The seen and the unseen: abstraction and the senses

I listened to three short talks today and found that they had something nice in common – they each show us how sensory experience (often vision) gives rise to mathematics that provides access to what cannot be seen, and clarifies what is seen.

The first of these talks was called Symmetry, reality’s riddle presented by Marcus du Sautoy.

It begins with a description of Galois’ famous death in a duel. Galois stayed up the whole night before the duel writing letters and outlining his mathematical ideas.  Marcus du Sautoy explains how Galois had found a new language, the language of symmetry, and goes on to take note of the ubiquitous presence of symmetry, showing it to us in, among other things, molecular structure, particle physics and art.  Du Sautoy explains how Galois saw that it is not just the symmetries we see in an object that characterize the object, but also the way the symmetries of an object interact with each other. With this insight, he develops a language that can identify the substance of the things unseen.  The symmetry that underlies a physical object is captured by a number.  And du Sautoy makes the point that this language, expressed in number, now makes it possible for him to create symmetrical objects in high dimensional spaces.  This is the mathematical idea of group theory.

The second was a talk given by David Deutsch: A new way to explain explanation. While I didn’t exactly enjoy his critique of pre-scientific ideas, he made some clarifying points about empiricism and the scientific method. He observes that science is originally distinguished by the conviction that all knowledge is derived from the senses.  While, he says, this helped by promoting observation and experiment, “it was obvious that there was something horribly wrong with it.”

Empiricism is inadequate because, well, scientific theories explain the seen in terms of the unseen.  And the unseen, you have to admit, doesn’t come to us through the senses.  We don’t see the origin of species.  We don’t see the curvature of space-time, and other universes.  But we know about those things.  How?

What we see, in all these cases, bears no resemblance to the reality that we conclude is responsible – only a long chain of theoretical reasoning and interpretation connects them

Deutsch makes the argument that testability is only valuable when the theory to be tested is “hard to vary, because every detail plays a functional role.”  By way of an example, he identifies all of the functional relationships in the modern explanation of the seasons – “that surfaces tilted away from radiant heat are heated less, that a spinning sphere, in space, points in a constant direction, that tilt also explains the sun’s angle of elevation at different times of year, and that it predicts that the seasons will be out of phase in the two hemispheres.”

Finally, it was Benoit Mandelbrot’s talk: Fractals and the art of roughness that I most enjoyed.

Mandelbrot also found a number that characterized an observation, a number that denoted the roughness of a surface.  In an engaging informal style he just tells us stories.

Humanity had to learn about measuring roughness…..Very few things are very smooth…What’s the length of the coastline, which seems to be so natural because it’s given in many cases, is, in fact a complete fallacy; there’s no such thing. You must do it differently.

What good it that, to know these things?  Well, surprisingly enough, it’s good in many ways…Now a lung is something very strange…The volume of a lung is very small, but what about the area of a lung?…Anatomists were arguing very much about that. Some say that a normal male’s lung has an area of the inside of a basketball court. And others say, no, five basketball courts…The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical consideration: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, or a man and or a little rodent….And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all of these branching systems, for which there is no geometry.   So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry.

Mandelbrot very gently makes clear that the objects conjured up by 19th century mathematicians, functions that have been called monstrous and used to demonstrate the break between mathematics and visible reality, are now used to describe some aspects of nature’s complexity. He shows us the appearance of a “fractal-to-be” in an 18th century Japanese painting and notes the intuitive grasp of fractals in the work of engineer Gustave Eiffel and in the tower that bears his name.  He tells us that halfway through his career he decided to test himself.  “Could I just look at something which everybody had been looking at for a long time and find something dramatically new?”   So he looked at “these things called Brownian motion,”

Then I was telling my assistant, “I don’t see anything.  Can you paint it?”  So he painted it, which means he put inside everything.  “Well, this thing came out…,” he said. And I said, “Stop! Stop! Stop! I see; it’s an island.”

Mandelbrot could see that this island had a fractal dimension but it was his friends who, 20 years later, won the Fields medal when they proved it.

Here, again, a sensory idea (roughness) is characterized by a number which takes some things away (like the length of a coastline) and brings other things into view (like a refined understanding of the lungs).

This talk is very pleasant to listen to and I recommend it if you haven’t seen it. Mandelbrot gives a sketchy description of the Mandebrot set but successfully describes the simplicity of its starting point. And he concludes with these very nice words:

Bottomless wonders spring from simple rules, which are repeated without end.


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