Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.
Interpretation
Mathematical reasoning involves both intuitive insights and creative problem-solving skills.
In this quote, Alan Turing highlights the dual nature of mathematical reasoning, suggesting that successful problem-solving in mathematics requires both intuition, which allows for understanding underlying concepts, and ingenuity, which fosters innovative approaches to challenges. This interplay between instinctive understanding and creative thinking is fundamental to the discipline of mathematics and contributes to its beauty and complexity.
In practice
In a classroom discussion about problem-solving strategies, a teacher may quote Turing to emphasize the importance of both intuition and creativity.
Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself.
I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.
Machines take me by surprise with great frequency.
Sometimes it is the people no one imagines anything of who do the things that no one can imagine.
I am not very impressed with theological arguments whatever they may be used to support. Such arguments have often been found unsatisfactory in the past. In the time of Galileo it was argued that the texts, 'And the sun stood still... and hasted not to go down about a whole day' (Joshua x. 13) and 'He laid the foundations of the earth, that it should not move at any time' (Psalm cv. 5) were an adequate refutation of the Copernican theory.
It seems probable that once the machine thinking method had started, it would not take long to outstrip our feeble powers⦠They would be able to converse with each other to sharpen their wits. At some stage therefore, we should have to expect the machines to take control.
the knowledge of impermanence that haunts our days is their very fragrance.
We are drowning in information, while starving for wisdom. The world henceforth will be run by synthesizers, people able to put together the right information at the right time, think critically about it, and make important choices wisely.
Prayer cannot bring water to parched fields, or mend a broken bridge, or rebuild a ruined city; but prayer can water an arid soul, mend a broken heart, and rebuild a weakened will.
As a reader and a writer, I'm happiest when apparently mutually exclusive states can somehow coexist.
Rage cannot be hidden, it can only be dissembled. This dissembling deludes the thoughtless, and strengthens rage and adds, to rage, contempt.
You only go around once, but if you play your cards right, once is enough.
Subscribe for the occasional hand-picked quote. No noise.