It's possible that Euclidian geometry is merely a local approximation, and one of those other geometries is the one that best describes the natural universe.

It wouldn't be "one of those other geometries" that I've read about, anyway. They are not compatible with Euclidean geometry. Perhaps they are local approximations in other parts of the natural universe. The geometry of which these are all local approximations would have to be something much more complex than any of them.

I don't know about that. So far Euclidean geometry has explained the increasingly large portion of the universe that we've been able to observe fairly well.

What troubles me is some of this wacky new cosmology. The outer edge of the visible universe is farther away than those stars could have traveled at lightspeed since the Big Bang. So, to explain this the cosmologists tell us that

*the empty space in between the stars is expanding.* Uh, "space" is another word for "nothing," so it sounds to me like they're saying "nothing is expanding," which is a rather meaningless statement and hardly explains the problem. Or if it isn't empty, and it's full of photons and gravitons and other subatomic particles that are the plumbing infrastructure of the universe, what happens to those particles, that plumbing, when the "space" they occupy is stretched? If they retain their original size, then the space between them is expanding in a really complex way. But then, so is the space between the stars, if the stars themselves are not expanding proportionally.

I think this theory has a long way to go. Perhaps it requires a trans-Euclidean geometry to really make sense.

In mathematics there are real numbers which can be used to talk about real things, and imaginary numbers which don't have a counterpart in the real world. The imaginary numbers make it possible to solve problems that cannot be solved just by using real numbers. How can imaginary numbers inform us about situations in the real world that real numbers cannot? It's certainly a good philosophical question.

No, it's a scientific question. Real numbers measure distances and other physical attributes of the universe. Imaginary numbers (and wouldn't every scientist like to shoot Descartes for being so famous that his dismissive name for a concept he initially rejected stuck, so we have to constantly explain it) are intermediate values that appear in calculations whose end purpose is to predict those measurements.

In electrical circuit theory, for example, voltage, current, resistance, inductance, frequency, and every other dimension of a circuit is measured in real numbers. There's no such thing as a device with an impedance of 6+

*j* ohms. (Electrical engineers use

*j* for the square root of -1 since

*i* already stands for current.) But the calculations for the way complex circuits work are full of

*j*'s, they just all resolve into real numbers by the time you get to the end of the calculation. They're just intermediate values with no analog in the natural universe and no analog is necessary.

It's like infinity. Infinity is a "number" we have to have because it's what we get when we divide something by zero, and it serves a purpose in many calculations, e.g. "infinite" series. But it doesn't exactly represent anything in the natural universe.

But what if our imaginary numbers are their real numbers, and vice versa?

The reason nobody has actually shot Descartes for naming them "imaginary" (besides the fact that the dude is already dead) is that the term is not entirely inappropriate. He of course called them that because he didn't believe they would prove necessary in the mathematics of his future. But in a context like this it's useful to remember that "imaginary" numbers are not representations of

*any measurements* in the natural universe.

We may wind up making wholesale revisions to relativity in order to explain the expanding universe, discovering an amazingly complicated relationship between gravity and electromagnetism, or finding that particle physics and uncertainty fall neatly in place when the Next Big Thing After String Theory comes along... but none of these steps forward in science will change the fact that the dimensions of the "real" universe are measured in "real" numbers.

Just look at the way complex numbers are graphed: there's a real axis and an imaginary access. They are not dimensions because they have two dimensions by definition!

Remember that mathematics is a tool of science, not a science itself, so all the rules of science don't apply to it. Mathematical theories are derived from pure abstractions rather than empirical observation. Therefore unlike scientific theories they are

*proven true.* No subsequent observation in the natural universe will prove them false.