What we call geometry is nothing but the study of formal properties of a certain continuous group; so that we may say, space is a group. The notion o… - Henri Poincaré

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.

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. ...we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it.

... les traités de mécanique ne distinguent pas bien nettement ce qui est expérience, ce qui est raisonnement mathématique, ce qui est convention, ce qui est hypothèse.