Seems to me that either the temperature is rising or it is not. Perfectly suitable for one bit of information.
The reasoning for determining the value of said bit is another matter, but I think your argument does not make sense. Unless the climate is stuck in some sort of weird quantum entanglement.
Temperature of what??? Temperature is a function of space. If you fix the spatial coordinates, you can tell if the temperature increased at that point between time t1 and time t2. However, there are an infinite number of points, and thus you need an infinite number of bits. And to make it worse, all these bits only tell you what happened at time t2 relative to time t1.
No need to bring quantum entanglement to the discussion. If temperature is a function of space, it's an infinite-dimensional problem. You can discretize space and transform it into a finite-dimensional problem. Still, there are an enormous number of points at which you need to measure temperature.
Last but not least: the atmosphere is some 80 Km thick. You need measurement at high-altitudes, too. Measurements at the surface are easy to obtain, but they're only part of the picture.
I just assume climate researchers use a reasonable definition. Maybe "moving average of 100 years" or whatever (I don't know).
"However, there are an infinite number of points"
Not sure what you are saying? Seems to me a finite sample should be sufficient to get a statistical significant result. Are you saying "OK, the measurement at point x was 30°C, but how can you be sure that 1m north of it the temperature was not -10°C"? I think that would be a bit silly. Again, the average should be your friend. Maybe at some measuring spots you miss the temperature by 10°C. So what? On average, those errors should cancel out.
"Still, there are an enormous number of points at which you need to measure temperature."
As I said, I am not interested in the way to get there, just in the final information upon which to act or not. And that is one bit.
(Edit: although I also dispute your claim that there is an enormous number of points. To make up an extreme counter example, suppose you measure temperature in your home, and you notice it rising 10°C every year - would you really need other measurements to conclude that something is wrong? Like when it was at 100°C, boiling temperature for water, would you still say "but OK, it is still reasonably chilly at 80km altitude, so nothing to worry about"?).
"Last but not least: the atmosphere is some 80 Km thick. You need measurement at high-altitudes, too."
Again, I assume that climate researchers pick meaningful spots to measure. Maybe they measure at 80km altitude, maybe not. Maybe it is relevant to measure at 80km, maybe not. I don't see a point to argue about that kind of thing on a "popular science" level - that is up to the specialists. I suppose temperatures at 80km could be measured, why not? Maybe it can be done with telescopes or satellites - why do you assume it is not being done?
It just seems like a cop out to say "yeah, but it is so complicated we can never be sure". We are not 100% sure how the human body works, but we can still fix parts of it and detect some things when they are wrong.
Have you ever read about what that definition is on any discussion on global warming? I haven't. So they tell us the temperature is rising, but we don't know what temperature. What a joke...
"Seems to me a finite saple should be sufficient to get a statistical significant result."
"seems to me" is not very scientific, but you're right. Fortunately, physics makes it easy for us. We only need to sample every few hundreds of meters or so.
"Maybe at some measuring spots you miss the temperature by 10°C. So what? On average, those errors should cancel out."
The problem is that if you don't have enough information of the present state, you will not be able to predict the (near) future state reliably. A very coarse discretization is most likely useless for prediction purposes.
"As I said, I am not interested in the way to get there, just in the final information upon which to act or not. And that is one bit."
If you don't understand the process used to obtain the numbers, you don't understand how much the numbers can be trusted. Scientists study things and make predictions. It's not the scientists' role to decide what to do. That is the politicians' role. Funny how things seem to have been turned upside down.
