November 2001

# The Math of Stuff

What are we, and everything around us, made of? Since the dawn of modern mathematics, humans have tried to use its methods to help answer this question.

The ancients believed that the world was made up of four basic "elements": earth, water, air, and fire. Around 350 BC, the ancient Greek philosopher Plato, in his book Timaeus, theorized that these four elements were all aggregates of tiny solids (in modern parlance, atoms). He went on to argue that, as the basic building blocks of all matter, these four elements must have perfect geometric form, namely the shapes of the five "regular solids" that so enamoured the Greek mathematicians -- the perfectly symmetrical tetrahedron, cube, octahedron, icosahedron, and dodecahedron.

As the lightest and sharpest of the elements, said Plato, fire must be a tetrahedron. Being the most stable of the elements, earth must consist of cubes. Water, because it is the most mobile and fluid, has to be an icosahedron, the regular solid that rolls most easily. As to air, he observed, somewhat mysteriously, that "... air is to water as water is to earth," and concluded, even more mysteriously, that air must therefore be an octahedron. Finally, so as not to leave out the one remaining regular solid, he proposed that the dodecahedron represented the shape of the entire universe.

To modern eyes, it is hard to believe that an intellectual giant such as Plato could have proposed such a whimsical theory. What on earth led him to believe that the geometer's regular solids could possibly underlie the structure of the universe? In fact, although the particulars of his theory can easily be dismissed as whimsy, the philosophical assumptions behind it are exactly the same as those that drive present day science. Namely, that the universe is constructed in an ordered fashion that can be understood using mathematics. To Plato, as to many others, as Creator of the universe, God must surely have been a geometer. Or, as the great Italian scientist Galileo Galilei wrote in the seventeenth century, "In order to understand the universe, you must know the language in which it is written. And that language is mathematics."

Believing that the world was constructed according to mathematical principles, Plato simply took the most impressive, most perfect piece of mathematics known at the time. That was the proof (found in Euclid's classic mathematics text Elements) that there are exactly five regular solids -- solid objects for each of which all the faces are identical, equal-angled polygons that meet at equal angles.

As recently as the seventeenth century, the famous astronomer Johannes Kepler, who discovered the mathematical formula that describes the motion of the planets around the Sun in our Solar System, was likewise seduced by the mathematical elegance of the regular solids. There were six known planets in Kepler's time, Mercury, Venus, Earth, Mars, Jupiter, and Saturn, and a few years previously Copernicus had proposed that they all rotated in circular orbits with the Sun at the center. (Kepler would later suggest, correctly, that the orbits were not circles but ellipses.) Starting from Copernicus's suggestion, Kepler developed a theory to explain why there were exactly six planets, and why they were at the particular distances from the sun that he and other astronomers had recently measured. There were precisely six planets, he reasoned, because between each adjacent pair of orbits (think of the orbit as a circle going round a spherical ball in space) it must be possible to fit, snuggly, an imaginary regular solid, with each solid used exactly once. After some experimentation, he managed to find an arrangement of nested regular solids and spheres that worked: the outer sphere (on which Saturn moves) contains an inscribed cube, and on that cube is inscribed in turn the sphere for the orbit of Jupiter. In that sphere is inscribed a tetrahedron; and Mars moves on that figure's inscribed sphere. The dodecahedron inscribed in the Mars-orbit sphere has the Earth-orbit sphere as its inscribed sphere, in which the inscribed icosahedron has the Venus-orbit sphere inscribed. Finally, the octahedron inscribed in the Venus-orbit sphere has itself an inscribed sphere, on which the orbit of Mercury lies.

Of course, Kepler's theory was completely wrong. For one thing, the nested spheres and the planetary orbits did not fit together particularly accurately. (Having himself been largely responsible for producing accurate data on the planetary orbits, Kepler was certainly aware of the discrepancies, and tried to adjust his model by taking the spheres to be of different thicknesses, though without giving any reason why the thicknesses should differ.) But besides, as we now know, there are not six planets but eight (nine if you want to count Pluto, these days officially classified as not being a planet).

For all that Plato's theory of matter and Kepler's theory of the Solar System were incorrect, however, let me repeat my earlier point that the same underlying philosophy underlies all of present-day scientific theorizing about the universe: that the universe operates according to mathematical laws. Or, as the contemporary physicist Stephen Hawking has remarked, in developing mathematical theories about the nature and origin of the universe, we are seeking to know "the mind of God."

Until the early part of the twentieth century, our best modern theory of matter was the atomic theory, which viewed everything as being made up of atoms, miniature "solar systems" in which a number of electrons (the planets) orbited a central nucleus (the sun). The atomic theory was supported by experimental evidence and the mathematical details had been worked out with considerable accuracy (using some pretty sophisticated mathematics). Clearly, to physicists of the time, God must indeed have seemed to be a geometer.

But then, in the early 1900s, scientists observed phenomena that did not fit the neat geometric picture of the "solar system atom," eventually forcing them to accept that atomic theory had reached its limits and would have to be abandoned (or drastically modified). What they found to replace it was a far more complicated mathematical explanation known as quantum theory. At the heart of quantum theory was the assumption that there is a built-in uncertainty about matter. If you were to focus attention on a single particle, such as an electron, you would find that it was not in any one fixed and definite place at any moment, but was constantly flitting around in an unpredictable fashion that could only be described mathematical using probability theory. Although quantum theory is nowadays widely accepted, in the early days its dependence on probability theory led Albert Einstein to dismiss it with the remark that "God does not play dice with the universe."

Even if you accept it, however, quantum theory forces you to rely on purely mathematical descriptions of reality. For instance, the human mind simply cannot grasp, on an intuitive level, quantum theoretic entities that behave both like particles and waves. As the late Richard Feynman, one of the leading pioneers in the development of modern quantum theory, once remarked: "There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. ... After people read Einstein's paper a lot of people understood the theory of relativity in one way or other, certainly more than twelve. On the other hand I think I can safely say that nobody understands quantum mechanics." [Richard Feynman, The Character of Physical Law, Cambridge, MA: MIT Press, 1965, p.129.]

When it comes to quantum mechanics, physicists have to abandon their intuitions and rely on the mathematics to tell them what's going on. Mathematics began as a system to help us to understand the world and to add precision to our understanding. With the arrival of quantum theory in the twentieth century, mathematics became our only way to understand. Today, the quantum theoretic lens has been focused on matter at an ever finer scale than electrons, to reveal that everything in the world consists ultimately of tiny folds and ripples in space-time (the study of which requires still more new mathematics). This development has led the writer and broadcaster Margaret Wertheim to quip, "These days, God isn't a geometer, he does origami."

What will come next, if anything, is hard to say. The original urge is still there: to understand what the stuff is that we and our world are made of. Moreover, we still have the belief that the answer will be found using mathematics. And that means that we can expect to see the development of new mathematics to support our quest, as we continue to push against the limits of our knowledge.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.