For 2 indistinguishable particles, we take the wave function to be $$\psi\pm (r_1,r_2) = A[\psi_a (r1)\psi_b (r2) \pm \psi_b (r1)\psi_a (r2) ]$$ where fermions get a - sign and bosons get a +
But, if both the particles are indistinguishable, isn't $\psi_a (r1) = \psi_a (r2)$, cause they behave the same way. Why do we take them to be different?
Also, can we calculate by some means that bosons get a + and fermions a -? or are they just experimental observations?