After I had, in my last blog post, mauled his Scientific American essay in my usual uncouth Rambo style, Michael Barany responded with great elegance and courtesy in a spirited defence of his historical claims to which I now intend to add some comments, thus extending this exchange by a fourth part.

On early practical mathematicians Michael Barany acknowledges that their work is for the public good but argues correctly that that doesn’t then a “public good”. I acknowledge that there is a difference and accept his point however I have a sneaky feeling that something is only referred to as a “public good” when somebody in power is trying to put one over on the great unwashed.

Barany thinks that the *Liber Abbaci* and per definition all the other abbacus books, only exist for a closed circle of insider and not for the general public. In fact abbacus books were used as textbooks in so-called abbacus schools, which were small private schools that taught the basics of arithmetic, algebra, geometry and bookkeeping open to all who could pay the fees demanded by the schoolteacher, who was very often the author of the abbacus book that he used for his teaching. It is true that the pupils were mostly the apprentices of tradesmen, builders and artists but they were at least in theory open to all and were not quite the closed shop that Michael Barany seems to be implying. In this context Michael Barany says that Recorde’s *Pathway to Knowledge*, a book on elementary Euclidean geometry, is eminently impractical. However elementary Euclidean geometry was part of the syllabus of all abbacus schools considered part of the necessary knowledge required by artist and builder/architect apprentices. In fact the first Italian vernacular translation of Euclid was made by Tartaglia, an abbacus schoolteacher.

Michael Barany makes some plausible but rather stretched argument to justify his couterpositioning of Recorde and Dee, which I don’t find totally convincing but slips into his argument the following gem*. **If you don’t like Dee as your English standard bearer for keeping mathematics close to one’s chest, try Thomas Harriot*. Now I assume that this flippant comment was written tongue in cheek but just in case.

Michael Barany’s whole essay contrasts what he sees as two approaches to mathematics, those who see mathematics as a topic for everyone and those who view mathematics as a topic for an elitist clique. In the passage that I criticised in his original essay he presented Robert Recorde as an example of the former and John Dee as a representative of the latter. A contrast that he tries to defend in his reply, where this statement about Harriot turns up. Now his elitist argument is very much dependent on a clique or closed circle of trained experts or adepts who exchanged their arcane knowledge amongst themselves but not with outsiders. A good example of such behaviour in the history of science is alchemy and the alchemists. Harriot as an example of such behaviour is a complete flop. Thomas Harriot made significant discoveries in various fields of scientific endeavour, mathematics, dynamics, chemistry, optics, cartography and astronomy, however he never published any of his work and although he corresponded with other leading Renaissance scholars he also didn’t share his discoveries with these people. A good example of this is his correspondence with Kepler, where he discussed over several letters the problem of refraction but never once mentioned that he had already discovered what we now know as Snell’s Law. Harriot remained throughout his life a closed circle with exactly one member, not a very good example to illustrate Michael Barany’s thesis.

I claimed that there was no advance mathematics in Europe from late antiquity till the fifteenth century. Michael Barany counters this by saying: *This cuts, for instance, the rich history of Islamic court mathematics out of the European history in which it emphatically belongs*; it doesn’t cut it. Ignoring Islamic Andalusia, Islamic mathematics was developed outside of Europe and although it started to reappear in Europe during the twelfth and thirteen centuries during the translator period nobody within Europe was really capable of doing much with those advanced aspects of it before the fifteenth century, so I stand by my claim.

We now turn to Michael Barany’s defence of his original: *In the seventeenth century’s Scientific Revolution, the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty.* This he contrast with a, in his opinion, eighteenth century where mathematicians help sway over the scientific community. I basically implied that this claim was rubbish and I still stand by that to that, so what does Michael Barany produce in his defence.

