Light will travel in the fastest way possible under a certain set of well known constraints. I think trying to apply this same model to how 'mathematicians' cross the road is flawed, although it is a good approximation when the start and end point are not displaced too far along the road.
As a simple counter example, think of a laser pointing along the length of a very long piece of glass. If you angle the laser slightly down, so that it hits the glass at a very small angle of attack, the laser will travel for a long distance inside the glass before exiting. It will not travel straight through the glass, but neither will it 'cross' to the other side very quickly. Regardless of the difference in the refractive index of the air and glass, you can always point the laser at an angle that causes the laser beam to travel for an arbitrary length of time in the glass.
Compare this to a mathematician walking along a very long highway. The time they will take to cross the road is NOT dependent on JUST how far they are walking. If it has heavy traffic then they will cross just as soon as their is a suitable gap. Extending the length of their journey (equivalent to decreasing the angle of attack for the laser) does not continue to increase the time they take crossing the road unboundedly.
The model is flawed for this obvious counter example, but it is flawed in simpler situations as well, mostly because the reality is that every section of road and every moment in time are not as ideal as each other for crossing the road.
Light will travel in the fastest way possible under a certain set of well known constraints. I think trying to apply this same model to how 'mathematicians' cross the road is flawed, although it is a good approximation when the start and end point are not displaced too far along the road.
As a simple counter example, think of a laser pointing along the length of a very long piece of glass. If you angle the laser slightly down, so that it hits the glass at a very small angle of attack, the laser will travel for a long distance inside the glass before exiting. It will not travel straight through the glass, but neither will it 'cross' to the other side very quickly. Regardless of the difference in the refractive index of the air and glass, you can always point the laser at an angle that causes the laser beam to travel for an arbitrary length of time in the glass.
Compare this to a mathematician walking along a very long highway. The time they will take to cross the road is NOT dependent on JUST how far they are walking. If it has heavy traffic then they will cross just as soon as their is a suitable gap. Extending the length of their journey (equivalent to decreasing the angle of attack for the laser) does not continue to increase the time they take crossing the road unboundedly.
The model is flawed for this obvious counter example, but it is flawed in simpler situations as well, mostly because the reality is that every section of road and every moment in time are not as ideal as each other for crossing the road.