and a well developed network of private schools for commercial mathematics since the 14^th century, that is a well established and well developed mathematical culture. As a student of medicine at Pisa Galileo would have taken the same courses in mathematics, astronomy and astrology as every other medical undergraduate, course that he himself would then teach as professor of mathematics both at Pisa and Padua starting only a few years later. Hobbes grew up in England and graduated from Oxford University in 1609. The first university chairs for the mathematical sciences in England were established in Oxford in 1620! Something akin to the Italian abbacus schools for commercial mathematics only began to emerge in England during Hobbes’ youth thanks to the efforts of Robert Record and John Dee. In Hobbes’ youth England had no mathematical culture!

Peterson continues his campaign of misinformation with his account of the various forms of Renaissance mathematics. I can’t even begin to explain in the space of this review everything that’s wrong with this account other than to repeat that its aim is to justify his claims for Galileo’s mathematical uniqueness.

There then follows a section on science and mathematics in classical antiquity that I can only sum up with the phrase “half read, half understood, half forgotten!” It’s as if an undergraduate had half read the assigned literature for a course in classical studies, only really understood half of what he had read and the forgotten half of it anyway. The result is horrendous! I think the worst aspect is a sort of Animal Farm belief mantra that Peterson insinuates in his account of antiquity  “mathematics good, philosophy bad”. Aristotle hasn’t got a leg to stand on because he wasn’t a fan of mathematics and poor Ptolemaeus gets taken to task for daring to start a book on mathematical astronomy with a philosophical discussion. I would draw a veil over the whole sorry mess but there is one point that I have to illuminate.

In his writings Galileo praises the Pythagorean philosophy. Now Peterson is aware that modern scholars tend to write off the Pythagoreans as a rather weird religious sect who practiced a strange form of number veneration. His Galileo couldn’t possibly be a fan of something like that! So Peterson proposes that the ‘real’ Pythagoreans were the mathematical giants of antiquity, Euclid, Appolonius, Archimedes etc. and that they kept their identity as Pythagoreans secret. We have left the realm of popular history of science and entered the world of Dan Brown. Mr Peterson the explanation is much simpler and much more rational. Galileo’s Pythagoreans were not the historical sect in Southern Italy but an idealised Renaissance view of a mathematized world, very simple.

The next two chapters of the book also owe more to a Dan Brown view of the world than to a historical one. We get treated to Dante the geometer. This is actually stuff that Peterson has published before and is being recycled here. Peterson has decoded Dante’s Paradiso and discovered that it’s really a text about advanced geometry. Dante’s description of heaven is really a description of the 3-sphere! For the non-mathematicians amongst my readers, if you’re still awake at the back there, a 3-sphere in the geometrical equivalent to the sphere in four dimensions, as the sphere is the geometrical equivalent to the circle in three dimensions. What in fact happens is that Dante arriving at the boundary to heaven describes what he sees in a way than can be interpreted, and is so by Peterson, as a non-mathematical description of a 3-sphere. Thus so Peterson Dante is the discoverer of the 3-sphere! There is again a much simpler and much more rational explanation for Dante’s description. For a mediaeval Christian heaven is literally indescribable, it is beyond human comprehension so when Dante arrives at the boundary of heaven he does just that. He describes something impossible, something beyond comprehension, little imagining that a few hundred years in the future that, which he has described will become for all intents and purposes the mathematicians 3-sphere. Dante is not its discoverer. Peterson argues that Dante was a knowledgeable geometer because he quotes two Euclidean theorems in his Paradiso. Unfortunately for Peterson both the theorems quoted by Dante are from the very beginning of the Elements and are such that any undergraduate at a mediaeval university would have studied/learnt in his liberal studies course and do not demonstrate that “Dante knew his Euclidean geometry very well”.

Peterson goes even further with his Paradiso Code fantasies claiming that the closing stanzas of the Paradiso are actually an encoded version of Archimedes’ proof of the incommensurability of the circle! I wont do a blow-by-blow account of this bizarre claim but I will make one trenchant comment. Peterson writes:

In all Dante’s writings he never mentions the name Archimedes, but there is a surviving treatise of Archimedes called On the Measure of the Circle. It is only a few pages long, and thus easily copied. It was widely circulated in Dante’s day, and even well before. [An exaggeration on Peterson’s part] We have seen how intensely Dante was interested in the measure of the circle [One brief reference in an unfinished work] – it is inconceivable that he would not have known the Archimedes treatise.

It is perfectly conceivable and highly probable.

