In which the author probably reveals his ignorance about quantum mechanics,
A little knowledge is a dangerous thing
So it turns out, in Kaluza-Klein theory, it's quite easy to unify electromagnetics and gravity relativistically. And the math is beautiful. So beautiful as to be quite convincing, except for one detail. The theory needs five-dimensional space-time. Where's the extra dimension? Did it go AWOL on a binge in the local bar? I can't really point to a fourth spatial dimension sitting here in my dining room, and, as one physicist put it, he "can't imagine what a universe with two temporal dimensions would be like."
Not only that, but the string theorists have expanded the number of dimensions to ten, or eleven. Didn't I once hear twenty-four? Not only are string-theoretical calculations impossibly difficult for the current generation of mathematicians, but the string theorists have also got to explain away the extra dimensions.
Now maybe the way out is to figure out how multidimensional time could manifest itself.
When you perform a measurement, you can have multiple outcomes. Quantum mechanics teaches us that the outcome is indeterminate. Each different outcome can create a different future. Each different future would have its own single arrow of time with its own series of future events. And from the point of view of someone who has not yet performed the measurement, all of these arrows of time are possible. How many time dimensions do we need to accomodate all these arrows?
In Feynman diagrams, the arrow of time is irrelevant. You have an initial state, a final state, and the particles perform all possible dances getting from the initial state to the final state. What you get to observe is a kind of superposition, a kind of statistical composite (but with complex probabilities) of all the ways of getting from one to the other. How do you fit all those alternatives into one time dimension? Do we even need to? The mathematics is just as happy figuring the probabilities of the initial state given the final state as the other way around.
Perhaps the extra dimensions of Kaluza-Klein and string theories are temporal, and provide room for all these futures.
Perhaps the limitations to two, or seven, or eight, or twenty-one time dimensions aren't even the real constraint on dimensions, but just the minimum number needed for the mathematics.
Foe example, if we take a two-dimensional curved surface, it's cleanly embeddable (locally, at least) in a three-dimensional Euclidean space, faithfully preserving the intrinsic curvature. There's nothing stopping it from being embedded faithfully in a higher dimensional space, but from the viewpoint of the math, three is all that one can derive from the surface geometry.
And I seem to remember hearing of a theorem that if you start with a three (or was it four) dimensional curved hypersurface, it was locally embeddable in a ten-dimensional space.
Perhaps the restriction of, say, ten dimensions is akin to this -- it may only the number of dimensions that actually make a difference, locally.
Perhaps there are an infinite number of these temporal dimensions (whatever "there are" means in sentences like this). But if that is so, we'd have to consider an infinite number of spatial dimensions, too.