Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures. - Henri Poincaré

Mathematicians are born, not made.

We have then many kinds of intuition; first, the appeal to the senses and the imagination; next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning.

John Stuart Mill understood the word existence in a material and empirical sense; he meant that in defining a circle we assert that there are round things in nature.
In this form his opinion is inadmissible. Mathematics is independent of the existence of material objects. In mathematics the word exist can only have one meaning; it signifies exemption from contradiction. Thus rectified, Mill's thought becomes accurate. In defining an object, we assert that the definition involves no contradiction.