No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.

It follows that the word probability, in its mathematical acceptance, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it will vary. Probability is the expectation founded upon partial knowledge.

Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave nether room nor demand for a theory of probabilities.

The general laws of Nature are not, for the most part, immediate objects of perception.

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.

To unfold the secret laws and relations of those high faculties of thought by which all beyond the merely perceptive knowledge of the world and of ourselves is attained or matured, is a object which does not stand in need of commendation to a rational mind.

It is not of the essence of mathematics to be conversant with the ideas of number and quantity.