In my original post I listed seven leading scholars of the seventeenth century who were mathematicians and whose very substantive contributions to the so-called scientific revolution was mathematical, on this Barany writes:

*Thony pretends that naming some figures remembered today both for mathematics and for their contributions to the scientific revolution contradicts this well-established historical claim.*

The, without any doubt, principle figures of the so-called scientific revolution are just *some figures*! Interesting? So what is Michael Barany’s *well-established historical claim*? We get offered the following:

*Following Steven Shapin and many who have written since his classic 1988 article on Boyle’s relationship to mathematics, I chose to emphasize the conflicts between the experimental program associated with the scientific revolution and competing views on the role of mathematics in natural philosophy.*

What we have here is an argument by authority, that of Steven Shapin, whose work and the conclusions that he draws are by no means undisputed, and one name Robert Boyle! Curiously a few days before I read this, science writer, John Gribbin, commentated on Facebook that Robert Hooke had to work out Boyle’s Law because Boyle was lousy at mathematics, might this explain his aversion to it? However Michael Barany does offer us a second argument:

*But to take just his most famous example, Newton’s prestige in the Royal Society is generally seen today to have had at least as much to do with his Opticks and his other non-mathematical pursuits as with his calculus, which contemporaries almost uniformly found impenetrable.*

Really? I seem to remember that twenty years before he published his *Opticks,* Old Isaac wrote another somewhat significant tome entitled *Philosophiæ Naturalis Principia Mathematica* [my emphasis], which was published by the Royal Society. It was this volume of mathematical physics that established Newton’s reputation, not only with the fellows of the Royal Society, but with the entire scientific community of Europe, even with those who rejected Newton’s central concept of gravity as action at a distance. This book led to Newton being elected President of the Royal Society, in 1704, the same year as the

*Opticks*was published. The

*Opticks*certainly enhanced Newton’s reputation but he was already considered almost universally by then to be the greatest living natural philosopher.

Is the *Opticks* truly non-mathematical? Well, actually no! When it was published it was the culmination of two thousand years of geometrical optics, a mathematical discipline that begins with Euclid, Hero and Ptolemaeus in antiquity and was developed by various Islamic scholars in the Middle Ages, most notably Ibn al-Haytham. One of the first mathematical sciences to re-enter Europe in the High Middle Ages it was propagated by Robert Grosseteste, Roger Bacon, John Peckham and Witelo. In the seventeenth-century it was one of the mainstream disciplines contributing to the so-called scientific revolution developed by Thomas Harriot, Johannes Kepler, Willebrord van Roijen Snell, Christoph Scheiner, René Descartes, Pierre Fermat, Christiaan Huygens, Robert Hooke, James Gregory and others. Newton built on and developed the work of all these people and published his results in his *Opticks* in 1706. Yes, some of his results are based on experiments but that does not make the results non-mathematical and if you bother to read the book you will find more than a smidgen of geometry there in.

In my opinion trying to recruit Newton as an example of non-mathematical experimental science is an act of desperation.

To be fair to Michael Barany the division between those who favoured non-mathematical experimental science and the mathematician really did exist in the seventeenth century, however it was largely confined to England and most prominently in the Royal Society. This is the conflict between the Baconians and the Newtonians that I have blogged about on several occasions in the past. Boyle, Hooke and Flamsteed, for example, were all Baconians who, following Francis Bacon, were not particularly fond of mathematical proofs. This conflict has an interesting history within the Royal Society, which led to disadvantages for the development of the mathematical sciences in England in the eighteenth century.

When the Royal Society was initially founded some mathematician did not become members because of the dominance of the Baconians and that despite the fact that the first President, William Brouncker, was a mathematician. Later under Newton’s presidency the mathematicians gained the ascendency, but first in 1712 after an eight-year guerrilla conflict between Newton and Hans Sloane, a Baconian and the society’s secretary. Following Newton’s death in 1727 (ns) the Baconians regained power and the result was that, whereas on the continent the mathematical sciences flourished and evolved throughout the eighteenth century, in England they withered and died, leading to a new power struggle in the nineteenth century featuring such figures as Charles Babbage and John Herschel.

