The excellent new book by Mervyn King, former governor of the Bank of England, is inevitably noticed mainly for its views on banking regulation and the outlook for the euro zone. For me the most important message of The End of Alchemy is its emphasis on radical uncertainty – or, to quote Donald Rumsfeld, former US defence secretary: "The things we do not know we do not know."

That emphasis reflects the parallel intellectual paths Lord King and I have taken since we were young dons 40 years ago.

In a book published in 1976, economist Milton Friedman disparaged a tradition that "drew a sharp distinction between risk, as referring to events subject to a known or knowable probability distribution, and uncertainty, as referring to events for which it was not possible to specify numerical probabilities".

Probability

Friedman went on: “I have not referred to this distinction because I do not believe it is valid. We may treat people as if they assigned numerical probabilities to every conceivable event.” Asked, “Who will win the war?”, Churchill might have responded, “Britain, with probability 0.7”; and Hitler with a similar answer but perhaps different number.

However absurd, this is what we were taught and what we passed on to the next generation of students. It seemed an exciting time for young turks in finance; insider trading in an old-boy network was to be superseded by a new generation of quants and rocket scientists. We had the mathematical tools to revolutionise investment banking. Our theory came to underpin the risk models used in financial institutions and imposed by regulators.

Statistics

But Friedman was wrong. There really are limits to the range of problems susceptible to the mathematics of classical statistics. He was, erroneously, rejecting the concept of radical uncertainty described 50 years earlier by the economists John Maynard Keynes and Frank Knight.

“By uncertain knowledge,” wrote Keynes in 1921, “I do not mean merely to distinguish what is known for certain from what is only probable. The sense in which I am using the term is that in which the prospect of a European war is uncertain . . . There is no scientific basis to form any calculable probability whatever. We simply do not know.”

While the long-term future of interest rates or copper prices, about which Keynes also speculated, might be approached probabilistically, questions about the social system 50 years hence are too open-ended, and the outcomes too varied and insufficiently specific, to be described in probabilistic terms.

A recent book on superforecasters, co-written by Philip Tetlock, illustrates the point well. By trying to turn multi-faceted questions into ones precise enough to enable those who proffer answers to be assessed for their accuracy, he makes the questions narrow and uninteresting: "How will the Syrian war develop" and "How will Europe manage its refugee crisis?" become: "How many Syrian refugees will land in Europe in 2016?"

More fundamentally there are things we do not know because we cannot imagine them. If you had described your smartphone to Friedman in 1976 he would not have understood what you were talking about, far less been able to speculate intelligently on the probability it would be invented or bought. These are the "black swans" Nassim Taleb has described.

‘Black swans’

The reader who once asked me which black swans were most likely to materialise in the next five years could not have missed the point more comprehensively.

There is a world of difference between low-probability events drawn from the tail of a known statistical distribution and extreme events that happen but had not previously been imagined. And it is usually the latter that give rise to crises – and opportunities.

– Copyright The Financial Times Limited 2016