The importance of examples

Examples are essential for the stimulation of our thoughts.

– An Introduction to Mathematics by Alfred North Whitehead

The last thing to be discovered in any science is what the science is really about

Mathematician studied for centuries the abstract problem of drawing tangents to curves before Newton finally discovered what the subject really is about — rate of change.

It is a well-founded historical generalization that the last thing to be discovered in any science is what the science is really about.

Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last some great truth is loosened.

– An Introduction to Mathematics by Alfred North Whitehead

Joint Discoveries Are Common

Joint discoveries are quite common in science.

Discoveries are not in general made before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea.

Examples of joint discoveries:

• Law of natural selection: Darwin and Wallace
• Discovery of Neptune: Adams and Leverrier
• Creation of differential calculus: Newton and Leibniz

– An Introduction to Mathematics by Alfred North Whitehead

Progress of the History of Thought

The progress of science is divided between

• periods characterized by a slow accumulation of ideas, and
• periods when, owing to the material of thought that has been patiently collected, some genius by the invention of a new method or point-of-view suddenly transforms the whole subject on to a higher level.

The genius who has the good fortune to produce the final idea which transforms a whole region of thought does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In considering the history of science it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances toward a new epoch.

– An Introduction to Mathematics by Alfred North Whitehead

See what is general in what is particular and what is permanent in what is transitory

Consider how all events are interconnected.  When we see the lightening, we listen for the thunder; when we hear the wind, we look for the waves on the sea; in the chill autumn, the leaves fall.  Everywhere order reigns, so that when some circumstances have been noted we can foresee that others will also be present.

The progress of science consists in observing these interconnections and in showing with a patient ingenuity that the events of this ever-shifting world are but examples of a few general connections or relations called laws.  To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought.

– An Introduction to Mathematics by Alfred North Whitehead