Having supposedly dealt with geometry in poetry Peterson now moves on to painting and to what I consider to be the only halfway good section of his book. Here he deals with the Renaissance discovery of linear perspective, which is indeed an important episode in the development of Renaissance mathematics. However here he can’t refrain from falsifying history to suit his prejudices. Alberti who actually wrote the first treatise on linear perspective is dismissed as not really understanding what he’s writing about and Luca Pacioli is dismissed as a showy braggart who’s not really a mathematician. This being the author of the Summa de arithmetica, geometria, proportioni et proportionalità one of the most important mathematics books published in the period as well as being Leonardo’s mathematics teacher. The purpose of these degradations is to raise the profile of Piero della Francesca who is the main subject of the section. Once again Peterson is recycling work that he has already published elsewhere but it is nice to see somebody giving both della Francesca’s work in linear perspective and in mathematics the attention it richly deserves. However it should be noted that Peterson contributes nothing original here he is merely recycling the researches of others, most notably Martin Kemp and Judith Field. This section closes with a real clanger, having explained how Pacioli plagiarised della Francesca’s work on the Platonic solids Peterson opinions the following about Leonardo who provided the wonderful geometrical illustrations for Pacioli’s work. He writes:

It is painful to think how vulnerable Leonardo might have been to the seductions of Pacioli’s geometry. In the end, Leonardo produced sixty careful drawings of the Platonic solids and other polyhedra, representing physical models of these figures fashioned as hollow frameworks, so you could see through them to the back. Leonardo was at the height of his powers – the Mona Lisa was still to come – but here he is doing something utterly mechanical, and almost embarrassing.

 Here Peterson displays an ignorance of Renaissance art at the beginning of the 16^th century that is almost embarrassing. After the discovery of linear perspective, starting with della Francesca, the correct perspective presentation of three-dimensional geometrical figures became almost a fetish amongst artists who were proud to demonstrate their mastery of perspective with such studies. Far from being embarrassing Leonardo would have revelled in the chance to demonstrate he very obvious superior skills as a draughtsman and if Peterson thinks that such drawing exercises are mechanical I would suggest that he tries to draw a copy of one of Leonardo’s figures.

Following painting we are presented with music another area where mathematics was applied in the Renaissance and for the first time in many pages we encounter Galileo again. This time we are more concerned with his father Vincenzo who was one of the leading music theorists of the period and involved in an infamous dispute over the divisions of the scale and the size of musical intervals. Again Peterson adds nothing new to a story that has been told many times although for the first time in the book we have Galileo the experimental physicist working together with his father on the laws governing the pitches of stretched strings, experiments that found their way into his Two New Sciences.

Music is used to introduce us to Johannes Kepler whose magnum opus was famously entitled The Harmony of the World and actually contains an extensive discussion of the dispute involving Vincenzo Galilei. We get a fairly standard account of the handful of letters the two men exchanged laced with Peterson’s attempt to denigrate Kepler’s mathematical achievements whilst praising Galileo’s:

Although higher mathematics in 1610 was still entirely identified with astronomy, [it wasn’t of which more later] Galileo meant to extend mathematics to earthly things, a philosophical revolution for which he needed the title Court Philosopher as much as Court Mathematician. For Kepler earthly mathematical problems simply could not compete with astronomical ones for beauty and importance.

 Peterson completely ignores the brilliant general mathematical advances in Kepler’s astronomical works, his presentation and analysis of the 13 semi-regular Archimedean solids, he demonstration that the conic sections are actually all the same general function, his extraordinary use of proto-integration in his ‘proof’ of his second planetary law. Instead we are treated with a bizarre presentation of his Dioptrice, his Six-Cornered Snowflake and his pamphlet on measuring wine barrels. Of the Dioptrice Peterson writes:

Kepler returns in Dioptrice to a problem he had omitted from his 1604 Optics, the use of lenses in astronomy, and especially the theory of the telescope, since the telescope had turned out to be useful in astronomy after all. His treatment of the subject is a practical, semi-quantitative account of what you see when you look through two lenses, undeniably the result of systematic experimental work, not something one usually thinks of in connection with Kepler.

 There is so much wrong with this passage it is hard to know where to begin. The Optics from 1604 does include the first account of the geometrical optics of lenses, the first such account ever to be published and it was this that enabled Kepler to publish his Dioptrice, a definitive account of the optics of telescopes so soon after Galileo had published his Sidereus Nuncius. His Optics had not included an account of the telescope because it didn’t exist in 1604 being first invented in 1608!  In the Dioptrice his treatment is totally theoretical displaying Kepler’s total mastery of the theory of geometrical optics, not a single hint of an experiment anywhere. The Dioptrice also includes accounts of optical systems with more than two lenses most notably the so-called terrestrial telescope with its third inverter lens one of Kepler’s three significant inventions in the book; the other two are the astronomical telescope and the telephoto lens.

