Let's consider an ideal gas, of some huge number of particles. The Maxwell-Boltzmann distribution describes the probability of measuring a particle speed in a range of speeds, through integration.
Why are we justified in assuming that there is a continuum of speeds to integrate over? Certainly we have a huge number of particles in the gas, but there are still only finitely many - 10^30 or 10^40 or whatever particles is huge, but still doesn't comprise a continuum. No matter how many particles there are, surely the speed distribution should remain discrete?
I understand that this is some approximation, but then how many particles is 'big enough' for this approximation to hold? 10^20? 10^30? These sort of numbers are huge by everyday experience, but don't compare to some sort of uncountable continuum. I'd be interested to know what people think!