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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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Volcanoes A Beginners Guide by Rosaly LopesVolcanoes: A Beginner’s Guide, Rosaly Lopes (Oneworld 2010)

My first introduction to volcanoes was fictional: Willard Price’s Volcano Adventure (1956), which stands out in his Adventure series because it centres on something inanimate, not on animals like lions or gorillas or elephants. This book by the NASA scientist Rosaly Lopes is factual but equally enjoyable. And some of it would fit well into Volcano Adventure anyway:

[V]olcanoes come with different sizes, shapes and temperaments. It is fascinating to study what causes these differences and understand that, while generalizations are possible, each volcano has its distinct quirks, just like people. We could also compare volcanoes to cats: with few exceptions, they spend most of their lives asleep. (ch. 1, “What are volcanoes?”, pg. 1)

When a volcano wakes, look out. They’ve slain cities, devastated eco-systems and shaped landscapes. They’re also shaped cultures. Like a thunderstorm or earthquake, an erupting volcano raises a big question in the minds of human observers: What caused something so powerful and impressive? Our explanations began with myth and moved to science. And they moved a long time ago: the ancient philosopher Anaxagoras “proposed that volcanic eruptions were caused by great winds within the Earth, blowing through narrow passages” (pg. 5) and becoming hot by friction. Two-and-a-half millennia later, scientists are plotting “silica (SiO2) content” against “alkali content” as they classify “different volcanic rocks” (ch. 2, “How volcanoes erupt”, pg. 15).

But Anaxogaras’ principles are still at work: seek the explanation in mindless mechanism, not in supernatural mind. Classification is another essential part of science. In vulcanology, the scientific study of volcanoes, magmas are classified and so are eruptions, from subdued to spectacular: Icelandic and Hawaiian are on the subdued side, Peléean, Plinian and Ultraplinian on the spectacular, with Strombolian and Vulcanian in between. Some eruptions are easy to understand and investigate. Some are difficult. Volcanoes can be as simple or complicated as their names. Compare Laki, on Iceland, with Eyjafjallajökull, also on Iceland.

Laki is an example of an eco-slayer:

Although the eruption did not kill anyone directly, its consequences were disastrous for farmland, animals and humans alike: clouds of hydrofluoric acid and sulphur dioxide compounds caused the deaths of over half of Iceland’s livestock and, ultimately, the deaths – mostly from starvation – of about 9,000 people, a third of the population. The climatic effects of the eruption were felt elsewhere in Europe; the winter of 1783-4 as noted as being particularly cold. (ch. 3, “Hawaiian and Icelandic eruptions: fire fountains and lava lakes”, pg. 31)

Lopes goes on to look at city-slayers like Mount Pelée and Vesuvius, but they can be less harmful to the environment. A spectacular eruption can be over quickly and release relatively little gas and ash into the atmosphere. And death-dealing is only half the story: volcanoes also give life, because they enrich the soil. They enrich experience too, not just with eruptions but with other phenomena associated with vulcanism: geysers, thermal springs, mudpools and so on.

And that’s just the planet Earth. Lopes also discusses the rest of the solar system, from Mercury, Venus and Mars to the moons of gas giants like Jupiter and Saturn. The rocky planets have volcanoes more or less like those on earth, but the moons of the gas giants offer an apparent paradox: cryovolcanoes, or “cold volcanoes”, which erupt ice and water, not superheated lava. On Neptune’s moon Triton, whose surface is an “extremely cold” -235ºC, cryovulcanism may even involve frozen nitrogen. The hypothesis is that under certain conditions, it’s heated by sunlight, turns into a gas and “explodes” in the “near-vacuum of Triton’s environment” (ch. 11, “The exotic volcanoes of the outer solar system”, pg. 138).

Hot or cold, big or small, on the earth or off it, volcanoes are fascinating things and this is an excellent introduction to what they do and why they do it.

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