Why isn't time just a function of the speed of light being finite Time is generally referred to as the fourth dimension and seems frequently to be treated as an equivalent to the 3 spatial dimensions. But then people seem to get tied in knots as to why time has an arrow, i.e. it progresses in one direction only but we can move freely in the 3 spatial demotions.
The thing I can't get my head around is why time is anything more than a function of the speed of light, i.e. time is defined by c rather than c being defined as distance/time. Consider a photon leaving the sun, because it cannot travel to an observer instantly  there is a "propagation of change" across space at the rate c.
It would great if someone could explain why time is a dimension and not just a function of the "speed" of light?
 A: Time is a physical quantity which cannot be defined, just like mass and length. They happen to be the fundamentals of our knowledge regarding understanding of nature. Nobody in this world can define time.
Moreover, c ,that is, the speed of light in vacuum is used to define the unit of time second.
This is because speed of light in vacuum has been observed to remain the same over centuries and is immune to change under all circumstances. So second is defined with the help of c.
We use time as a parameter to analyse various physical phenomena like motion of macroscopic and microscopic objects, radioactivity, occurrence and duration of an event etc. When we calculate the time period of a second's pendulum, or the time of existence of subatomic particles viewed through a particle accelerator, we do not use c to calculate it, rather pico and femtoseconds are used.
Time hence is justifiably a dimension only as the notion of time came long,long back, nobody knows when. It was only after that that man had the concept of distance, displacement,speed and velocity and proceeded to calculate the speed of light.
*c=3*10^8 m/s and 1 sec=time taken by light to travel 3*10^8 metres*.
But time is surely not a function of the speed of light.  
A: Newtonian mechanics, with no upper limit on velocities, is perfectly consistent and has no problem talking about time.  
A: Time is defined as the change we experience. Which is things being different in two different places in time. Which is annoyingly circular.
Why can't we freely move back and forth in time like spatial dimensions? Because then it wouldn't conform to our idea of time. It's perfectly plausible though, to have another kind of being who perceives one of our spatial dimensions as time, while perceiving our temporal dimension as space. In order to live in our universe, it would be a pretty weird physiology, but it's not inherently impossible. There could be universes where two beings intersect at some spacetime. One being's past is the other being's left and for just an instant they are the same physical thing while perceiving their universe in totally different ways.
If light moved at infinite speeds, there would still be time (well, there could be, if all the other changes to our universe necessary to allow infinite light speed didn't break time somehow). It would just be like a video game: instead of seeing distance galaxies as they were billions of years ago, we would see the light from their current positions. But there would still be a limited number of photons, so it wouldn't be infinite light densities. Also, other objects would still be hindered by their mass, so they would still require time to change or move.
At the end of the day, "time" is just a word to describe the way we perceive the universe. It doesn't mean it's the only possible perception, or that our perception is necessarily true in any absolute sense (if absolutes are even possible). Similarly, the "time is a fourth dimension" model may not be the only valid model. But it works very well at describing our universe, so it's not all wrong either.
A: 
Why isn't time just a function of the speed of light being finite

In a way it is. If the speed of light was infinite, everything would happen at once. And it doesn't. But more generally I think it's better to say time is a function of motion. The mechanism of a clock is called a movement. A clock doesn't literally measure the flow of time like it's some kind of cosmic gas meter. A clock "clocks up" some kind of regular cyclical motion and displays a cumulative result called the time. 

Time is generally referred to as the fourth dimension 

Yes. Like Jim said you need the four coordinates to specify an event. 

and seems frequently to be treated as an equivalent to the 3 spatial dimensions.

It isn't the equivalent. There's a minus sign in the t term in the spacetime interval. 

But then people seem to get tied in knots as to why time has an arrow, i.e. it progresses in one direction only but we can move freely in the 3 spatial demotions.

They do get tied up in knots with this. IMHO the point they don't appreciate is that the time dimension is a dimension in the sense of measure, not in the sense of freedom of motion. I can hop forward a metre but you can't hop forward a second. 

The thing I can't get my head around is why time is anything more than a function of the speed of light, i.e. time is defined by c rather than c being defined as distance/time. 

In a way it is. The "coordinate" speed of light varies with gravitational potential, and optical clocks go slower when they're lower. Then we talk about gravitational time dilation. But these clocks don't go slower because time goes slower. There is no actual time flowing inside these clocks. This book about Gödel and Einstein is worth reading. It's heavy going at times, but I liked it. Note though that the blurb is misleading. Time exists like heat exists, and a hundred years will kill you just as surely as a hundred degrees C. 

Consider a photon leaving the sun, because it cannot travel to an observer instantly there is a "propagation of change" across space at the rate c.

Yes, we say it takes 8 minutes. But think about what we're really doing: we're comparing the motion of something inside the clock with the motion of the photon through space. And our clock could have been an optical clock.   

It would great if someone could explain why time is a dimension and not just a function of the "speed" of light?

I think Jim did that well enough. Check out the word dimension in the online etymology dictionary:
"dimension (n.) 
late 14c., "measurement, size," from Latin dimensionem (nominative dimensio) "a measuring," noun of action from past participle stem of dimetri "to measure out," from dis- (see dis-) + metiri "to measure" (see measure). Meaning "any component of a situation" is from 1929. Related: Dimensional; dimensions."
It's to do with measurement. Time is a dimension in the sense of measure. That's what the word meant: "a measuring". Temperature used to thought of as a dimension too, but it isn't any more. The meaning of the word has gradually changed, and now people think of a dimension as something that offers freedom of motion, not something to do with measurement. 
A: Time such a pesky pest
To leave it alone will be the best
Cause its sure not to let you rest
Until your thoughts explode in chest.
What you say cannot be refuted. But speed of light alone cannot define it
But you are accepting the notion of space , then of course space and motion will together be able to define time. Similarly c can be thought of as a conversion factor between space and time.
Time has only one direction for us because as you know pretty well you cannot just perform the murder of your great great grandfather, or say following your definition,cannot go to the state when the sun emitted the zillionth photon from the state when the sun emits (zillion +1)th photon. It is defined as change of something, of course, but whatever the reason, in macroscopic world the change can take place only in one direction(in QM the notion of time .....well lets leave it for now)
Now if you only consider t to be function of motion its already been done in special relativity(at the same time length also depends on reference frame)
Treating time as fourth dimension alongside space gives rise to space time continuum which can be affected by energy such that it gives rise to the force of gravity,as seen in general relativity. So you see there's no such inconsistency in treating time as fourth dimension.. What you are thinking is more of a philosophical question rather than physical one. But of course you can call head as "the other part of body except tail". It's not wrong.
Dimensionally also, t cannot be a function of c, whatever dimensionless constant you may use. You must use other universal constants. Again, the constants don't change, so you must introduce some variable that changes to grasp the notion of how time change. According to Occam's Razor, instead of complicating things, if time is considered to be fundamental, and it simplifies things, then the latter is preffered over the former.
