Posts Tagged ‘mathematicians’

The Invention of Science by David WoottonThe Invention of Science: A New History of the Scientific Revolution, David Wootton (Allen Lane 2015)

I picked up this book expecting to start reading, then get bored, start skimming for interesting bits, and sooner or later give up. I didn’t. I read steadily from beginning to end, feeling educated, enlightened and even enthralled. This is intellectual history at nearly its best, as David Wootton sets out to prove what is, for some, a controversial thesis: that “Modern science was invented between 1572, when Tycho Brahe saw a new star, and 1704, when Newton published his Opticks” (introduction, pg. 1).

He does this in a clever and compelling way: by looking at the language used in science across Europe. If there was indeed a scientific revolution and science was indeed a new phenomenon, we should expect to see this reflected in language. Were old words given new meanings? Did new words and phrases appear for previously inexpressible concepts? They were and they did. “Scientist” itself is a new word, replacing earlier and less suitable words like “naturalist”, “physiologist”, “physician” and “virtuoso”. The word “science” is an example of an old word given a new meaning. In Latin, scientia meant “knowledge” or “field of learning”, from the verb scire, “to know”.

But it didn’t mean a systematic collective attempt to investigate and understand natural phenomena using experiments, hypotheses and sense-enhancing, evidence-gathering instruments. Science in that sense was something new, Wootton claims. He assembles a formidable array of texts and references to back his thesis, which is part of why this book is so enjoyable to read. As Wootton points out, the “Scientific Revolution has become almost invisible simply because it has been so astonishingly successful.” Quotations like this, from the English writer Joseph Glanvill, make it visible again:

And I doubt not but posterity will find many things, that are now but Rumors, verified into practical Realities. It may be some Ages hence, a voyage to the Southern unknown Tracts, yea possibly the Moon, will not be more strange then one to America. To them, that come after us, it may be as ordinary to buy a pair of wings to fly into remotest Regions; as now a pair of Boots to ride a Journey. And to conferr at the distance of the Indies by Sympathetick conveyances, may be as usual to future times, as to us in a litterary correspondence. (The Vanity of Dogmatizing, 1661)

Glanvill’s prescience is remarkable and he’s clearly writing in an age of pre-science or proto-science. He wasn’t just a powerful thinker, but a powerful writer too. So was Galileo and Wootton, who has written a biography of the great Italian, conveys his genius very clearly in The Invention of Science. You can feel some of the exhilaration of the intellectual adventure Galileo and other early scientists embarked on. They were like buccaneers sailing out from Aristotle’s Mediterranean into the huge Atlantic, with a new world before them.

Wootton also emphasizes the importance of Galileo’s original speciality:

The Scientific Revolution was, first and foremost, a revolt by the mathematicians against the authority of the philosophers. The philosophers controlled the university curriculum (as a university teacher, Galileo never taught anything but Ptolemaic astronomy), but the mathematicians had the patronage of princes and merchants, of soldiers and sailors. They won that patronage because they offered new applications of mathematics to the world. (Part 2, “Seeing is Believing”, ch. 5, “The Mathematization of the World”, pg. 209)

But there’s something unexpected in this part of the book: he describes “double-entry bookkeeping” as part of that mathematical revolt: “the process of abstraction it teaches is an essential precondition for the new science” (pg. 164).

He also has very interesting things to say about the influence of legal tradition on the development of science:

Just as facts moved out of the courtroom and into the laboratory, so evidence made the same move at around the same time; and, as part of the same process of constructing a new type of knowledge, morality moved from theology into the sciences. When it comes to evidence, the new science was not inventing new concepts, but re-cycling existing ones. (Part 3, “Making Knowledge”, ch. 11, “Evidence and Judgment”, pg. 412)

Science was something new, but it wasn’t an ideology ex nihilo. That isn’t possible for mere mortals and Wootton is very good at explaining what was adapted, what was overturned and what was lost. Chapter 13 is, appropriately enough, devoted to “The Disenchantment of the World”; the next chapter describes how “Knowledge is Power”. That’s in Part 3, “Birth of the Modern”, and Wootton wants this to be a modern book, rather than a post-modern one. He believes in objective reality and that science makes genuine discoveries about that reality.

But he fails to take account of some modern scientific discoveries. The Invention of Science is a work of history, sociology, philology, and philosophy. It doesn’t discuss human biology or the possibility that one of the essential preconditions of science was genetic. Modern science arose in a particular place, north-western Europe, at a particular time. Why? The Invention of Science doesn’t, in the deepest sense, address that question. It doesn’t talk about intelligence and psychology or the genetics that underlie them. It’s a work of history, not of bio-history or historical genetics.

