Dec 292014

Like all children, little Kurt Gödel kept asking ‘why‘ – so much so that his parents called him Der Herr Warum, Mr Why (Goldstein, p. 54). Unlike most children, however, he was hard to satisfy with a ‘That’s the way it is’ answer. Throughout his life, he was fixated on the Principle of Sufficient Reason: everything has a cause or, as he put it, Die Welt ist vernünftig: the world is intelligible. Like Laplace, Spinoza and Leibniz, going back to the ancient Greeks through the mediaeval Scholastics, Gödel saw the principle as self-evident. There is no such thing as chance: everything that happens was destined to become, according to the ‘great laws of nature’. Chance is just a measure of mankind’s ignorance of those laws, which are far too complex for us to comprehend. But for Laplace’s demon ‘nothing would be uncertain and the future, as the past, would be present to its eyes’.

Every event has a logical explanation, not simply of why it happened, but of why it was necessarily going to happen in no other possible way than the way it did. Gödel – one of the smartest men ever on earth, alongside an equally impressive list of predecessors – found it obvious. I find it crazy. There might well be a logical explanation for it, but I can’t help seeing it as a misguided principle and the source of a most treacherous pitfall: once we are convinced that there must be a cause, we are bound to find one, irrespective of how much evidence we gather to back it up.

Gödel started early. According to his brother, at the age of eight he suffered from joint rheumatism and high fever, which he learnt could cause permanent heart damage. Since then, and throughout his life, he remained convinced, based on no evidence, that he had an injured heart (Goldstein, p. 56). I remember when I was a child my father bought a one-volume health encyclopaedia – the latest stuff from America – which he soon came to hate and laugh about, because for any symptom he looked up there always was at least one horrible, graphically illustrated cause. This is the earliest memory I have of what has become my fixation: the probability of a hypothesis given some evidence is not the same as the probability of the evidence given the hypothesis. Astonishingly, Gödel didn’t get it. Perhaps, in keeping with his Platonism, he thought that, as there is no such thing as chance, there is no such thing as probability – not a deduction that Laplace would have shared. Be as it may, things got no better as he grew older. Since there is no chance, Gödel did not believe in Darwinian evolution: “You know Stalin didn’t believe in evolution either, and he was a very intelligent man” was his jaw-dropping conversation stopper with Thomas Nagel (Goldstein p. 32). In middle age, he came to believe in ‘a vast conspiracy, apparently in place for centuries, to suppress the truth “and make men stupid”‘ (Goldstein p. 48). The same men, as Karl Menger recalled, who were responsible for destroying Leibniz’s manuscripts. “Who could have an interest in destroying Leibniz’s writings?” Menger had queried. “Naturally, those people who do not want men to become more intelligent,” was the logician’s reply (Goldstein p. 247). In the end, it all sadly turned into full-blown paranoia: to Oskar Morgenstern he ‘reported his suspicions that there were those who were trying to kill him, that his wife Adele had given away all his money, and that his doctors understood nothing of his case and were conspiring against him (Goldstein, p. 248).

One more proof that intelligence is not a one-dimensional affair. The Principle of Sufficient Reason is not just crazy: it can make one crazy, by luring him into the fabrication of baseless explanations.

In our framework, the principle can be expressed as: for any hypothesis H, there must be some evidence E such that P(H|E)=1 or 0. We call it conclusive evidence, of which there are four types. Conclusive evidence is sufficient to prove that H is certainly true or false: a Smoking Gun (positive evidence) or a Strangler Tie (negative evidence) are sufficient to prove that the hypothesis is true; a Perfect Alibi (positive evidence) or a Barking Dog (negative evidence) are sufficient to prove it false. Conclusive evidence reveals a causal relationship between the evidence and the hypothesis. Let’s take for example the first type: the gun is smoking because the suspect is guilty. That is: it is sufficient to see the effect of a Smoking Gun to conclude that a guilty suspect must be its cause – the efficient cause in Aristotle’s terminology. Notice that it is not necessary: the suspect may be guilty even without a Smoking Gun. But if there is one, he must be guilty. Likewise, assuming there is no smoke without fire, then smoke is conclusive evidence of fire. There is smoke because there is a fire. It is sufficient to see the effect of smoke to conclude that it was caused by a fire. It is not necessary: there can be a fire without smoke. But if there is smoke, there must be a fire. Finally, the ball landing on the floor is conclusive evidence that I dropped it. The ball landed on the floor because I dropped it. It is sufficient to see the effect of the ball landing on the floor to conclude that I caused it to drop. But in this case it is also necessary: the ball could not have landed on the floor unless I dropped it. Dropping the ball is not only conclusive evidence of it landing on the floor: it is perfect evidence. The ball lands on the floor if and only if I drop it.

This is the great thing about conclusive evidence: it brings certainty. As evidence accumulates multiplicatively, even a single piece of conclusive or, even better, perfect evidence allows us to resolve the evidential tug of war in one fell swoop: the hypothesis is certainly true or certainly false, irrespective of initial priors and any amount of evidence accumulated on the other side.

Hence we can see the allure of the Principle of Sufficient Reason: it says that there is conclusive evidence for any hypothesis. The evidence must be there somewhere: if we find it, even a single piece of it is enough to attain unassailable certainty. On the one hand, this is great: it spurs us into asking more and deeper questions in search for the ultimate answer. But on the other hand it is a great menace: the stronger is our desire for conclusive evidence, the higher is the risk that we dream it up. It is a common pitfall, where Gödel’s obsessions share room with Conan-Doyle’s naivety and the outright wackiness of assorted conspiracists.

Everybody likes certainty and, pace Benjamin Franklin, there are many things we are completely certain about beyond death and taxes. Also, to a greater or lesser extent, we all dislike uncertainty. Some positively hate it, some other are quite comfortable with it, and in some circumstances might even enjoy it. But, generally speaking, we all prefer certainty to uncertainty: we want to know.

Alas, very often we can’t. Most evidence is inconclusive. Not only about the future, but also about the present as well as the past. Will my child become a football champion? Is Linda a Greenpeace supporter? Was the cab in the accident green or blue? Most hypotheses are torn in a tug of war between confirmative and disconfirmative evidence, where neither side can prevail. When certainty is unattainable, we do not know if something is true or false: we can only believe it is probably true, and therefore probably false.

Such state of uncertainty goes well beyond our convenient analytical examples. Uncertainty is abundant and pervasive in most matters, from the trifling to the weightiest. Even if the Principle of Sufficient Reason were true, only Laplace’s demon, who can see perfect and timeless evidence, would be certain about them. The rest of us need to come to terms with our ignorance and, following in Laplace’s footsteps, acknowledge uncertainty and deal with it.

We can refuse the challenge, take comfort in the Principle of Sufficient Reason, resolve that there must be a cause and proceed to make it up. It is amazing what even very intelligent people can regard as self-evident. Less blatantly, we can pick and choose the evidence that best fits our dispositions. Or we can accept uncertainty, gather all the evidence that we can see, properly balance it and try our best to come up with well-calibrated probabilities.

Contrary to a common misconception, being comfortable with uncertainty is the very ethos of science. Science is not the repository of incontrovertible truths, “scientifically proven” on the basis of conclusive evidence. As Frank Hahn wrote on the front page of a book of his that I had asked him to sign in my student days: ‘You must regard everything in this book as provisional and not as “science”‘. An unforgettable piece of advice to a young graduate eager to find the truth.

One day every human being will realise that everything that has ever been written in any book has been written by us. The evidence is conclusive.

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