19th-century Mathematician Quotes
The infinite time of the geologists is in the past; and most of their speculations regarding this subject seem to imply the absolute infinity of time, as if the human imagination was unable to grasp the period of time requisite for the formation of a few inches of sand or feet of mud, and its subsequent consolidation into rock.
Mathematics deals mainly with realities and stern facts, and it arms those whose life work is beset with physical difficulties. Hence it appeals especially to the leaders of the common people, and it performs a fundamental function in the education of those who aim to excel in overcoming material difficulties.
The purely formal sciences, logic and mathematics, deal with those relations which are, or can be, independent of the particular content or the substance of objects. To mathematics in particular fall those relations between objects which involve the concepts of magnitude, of measure and of number.
The application will then, sometimes directly, but more frequently through a chain of intermediate stages, have significant bearing upon the content of human experience and furnish results which may be called true. Their truth then gives, retroactively, a sound basis for belief in the validity of the conclusions of mathematics.
Mathematicians have never been in full agreement on their science, though it is said to be the science of self-evident verities — absolute, indisputable and definitive. They have always been in controversy over the developing aspects of mathematics, and they have always considered their own age to be a period of crisis.
The subject for which I am asking your attention deals with the foundations of mathematics. To understand the development of the opposing theories existing in this field one must first gain a clear understnding of the concept "science"; for it is as a part of science that mathematics originally took its place in human thought.