Physicist: The best way to think about it is; there is a speed (C) that is the fastest speed and, by the way, light goes that fast. There’s nothing special about light, it’s just a useful way of describing C (“the speed of light”). Photons are just another podunk massless particle, whipping around the universe as fast as fast can be.
Historically, the derivation of the strange properties of C (relativity) relies on a pretty straight forward piece of Einsteinian logic, based in part on an understanding of light.
1) All the laws of physics work the same, whether you’re moving or not. There is no experiment that can tell you whether or not you’re moving.
2) Light is an electromagnetic wave, and the velocity of these waves can be derived from Maxwell’s laws.
3) Maxwell’s laws, like all physical laws, are independent of how fast you’re moving. So the speed of light must also be independent of how fast you’re moving.
4) So, there exists a speed (the speed that light travels at) that is the same to everyone, no matter how fast they themselves are moving. Holy crap! There’s your special relativity!
So when you see equations like 
So why is C the fastest speed? A good way to think of it is to first ask; how do you know when you’re moving faster than something else? If you’re driving down the highway and you’re moving faster than the car in front of you, then eventually you’ll pass that car. However, C is the same to everyone, no matter what. So, say a photon goes past you, and you try to catch up. But no matter how much you speed up, the photon will always be moving away at the speed of light. You can never catch up (or even come close to starting to catch up). So, regardless of perspective, the photon is always moving faster than you.
Some of this may seem seem contradictory, but surprisingly, it’s all self consistent. Very surprisingly.
A related question: Why is C a finite value, i.e. why is it not infinite?
Or maybe another form of the question: is C just a fundamental constant the value of which cannot be explained, or can it be derived from other more fundamental constants/principles?
OK you can say according to Maxwell c = sqrt(1/e0µ0), but this then just extends the question to e0 and µ0
Pretty much. There are many physical constants, and not as many equations relating them, so you’ll always end up with at least a few (19) variables that just are what they are. Your example above is about the best example. Of the magnetic permittivity of space, electric permittivity of space, and the speed of light, which is fundamental? There’s no way to tell for sure, but I cast my vote for C, and the electric permittivity.
Magnetism can be derived from electric forces and relativity, but we could have it backwards, or it may not matter to the universe at all. In practice we have these constants measured, so we just use what we have, and don’t stress terribly on what came first (after all, you can eat eggs and chicken without too much trouble).
Thanks! Also for the subsequent post
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Hi. This isn’t exactly related to the post, but regardless, I’m perplexed. What happened in this experiment that the scientists claim they’ve sent light faster than c? Is there no “universal speed limit”?
http://www.cbsnews.com/stories/2000/07/19/tech/main216905.shtml
This is pretty tricky, but it comes down to a distinction between “phase velocity” and “group velocity”. Without going into detail (a long post) the experiment essentially requires a “standing wave” with waves at many wavelengths. These waves interact in such a way that they create peaks, like when you pluck two slightly-out-of-tune guitar strings and you hear a pulsing sound.
These peaks move around in the standing wave region, and you can more or less keep track of where they are. In the middle of the region is a specially constructed cloud with extremely weird optical properties, that changes how fast light of various frequencies move through it (slowing only, but to various degrees).
By changing the different frequencies you change how the peaks move around (making them faster than light in this case), however nothing is actually moving faster than light, just a non-information-carrying effect.
Another example of this sort of thing is the “scissor paradox”. When you close a pair of scissors the point where the blades intersect is moving much faster than either blade, and if you were to construct a pair of scissors large enough then the point of intersection would move faster than light, but it wouldn’t be able to carry any kind of information (in part) because you’d already see the slower-than-light blades moving long before the intersection point got to you.
This pulse thing is a similar “paradox”.
Hmm… kinda crazy. And cool. Thank you!
Update! The long reply above has been expanded into a post:
http://www.askamathematician.com/?p=4037
Ok, this might sound a bit quirky, but how did Einstein really did get to know or derive or arrive at the conclusion that the fastest achievable speed is the light speed ? (1)
(2)Well, he had a little doubt that what would happen if he were to travel next to a photon at its own speed. Let us consider that I can travel at the speed of light, even if relativity doesn’t permit me to. Then travelling next to a photon at its own speed, will I be able to see the oscillating electric and magnetic vibrations?
There were a series of experiments, most notably Michelson-Morley, that demonstrated that the speed of light seems to be the same regardless of how the experimental apparatus is moving. In the case of Michelson-Morley, they waited until the Earth was in different parts of its orbit (and thus moving fast in different directions), and even did the experiment on a boat so they could move around at will.
The result was ultimately inescapable: the speed of light is the same regardless of how you’re moving. As a result, as you speed up, you’ll find that light is still passing by at the speed of light. You can never even get close!