Six maps that will make you rethink the world. Not only are the maps beautiful, but the interview is pretty interesting too.
The math behind computational learning is hard, but Christopher Olah makes understanding it really easy. Every article on his blog is worth reading.
Forty-four percent of the way through the complete Sherlock Holmes works on Kindle and Mycroft is finally introduced:
“My dear Watson,” said he, “I cannot agree with those who rank modesty among the virtues. To the logician all things should be seen exactly as they are, and to underestimate one’s self is as much a departure from truth as to exaggerate one’s own powers. When I say, therefore, that Mycroft has better powers of observation than I, you make take it that I am speaking the exact and literal truth.”
“You wonder,” said my companion, “why it is that Mycroft does not use his powers for detective work. He is incapable of it.”
“But I thought you said—”
“I said that he was my superior in observation and deduction. If the art of the detective began and ended in reasoning from an arm-chair, my brother would be the greatest criminal agent that ever lived. But he has no ambition and no energy. He will not even go out of his way to verify his own solutions, and would rather be considered wrong than take the trouble to prove himself right. Again and again I have taken a problem to him, and have received an explanation which has afterwards proved to be the correct one. And yet he was absolutely incapable of working out the practical points which must be gone into before a case could be laid before a judge or jury.”
From the chapter on Leonardo da Vinci in Visari’s Lives:
Da Vinci would buy and the immediately release birds…
He was so pleasing in conversation, that he attracted to himself the hearts of men. And although he possessed, one might say, nothing, and worked little, he always kept servants and horses, in which latter he took much delight, and particularly in all other animals, which he managed with the greatest love and patience; and this he showed when often passing by the places where birds were sold, for, taking them with his own hand out of their cages, and having paid to those who sold them the price that was asked, he let them fly away into the air, restoring to them their lost liberty. For which reason nature was pleased so to favor him, that, wherever he turned his thought, brain, and mind, he displayed such divine power in his works, that, in giving them their perfection, no one was ever his peer in readiness, vivacity, excellence, beauty, and grace.
Was it karma? In another story, a Prince had commissioned a work from da Vinci, but da Vinci was not working on it, and the Prior grew worried…
Leonardo, knowing that the intellect of that Prince was acute and discerning, was pleased to discourse at large with the Duke on the subject, a thing which he had never done with the Prior: and he reasoned much with him about art, and made him understand that men of lofty genius sometimes accomplish the most when they work the least, seeking out inventions with the mind, and forming those perfect ideas which the hands afterwards express and reproduce from the images already conceived in the brain.
Rulers, statesmen, and nations are told that they ought to learn from the experience of history. Yet what experience and history teach us is this, that nations and governments have never learned anything from history, nor acted in accordance with the lessons to be derived from it. Each era has such particular circumstances, such individual situations, that decisions can only be made from within the era itself. In the press of world events, there is no help to be had from general principles, nor from the memory of similar conditions in former times—for a pale memory has no force against the vitality and freedom of the present. In this respect, nothing is more trite than the repeated appeal to Greek and Roman examples, which was so commonplace at the time of the French Revolution. No difference could be greater than that between the nature of those ancient peoples and our own time.
From Chapter 1 of Hegel’s Introduction to the Philosophy of History. The translator, Leo Rauch, footnotes an interesting quote from Hume:
See Hume’s Enquiry Concerning Human Understanding, Section VIII, Part I, “Would you know the sentiments, inclinations, and course of life of the Greeks and Romans? Study well the temper and actions of the French and English. … Mankind are so much the same, in all times and places, that history informs us of nothing new or strange in this particular.”
It’s worth noting that Hume lived in rather extreme poverty until he published a multi-volume history of England, after which he skyrocketed to fame and fortune in the intellectual atmosphere of his time. Surely such a deep study of history helped to inform his other writings.
The case for using computer algebra
Many academics and researchers get annoyed when students use computer algebra programs such as Mathematica to evaluate simple integrals that they maintain should be done by hand. The question I ask is “At what point do you expect your students to switch over to using a computer?”. Most mathematical examples are artificial in that closed form expressions exist. However, in nearly any real problem, this is not the case.
I learnt mathematics using slide rules and tables, then calculators, then computers, then symbolic algebra. To me, this is a valid progression — but not one that everyone should have to go through. I feel that the only way true progress can be made is if we don’t have to learn a whole set of rules. If we had to do calculus using Newton’s geometrical constructs then progress would be very slow. The real question is what are the essential tools and lessons. To me, knowing what a derivative and integral mean “physically” is far more important than knowing how to compute a specific integral.
Many people feel that reliance on computer algebra means that students can’t do calculus by hand and hence the really don’t understand what’s going on, just how to get the answer by computer. Calculus concepts are subtle. However, just knowing the mechanics of computing an integral or derivative does not imply understanding. I believe that it is possible to have true understanding without computation.
From a list of pet peeves of Paul Abbot, physics professor at University of West Alabama.