What are mathematicians doing?

Let me emphasize the point I have been trying to make. The mathematician's playing with the roots of equations, a play which had no practical motivations and almost no possibilities of practical application, led to the recognition of the importance of symmetry and groups. The study of theory of groups led to mathematical discoveries in geometry and differential equations, and finally to prediction of the existence of a new elementary particle. Surely a surprising outcome for the ivory-tower speculations of an impractical mathematician! Despite my professional bias, I must acknowledge that the importance of symmetry was recognized before mathematicians invented the theory of groups. In 1794 William Blake wrote: Tiger, Tiger, burning bright In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry? However, to the mathematicians must be given the credit of recognizing that, to understand symmetry, you must study the theory of groups. I can now answer my original question, What are mathematicians doing? They are trying to make precise the intuitions of poets.