Lies? That seems to be overstating the case quite a bit since you use math concepts later in your post.
Math, in science, is used as a language. You may as well say "...you use French concepts later in your post." Further, I was explaining why the concept of a mean doesn't necessarily represent any individual in a population, particularly in biology--I need to use math concepts, because I'm discussing a math concept. I mean, try to explain a medical term without using medical concepts. The idea is nonsensical.
Secondly, I didn't say means were lies. I said they were USEFUL lies. The modifier makes it a different concept, and you cannot treat "useful lies" as equivalent to "lies". Useful lies are all around us, particularly in education. The whole "First assume a spherical chicken" joke highlights the useful lies physicsts use, even the top theoretical physicsts. Useful lies aren't necessarily wrong; the parts they're simply simplifications of concepts to make them easier for our minds to work with. Means are exactly that. It's very, very hard to handle an entire population. The mean is a single number (and the mean member is the mean for all the measurements). That's MUCH simpler to work with.
Ask yourself this: why do we include error bars in means of populations where we know all the members? The answer is that, fundamentally, those error bars are our admission that what we're working with isn't the population itself, and we're accepting a certain amount of screwyness in our data in exchange for making the calculations easier. (Yes, I know the justification for error bars in many fields; I'm not addressing them. I'm addressing the existence of error bars around a mean in a population where all members have been measured. Even the best-case scenario for means includes error.)
Why would a mathematical concept have to have a physical expression?
Depends on the concept. The mean height of a population, for example, really should have physical experession--we're dealing with discrete data here, individual organisms to be specific. If the mean doesn't exist, in what sense is it true? If no human has the average height, the average height is entirely fictional. It's USEFUL, but it's fictional. It's a very serious error to confuse the simplifications with reality, one that gets you into all kinds of trouble once you try to use them to deal with reality. Maybe not so much in physics, but in biology and geology those errors can quickly add up, until you realize that your concept of what the world should look like is completely different from what it is.