Dr. Who

Who doesn’t know Doctor Who, the 60 year old BBC sci-fi franchise? But the Doctors’ universe  seems very different from ours. For one thing, it supports something called a TARDIS. For the  uninitiated, a TARDIS is an old fashioned phone booth (or ‘box’) with one very unusual feature: the volume of its interior is much, much larger than the volume of its exterior. 

According to Heidegger, a work of art shows us something new about our world. On the basis of  that definition, Doctor Who is museum grade.  

When we were in school, we complained about having to learn arithmetic. Now, later in life,  we’re glad we learned at least the basics. Whether we know it or not, we use arithmetic every  day in a million different ways. Whenever we take a trip, bake a cake, budget our income, save  for retirement, we rely on arithmetic. Thank God we live in a universe where it works. 

But does it? And do we? 2500 years ago, the paradoxical Zeno proved that it doesn’t  and/or that we don’t. He used a hypothetical foot race between the fastest man in the world,  (no, not Adrian Bolt, but Achilles) and one of nature’s slowest creatures, a lowly tortoise. Of  course, we all know that in our world, Achilles would have beaten the reptile with ease. 

However, Zeno uses the rules of arithmetic to prove that Achilles can’t win (as long as the  tortoise has a head start). Zeno’s proof does not invalidate Achilles’ still unbroken track and field  records, nor does it fit the tortoise with supernatural jets. Instead, Zeno ‘simply’ proves that the  laws of arithmetic do not adequately model our actual world. 

Zeno’s paradoxes have had a rocky history. Turns out, human beings don’t like to be told they’re  wrong. We’ll do just about anything to escape that obvious conclusion. Newton and Leibniz  went so far as to invent the Calculus, the scourge of high school seniors and college freshmen  everywhere, just to prove Zeno wrong. 

Bertrand Russell resolved the dilemma: calculus notwithstanding, Zeno’s paradox holds. Why?  Because spacetime is discrete, not continuous as arithmetic assumes. Of course, our world for  the most part acts as though it were continuous, even though it isn’t, and so arithmetic works,  even though it doesn’t…for the most part. 

Zeno notwithstanding, calculus took us the Moon and back; but that doesn’t change the fact that arithmetic is a hack! That’s 6 years of primary school education down the drain! 

Somewhere between arithmetic and calculus most of us were introduced to Geometry. We  learned a system that was developed 2400 years ago by a Greek mathematician named Euclid.  Just as real number arithmetic only works in a continuous universe, Euclidean geometry only  works in a flat universe. Trouble is, the world we live in is neither continuous nor flat. 

Of course, we can always just pretend. We’re very good at that you know. We can pretend that  the universe is flat just as we pretend that it is continuous. In reality, the universe is  fundamentally discrete…and curved. Continuity and flatness are approximations that work in a  wide range of circumstances; but they don’t accurately represent the real world. 

It was not until 1800 CE that scholars realized that we need a geometry that describes curved  surfaces and spaces; if you don’t believe me, try steering your starship in the neighborhood of a  black hole and see how far you get with just a Triple A Trip-tick. And so ‘non-Euclidean’  geometry was born. 

But even non-Euclidean geometries make certain questionable assumptions about the nature of  the real world. As in the case of arithmetic, the issue has to do with continuity and order. Turns  out, even non-Euclidean geometries may not go far enough to capture the enigmatic nature of  reality. We need a geometry that takes us even further from Euclid’s world. 

Both arithmetic and Euclidean geometry rely on the set of real numbers, augmented by  imaginary and complex numbers. But the set of real numbers may not be enough. Between any  two real numbers, there are an infinite number of hyperreal infinitesimals. Adding hyperreals to  the domain results in a new species of ‘continuity and order’ that is very different from what is  suggested by real numbers alone. Mathematicians call geometries that incorporate this expanded definition of ‘number’, non-Archimedean.  

In standard geometries, Euclidean or non, certain quantitative and spatial relations hold and are  conserved. In non-Archimedean geometries, not so much! In fact, these geometries have three  highly counter-intuitive features in common: (1) all triangles are either isosceles or equilateral;  (2) any point in a sphere can be treated as its center; and (3) any two spheres are either disjoint or they contain one another entirely. There is no such thing as tangency or overlap.  

These principles of non-Archimedean geometry work together to produce an astonishing result:  If a Sphere A contains a Sphere B, the volume of B may be greater than the volume of A. That is  the principle behind TARDIS whose interior volume exceeds its exterior volume…by a lot. And  we have the creators of Doctor Who to thank for introducing us Muggles to this revelation. 

The contingency (vs. necessity) of Archimedean Space was brought home to me recently when I  watched an episode of Doctor Who with a particularly precocious pre-tween. I hazarded a  comment, “Isn’t it odd the way the inside of the box is bigger than the outside?” My question  was unexpectedly met with a blank stare, followed by: “Why is that odd?” 

Bingo! So Euclidean geometry is something we learn, not something we intuit. Here I was,  thinking I was about to teach my young friend something about geometry, only to have him school me. (What else is new!) At my age, I have apparently lost the ability to navigate through  several different configurations of space simultaneously. Fortunately, my much younger  companion has not. Bravo…and thanks!

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Marcel Duchamp. The Bride Stripped Bare by Her Bachelors, Even (The Large Glass), 1915–1923. Oil, varnish, lead foil, lead wire, and dust on two glass panels. 109.25 inches by 69.25 inches. Philadelphia Museum of Art, Philadelphia.