In Djursholm, an elegant suburb of Stockholm, is the headquarters of the Mittag-Leffler Institute. The mansion was built in the 1890s by the Swedish mathematician Gösta Mittag-Leffler (1846–1927), whose marriage to the wealthy heiress Signe af Lindfors had provided the necessary means to allow himself and his family a refined residence.
In 1916, the couple donated the property, including its excellent library, to the Swedish Academy of Sciences, so that a mathematics institute could be built there. The donation included a good amount of money, but it was devalued during the 1st World War, which made the project unfeasible. The institute would only be created in 1969, becoming a pole of attraction for mathematicians from all over the world.
On the ground floor of the institute is a solid fireplace in lead-gray granite. Engraved in the stone is an inscription in ancient writing that a Swedish friend deciphered for me with some difficulty: “The mind reaches no further than the word.” An elegant way of saying that what we can’t explain to others we don’t really know.
It’s an important idea for someone like me who trains students to discover, understand, and communicate mathematical ideas. That’s why I often repeat the Mittag-Leffler phrase to my students. Way of saying that if your thesis is not well written, it’s because you still don’t understand the subject properly.
But not everyone agrees. In the book “Tacit Dimension”, published in 1966, the British philosopher of Hungarian origin Michael Polanyi (1891–1976) pointed out that human knowledge of the world and of ourselves is, to a large extent, beyond our capacity for expression. “We know more than we can say,” he said.
Knowing how to drive is much more than following the basic instructions (release the handbrake, etc) that we received from the driving school instructor: if it weren’t so, it would be enough to listen. But this much more, which we acquire by doing, we are not able to describe. Recognizing a face, playing chess or speaking a foreign language are other examples of things we know how to do but are unable to express how we do it.
At the time, the “Polanyi paradox” was seen as a deep blow to the idea of artificial intelligence: if we don’t know how to explain how we perform such tasks, how can we make machines do them? I will continue next week.