To claim as Michael Barany does that this conflict within the English scientific community meant that mathematics played an inferior role in the seventeenth century is a bridge too far and contradicts the available historical facts. Yes, the mathematization of nature was not the only game in town and interestingly non-mathematical experimental science was not the only alternative. In fact the seventeenth century was a wonderful cuddle-muddle of conflicting meta-physical views on the sciences. However whatever Steven Shapin might or might not claim the seventeenth century was a very mathematical century and mathematics was the principle driving force behind the so-called scientific revolution. As a footnote I would point out that many of the leading experimental natural philosophers of the seventeenth century, such as Galileo, Pascal, Stevin and Newton, were mathematicians who interpreted and presented their results mathematically.

Thony, not wanting to take away anything from your spirited and just defence of the role of mathematical philosophy in late C17, but Newton’s Opticks was published in 1704, not 1706.

Oops! Had the date of the first Latin edition in my head and not the original English one. Must fact check before publishing!

Hi Thony,

Thanks for your re-reply! I think my re-response is brief enough to deposit here rather than a separate page.

You’ve made my point about abbacus schools for me: my entire argument turns on the importance of distinguishing access *in principle* from access *in practice*.

We’ll just have to disagree about the shape of Europe, but I hope you’ll acknowledge that, semantics aside, “European history” is much richer and more complete if one includes the parts of the Islamic world that were in close contact with those lands with the present poltiical classification of “Europe”. To rule out Baghdad, e.g., because it doesn’t line up the right way on a map, strikes me as narrow and anachronistic, especially since historical conceptions of the boundaries of Europe have always been in flux.

Perhaps I wasn’t being clear about Harriot: the man you describe is the one I had in mind. Surely you don’t think his mathematics was *everywhere*?

But we really *really* seem to disagree about the scientific revolution. That’s ok; we wouldn’t be the first pair to have fundamental disagreements about it! I do indeed rely a great deal on authorities who have studied the period in greater detail than I have, and as those go Shapin is no slouch. This is not the place, however, to rehearse his CV, nor to give a full bibliography of the many other authorities who have informed my own understanding of the period.

Sure, Newton was admired and respected for his Principia, but who actually read and understood it? According to the historiography I know, not many people. You know this, of course, and you also know how controversial many claims about mathematics were in and beyond the Royal Society.

You can also rest assured that I have no quarrel with your claim that mathematics was important in the 17th century. I know enough about the period to know that you’ve necessarily left out an enormous pile of evidence about the uses and roles of mathematics. But look again at the claim you imply this refutes, that “the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty.” The fact that not everyone was dubious of mathematics does not contradict my own assertion that a good many promoters of the new experimental science were indeed dubious. In fact, your own post backs me up on the latter point. So we agree after all!

In that case, I’d encourage you to revisit the rest of my Scientific American essay, which describes how those (including the intellectual heirs of several you note here) who took a different approach to mathematics than the Boyleans and Baconians came to use mathematics to secure their positions as elites. Their legacy continues to shape mathematics’ place in society today, and it’s important to recognize how, as well as what that implies for society’s engagement with this powerful field of knowledge.

– Michael

I have always been curious about the claim that Harriot had the sine law, which never struck me as very well established by those making the claim. Then I found this article by Robert Goulding that cleared things up:

http://link.springer.com/article/10.1007/s00407-013-0125-1#Fn41

From what I gather from this and Lohne’s 1959 article, Harriot had the sine law in the sense that he figured out through experiment that for any two substances, when light passes through one into another the ratio of the sines of the angle of incidence and of refraction is a constant. But that constant would be different for any two substances (air and water, air and glass etc) and would have to be re-established experimentally each time.

Would his discovery have struck him as a “law of nature” the way it does for us? Or just a better way to calculate than had Witelo suggested? I just mean I’m not sure his not sharing it was a matter of different priorities rather than secrecy.

How is this different from what one usually calls the sine law of refraction? The ratio of the sines does depend on the two media. Do you mean he thought the law applied only to air and glass, and wouldn’t work for other materials even with a different constant?

(I had access to the abstract of the article you cited, not the full text.)

