Philosophical knowledge is the knowledge gained by reason from concepts ; mathematical knowledge is the knowledge gained by reason from the construction of concepts.
Most men use their knowledge only under guidance from others because they lack the courage to think independently using their own reasoning abilities. It takes intellectual daring to discover the truth.
Deaths, births, and marriages, considering how much they are separately dependent on the freedom of the human will, should seem to be subject to no law according to which any calculation could be made beforehand of their amount; and yet the yearly registers of these events in great countries prove that they go on with as much conformity to the laws of nature as the oscillations of the weather.
Ours is an age of criticism, to which everything must be subjected. The sacredness of religion, and the authority of legislation, are by many regarded as grounds for exemption from the examination by this tribunal, But, if they are exempted, and cannot lay claim to sincere respect, which reason accords only to that which has stood the test of a free and public examination.
Reason must approach nature in order to be taught by it. It must not, however, do so in the character of a pupil who listens to everything that the teacher chooses to say, but of an appointed judge who compels the witness to answer questions which he has himself formulated.
All appearances have a determinate magnitude (the relation of which to another assignable). The infinite does not appear as such, likewise not the simple. For the appearances are included between two boundaries (points) and are thus themselves determinate magnitudes.
The instruction of children should aim gradually to combine knowing and doing. Among all sciences mathematics seems to be the only one of a kind to satisfy this aim most completely.