Peterson will have us believe that Kepler was unaware of the mathematical and scientific significance of his snowflake pamphlet because he writes it in a deprecating humorous style. I beg to differ. Peterson also wilfully ignores or is ignorant of the fact that Kepler’s pamphlet on measuring wine barrels in an important early example of the use of integration to determine volumes.

Kepler was a vastly superior mathematician in comparison to Galileo and his contributions to the evolution of modern mathematics are manifold but Peterson tries desperately to play this fact down because it totally contradicts his central theme than Galileo was unique in the period in his use of mathematics to solve problems in physics.

Next up in Peterson’s survey of Renaissance arts and crafts is architecture and the Renaissance artist engineers adherence to the Pythagorean theories of harmonic proportions in design. His presentation is very superficial and not very illuminative but he still manages to make a serious failure. At the end of the section he discusses Copernicus’ claim as to why his heliocentric system was superior to the Ptolemaic geocentric one, which Copernicus frames in terms of the natural harmony of his system. Peterson writes:

There is nothing in the Ptolemaic description to determine the third dimension, however – the distance from us to the various planets.

 Although the relative size of the planetary orbits is automatically dictated by the heliocentric system it is true that this is not the case with the geocentric system. However the geocentric astronomy did have a convention for determining those distances so to say, as Peterson does, “there is nothing” is historically false. Copernicus argument is that in his system the determination follows from the system whereas in the Ptolemaic one it is only decided by convention.

This introduction to Renaissance architectural theory is only a lead in to what Peterson obviously regards as one of the high points of his book, his discussion of Galileo’s public lecture on the dimensions of hell in Dante’s Inferno. Here Peterson thinks that he has made an important discovery that Galileo uses physical scaling arguments in his description of the structure of hell. This, according to Peterson, is an important major new scientific method that Galileo is introducing here. I think he is over egging the cake. Apart from anything else as he himself admits Galileo got the argument wrong.

We are now approaching the conclusions of the book and they are prefaced with a survey of “mathematics old and new”. This survey as with all the accounts of Renaissance mathematics in the book is to put it mildly highly inaccurate and inadequate. First we get a rather tired retelling of the Tartaglia – Cardano episode and the solution of the cubic equation at the end of which we are told that Galileo had no interest in the new algebra. Other workers in the field such as Bombelli or Stiffel don’t even get a mention. The section on geometry starts rather bizarrely with the Gregorian calendar reform and we are informed that Peuerbach and Regiomontanus did their work in trigonometry as a contribution to this reform, which is complete historical rubbish. Having described their contributions to the development of trigonometry, Peterson then dismisses them as irrelevant and insignificant. Again we have no mention of the others who made contributions to the evolution of trigonometry in the 16^th century; contributions that led Ivor Gratten-Guiness, one of the leading historians of mathematics, to label the period “the age of trigonometry”.  This chapter on mathematics closes with a section on translations, highly appropriate as the Renaissance is all about the recovery of original Greek and Latin texts and here Peterson drops one of the biggest clangours in the entire book whilst discussing the first printed edition of Archimedes. This is important to him as he presents Galileo, quite correctly, as taking Archimedes as his mathematical role model. Peterson writes:

Tartaglia produced the first printed Archimedes in 1543, the one that Galileo studied so intensively on the advice of Ostilio Ricci, but it was just the thirteenth-century translation of Wilhelm Moerbeke. (The comment about the old translation is that it is therefore a highly defective translation.)

 In 1543 Tartaglia published the first Italian translation of Euclid’s Elements, from the Latin and not the Greek original, but he never published an edition of Archimedes in any form what so ever. The first printed edition of the works of Archimedes was published by Johann Herwagen in Basil in 1544. It is a bilingual edition Greek with the Latin translation of Gerard of Cremona edited and corrected from the Greek by Regiomontanus and then edited for publication by Thomas Geschauff known as Venatorius, a cleric and scholar from Nürnberg. I really don’t understand how Peterson could have got it so completely wrong.

The penultimate chapter now promises to reveal Peterson’s main claim the uniqueness of Galileo’s contribution to mathematics. I have reread the chapter several times and I can’t find anything what so ever to justify this claim. He also here wants to demonstrate how Galileo’s training as an artist influenced his scientific discoveries. On this the only real evidence he produces is Galileo’s telescopic illustration of the moon a subject that has been dealt with exhaustively by Albert van Helden in numerous papers and Horst Bredekamp in the very thick book mention very early on in this review. The New Scientist reviewer of Peterson’s book even wrote the following:

All this is true, but Peterson does not show in any detail how this cultural background led Galileo to make the two great discoveries for which he is famous in physics: the law of freely falling bodies, and the fact that the trajectory of a projectile is a parabola.

