Mathematically, I understand why proper time, $\tau$ is an invariant quantity, since it is defined in terms of the spacetime metric $d\tau=\sqrt{-ds^{2}}$ (using the signature $(-+++)$ and with $c=1$). More abstractly, $\tau$ simply parametrises the length between two points along a worldline and hence is "obviously" invariant in this sense.
However, putting this aside for a moment, intuitively I'm less certain how to provide an answer to the question: why it is the case that proper time is physically an invariant quantity?
Consider a particle in Minkowski spacetime. If two different observers, Alice and Bob, are moving at different velocities with respect to the particle and with respect to one another, and each measures the elapsed time for the particle to propagate from one point to another, then they will measure different time intervals to one another. However, they will both agree on the elapsed proper time of the particle. Is the reason why this is the case because the question, "what is the time 'experienced' by the particle?", is a frame independent question – the proper time is a measure of the amount of "physical process" that the particle undergoes as it "moves" along its worldline, and this is a physical (coordinate independent) phenomenon? If Alice and Bob disagreed on the amount of elapsed proper time then they would be disagreeing with the particle on how much time has elapsed for the particle which would be nonsense?!
Apologies for such a basic question, I'm hoping someone can clear up any confusion for me.
Best Answer
Let's be precise here. The 'invariance' in question is invariance of the spacetime interval under Lorentz transformations. Lorentz transformations here relate the coordinates of an event as measured by Alice with that of Bob, where they have a boost velocity with respect to each other. As such, Alice measures some time $t_A$ and Bob $t_B$. The transformations do the job of taking you from one observer to the other to see what it's like on their side of the world; it's like saying that Alice puts herself in Bob's shoes or vice versa.
But when you are talking about proper time, you are, by definition, adhering to only one observer: the particle itself. It doesn't make sense to say that the particle is boosted with respect to itself. There is no ambiguity in choosing a reference frame before deciding on performing a measurement, because the reference frame/observer has been chosen, a priori.