Friday, October 11, 2019
Friday, October 4, 2019
1543
Two books published in 1543 marked a turning point, the beginning of the scientific revolution. In that year, the Flemish doctor Andreas Vesalius reported the results of his dissections of human cadavers, a practice that had been forbidden in earlier centuries. His findings contradicted fourteen centuries of received wisdom about human anatomy. In that same year, the Polish astronomer Nicolaus Copernicus finally allowed publication of his radical theory that the Earth moved around the sun. He’d waited until he was near death (and died just as the book was being published) because he’d feared that the Catholic Church would be infuriated by his demotion of the world from the center of God’s creation. He was right to be scared. After Giordano Bruno proposed, among other heresies, that the universe was infinitely large with infinitely many worlds, he was tried by the Inquisition and burned at the stake in Rome in 1600.
Archimedes
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| https://www.sapaviva.com/archimedes-of-syracuse/ |
For one thing, there are a lot of funny stories about him. Several portray him as the original math geek. For example, the historian Plutarch tells us that Archimedes could become so engrossed in geometry that it “made him forget his food and neglect his person.” (That certainly rings true. For many of us mathematicians, meals and personal hygiene aren’t top priorities.) Plutarch goes on to say that when Archimedes was lost in his mathematics, he would have to be “carried by absolute violence to bathe.” It’s interesting that he was such a reluctant bather, given that a bath is the setting for the one story about him that everybody knows. According to the Roman architect Vitruvius, Archimedes became so excited by a sudden insight he had in the bath that he leaped out of the tub and ran down the street naked shouting, “Eureka!” (“I have found it!”)
Other stories cast him as a military magician, a warrior-scientist / one-man death squad. According to these legends, when his home city of Syracuse was under siege by the Romans in 212 BCE, Archimedes — by then an old man, around seventy — helped defend the city by using his knowledge of pulleys and levers to make fantastical weapons, “war engines” such as grappling hooks and giant cranes that could lift the Roman ships out of the sea and shake the sailors from them like sand being shaken out of a shoe. As Plutarch described the terrifying scene, “A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.”
In a more serious vein, all students of science and engineering remember Archimedes for his principle of buoyancy (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced) and his law of the lever (heavy objects placed on opposite sides of a lever will balance if and only if their weights are in inverse proportion to their distances from the fulcrum). Both of these ideas have countless practical applications. Archimedes’s principle of buoyancy explains why some objects float and others do not. It also underlies all of naval architecture, the theory of ship stability, and the design of oil-drilling platforms at sea. And you rely on his law of the lever, even if unknowingly, every time you use a nail clipper or a crowbar.
Archimedes might have been a formidable maker of war machines, and he undoubtedly was a brilliant scientist and engineer, but what really puts him in the pantheon is what he did for mathematics. He paved the way for integral calculus. Its deepest ideas are plainly visible in his work, but then they aren’t seen again for almost two millennia. To say he was ahead of his time would be putting it mildly. Has anyone ever been more ahead of his time?
Two strategies appear again and again in his work. The first was his ardent use of the Infinity Principle. To probe the mysteries of circles, spheres, and other curved shapes, he always approximated them with rectilinear shapes made of lots of straight, flat pieces, faceted like jewels. By imagining more and more pieces and making them smaller and smaller, he pushed his approximations ever closer to the truth, approaching exactitude in the limit of infinitely many pieces. This strategy demanded that he be a wizard with sums and puzzles, since he ended up having to add many numbers or pieces back together to arrive at his conclusions.
His other distinguishing stratagem was blending mathematics with physics, the ideal with the real. Specifically, he mingled geometry, the study of shapes, with mechanics, the study of motion and force. Sometimes he used geometry to illuminate mechanics; sometimes the flow went in the other direction, with mechanical arguments providing insight into pure form. It was by using both strategies with consummate skill that Archimedes was able to penetrate so deeply into the mystery of curves.
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Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later. This essential role of intuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics.