Well, the Cartesian / modern form holds that the ratio of the sines is as to the ratio of the speeds of light travelling through the media. Even though Descartes’ “proof” of this was bogus, it certainly met his and his readers’ critieria for a fundamental “law of nature”, and derivable from knowledge of the nature of light and matter, even if it would have to be empirically established for any two media because you can’t know the speed of light moving through them. Given how empirical options was in the 16th century, I’m not sure Harriot would have seen his “law” in the same light, as opposed to just a useful shortcut for analyzing refraction through different media. For example, Harriot is much more grandiose in claiming to know the “secrets” of the rainbow, a big question in natural philosophy at the time.

Oh for an edit function “…how empirical optics was in the 16th century…”

Actually, the ratio of the sines is the inverse of the ratio of the speeds; Descartes incorrectly said it was the ratio. (Wikipedia notes that Descartes derivation was inconsistent with his belief that light has infinite speed. Also, Wikipedia claims that the first guy to come up with the law was Ibn Sahl in 984.)

To the best of my knowledge, Fermat was the first to claim the correct form. (It follows from his principle of least time.)

Anyway, I wouldn’t understand “the sine law” to include that bit about the speeds, especially since the first measurements of the speed of light in media like glass didn’t come until the mid-19th century (Fizeau? Foucault?), and we don’t call it the Fermat-Fizeau-Foucault law. Likewise, Kepler gets credit for his three laws, even though Newton was the first to derive them from a theory that’s still taught in physics departments.

I’m not talking about what we think is a law or not or arguing about which historical figure was the better scientist; I’m talking about what Harriot (and Descartes) would have thought was a law of nature and more generally what that concept would have meant to a 17th century natural philosopher. It seems to me that Harriot didn’t publicize his great result, the sine law of refraction, quite possibly because he didn’t see it as the great result, the “law of nature” it is to us (and would be to Descartes and Fermat), but simply a method of improving his calculations on route to understanding things like the rainbow or the burning mirror (the subject of the Goulding article).

Also, you might be interested in A Mark Smith’s monograph on Descartes’ derivation of the sine law and the principles he used to derive it. Smith at least argues Descartes’ proof was consistent with his understanding of light and of perspectivist optics. (As I read it, in very loose terms, for a lot of renaissance thinkers, influenced by neoplatonist thought about light and the perspectivists, light traveled at an infinite speed but in a limiting sense: it had no mass and in a sense was at the limit of matter. But it had other properties of matter; for instance it existed independently of the medium through which it travelled, which was not true for Aristotle. So you could treat it as matter, with finite speed, for purposes of thought experiments.)

(lol, Aristotle’s view was that light did not exist independently of the medium, also arguably true for Al Hazen et al. Aristotle himself definitely existed independently of the medium. Also, I should have said you can treat light as being very rare matter (with infinitesimal mass and very very large speed for purposes of thought experiments, in which case its speed would depend on the medium) for purposes of thought experiments.

All interesting comments. My impression is that Harriot was bad about publishing

anyof his discoveries. How come you don’t have your own website/blog? (I should talk.) Or do you?Very true. But Harriot did request for his algebra to be published posthumously and bragged up a storm to Kepler about his deep secret understanding of the rainbow in Della Porta-ish language. (Kepler then shared with Harriot his own revised view of the rainbow which is now more famous. Harriot definitely did himself no favors by being so cagey.) I’m just talking about his relative silence / modesty about the sine law which modern historians consider his piece to resistance. I’m not sure *he* did or would have seen it that way.

As for blogging, I’ve wasted an hour on here today already! If I had my own blog I’d never get anything done. I’m sure you can relate 😉

Ibn Sahl did indeed have a geometrical understanding of refraction that was equivalent to the sine law

Just to repeat what I wrote above and that which I wrote a whole blog post about earlier, Harriot quite literally published none of his scientific research and we don’t really know why. He did make arrangements to have his algebra published posthumously but the people who edited it for publication really didn’t understand what Harriot had achieved and removed a lot of his most important innovations before publication.

As to Harriot’s views on the metaphysical status of the law of refraction I’m afraid I have no idea and would have to go back to the literature on Harriot to find the answer and I’m not even sure that I would find it there.

