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[Learning is] the actual process of broadening yourself, of knowing there's a little extra facet of the universe you know about and can think about and can understand. It seems to me that when it's time to die, and that will come to all of us, there'll be a certain pleasure in thinking that you had utilized your life…
The young specialist in English Lit, having quoted me, went on to lecture me severely on the fact that in every century people have thought they understood the Universe at last, and in every century they were proved to be wrong. It follows that the one thing we can say about our modern 'knowledge' is that it is wrong…
People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual…
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
The world of mathematics and theoretical physics is hierarchical. That was my first exposure to it. There's a limit beyond which one cannot progress. The differences between the limiting abilities of those on successively higher steps of the pyramid are enormous. I have not seen described anywhere the shock a talented…
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt…
The sun's rays proceed from the sun along straight lines and are reflected from every polished object at equal angles, i.e. the reflected ray subtends, together with the line tangential to the polished object which is in the plane of the reflected ray, two equal angles. Hence it follows that the ray reflected from the…
There is no thing as a man who does not create mathematics and yet is a fine mathematics teacher. Textbooks, course material—these do not approach in importance the communication of what mathematics is really about, of where it is going, and of where it currently stands with respect to the specific branch of it being…
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
[P]olitical and social and scientific values … should be correlated in some relation of movement that could be expressed in mathematics, nor did one care in the least that all the world said it could not be done, or that one knew not enough mathematics even to figure a formula beyond the schoolboy s=(1/2)gt2. If…
My investigation asked the question of whether there is a secret formula in tree design and whether the purpose of the spiral pattern is to collect sunlight better. After doing research, I put together test tools, experiments and design models to investigate how trees collect sunlight. At the end of my research…
Studies show that, compared with controls, kids allowed a specific type of open-ended play time were: • More creative. On average they came up with three times as many nonstandard creative uses for specific objects (a standard lab measure) as did controls. • Better at language. The children’s use of language was more…
Many well-meaning moms and dads think their child’s brain is interested in learning. That is not accurate. The brain is not interested in learning. The brain is interested in surviving. Every ability in our intellectual tool kit was engineered to escape extinction. Learning exists only to serve the requirements of…
Even in an emotionally stable home, one without regular marital hostility, there will be fights. Fortunately, research shows that the amount of fighting couples do in front of their children is less damaging than the lack of reconciliation the kids observe. Many couples will fight in front of their children but…
There remains simple experience which, if taken as it comes, is called accident; if sought for, experiment. But this kind of experience is no better than a broom without its band, as the saying is — a mere groping, as of men in the dark, that feel all round them for the chance of finding their way, when they had much…
One of the more startling findings about early enrichment is the effect of music. You can hardly pick up a newspaper without seeing some kind of reference to how Mozart makes people smarter. The governor of Georgia recently proposed spending $105,000 of state money to provide every newborn baby with a compact disc of…
The similarity between different vertebrate embryos is indeed remarkable. Since the early 1800s, embryologists have been struck by the parallel between early development in various animal species and their evolutionary relationship, a resemblance conveniently abbreviated by the saying "ontogeny recapitulates…
when they are more used to sleeping alone, sleep differently when with when they are more used to sleeping alone, sleep differently when with their mothers. The babies seem to spend a greater percentage of their sleep time in levels 1-2 and less time at the deeper levels, exhibit more REM sleep, and are awake longer…
Let me suggest a new axiom: juxtaposition is the spice of life. Humanity’s biggest talent, unique to us, is juxtaposing, finding and operating novel relationships between things or ideas... Recent ideas on neural activity suggest that the brain operates in a very associative way, with small neuron clusters containing…
One, two, three ragged files of Canada geese skim the treetops, preceded and followed by their honking chorus. I freeze in my tracks to watch them pass, heading south, feathers ruffled by the last warm breezes of the season. When their honks have faded into silence, I notice a chill in the air. The spinning planet has…