Mathematicians have defined the infinite as a quantity which is always greater than any assignable quantity, and the infinitesimal as a quantity that is always smaller than any assignable quantity, and have used these conceptions to deal with quantities which vary continuously. More recent mathematical thought has defined the infinite number by certain characters which are not possessed by finite numbers and have in this way been able to formulate the conception of compactness, which enables the mathematician to conceive of the continuum in terms of the discrete. In practice the infinite has been a conception by means of which men have been able to deal with certain problems of the continuum and the discrete, such as that of a continuously changing velocity, the relation of curves to inscribed and circumscribed lines, the relations of points and instants to continuous space and time and many others. In a manner not logically unlike this Hegel undertook to state the positive character of the infinite as the reference of being that binds it to itself when passing into the other. This Hegel called the true infinity and illustrated it by the circle as contrasted to the indefinite straight line. But it cannot be said that in philosophy or elsewhere in thought Hegel's undertaking has solved any problems. So far as a positive character appears in our thought of the infinite, it is emotional not discursive. Ancient Greek thought found the highest perfection in the finite, the defined. It remained for the latest phase of Greek speculation and modern theology and philosophy to identify perfection and the highest reality with the infinite.


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