I think you’re overly snifffy about “public good” and the difference between “a” and “the”. its a perfectly sensible and useful concept. Just for reference see https://en.wikipedia.org/wiki/Public_good though doubtless you already have.

Just to add some fuel to the fire, here are some quotes from a post over at Ether Wave Propaganda, “Scientists and the History of Science: The Shapin View”:

What I think needs to be understood is that the vision of a histrionic, ideologically entrenched scientific community clinging to unjustified social authority may well itself be a myth created by historians and sociologists in an attempt to define a more heroic role for themselves. Shapin has played an important role in generating a mythology around this myth. …

While science studies has certainly been criticized by scientists for its supposed failings, no one, to my knowledge, has seen fit to lodge the much more damaging complaint that the discipline is running a (comparatively miniature and less effective) hype machine of its own, continually boasting of the superiority of its critical tools, without those tools ever being subjected to serious independent critical analysis.

I see that Michael Barany himself contributed a couple of comments to that post.

Yup, we need an edit key. Everything between the link and the last sentence is a quote from Will Thomas’s post.

After Robert Recorde’s death, in 1558, John Dee became the leading scientist and scientific teacher in England. In 1561, and again in 1566, John Dee republished Recorde’s popular The Grounde of Artes, with minor changes. The most obvious change Dee made in 1561 was to include the following on the title page; ‘and now of late ouerseen & augmented with new & necessarie Additions. I. D.’ Unfortunately, Recorde’s clear explanations of addition and multiplication of fractions, introduced in the 1558 edition, were changed by Dee, making the explanation confusing.

Both Dee and his French friend, Petrus Ramus, were ardent believers in making practical mathematics, arithmetic in particular, available in the vernacular, for the benefit of the masses.

BTW the 1570 edition of Grounde of Artes cross promoted Billingsley’s English edition of Elements, published the same year. Dee wrote the English introduction to the first English Euclid. Not something someone who didn’t want mathematics to become widespread would have done!

Because of John Dee’s early intervention (and additions), the Grounde of Artes continued to be printed until 1699! So John Dee was,, in my opinion, a wonderful contributor towards the mathematical education of the masses.

REFERENCES:

The Early Editions of Robert Recorde’s Ground of Artes, Joy B. Easton, p. 529, Isis, Vol. 58, No. 4, 1967. NOTE The uppercase j for John commonly appeared as I.

Robert Recorde Tudor Polymath, Expositor and Practitioner of Computation, Jack Williams, p. 87, Springer-Verlag, London, 2011

Those who want to read Steven Shapin’s “classic” article on Boyle and mathematics will find it here: http://tinyurl.com/hhsy2ax. It claims Boyle spoke for the experimental community, but provides no examples of anyone agreeing with him on mathematics. Lumping Hooke and Flamsteed with Boyle, as Thony does, seems odd to me — both Hooke and Flamsteed would have been regarded as mathematicians by contemporaries (and of course definition of what constitutes mathematics have changed over time — in the seventeenth century all astronomers were “mathematicians”); and the claim that there is a long-lasting and distinctively English Baconian tradition (on which Kuhn laid emphasis in an article which really is a classic, published in 1976: “Mathematical versus experimental traditions in the development of physical science”) depends on a view of the differences between French and English science which is, I think, somewhat problematic — Kuhn, who apparently had good German, may not have had good French, as he doesn’t seem to have gasped the differences between the French and English vocabularies for discussing science.

Barany sees Boyle as critical of mathematical reasoning in the experimental sciences. Certainly he sometimes seems to be. But here are some examples that suggest a rather different view — Shapin himself quotes Boyle agreeing with Plato (or rather Galileo) that the universe is a text written in the language of mathematics, but I don’t remember these passages being quoted in Shapin’s “classic” essay:

New Experiments, 1660:

“Ever since I discern’d the usefulness of speculative Geometry to Natural Philosophy, the unhappy Distempers of my Eyes, have so far kept me from being much conversant in it, that I fear I shall need the pardon of my Mathematical Readers, for some Passages, which if I had been deeply skill’d in Geometry, I should have treated more accurately.”

