Mathematics and Science Quotes

“All the researches so far mentioned have a common characteristic. They propose to teach mathematics to a pupil who does not yet know the difference that exists between infinity and the finite; they do not hasten to teach him what this difference consists of; they begin by teaching him all that can be known about infinity without being concerned about this distinction. Then in a remote region of the field in which they made him wander, they show him a small corner where the finite numbers are hidden.
This seems to me psychologically false; the human mind does not proceed naturally in this manner, and even though we might extricate ourselves without too many antinomical mishaps, this method would be no less contrary to sound psychology.
Mr. Russell will tell me no doubt that it is not a question of psychology, but of logic and epistemology; and I shall be led to answer that there is no logic and epistemology independent of psychology; and this profession of faith will probably close the discussion.”
― Henri Poincaré, Mathematics and Science: Last Essays

“The antinomies to which certain logicians have been led arise from the fact that they have been unable to avoid certain vicious circles. This happened when they considered finite collections, but this happened much more often when they laid claim to treating of infinite collections. In the first case, they could easily have avoided the trap into which they fell; or, more exactly, they themselves laid the trap into which they chose to fall, and they were even obliged to be very careful not to miss the trap; briefly, in this case, antinomies are merely toys. Very different are those generated by the notion of infinity; it often happens that the logicians fall into it without doing it on purpose, and even when forewarned, they are still not at ease.”
― Henri Poincaré, Mathematics and Science: Last Essays

“All the researches so far mentioned have a common characteristic. They propose to teach mathematics to a pupil who does not yet know the difference that exists between infinity and the finite; they do not hasten to teach him what this difference consists of; they begin by teaching him all that can be known about infinity without being concerned about this distinction. Then in a remote region of the field in which they made him wander, they show him a small corner where the finite numbers are hidden. This seems to me psychologically false; the human mind does not proceed naturally in this manner, and even though we might extri cate ourselves without too many antinomical mishaps, this method would be no less contrary to sound psychology. Mr. Russell will tell me no doubt that it is not a question of psychology, but of logic and epistemology; and I shall be led to answer that there is no logic and epistemology independent of psychology; and this profession of faith will probably close the discussion.”
― Henri Poincaré, Mathematics and Science: Last Essays

“Formal logic is nothing but the study of the properties common to all classifications; it teaches us that two soldiers who are members of the same regiment belong by this very fact to the same brigade, and consequently to the same division; and the whole theory of the syllogism is reduced to this.”
― Henri Poincaré, Mathematics and Science: Last Essays