Philip J Davis - Collected Quotes

"One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.” (Philip J Davis, "Number", Scientific American, No 211 (3), 1964)

“It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands.” (Philip J Davis, “The Mathematics of Matrices”, 1965)

“[…] how completely inadequate it is to limit the history of mathematics to the history of what has been formalized and made rigorous. The unrigorous and the contradictory play important parts in this history.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

“Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1995)

"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, “The Mathematical Experience”, 1985)

"Some order begins to emerge from this chaos when the primes are considered not in their individuality but in the aggregate; one considers the social statistics of the primes and not the eccentricities of the individuals." (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1985)

“That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects (at a later stage, of marks on paper or even mental images).” (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1985)

“The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights.”  (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

“Where is the place of mathematics? Where does it exist? On the printed page, of course, and prior to printing, on tablets or on papyri. Here is a mathematical book - take it in your hand; you have a palpable record of mathematics as an intellectual endeavor. But first it must exist in people's minds, for a shelf of books doesn't create mathematics.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)