The deepest and most revolutionary insights arose from quantum theory. They are at two levels. First is the very basic fact that matter is built up from vast numbers of copies of a few fundamental components (such as electrons, quarks, photons and gluons). The properties of these elementary building-blocks are always and everywhere the same — universal. If this were not the case, there could be no laws of chemistry, because every atom would have its own quirky properties.
In quantum mechanics, the view is fundamentally shifted, as there is a radical difference between two particles being precisely identical and being merely similar. If A and B are identical, then when A travels from position x1 to y1 and B from x2 to y2, the final result is the same as when A goes to y2 and B to y1. In quantum mechanics, the basic goal is to calculate amplitudes, the square of which gives the probability of an event. To get the total amplitude to find particles at y1,y2, we must add (for identical bosons) or subtract (for identical fermions) the amplitudes for these two possibilities, and then square to get the probability. If the particles are distinguishable, then so are the two final states, so we must square first, then add.