In 2016, that isn’t a great failing. History of science hasn’t yet been revolutionized by science. But I would like to see the thesis of this book re-visited in the light of books like Gregory Clark’s A Farewell to Alms (2007), which argues that the Industrial Revolution in England had to be preceded by a eugenic revolution in which the intelligent and prudent outbred the stupid and feckless. The Invention of Science makes it clear that Galileo was both a genius and an intellectual adventurer. But why were there so many others like him in north-western Europe?

I hope that historians of science will soon be addressing that question using genetics and evolutionary theory. David Wootton can’t be criticized for not doing so here, because bio-history is very new and still controversial. And he may believe, like many of the post-modernists whom he criticizes, in the psychic unity of mankind. The Invention of Science has other and less excusable flaws, however. One of them is obvious even before you open its pages. Like Dame Edna Everage’s bridesmaid Madge Allsop, it is dressed in beige. The hardback I read does not have an inviting front cover and Wootton could surely have found something equally relevant, but more interesting and colourful.

After opening the book, you may find another flaw. Wootton’s prose is not painful, but it isn’t as graceful or pleasant to read as it could have been. This is both a pity and a puzzle, because he is very well-read in more languages than one: “We take facts so much for granted that it comes as a shock to learn that they are a modern invention. There is no word in classical Greek or Latin for a fact, and no way of translating the sentences above from the OED [Oxford English Dictionary] into those languages.” (Part 3, “Facts”, pg. 254)

He certainly knows what good prose looks like, because he quotes a lot of it. But his own lacks the kind of vigour and wit you can see in the words of, say, Walter Charleton:

[I]t hath been affirmed by many of the Ancients, and questioned by very few of the Moderns, that a Drum bottomed with a Woolfs skin, and headed with a Sheeps, will yeeld scarce any sound at all; nay more, that a Wolfs skin will in short time prey upon and consume a Sheeps skin, if they be layed neer together. And against this we need no other Defense than a downright appeal to Experience, whether both those Traditions deserve not to be listed among Popular Errors; and as well the Promoters, as Authors of them to be exiled the society of Philosophers: these as Traitors to truth by the plotting of manifest falsehoods; those as Ideots, for beleiving and admiring such fopperies, as smell of nothing but the Fable; and lye open to the contradiction of an easy and cheap Experiment. (Physiologia Epicuro-Gassendo-Charltoniana, 1654)

The Invention of Science is also too long: its message often rambles home rather than rams. If Wootton suffers from cacoethes scribendi, an insatiable itch to write, then I feel an itch to edit what he wrote. It’s good to pick up a solid book on a solid subject; it would be even better if everything in the book deserved to be there.

But if the book weren’t so good in some ways, I wouldn’t be complaining that it was less than good in others. In fact, I wouldn’t have finished it at all and I wouldn’t be heartily recommending it to anyone interested in science, history or linguistics. But I did and I am. The Invention of Science is an important book and an enjoyable read. I learned a lot from it and look forward to reading it again.


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Numericon by Marianne Freiberger and Rachel ThomasNumericon, Marianne Freiberger and Rachel Thomas (Quercus Editions 2014)

An easy-to-read introduction to some profound and difficult ideas. You can start climbing the foothills of Mount Maths here and train for the icy cliffs and avalanche-scoured slopes above. And perhaps the mountain rises for ever. There may be important mathematical questions that are impossible to answer. But some answers we already have, like the irrationality of √2 and the infinitude of the primes, are astonishing to contemplate. How are finite creatures able to discover facts about infinity? Logic is a lever that can move more than the universe. Euclid described 2,300 years ago how it could be used to prove that primes never ended: for every integer, there is a prime.

Numericon explains his proof, then goes on to describe how the German Georg Cantor (1845-1918) used logic to prove that there is more than one kind of infinity. In fact, there’s an infinite hierarchy of infinities, climbing endlessly into the metaphysical empyrean. That seems like an insane idea and Cantor did end his life in a sanatorium. He’s an extreme example of something Freiberger and Thomas note in their introduction: that mathematicians are often eccentric. It’s a strange subject that doesn’t come naturally to human beings. We can’t escape it, because our brains and bodies are governed by mathematics, but we don’t need to apply it explicitly and consciously to get through life. Those who lift the surface of reality and gaze upon its brilliant mathematical core can find that the light troubles and even subverts their brains.

