Is time nothing but the speed of light (or the light itself)? With regard to relativistic effects on time, all the examples and explanations revolve around light and its speed. Especially in explanatory situations that explain this using photon clock, it seems that when the clock moves near the speed of light, the photon has to take a long path to reach the other mirror.
So all of these explanation are driving me to conclude or believe that the time is nothing but the speed of light (or the light itself) and all this special relativistic effects are built around the assumptions that only light (or its speed) are superior in this universe.


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*What if there is another thing (probably invisible so we never found it?) which travels faster than light? Would we need to change all our laws of relativity?

*Coming back to the photon clock example, who decided that only photons traveling between two mirrors can be used to define a tick (I mean one second)? Would we replace the photon with some other thing if it is found in the future that the thing actually moves faster that light?
More questions:
- Is it possible to understand this with out using any Math? 


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*It appears to me that constancy of $c$ is a result of our Math. So universal properties (like the speed of light never changes) are decided using Math? What if an Alien Math equation differs from ours theory?


I think I'm not able to understand a simple key between time and its relation between speed of light (and probably accelerating bodies with respect to other bodies). 


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*And in the problem of twin paradox, the person who travels near speed of light and comes back to Earth would actually see his friend aged more than him. So how does the speed/acceleration would slows down his aging process?


I would be very thankful if you can explain this with out using any Math. Even Einstein must started with a thought process before using Math to solve/prove this problem.
 A: Special relativity is often introduced to students using light clocks because this is a reasonably accessible way to understand that phenomena like time dilation and length contraction must occur. However you should not be mislead into thinking that we use light clocks to define special relativity. The fundamental principle of special relativity (and in fact general relativity too) is that the geometry of spacetime is defined by a metric (in SR this is the Minkowski metric) that gives us an invariant line element:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{1} $$
The answers to the recent question What is the intuition behind the Lorentz factor from Special Relativity explore this issue.
The fact that the speed of light is a constant, and the maximum speed possible, is derived from the fact that $c$ in equation (1) above is a constant.
A: Time is that which is measured by clocks. How clocks behave when they are being moved relative to each other is simply a collection of experimental facts. At no time do we need any light to do those experiments, and it totally doesn't matter to moved clocks if we are moving them during the day or the night. 
So what, exactly is your question? Is it why Einstein talked a lot about light in his famous paper "Ueber die Elektrodynamik bewegter Koerper"? Einstein clearly says that in the preface of the paper itself:
"...Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies."
The paper is basically an elaboration on how measurements that were well established at Einstein's time can be understood in a modified framework of dynamics. He uses light signals to "enlighten" the reader about the new framework, in which the speed of light is a constant, but that is methodologically not necessary and is not done that way in modern ways of teaching relativity. One can completely exploit Einstein's postulates, which are simply often measured physical facts, without ever mentioning light. 