Archimedes concludes with the hope that “there will be some among the present as well as future generations who by means of the method here explained will be enabled to find other theorems which have not yet fallen to our share.” That almost brings a tear to my eye. This unsurpassed genius, feeling the finiteness of his life against the infinitude of mathematics, recognizes that there is so much left to be done, that there are “other theorems which have not yet fallen to our share.” We all feel that, all of us mathematicians. Our subject is endless. It humbled even Archimedes himself.
Limit (in calculus)
A limit is like an unattainable goal. You can get closer and closer to it, but you can never get all the way there.
the unattainability of the limit usually doesn’t matter. We can often solve the problems we’re working on by fantasizing that we can actually reach the limit and then seeing what that fantasy implies. In fact, many of the greatest pioneers of the subject did precisely that and made great discoveries by doing so. Logical, no. Imaginative, yes. Successful, very.
A limit is a subtle concept but a central one in calculus. It’s elusive because it’s not a common idea in daily life. Perhaps the closest analogy is the Riddle of the Wall. If you walk halfway to the wall, and then you walk half the remaining distance, and then you walk half of that, and on and on, will there ever be a step when you finally get to the wall?
The answer is clearly no, because the Riddle of the Wall stipulates that at each step, you walk halfway to the wall, not all the way. After you take ten steps or a million or any other number of steps, there will always be a gap between you and the wall. But equally clearly, you can get arbitrarily close to the wall. What this means is that by taking enough steps, you can get to within a centimeter of it, or a millimeter, or a nanometer, or any other tiny but nonzero distance, but you can never get all the way there. Here, the wall plays the role of the limit. It took about two thousand years for the limit concept to be rigorously defined. Until then, the pioneers of calculus got by just fine with intuition. So don’t worry if limits feel hazy for now. We’ll get to know them better by watching them in action. From a modern perspective, they matter because they are the bedrock on which all of calculus is built.
If the metaphor of the wall seems too bleak and inhuman (who wants to approach a wall?), try this analogy: Anything that approaches a limit is like a hero engaged in an endless quest. It’s not an exercise in total futility, like the hopeless task faced by Sisyphus, who was condemned to roll a boulder up a hill only to see it roll back down again over and over for eternity. Rather, when a mathematical process advances toward a limit (like the scalloped shapes homing in on the limiting rectangle), it’s as if a protagonist is striving for something he knows is impossible but for which he still holds out the hope of success, encouraged by the steady progress he’s making while trying to reach an unreachable star.
Calculus is more than a language
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| Photo by Roman Mager on Unsplash |
Calculus, like other forms of mathematics, is much more than a language; it’s also an incredibly powerful system of reasoning. It lets us transform one equation into another by performing various symbolic operations on them, operations subject to certain rules. Those rules are deeply rooted in logic, so even though it might seem like we’re just shuffling symbols around, we’re actually constructing long chains of logical inference. The symbol shuffling is useful shorthand, a convenient way to build arguments too intricate to hold in our heads.
If we’re lucky and skillful enough — if we transform the equations in just the right way — we can get them to reveal their hidden implications. To a mathematician, the process feels almost palpable. It’s as if we’re manipulating the equations, massaging them, trying to relax them enough so that they’ll spill their secrets. We want them to open up and talk to us.
In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity. That might come as a surprise to you, given that calculus has a reputation for being complicated. And there’s no denying that some of its leading textbooks exceed a thousand pages and weigh as much as bricks. But let’s not be judgmental. Calculus can’t help how it looks. Its bulkiness is unavoidable. It looks complicated because it’s trying to tackle complicated problems. In fact, it has tackled and solved some of the most difficult and important problems our species has ever faced.
Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity. Instead of cutting a big problem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them. Once that’s done, it solves the original problem for all the tiny parts, which is usually a much easier task than solving the initial giant problem. The remaining challenge at that point is to put all the tiny answers back together again. That tends to be a much harder step, but at least it’s not as difficult as the original problem was.
Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.
