Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
Freeman DysonRelated topics
context mathematics meaning paper pure study theorem truth work reductionismRelated quotes
What seems certain is that Pythagoras developed the idea of mathematical logic... He realized that numbers exist independently of the tangible world and therefore their study was untainted by inaccuracies of perception. This meant he could discover truths which were independent of opinion of prejudice and which were more absolute then any previous knowledge.
Simon Singh
"What is the meaning of life?" This question has no answer except in the history of how it came to be asked. There is no answer because words have meaning, not life or persons or the universe itself. Our search for certainty rests in our attempts at understanding the history of all individual selves and all civilizations. Beyond that, there is only awe.
Julian Jaynes
The objective psychologist, hoping to get at the physiological side of behavior, is apt to plunge immediately into neurology trying to correlate brain activity with modes of experience ... The result in many cases only accentuates the gap between the total experience as studied by the psychologist and neural activity as analyzed by the neurologist.
Roger Wolcott Sperry
Take the sum of human achievement in action, in science, in art, in literature -- subtract the work of the men above forty, and while we should miss great treasurers, even priceless treasures, we would practically be where we are today. . . . The effective, moving, vitalizing work of the world is done between the ages of twenty-five and forty.
William Osler
We must not forget that when radium was discovered no one knew that it would prove useful in hospitals. The work was one of pure science. And this is a proof that scientific work must not be considered from the point of view of the direct usefulness of it. It must be done for itself, for the beauty of science, and then there is always the chance that a scientific discovery may become like the radium a benefit for humanity.
Marie Curie