A Quote by Imre Lakatos
Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.
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Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
If we once admit that our life is here for the purpose of race-improvement, then we question any religion which does not improve the race, or the main force of which evaporates, as it were, directing our best efforts toward the sky.... Improvement in the human race is not accomplished by extracting any number of souls and placing them in heaven, or elsewhere. It must be established on earth, either through achievement in social service, or through better children.
What exactly is mathematics? Many have tried but nobody has
really succeeded in defining mathematics; it is always something
else. Roughly speaking, people know that it deals with numbers,
figures, with relations, operations, and that its formal procedures
involving axioms, proofs, lemmas, theorems have not changed
since the time of Archimedes.
It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself.
Proofs exist only in mathematics and logic, not in science.
Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!"
To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples.
But in the past, US companies have been able to increase their profits through downsizing in the US, through colonizing other people's resources, and through the increase of globalization.
Every church needs to grow warmer through fellowship, deeper through discipleship, stronger through through worship, and larger through evangelism.
Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.
To be aflame with silence, with joy, is wisdom. It is not through logic but through love. It is not through words but through a wordless state called meditation or a state of no-mind, satori, samadhi.
It does you no good to see the number two or number three man in the corporation-you have to get through to number one.
the 'total overpaintings' developed... through incessant reworking. The original motif peeped through the edges. Gradually it vanished completely.
Formal logic is mathematics, and there are philosophers like Wittgenstein that are very mathematical, but what they're really doing is mathematics - it's not talking about things that have affected computer science; it's mathematical logic.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
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