Quotable Mathematics

"A single observation that is inconsistent with some generalization points to the falsehood of the generalization, and thereby 'poi...

"For group theoretic reasons the most impressive paradoxical decompositions occur in dimension at least three, but there are also inter...

"It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed with in the field of ...

"As for me (and probably I am not alone in this opinion), I believe that a single universally valid principle summarizing an abundance ...

"[It used to be that] geometry must, like logic, rely on formal reasoning in order to rebut the quibblers. But the tables have turned. ...

"If a fact is to be a picture, it must have something in common with what it depicts. […] What a picture must have in common with reali...

"A system of variables is 'interrelated' if an action that affects or meant to affect one part of the system will also affect o...

"Among all the liberal arts, the first is logic, and specifically that part of logic which gives initial instruction about words. [...]...

"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question....

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which ...

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the...

 "[…] a conceptual model is a diagram connecting variables and constructs based on theory and logic that displays the hypotheses to be ...

"Simple observation generally gets us nowhere. It is the creative imagination that increases our understanding by finding connections b...

"On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formul...