Bayesland

(Sorry — a few complicated months)

Speaking of best books I have ever read, a more recent one is David Wootton’s The Invention of Science.

Wootton does a marvellous job explaining mankind’s transition from a worldview based on authority to one based on evidence.

As reprised in Steven Pinker’s latest book (p. 9), a typical well-educated Englishman in 1600 believed in demons, witches, werewolves, magicians, alchemy, astrology and other nonsense (p. 6). But a mere century and a quarter later his whole perspective had changed:

Between 1600 and 1733 (or so — the process was more advanced in England than elsewhere) the intellectual world of the educated elite changed more rapidly than at any other time in previous history, and perhaps than at any time before the twentieth century. Magic was replaced by science, myth by fact, the philosophy and science of ancient Greece by something that is still recognizably our philosophy and our science, with the result that my account of an imaginary person in 1600 is automatically couched in terms of ‘belief’, while I speak of such a person in 1733 in terms of ‘knowledge’. (p. 11–12).

Commonly referred to as the ‘Scientific Revolution’, this transition is not easy to understand. The images we have in mind are of sinister cardinals persecuting Galileo and of barmy philosophers refusing to look into his telescope. In the same vein, Wootton quotes Joseph Glanvill, an early advocate of the revolution, who derided the view that telescopes and microscopes were

all deceitful and fallacious. Which Answer minds me of the good Woman, who when her Husband urged in an occasion of difference, I saw it, and shall I not believe my own Eyes? replied briskly, Will you believe your own Eyes, before your own dear Wife? (p. 74, Italics and bold in the original).

(I find this particularly funny, wondering about how essentially the same joke found its way down to Richard Pryor, through Groucho Marx’s Duck Soup. An equivalent joke my friend Peter told me many years ago is that of the English aristocrat, which I used here).

Obviously, such hilarious caricatures leave much to explain. Educated people in 1600 and earlier were no dimwits. So why did they hold what to our eyes seem such outrageously weird beliefs? This is a focal theme in the Bayes blog. Hence I was intrigued to find out that Wootton’s book is centred on the same key concepts.

Following Aristotle, a seventeenth century educated person was taught to think deductively: draw necessary conclusions from undisputable premises. It would be a mistake, however, to imply that he ignored evidence. As we have seen, there is no such thing as a priori knowledge, independent of evidence. Knowledge cannot but be based on some form of evidence — empirical, as it is plain to our eyes; or axiomatic, as it was common before the Scientific Revolution, all the way back to ancient Greece and beyond. Episteme was absolute, irrefutable, self-evident knowledge. And even the wackiest myths and legends of primordial peoples were not haphazard fantasies but elaborations of authoritative evidence, perhaps in the form of dreams by elderly sages and wise men, who interpreted them as divine revelations they were called upon to proclaim and propagate.

Aristotelian principles were self-evident truths. Such as: All bodies move towards their natural place. Therefore, as stars rotate around it and every object falls towards its core, the earth must be the centre of the universe. Or: Heavier objects fall faster than lighter ones. Therefore, a two-kilo bag of sugar falls faster than a one-kilo bag (Wootton, p. 70). Or: Hard substances are denser and heavier than soft substances. Therefore, ice is heavier than water (p. 71).

These are what we call extreme priors: beliefs that are seen as so obviously self-evident that it is considered pointless to test them through menial experimentation (p. 319). As obviously, however, they are — they cannot but be — the product of evidence. I see stars rotate around the earth and objects fall towards its core: therefore, I infer that all bodies move towards their natural place. I see that a two-kilo bag of sugar falls faster than a one-kilo bag: therefore, I infer that heavier objects fall faster than lighter ones. I see that ice is heavier than water: therefore, I infer that hard substances are heavier than soft ones. The evidence is all wrong, hence the inferences are wrong. But how do I know that? Remember: the closer our priors are to the extreme boundaries of Faith, the stronger must be the evidence required to change them. And, as with Glanvill’s husband, little it matters if the evidence is right in front of our eyes. It is plain to see, for instance, that ice floats on water and, as Archimedes — whose writings had been translated in Latin since the twelfth century — had found out in 250 BCE, this is only possible if ice is lighter than the water it displaces. But hey, who is a mere mathematician compared to the supreme father of natural philosophy? Aristotle had figured out that hard substances are heavier. So there must be another reason why ice floats. Well, it is because of its shape: flat objects cannot penetrate water and therefore remain on the surface. Galileo would patiently prove this was nonsense (p. 315), but philosophers remained unimpressed. In the same vein, when Galileo asked his philosopher friend and colleague Cremonini to look at the mountains on the moon through his telescope, Cremonini refused, not because he was a blockhead — far from it: he was a highly respected professor of natural philosophy for sixty years and earned twice as much as Galileo — but because he did not trust the evidence: he did not regard it as strong enough to dent his Aristotelian belief that the moon was a perfect, unblemished sphere.