 Peterson even manages to confuse these “two great discoveries”. He writes that Pierre Duhem says that the parabola law had already been discovered in the fourteenth century. What in fact Duhem and others say, perfectly correctly, is that the mean speed theorem, the core element of the laws of fall, was discovered by the Oxford mathematician William Heytesbury in the fourteenth century and that this was demonstrated by the Paris physicist Nicolas Oresme, in the same century, with a graph identical to the one used by Galileo to demonstrate the same law. Peterson doesn’t even mention the fact that almost the complete laws of fall were published by the Italian mathematician Benedetti in the 1550s. Peterson also completely ignores the fact that the Dutch mathematician Simon Stevin was doing very similar work contemporaneously to Galileo in mathematical physics; facts that rather spoils his uniqueness claims for Galileo. This section also contains what I regard as the most revealing mistake in the book. In his attempt to negate the achievements of Galileo’s mediaeval predecessors Peterson writes:

The medieval theory was not so much a theory of motion as it was a theory of any quality that varied with position or time, the quality of color for example.

 Remember that earlier quip about “half read, half understood, half forgotten!” Here we have a perfect example. In his philosophy Aristotle did in fact regarded terrestrial motion, celestial motion is completely different, as one example of change. The earthly or sub-lunar sphere being characterised by impermanence, decay and change as opposed to the supra-lunar sphere which was permanent, eternal, unchanging. All forms of change were therefore properties of the sub-lunar sphere. However when discussing change he did not put up a general theory but dealt with the different forms of change separately. His theory of motion dealt with motion and only with motion. This theory was highly unsatisfactory and was already challenged in the Early Middle Ages by John Philliponus and in the High Middle Ages by both Islamic and European scholars who developed the so-called impetus theory of motion; a theory that even Galileo used in his early investigations of motion. The mathematical treatment of motion carried out by the Oxford Calculatores and the Paris Physicists in the 14^th century was just that, a mathematical theory of motion and not as Peterson claims a general Aristotelian theory of change.

Following, what is, a rather disappointing conclusion to his stated claims for his book Peterson closes with a rather strange chapter in which he thinks he can prove that an oration on the virtues of mathematics held by a student of a student of Galileo’s was actually written by Galileo. Here his arguments are somewhat less than convincing and even if he is right the oration is nothing more than a puff piece of no real significance so one is left asking, “so what?”

The book closes with an epilogue, which contains probably the best conclusion by Peterson concerning Galileo in the whole book. He writes:

It is not easy to write about Galileo and yet ignore the Copernican controversy, as I have done until now. In the end I feel a responsibility to record my view, namely that the importance of the Copernican controversy in Galileo’s biography is overstated. [my emphasis]

I couldn’t agree more.

It is actually a very important historical fact to realise that Galileo was actually a typical Renaissance artist-engineer in the very strong tradition of such people as Brunelleschi, Alberti, Leonardo, Michelangelo, to a lesser extent Dürer and many lesser known figures such as Ignazio Danti or even, later than Galileo, Nicholas Mercator rather than the “modern scientist” as he is most often presented. It is also very important if we are ever going to really understand the emergence of modern science in the Early Modern Period to study the working practices of these artisans and to see where their methodologies led to those that we now recognise as scientific. Galileo, given his contributions to the evolution of science, would seem to be a perfect object for such a study and that is what Peterson’s book seemed to be promising. Maybe I’m being to hard in my judgement but I think he has failed totally to do so. Even more worrying in his attempt to achieve his second stated aim, that is to show that Galileo’s use of mathematics was ground breaking and in some way unique, he has presented a warped, distorted and I’m afraid to say deliberately false picture of mathematics and its use in the Early Modern Period before Galileo. I cannot honestly recommend this book to anyone.

I said that a study of Galileo as a Renaissance artist-engineer and how this affected his scientific work would be very interesting and although Peterson has in my opinion failed to deliver somebody else has. Appearing to late for Peterson to have read it whilst writing his own book is Matteo Valleriani, Galileo Engineer, Springer, 2010. Unlike Peterson’s book this is not aimed at the popular reader but is in fact the author’s doctoral thesis. It is also not made more readable by the fact that the author, an Italian, wrote it in English and did not allow anybody to, shall we say, smooth out his language. However the book is actually fairly readable and for anybody who is really interested in how the training of a Renaissance artist-engineer translated into scientific methodology I heartily recommend Valleriani’s book.

 

 

 

 

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[1] One of the reviews on the cover of Peterson’s book is by a professor of Italian.