And some avert their gaze. The German Carl Friedrich Gauss (1777-1855) may have been the greatest mathematician who ever lived, but he declined to publish his revolutionary ideas about Euclid’s fifth axiom: that parallel lines never meet. New geometries arise if the axiom is eliminated, but Gauss didn’t claim the credit for discovering them. As this book describes, it was the younger mathematicians János Bolyai (1802-60) and the Russian Nikolai Lobachevsky (1792-1865). And Bolyai’s mathematician father had tried to warn his son off.

Mathematicians need a sense of adventure. They need recognition and support too, but sometimes even the great ones find both hard to win. Sometimes their greatness is part of the problem, because the work they produce is too new and powerful to be properly appreciated. The Norwegian Niels Henrik Abel (1802-29) died at 26 “from tuberculosis and in abject poverty” (pg. 183). The Frenchman Évariste Galois (1811-32) died even younger, at 21, fatally wounded in a duel. Perhaps it was an affair de cœur, perhaps a plot by his enemies: Galois was revolutionary in both his mathematics and his politics. The Indian Srinivasa Ramanujan (1887-1920) is another great who died young after making discoveries of permanent value.

This book discusses all three and many more, but mathematicians aren’t truly important. Mathematics is necessary; mathematicians are contingent. If advanced civilizations exist on other planets or in parallel universes, their art and literature might be entirely different from our own or might not exist at all. But their mathematics would be recognizable, whatever the nature of the brains that had discovered it. Advanced civilization is impossible to imagine without mathematics and although we can ignore mathematics if we choose, we’re missing something central to existence if we do. As Darwin said: it seems to give one an extra sense. It’s a mind’s eye that can see into infinity – and into infinitesimality.

As though to reflect the importance of mind in maths, this book has relatively few and simple illustrations. Maths can yield gorgeous complexity for real eyes, but the mind’s eye is more important. So this book is eye-candy for the mind. Eye-brandy too: maths can both dazzle you and make you drunk. It’s appropriate that Numericon echoes Necronomicon, because new worlds wait within the pages of both.

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Front cover of A Mathematician's Apology by G.H. HardyA Mathematician’s Apology, G.H. Hardy (1940)

The World Wide Web is also the Random Reading Reticulation – the biggest library that ever existed. Obscure texts and ancient manuscripts are now a mouse-click away. A Mathematician’s Apology is neither obscure nor ancient, but it wasn’t easy to get hold of before it became available online. I’ve wanted to read it for a long time. And now I have.

Alas, I was disappointed. G.H. Hardy (1877-1947) was a very good mathematician, but he’s not a very good writer about mathematics. And in fact, he didn’t want to be a writer at all, good, bad or indifferent:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. (Op. cit., Section 1)

But is philosophy of mathematics work for second-rate minds? At the highest level, I don’t think it is. The relation of mathematics to reality, and vice versa, is a profoundly interesting and important topic, but Hardy doesn’t have anything new or illuminating to say about it:

It may be that modern physics fits best into some framework of idealistic philosophy — I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way. (Section 24)

He experienced and explored that mathematical reality, but he can’t communicate the excitement or importance of doing so very well. I wasn’t surprised by his confession that, as a boy: “I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively” (sec. 29). He says it wasn’t until he had begun his degree at Cambridge that he “learnt for the first time … what mathematics really meant” (ibid.).

In this, he was very different from someone he helped make much more famous than he now is: an unknown and struggling Indian mathematician called Srinivasa Ramanujan, who sparked Hardy’s interest by sending him theorems of startling originality and depth before the First World War. Hardy brought Ramanujan to England, but barely mentions his protégé here. All the same, his respect and even perhaps his affection are still apparent:

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one thing you could never have done, and that is to have collaborated with both [J.E.] Littlewood and Ramanujan on something like equal terms.’ (sec. 29)

Very few people could have done that: a mere handful of the many millions who lived at the time. So it would be wrong to expect that Hardy could both ascend to the highest peaks of mathematics and write well about what he experienced there. He couldn’t and A Mathematician’s Apology supplies the proof. That’s a shame, but the text is short and still worth reading. Hardy had no false modesty, but he had no delusions of grandeur either:

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created is undeniable: the question is about its value.

It was valuable: that is why Hardy is still remembered and celebrated, sixty-seven years after his death. He is also still famous as an atheist, but you could say that he spent his life in the service of Our Lady – Mathematica Magistra Mundi, Mathematics Mistress of the World.

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