Excellency of Theology, 1674:

“And I need not tell you, that since Him [Bacon], Des-Cartes, Gassendus, and others, having taken in the Application of Geometrical Theorems, for the Explication of Physical Problems; He, and They, and Other Restorers of Natural Philosophy, have brought the Experimental and Mathematical way of Inquiring into Nature into at least as high and growing an Esteem, as ever it possess’d when it was most in Vogue among the Naturalists that preceded Aristotle.”

Medicina Hydrostatica, 1690:

“I may have somewhat more cause to Apologize for this; That I have not cast a Treatise about a Subject wherein Mechanicks are so much imployed, into the Form of Propositions; and given it a more Mathematical Dress. But I was unwilling by that means to discourage those many, who, when they meet with a Book, or Writing, wherein the Titles of Theoreme, Probleme, and other Terms of Art, are conspicuously placed, use to be frighted at them; and thinking them to be written only for Mathematical Readers, despair of understanding it, and therefore lay it aside, as not meant for the use of such, as they.”

The impression I carry away from passages such as this is (1) Boyle thought mathematical arguments were often a valid way of conducting natural knowledge — see, for example, his praise of Stevin; (2) He thought mathematical arguments often put off many readers; and (3) he thought that “the Experimental and Mathematical way of Inquiring into Nature” was one single enterprise, generally conducted by the same people, even if they differed sometimes in the type of argument they chose to prefer. He did not think that mathematics and the experimental method were at odds, but that they were complementary, and this for the very simple reason that he was an advocate of the mechanical philosophy, and mechanics was everywhere and by everyone accepted to be a branch of mathematics. Here he is explaining the relationship between chemistry and the mechanical philosophy in 1674:

“to affirm, that, because the Furnaces of Chymists afford a great number of uncommon Productions and Phaenomena, there are Bodies or Operations amongst things purely Corporeal, that cannot be deriv’d from, or reconcil’d to, the comprehensive and pregnant Principles of the Mechanical Philosophy, is, as if, because there are a great number and variety of Anthems, Hymns, Pavins, Threnodies, Courants, Gavots, Branles, Sarabands, Jigs, and other (grave and sprightly) Tunes to be met with in the Books and Practises of Musitians, one should maintain, that there are in them a great many Tunes, or at least Notes, that have no dependence on the Scale of Music; or, as if, because, besides Rhombusses, Rhomboids, Trapeziums, Squares, Pentagons, Chiliagons, Myriagons, and innumerable other Polygons, Re|gular and Irregular, one should pre|sume to affirm, that there are among them some Rectilinear Figures, that are not reducible to Triangles, or have Affections that will overthrow what Euclid has taught of Triangles and Polygons.”

For Boyle, the defence of the mechanical philosophy was, quite simply, a defence of the claim that the book of nature was written in the language of mathematics; that this would increasingly be recognized to be the case he had no doubt. Boyle believed that over time a tighter and tighter connection would be established between mathematical reasoning and the experimental method. It would thus seem entirely mistaken present an account of his views which suggests he was hostile to mathematical reasoning, even though he did indeed think mathematicians were often overconfident, and sometimes almost incomprehensible.

“Michael Barany’s whole essay contrasts what he sees as two approaches to mathematics, those who see mathematics as a topic for everyone and those who view mathematics as a topic for an elitist clique.”

This introductorary disclaimer springs to mind.

“No simple generalisation will cover all cases but there can be no doubt that, in all of the contexts discussed in the following chapters, we should beware of veeering erratically between the view of a literate elite narrowly defined by the limited spread of writing skills and any unrealistic notion of a broad popular litracy in the ancient world.

A.k. Bowman & G.Woolf, Literacy and Power in the Ancient World

Literacy rather than maths but in terms of contemporary historical issues and approaches the subjects seem rather entwined.

I just want to rah-rah this discussion. Bravo, all! This is really cool.