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Posts Tagged ‘Thomas Aquinas’

Francis Walker’s Aphids, John P. Doncaster (British Museum 1961)

Is this a candidate for Russell Ash’s and Brian Lake’s classic collectors’ guide Bizarre Books (1985)? Yes, I’d say so. It’s not as outré or eccentric as Who’s Who in Barbed Wire (“Containing ‘Names and addresses of active barbed wire collectors’”) or Walled Up Nuns and Nuns Walled In (“With Twenty Illustrations”), but few books are. I’ve certainly never seen a book about aphidology before.

I didn’t even know the word existed. Do aphids deserve a discipline of their own? I’ll let Thomas Aquinas answer that:

[C]ognitio nostra est adeo debilis quod nullus philosophus potuit unquam perfecte investigare naturam unius muscæ: unde legitur, quod unus philosophus fuit triginta annis in solitudine, ut cognosceret naturam apis. – Expositio in Symbolum Apostolorum (1273).

Our understanding is so weak that no philosopher can understand the nature of a single fly; whence it is read, that one philosopher was thirty years in the wilderness, that he might understand the nature of the bee.

For apis read aphis. The philosophus in this case may have begun his obsession like this:

Francis Walker seems first to have turned his attention to the study of aphids in the autumn of 1846 when he observed them swarming and ovipositing on furze. In the summer and autumn of the following year he made copious and systematic collections of such species as he could find in the neighbourhood of his home in Southgate, at that time a country town a few miles north of London. (“Walker’s Aphid Studies”, pg. 1)

Walker was employed as an entomologist at the British Museum and this book is an attempt to analyse what he collected and named. It’s very detailed and might seem very dry. But there’s a lot of food for the historic imagination in descriptions like this:

Aphis particeps Walker = Myzus persicae (Sulzer)

1848 Zoologist, 6, 2217.

1852 List Homopt. Ins. Brit. Mus., 4, 1011.

Collected with four other species from Cynoglossum officinale near Fleetwood, Lancashire, in October, and described as follows:

The wingless viviparous female. The body is pale brown, small, oval, shining, and rather flat; the antennae are pale yellow and longer than the body; the rostrum is pale yellow; its tip and the eyes are black: the tubes are pale yellow and rather more than one-fourth of the length of the body; the legs are pale yellow; the tips of the tarsi are black. (pg. 103)

Cynoglossum officinale is a purple-flowered, sand-growing wildflower whose common name is hound’s-tongue. The officinale of its specific name is a reference to its use in herbal medicine. In Anglo-Saxon times and the Middle Ages, herbalists or magicians would have been picking its leaves; in the nineteenth century, a scientist called Francis Walker was picking aphids off it.

There’s a vignette like that with many of the other descriptions, as Walker simultaneously collects aphids and moments of his own life. I think he must have been an odd and obsessive man, but he had colleagues, even although aphidology can never have been a crowded profession. The description for “Aphis bufo Walker = Iziphya bufo (Walker)” notes that this species was

Found in the beginning of October by the sea-shore near Fleetwood [Lancashire] on Lycopsis arvensis, the small bugloss; also by Mr. Hardy near Newcastle on Carex arenaria, sand reed, and by Mr. Haliday near Belfast. (pg. 37)

Were Walker, Hardy and Haliday rivals as much as colleagues? I like the idea of obsessive aphidologists racing each other to find and record new species. Francis Walker could have been a character in a story by Arthur Conan Doyle or H.G. Wells. Ernest Rutherford is said to have divided science into two branches: physics and stamp-collecting. That’s unfair, but aphidology and other branches of entomology and natural history are like subtler and stranger forms of stamp-collecting.

The similarities were stronger in Victorian times, before biology began to merge with chemistry and mathematics. Indeed, Walker began his collecting well before Darwin published The Origin of Species (1859) and perhaps he didn’t like the new science. The preface to this book notes that “Walker’s name has come to be a by-word among insect taxonomists for his inaccuracy and superficiality”, but praises him for making a “significant and important advance in aphidological knowledge” and says that his “catalogues and lists formed the nucleus [of] the vast collections of today”.

“Today” was 1961, but this is a very neat and well-printed book in a solid green binding. I hope Francis Walker would have been pleased by it and by the thought that he’s inspired someone in the twenty-first century to look at aphids with new interest and wonder.