The idea that Aristotle had it all figured out and that all ‘natural philosophy’ logically descended from his principles was at the core of the seventeenth century’s worldview. As Wootton puts it (reprising Borges), Shakespeare had no real sense of progress. He treated his characters in the Roman plays as if they were his contemporaries. ‘History did not exist for him’ (p. 5). The governing assumption was that, as in Ecclesiastes (1:9), there was ‘nothing new under the sun’ (p. 63). The event that triggered a seismic change in this view and initiated the Scientific Revolution was the discovery of America at the end of the fifteenth century. That’s where Wootton places what he expressively calls ‘the discovery of discovery’ (Chapter 3). There is arguably no better way to convey this concept than through Hamlet’s immortal words to Horatio, which Wootton does not quote, probably because they are so well-known and overused — although he hints at them in the title of Part One. So I will do it for him: ‘There are more things in heaven and earth, Horatio, Than are dreamt in your philosophy’ (Act I, Scene V).

The discovery of the New World showed mankind that in fact there was plenty new under the sun (including black swans, although for those we had to wait until the end of the seventeenth century) and gave rise to an explosive search for new evidence, which continues unabated, in fact accelerating, to our days. Over the following two centuries, curiosity — which theologians, reigning supreme above philosophers in the hierarchy on medieval science, regarded as a sin — became the mighty fuel of progress that it still is.

From their perspective, theologians were right: as long as knowledge is anchored to the two extreme boundaries of Faith, it remains impervious to evidence. Episteme above Doxa, truth above opinion, knowledge above experience, demonstration above persuasion. The discovery of discovery changed all that: it instilled in the minds of educated people ‘the idea that experience isn’t simply useful because it can teach you things that other people already know: experience can actually teach you that what other people know is wrong. It is experience in this sense — experience as the path to discovery — that was scarcely recognized before the discovery of America’ (p. 81).

This is the true sense of experience: exposure to the peril of being wrong. As curiosity compelled people to leave the secure shores of Aristotelian self-evidence, it encouraged them to embrace Cromwell’s rule, which we might as well rename Glanvill’s rule: Believe Your Own Eyes. This was no blanket surrender to evidence at face value. People remained wary — as we are — that evidence can be deceitful. But they opened their mind to the possibility that, in the right amount and shape, it might be capable of changing and even overturning their prior beliefs. Like Cremonini, they still suspected — and rightly so — that eyes can lie. But, unlike him, they gave them a chance: they were ready to answer Popper’s question.

This was the task that natural philosophers — as they were commonly known until the nineteenth century, when William Whewell coined the term ‘scientist’ (p. 28) — set out to accomplish: accumulate enough evidence to prove hypotheses true or false. They did so through carefully crafted experiments, which — precisely because they were well aware of the fallibility of evidence — they persistently reproduced, shared and challenged, provando e riprovando (p. 300), with the ultimate goal of devising the experimentum crucis (p. 381) which, by yielding conclusive evidence (p. 194), could allow them to proclaim a consensual winner of the evidential tug of war. Thus Truth, until then the preserve of infallible self-evident axioms, became a destination, to be travelled to through fallible empirical evidence. Prior Faith became posterior Certainty.

Reverend Thomas Bayes was born in the midst of this journey and lived through it a quiet and secluded life. He was by no means a protagonist of the Scientific Revolution — so much so that he doesn’t even earn a mention in Wootton’s book. Yet he was very much a man of his time, and his theorem encapsulates so well the ethos of the revolution that we can surely call the journey’s destination ‘Bayesland’.

(Wootton does mention Laplace’s dictum, attributing it to The Logic of Port-Royal, which ‘had acknowledged that the more unlikely an event the stronger the evidence in favour of it would have to be in order to ensure that it was more unlikely that the evidence should be false than that the event should not have occurred’ (p. 465)).

Bayesland is where we live and where we have always lived — Archimedes and Aristotle, Galileo and Cremonini, Shakespeare and Groucho Marx, you and I and all living creatures. We learn by experience, updating our beliefs through a multiplicative accumulation of evidence. We all are and have always been Bayesian.

This has been the Scientific Revolution’s greatest achievement: show mankind that the way we have always learnt in practice was also valid in theory. Progress started when we stopped wasting time thinking we were doing something else. The effect of such a seemingly simple conceptual clarification has been breathtaking:

Of course, it was far from simple — as Wootton brilliantly shows. His book is a pleasure to read from beginning to end, including his thick jungle of notes. I warmly recommend it.

Originally published at Bayes.