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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.”

“Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.”

“The deep paradox uncovered by AI research: the only way to deal efficiently with very complex problems is to move away from pure logic.... Most of the time, reaching the right decision requires little reasoning.... Expert systems are, thus, not about reasoning: they are about knowing.... Reasoning takes time, so we try to do it as seldom as possible. Instead we store the results of our reasoning for later reference.”

“Philosophy, like science, consists of theories or insights arrived at as a result of systemic reflection or reasoning in regard to the data of experience. It involves, therefore, the analysis of experience and the synthesis of the results of analysis into a comprehensive or unitary conception. Philosophy seeks a totality and harmony of reasoned insight into the nature and meaning of all the principal aspects of reality.”

“One of the tragedies of our day is that too many boys are growing up without guidance of a father, or another man, to show them what it looks like to do away with that boyhood stuff. As a result, they often move into adolescence and then adulthood looking like men but still speaking, reasoning, and behaving like boys.”

“Let us confess a truth, humiliating to human pride; - a very small part only of the opinions of the coolest philosopher are the result of fair reasoning; the rest are formed by his education, his temperament, by the age in which he lives, by trains of thought directed to a particular track through some accidental association - in short, by prejudice.”

“The culture and educational system of the contemporary West are based almost exclusively upon the training of the reasoning brain and, to a lesser degree, of the aesthetic emotions. Most of us have forgotten that we are not only brain and will, senses and feelings; we are also spirit. Modern man has for the most part lost touch with the truest and highest aspect of himself; and the result of this inward alienation can be seen all too plainly in his restlessness, his lack of identity and his loss of hope.”

“The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.”

“Logical investigations can obviously be a useful tool for philosophy. They must, however, be informed by a sensitivity to the philosophical significance of the formalism and by a generous admixture of common sense, as well as a thorough understanding both of the basic concepts and of the technical details of the formal material used. It should not be supposed that the formalism can grind out philosophical results in a manner beyond the capacity of ordinary philosophical reasoning. There is no mathematical substitute for philosophy.”

“A saving knowledge of Christ crucified and risen is not the mere result of right reasoning about historical facts. It is the result of spiritual illumination to see those facts for what they really are: a revelation of the truth and glory of God in the face of Christ-who is the same yesterday today and forever.”

“Science has long been in the value business. Despite a widespread belief to the contrary, scientific validity is not the result of scientists abstaining from making value judgments; rather, scientific validity is the result of scientists making their best efforts to value principles of reasoning that link their beliefs to reality, through reliable chains of evidence and argument.”