Story of laser
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| Photo by Gerardo Barreto on Unsplash |
Using observation and experiment, scientists worked out the laws of change and then used calculus to solve them and make predictions. For example, in 1917 Albert Einstein applied calculus to a simple model of atomic transitions to predict a remarkable effect called stimulated emission (which is what the s and e stand for in laser, an acronym for light amplification by stimulated emission of radiation). He theorized that under certain circumstances, light passing through matter could stimulate the production of more light at the same wavelength and moving in the same direction, creating a cascade of light through a kind of chain reaction that would result in an intense, coherent beam. A few decades later, the prediction proved to be accurate. The first working lasers were built in the early 1960s. Since then, they have been used in everything from compact-disc players and laser-guided weaponry to supermarket bar-code scanners and medical lasers.
Saturday, September 21, 2019
Saturday, September 14, 2019
Karakterinize uymayan bir tepki verdiğinizde inandırıcılığınızı zedelemiş olursunuz
Evet, bir mü’min, Rabbisiyle münasebetleri yanında insanlarla olan muamelelerinde de dinin emirleri istikametinde hareket etmeyi karakter hâline getirmelidir.
Biraz daha açacak olursak, şayet bir insan, kim olduğuna bakmadan herkesi sevgiyle kucaklama, karşılaştığı herkese tebessüm yağdırma, muhtaçlara yardım etme, çevresindekilere izzet ü ikramda bulunma gibi güzel sıfatları tabiat ve karakter hâline getirememişse, bir gün beklemediği çirkin bir muameleyle karşılaştığında farkına varmaksızın hırçın ve haşin bir tavır sergileyebilir. Böyle biri karşılaştığı her kötü muamele karşısında mü’mine yakışır şekilde mukabelede bulunmayı iradesine havale edeceğinden ciddi mânâda zorlanacak ve bazen falso yaşamaktan kurtulamayacaktır.
Tavır ve davranışlarındaki bu zikzaklar ise onun inanılırlık ve güvenilirliğini zedeleyecektir. İnanan gönüller olarak eğer biz çevremizde inandırıcı ve güven vaat eden biri olmak istiyorsak, gerek ibadetleri, gerek haramlardan sakınmayı ve gerekse de muamelata ait hususları tabiatımızın bir buudu hâline getirmeliyiz.”
İçeriden veya dışarıdan yapılan en alçakça zulümler karşısında bile karakterden taviz verilmemelidir!..
“Her şeye rağmen, kimi zaman insanın karakterinde, hâdisenin şiddetine göre çatlama ve kırılmalar meydana gelebilir. Karakterindeki kırılma, o insanın gayret-i diniyesinden kaynaklanabileceği gibi bazen de birilerinin hiçbir insaf ölçüsü tanımayan iftira ve hakaretlerinden, onun dem ve damarına dokundurmasından da kaynaklanabilir. Bu durum karşısında insan hiç farkına varmaksızın bir anda olumsuz bir havaya girebilir. Karşılıklı atışmalar ve tartışmalar yaşanabilir; kalbler kırılabilir.
Fakat unutmamak gerekir ki, ne olursa olsun, karakterinize uymayan bir tepki verdiğinizde inandırıcılığınızı zedelemiş olursunuz. Bu itibarladır ki hakiki bir mü’min, en alçakça saldırı ve tecavüzler karşısında bile karakterinden taviz vermemelidir. Mukabele edecekse bile, bu, edep ve ahlâk âbidesi bir mü’mine yakışır şekilde olmalıdır.
Yüksek Karakterli Sabır Kahramanları karakter kırılması yaşamamalı, güveni sarsmamalı ve daha büyük yanlışlara yol vermemelidir!..
Osman Şahin, tr724.com
Utangaç Çocuklar
Utangaç bir çocuğun ebeveyni çocuğunun geleceği konusunda genellikle endişelidir. Bilim insanları da bu endişelerin yersiz olmadığını söylüyor, çünkü araştırmalar, çocuklardaki çekingenliğin ileri yaşlarda kaygı bozukluğuna dönüşme riski taşıdığını gösteriyor. Ebeveynlerin utangaç çocuklarını koruma çabalarının ise durumu daha da kötüleştirmesi de mümkün.
