A Quote by Felix Klein
Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.
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Fractions, decimals, algebra, geometry, trigonometry, calculus, mechanics - these are the steps up the mountain side. How high is one going to get? For me, the pinnacle was Projective Geometry. Who today has even heard of this branch of mathematics?
Today, we're taking a break from the concerns and the bustle of the work-a-day world. But we're also making a new beginning... Let us renew our faith that as free men and women we still have the power to better our lives, and let us resolve to face the challenges of the new year holding that conviction firmly in our hearts. That, after all, is our greatest strength and our greatest gift as Americans.
I guess I see 'Goo' half as a really New York record because I think there are a lot of really particular New York references on it, but I also see it, for us, as the first of our records that really opened up to the larger world around us.
In this age of space flight, when we use the modern tools of science to advance into new regions of human activity, the Bible... remains in every way an up-to-date book. Our knowledge and use of the laws of nature that enable us to fly to the Moon also enable us to destroy our home planet with the atom bomb. Science itself does not address the question whether we should use the power at our disposal for good or for evil. The guidelines of what we ought to do are furnished in the moral law of God.
Science is much more than a body of knowledge. It is a way of thinking. This is central to its success. Science invites us to let the facts in, even when they don't conform to our preconceptions. It counsels us to carry alternative hypotheses in our heads and see which ones best match the facts. It urges on us a fine balance between no-holds-barred openness to new ideas, however heretical, and the most rigorous skeptical scrutiny of everything - new ideas and established wisdom.
Improved perception of our somatic feelings not only gives us greater knowledge of ourselves but also enables greater somatic skill, facility, and range of movement that can afford our sensory organs greater scope in giving us knowledge of the world. Besides augmenting our own possibilities of pleasure , such improved somatic functioning and awareness can give us greater power in performing virtuous acts for the benefit of others, since all action somehow depends on the efficacy of our bodily instrument.
There is no royal road to geometry.
Sire, there is no royal road to geometry.
We have heard of a Society for the Diffusion of Useful Knowledge. It is said that knowledge is power, and the like. Methinks there is equal need of a Society for the Diffusion of Useful Ignorance, what we will call Beautiful Knowledge, a knowledge useful in a higher sense: for what is most of our boasted so-called knowledge but a conceit that we know something, which robs us of the advantage of our actual ignorance? What we call knowledge is often our positive ignorance; ignorance our negative knowledge.
I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.
The greatest discoveries of science have always been those that forced us to rethink our beliefs about the universe and our place in it.
We have no knowledge, that is, no general principles drawn from the contemplation of particular facts, but what has been built up by pleasure, and exists in us by pleasure alone. The Man of Science, the Chemist and Mathematician, whatever difficulties and disgusts they may have had to struggle with, know and feel this. However painful may be the objects with which the Anatomist's knowledge is connected, he feels that his knowledge is pleasure; and where he has no pleasure he has no knowledge.
God, the source of our knowledge, has been expelled from the classroom. He gives us His greatest blessing, life, and yet many would condone the taking of innocent life. We expect Him to protect us in a crisis, but turn away from Him too often in our day-to-day living. I wonder if He isn't waiting for us to wake up.
The only royal road to elementary geometry is ingenuity.
The library connects us with the insight and knowledge, painfully extracted from Nature, of the greatest minds that ever were, with the best teachers, drawn from the entire planet and from all our history, to instruct us without tiring, and to inspire us to make our own contribution to the collective knowledge of the human species.
Science is expanding, and with it our vision of the universe. although this new and constantly changing view may not always give us comfort, it does have the virtue of truth according to our most effective resources for acquiring knowledge. No philosophy, moral outlook, or religion can be inconsistent with the findings of science and hope to endure among educated people.
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