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Infinitesimal by Alexander AmirInfinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander (Oneworld 2014)

Infinitesimal is an entertaining read on a fascinating topic: the pioneers of a new form of mathematics and those who opposed them. Amir Alexander claims that “the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms unknown in human history” (Introduction, pg. 14).

It’s an extraordinary claim and I don’t think he manages to provide extraordinary proof for it. In fact, he probably gets cause-and-effect reversed. Is it likelier that new mathematics opened minds, dynamized science and transformed politics or that open minds created new forms of mathematics, science and politics? I’d suggest that support for the new mathematics was a symptom, not a cause, of a new psychology. But Alexander makes a good case for his thesis and there is no doubt that the world was changed by the willingness of mathematicians to use infinitesimals. Calculus was one result, after all. The book begins in Italy and ends in England, because the pioneers lost in Italy:

For nearly two centuries, Italy had been home to perhaps the liveliest mathematical community in Europe. … But when the Jesuits triumphed over the advocates of the infinitely small, this brilliant tradition died a quick death. With Angeli silenced, and Viviani and Ricci keeping their mathematical views to themselves, there was no mathematician left in Italy to carry on the torch. The Jesuits, now in charge, insisted on adhering close to the methods of antiquity, so that the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, England, France and Switzerland. (ch. 5, “The Battle of the Mathematicians”, pg. 178)

Why were the Jesuits involved in an esoteric mathematical dispute? You might say that de minimis curat Loyola – Ignatius Loyola (1491-1556), founder of the Jesuits, cared about anything, no matter how small, that might undermine the authority of the Church. In the view of his successors, the doctrine of indivisibles did precisely that: “in its simplest form, the doctrine states that every line is composed of a string of points, or ‘indivisibles’, which are the line’s building blocks, and which cannot themselves be divided” (Introduction, pg. 9).

Indivisibles must be infinitesimally small, or they wouldn’t be indivisible, but then how does an infinitesimal point differ from nothing at all? And if it isn’t nothing, why can’t it be divided? These paradoxes were familiar to the ancient Greeks, which is why they rejected infinitesimals and laid the foundations of mathematics on what seemed to them to be solider ground. In the fourth century before Christ, Euclid used axioms and rigorous logic to create a mathematical temple for the ages. He proved things about infinity, like the inexhaustibility of the primes, but he didn’t use infinitesimals. When Archimedes broke with Greek tradition and used infinitesimals to make new discoveries, “he went back and proved every one of them by conventional geometrical means, avoiding any use of the infinitely small” (Introduction, pg. 11).

So even Archimedes regarded them as dubious. Aristotle rejected them altogether and Aristotle became the most important pre-Christian influence on Thomas Aquinas and Catholic philosophy. Accordingly, when mathematicians began to look at infinitesimals again, the strictest Catholics opposed the new development. Revolutionaries like Galileo were opposed by reactionaries like Urban VIII.

But the story is complicated: Urban had been friendly to Galileo until “the publication of Galileo’s Dialogue on the Copernican system and some unfavourable political developments” (pg. 301). So I don’t think the mathematics was driving events in the way that Alexander suggests. Copernicus didn’t use them and the implications of his heliocentrism were much more obvious to many more people than the implications of infinitesimals could ever have been. That’s why Copernicus was frightened of publishing his ideas and why Galileo faced the Inquisition for his astronomy, not his mathematics.

But Amir’s thesis makes an even more interesting story: the tiniest possible things had the largest possible consequences, creating a new world of science, politics and art. In Italy, two of the chief antagonists were Galileo and Urban; in England, two were the mathematician John Wallis (1616-1703) and the philosopher Thomas Hobbes (1588-1679). Alexander discusses Wallis and Hobbes in Part II of the book, “Leviathan and the Infinitesimal”. Hobbes thought that de minimis curat rex – “the king cares about tiny things”. Unless authority was absolute and the foundations of knowledge certain, life would be “nasty, brutish and short”.

However, there was a big problem with his reasoning: he thought he’d achieved certainty when he hadn’t. Hobbes repeatedly claimed to have solved the ancient problem of the “quadrature of the circle” – that is, creating a square equal in size to a given circle using only a compass and an unmarked ruler. Wallis demolished his claims, made Hobbes look foolish, and strengthened the case for religious toleration and political freedom. But I don’t think this new liberalism depended on new mathematics. Instead, both were products of a new psychology. Genetics will shed more light on the Jesuits and their opponents than polemics and geometry textbooks from the period. Alexander’s theory is fun but flawed.

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