Psikologlar ve çocuk gelişimi uzmanları, utangaç çocukları desteklemenin yollarını arıyor. New York Üniversitesi’nden psikolog Sandee McClowry’ye göre yapılması gereken şey, çocukların temel yapısal özelliklerini değiştirmeye çalışmadan, onları kendilerini rahat hissettikleri bölgelerin dışına çıkmaya ikna etmek. Onları oldukları gibi kabul etmek, utangaç çocuklar için çok önemli.
Psikologlar utangaçlığı sosyallikten kaçınma, sosyal etkileşimlere maruz kalınması durumunda ise sıkıntı ve gerginlik hissedilmesi olarak tanımlamaktadır. Utangaçlık üzerine çalışan araştırmacılar, hem insanlarla tanıştıklarında, hem de ilk defa karşılaştıkları durumlarda kaygıları tetiklenen çocukları daha iyi teşhis etmek için daha geniş bir kavram olan davranış tutukluluğundan yararlanıyor.
Utangaçlık, çocuklarda bir karakter özelliğidir. Psikologlar bu gibi karakteristik özelliklerin oldukça ısrarcı olduğunu belirtiyor. 1988 yılında Child Development dergisinde yayınlanan bir araştırmada 4 yaşında çocukların davranışları incelenmiş, aynı çocuklar 7 buçuk yaşındayken bir inceleme daha yapılmış. Araştırma sonucunda 4 yaşındayken utangaç olan çocukların da girişken olan çocukların da 7 buçuk yaşına geldiklerinde yine aynı davranışları sergiledikleri gözlemlenmişti.
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Aşırı korumacı ebeveynlerin aslında onlara zarar verdiğini belirten McClowry, anne ve babaların davranışlarında belirli bir denge tutturması gerektiğini vurguluyor. “Yapı iskelesi” olarak adlandırılan teknik, utangaç çocukların ebeveynleri için oldukça uygun görünüyor. Eğitimde kullanılan “yapı iskelesi” tekniği, öğrencilere başta yoğun destek verip bu desteği yavaş yavaş ve düzenli olarak azaltarak onların daha bağımsız hale gelmesini sağlamak anlamına geliyor. Bu teknik, utangaç çocukların kabuklarından çıkmalarına yardımcı olabilir.
McClowry, bu teknikle ilgili olarak kamp örneğini veriyor. Örneğin bir çocuk kampa gitmek istiyor ancak geceyi evden uzakta geçirmekten korkuyorsa, anne ve baba işe, çocuğun arkadaşlarını evlerinde kalmaya davet etmekle başlayabilir. Ardından bir geceyi büyükannelerinin evinde geçirerek çıtayı yavaş yavaş yükseltebilirler. Elbette anne ve baba “yapı iskelesi” tekniğini uygularken çocuğun rahatsızlık hissedip hissetmediğini de kontrol etmeli, çocuk daha fazla katlanamaz hale gelirse onu zorlamamalıdır. Ayrıca çocukların daha büyük olduğu durumlarda ebeveynler bu tecrübeyi onunla konuşabilir, ona nasıl hissettiğini, neyin daha iyi hissetmesini sağladığını ve bu adımdan sonra ne yapmak istediğini sorabilir.
Kaynak: Herkese Bilim Teknoloji
My Sister The Serial Killer
"I am not angry. If anything, I am tired," Korede says, faced with yet another bloody crime scene to scour, yet another body to dump. The first few times, her beautiful sister Ayoola's self-defense claims seemed plausible, but the bodies have added up. And Korede Googled it: Three murders makes you a serial killer.
My Sister, the Serial Killer, the wry debut novel by Nigerian writer Oyinkan Braithwaite, tests the bonds of family, when family comes armed. The title says it all: Ayoola likes to kill her boyfriends. Korede can't quite bear to see her get caught: "Ayoola needs me; she needs me more than I need untainted hands." So, the gloves come on and the bleach comes out.(NPR)
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