Well, if mathematics is "God's copyright notice" on the world, then an equation that "comes out right" might very well be pointing to something real going on the the world, but real in a way that cannot be directly observed.
Professor Hill's reconciliation of the problem of a particle (e.g., a neutrino) with a faster-than-light velocity in terms of Special Relativity theory "rested on ignoring the speed of light’s status as an absolute limit, and instead, us[ed] the information where the relative velocity of two observers is infinite."
Once cannot imagine, from the standpoint of ordinary experience, what an "infinite velocity" is. Indeed, "infinite" is a mathematical term that is not ordinarily constructible in physics.
But then, the square root of the number –1 is also not constructible in physical terms. And yet this value, an imaginary number — i — is an extraordinarily useful number in the understanding and mathematical modeling of the real world.
All square roots of negative numbers are "imaginary." As Leonard Euler put it, "...of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible." [But there they are.]
According to George Gamow,
The family of imaginary numbers represents, so to speak, a fictitious mirror image of the ordinary or real numbers, and, exactly in the same way as one can produce all real numbers starting with the basic number 1, one can also build up all imaginary numbers from the basic imaginary unit [the square root of –1]....My point in getting into this description of imaginary numbers is that it seems possible to me that, like i, "infinite velocity" is also a purely "imaginary" term — and one that might prove highly useful in extending Einstein's special relativity theory, just as imaginary numbers extended and supplemented number theory.
...each ordinary real number has its imaginary double. One can also combine real and imaginary numbers to make single expressions.... Such hybrid forms are usually known as complex numbers.
For well after two centuries after imaginary numbers broke into the domain of mathematics they remained enveloped by a veil of mystery and incredibility until finally they were given a simple geometrical interpretation by two amateur mathematicians: a Norwegian surveyor by the name of Wessel and a Parisian bookkeeper, Robert Argand.
According to their interpretation a complex number, as for example 3+4i, may be represented [as showing that] 3 corresponds to the horizontal distance, and 4 to the vertical, or ordinate.
Indeed, all ordinary real numbers (positive or negative) may be represented as corresponding to the points on the horizontal axis [of a Cartesian plane], whereas all purely imaginary ones are represented by the points on the vertical axis. When we multiply a real number, say 3, representing a point on the horizontal axis, by the imaginary unit i we obtain the purely imaginary number 3i, which must be plotted on the vertical axis. Hence, the multiplication by i is geometrically equivalent to a counterclockwise rotation by a right angle.
It has become customary to regard the speed of light as the absolute "speed limit" of the universe. But how do we really know this is a fact? We're speeding right along with it, in our own relativistic frames of observation.
In a situation like that, it seems to me mathematics is your friend — for it points beyond what can be ordinarily perceived and directly experienced.
I'm looking forward to seeing how Professor Jim Hill and Dr. Barry Cox manage to integrate a faster-than-light particle into Einstein's theory (assuming they're able to do that, of course).
“My point in getting into this description of imaginary numbers is that it seems possible to me that, like i, “infinite velocity” is also a purely “imaginary” term”
Indeed, if there are more than 4 dimensions, then traveling in the direction of a higher dimensional axis could lead to instantaneous motion across distances in the lower dimensions, looking like infinite velocity. Yet, we wouldn’t have to accelerate; just make a turn in a direction that to an observer, would be “imaginary”.
(Think about the relationship of this discussion to our discussions on "God's 'Universal Now'")...
["Catch-up" FReepmail in the works...]
I'm looking forward to seeing how Professor Jim Hill and Dr. Barry Cox manage to integrate a faster-than-light particle into Einstein's theory (assuming they're able to do that, of course).
Here's another example to think about, namely knot theory and the unreasonable effectiveness of mathematics in the natural sciences.