A human group transforms itself into a crowd when it suddenly responds to a suggestion rather than to reasoning, to an image rather than to an idea, to an affirmation rather than to proof, to the repetition of a phrase rather than to arguments, to prestige rather than to competence.
There are three basic approaches to AI: Case-based, rule-based, and connectionist reasoning.
Preaching is compelling to young secular adults ... - not if preachers use video clips from their favorite movies and dress informally and sound sophisticated, - but if the preachers understand their hearts and culture so well that listeners feel the force of the sermon's reasoning, even if in the end they don't agree with it.
We put [young children] into kindergarten where their reasoning powers are ruined; or, if we can afford it, we buy Montessori outfits that were invented for semi-imbeciles in Italian slums; or we send them to outdoor schools and give them prizes for sleeping.
Theoretically, I grant you, there is no possibility of error in necessary reasoning. But to speak thus "theoretically," is to uselanguage in a Pickwickian sense. In practice, and in fact, mathematics is not exempt from that liability to error that affects everything that man does.
It is certain that the only hope of retroductive reasoning ever reaching the truth is that there may be some natural tendency toward an agreement between the ideas which suggest themselves to the human mind and those which are concerned in the laws of nature.
Our scientific world is our world of reasoning. It has its greatness and uses and attractions. We are ready to pay homage due to it. But when it claims to have discovered the real world for us and laughs at the worlds of all simple-minded men, then we must say it is like a general grown intoxicated with his power, usurping the throne of his king
God is intelligent; but in what manner? Man is intelligent by the act of reasoning, but the supreme intelligence lies under no necessity to reason. He requires neither premise nor consequences; nor even the simple form of a proposition. His knowledge is purely intuitive. He beholds equally what is and what will be. All truths are to Him as one idea, as all places are but one point, and all times one moment.
What sets us against one another is not our aims - they all come to the same thing - but our methods, which are the fruit of our varied reasoning.
Do not expect to arrive at certainty in every subject which you pursue. There are a hundred things wherein we mortals. . . must be content with probability, where our best light and reasoning will reach no farther.
Since I found that one could make a case shadow from a three-dimensional thing, any object whatsoever - just as the projecting of the sun on the earth makes two dimensions - I thought that by simple intellectual analogy, the fourth dimension could project an object of three dimensions, or, to put it another way, any three-dimensional object, which we see dispassionately, is a projection of something four-dimensional, something we are not familiar with.
Thought that accepts reality as given is no thought at all.
Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
While often being called transdisciplinary, theonomous reasoning is actually a first step back to ancient wisdom in which methodological sensation [or what we now know as science] has completely merged with methodological revelation [or totally known mystical knowledge in which every aspect of the occult has been overcome]. A true tradition has no occult or hidden phases left in its process. The creators and the audience are in perfect harmony.
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.
Mathematics is purely hypothetical: it produces nothing but conditional propositions.
A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds...how does it come about that so many persons are here refractory?
But as the work proceeded I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was not more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Do you think it's possible that things that seem to be discrete in three dimensions might all be part of the same bigger object in four dimensions? ...What if humanity- that collective noun we so often employ- really is, at a higher level, a singular noun? What it what we perceive in three dimensions as seven billion individual human beings are really all just aspects of one giant being?
The most careful reasoning characters are very often the most easily abashed.
The political spirit is the great force in throwing the love of truth and accurate reasoning into a secondary place.
Propriety of thought and propriety of diction are commonly found together. Obscurity and affectation are the two greatest faults of style. Obscurity of expression generally springs from confusion of ideas; and the same wish to dazzle, at any cost, which produces affectation in the manner of a writer, is likely to produce sophistry in his reasonings.
As an exercise of the reasoning faculties, pure mathematics is an admirable exercise, because it consists of reasoning alone and does not encumber the student with any exercise of judgment.
We have millions of questions that need answers because there are so many things that the reasoning mind cannot explain. It is not important if the answer is correct; just the answer itself makes us feel safe.
Mathematical Reasoning is not only exact; it has its own criteria of reality.
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