Quotable Mathematics

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our i...

"[…] there is an irreducible difference between the world and our experience of it. We as human beings do not operate directly on the w...

"Sometimes a thing is perceived [via sense-perception] when it is observed; then it is imagined, when it is absent [in reality] through...

  "The sole purpose of physical theory is to provide a representation and classification of experimental laws; the only test permitting...

"We have thus assigned to pure reason and experience their places in a theoretical system of physics. The structure of the system is th...

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data ...

"Cultivate that kind of knowledge which enables us to discover for ourselves in case of need that which others have to read or be told ...

"When we study the structure of the atom, we shall arrive at the conclusion that it is an immense reservoir of energy solely constitute...

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselv...

"The study of theory must go hand in hand with that of facts: and for dealing with most modern problems it is modern facts that are of ...

"Mathematical model making is an art. If the model is too small, a great deal of analysis and numerical solution can be done, but the r...

"A map is a graphical representation of geographical or astronomical features, but this may range from a sketch of a subway system, to ...

"A mental model is a collection of 'connected' autonomous objects. Running  a mental model corresponds to modifying the paramet...

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation th...

"Cartography is about representation. This statement may seem obvious, but it has been overlooked in our search for organizing principl...

"The sciences bring into play the imagination, the building of images in which the reality, of the past is blended with the ideals for ...

"This is the greatest degree of impoverishment; the [mental] image, deprived little by little of its own characteristics, is nothing mo...

"Observe this: the abstraction of the philosopher is meant to keep the object itself, with its perturbing suggestions, out of sight, al...

"Chemistry and physics are experimental sciences; and those who are engaged in attempting to enlarge the boundaries of science by exper...

"Mere deductive logic, whether you clothe it in mathematical symbols and phraseology or whether you enlarge its scope into a more gener...