Why is it 'difficult' to define probability?

A couple of weeks ago, we looked at Kolmogorov's axioms for probability as the definition of the probability function. But Kolmogorov's axioms only really describe the properties $\mathbb{P}(A)$ should have and don't go into much detail as to what $\mathbb{P}(A)$ is. And the lecture notes also mention that the question of what probability is doesn't really have a satisfactory answer. So why exactly is this the case? Can't we just say something like probability is the likelihood of an event happening or something like this?

What do you mean the likelihood of an event occurring though? What does it mean to say that there is a 60% chance that A will occur for example.

Thanks. This is exactly what I was thinking, but wasn't sure if it was the right train of thought. Although I guess you could say that you would expect 60% of n outcomes should be approx A given a large n.

Yeah that's one interpretation of probability. There are others.