Without Calculus

Photo by Shubham Sharan on Unsplash

Without calculus, we wouldn’t have cell phones, computers, or microwave ovens. We wouldn’t have radio. Or television. Or ultrasound for expectant mothers, or GPS for lost travelers. We wouldn’t have split the atom, unraveled the human genome, or put astronauts on the moon. We might not even have the Declaration of Independence.

It’s a curiosity of history that the world was changed forever by an arcane branch of mathematics. How could it be that a theory originally about shapes ultimately reshaped civilization?

The essence of the answer lies in a quip that the physicist Richard Feynman made to the novelist Herman Wouk when they were discussing the Manhattan Project. Wouk was doing research for a big novel he hoped to write about World War II, and he went to Caltech to interview physicists who had worked on the bomb, one of whom was Feynman. After the interview, as they were parting, Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn’t. “You had better learn it,” said Feynman. “It’s the language God talks.”

For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question. In any case, it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations. Such equations describe the difference between something right now and the same thing an instant later or between something right here and the same thing infinitesimally close by. The details differ depending on what part of nature we’re talking about, but the structure of the laws is always the same. To put this awesome assertion another way, there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses it.

Isaac Newton was the first to glimpse this secret of the universe. He found that the orbits of the planets, the rhythm of the tides, and the trajectories of cannonballs could all be described, explained, and predicted by a small set of differential equations. Today we call them Newton’s laws of motion and gravity. Ever since Newton, we have found that the same pattern holds whenever we uncover a new part of the universe. From the old elements of earth, air, fire, and water to the latest in electrons, quarks, black holes, and superstrings, every inanimate thing in the universe bends to the rule of differential equations. I bet this is what Feynman meant when he said that calculus is the language God talks. If anything deserves to be called the secret of the universe, calculus is it.

By inadvertently discovering this strange language, first in a corner of geometry and later in the code of the universe, then by learning to speak it fluently and decipher its idioms and nuances, and finally by harnessing its forecasting powers, humans have used calculus to remake the world.

**

When I said earlier that without calculus we wouldn’t have computers and cell phones and so on, I certainly didn’t mean to suggest that calculus produced all these wonders by itself. Far from it. Science and technology were essential partners — and arguably the stars of the show. My point is merely that calculus has also played a crucial role, albeit often a supporting one, in giving us the world we know today.

Take the story of wireless communication. It began with the discovery of the laws of electricity and magnetism by scientists like Michael Faraday and André-Marie Ampère. Without their observations and tinkering, the crucial facts about magnets, electrical currents, and their invisible force fields would have remained unknown, and the possibility of wireless communication would never have been realized. So, obviously, experimental physics was indispensable here.

But so was calculus. In the 1860s, a Scottish mathematical physicist named James Clerk Maxwell recast the experimental laws of electricity and magnetism into a symbolic form that could be fed into the maw of calculus. After some churning, the maw disgorged an equation that didn’t make sense. Apparently something was missing in the physics. Maxwell suspected that Ampère’s law was the culprit. He tried patching it up by including a new term in his equation — a hypothetical current that would resolve the contradiction — and then let calculus churn again. This time it spat out a sensible result, a simple, elegant wave equation much like the equation that describes the spread of ripples on a pond. Except Maxwell’s result was predicting a new kind of wave, with electric and magnetic fields dancing together in a pas de deux. A changing electric field would generate a changing magnetic field, which in turn would regenerate the electric field, and so on, each field bootstrapping the other forward, propagating together as a wave of traveling energy. And when Maxwell calculated the speed of this wave, he found — in what must have been one of the greatest Aha! moments in history — that it moved at the speed of light. So he used calculus not only to predict the existence of electromagnetic waves but also to solve an age-old mystery: What was the nature of light? Light, he realized, was an electromagnetic wave.

Maxwell’s prediction of electromagnetic waves prompted an experiment by Heinrich Hertz in 1887 that proved their existence. A decade later, Nikola Tesla built the first radio communication system, and five years after that, Guglielmo Marconi transmitted the first wireless messages across the Atlantic. Soon came television, cell phones, and all the rest.

Clearly, calculus could not have done this alone. But equally clearly, none of it would have happened without calculus. Or, perhaps more accurately, it might have happened, but only much later, if at all.

Creativity vs Rules... and Math

You can make the case that all creative endeavors are about pushing against constraints. In the words of physicist Richard Feynman, “Creativity is imagination in a straitjacket.” Take the sonnet, whose tight formal restrictions—Follow this rhythm! Adhere to this length! Make sure these words rhyme! Okay… now express your love, lil’ Shakespeare!—don’t undercut the artistry but heighten it. Or look at sports. Humans strain to achieve goals (kick the ball in the net) while obeying rigid limitations (don’t use your hands). In the process, they create bicycle kicks and diving headers. If you ditch the rulebook, you lose the grace. Even the wacky, avant-garde, convention-defying arts—experimental film, expressionist painting, professional wrestling—draw their power from playing against the limitations of the chosen medium.

Creativity is what happens when a mind encounters an obstacle. It’s the human process of finding a way through, over, around, or beneath. No obstacle, no creativity.

But mathematics takes this concept one step further. In math, we don’t just follow rules. We invent them. We tweak them. We propose a possible constraint, play out its logical consequences, and then, if that way leads to oblivion—or worse, to boredom—we seek a new and more fruitful path.

Öğrenmeyi ve Hatırlamayı Kolaylaştıran Yöntem: Feynman Tekniği

İki tip bilgi vardır. Bir şeyin adını bilmeye yönelik olan bilgi ve o şeyi bilmeyi temel alan bilgi. Nobel ödüllü fizikçi Richard Feynman aralarındaki farkı şu çarpıcı anekdotta anlatır:

“Şu kuşu görüyor musun? Bu bir kahverengi gerdanlı ardıç kuşu, ona Almanya’da halzenfugel ve Çin’de ise chung ling deniyor. Ona verilen tüm bu adları bilsen bile yine de bu kuş hakkında hiçbir şey bilmiyor olursun. Bildiğin sadece insanlar hakkında bir şey olur, yani kuşa ne ad verdikleri. Şimdi bu kuş ötüyor, yavrularına uçmayı öğretiyor ve yazın ülkenin bir ucundan diğer ucuna kilometrelerce uçuyor ve kimse yolunu nasıl bulduğunu bilmiyor.”

Buradan da anlayabileceğiniz gibi bir şeyin adını/tanımını bilmek onu anladığınız anlamına gelmez hiçbir zaman. Bir fikri gerçekten anlıyor musunuz yoksa bu fikrin tanımını biliyorsunuz, bunu sınamanın bir yolu var. Buna Feynman Tekniği deniyor.

Sizler de Feynman Tekniğini,

1. Gerçekten anlamadığınız konuları/fikirleri anlamak için  

2. Anladığınız fakat sınavlarda unuttuğunuz konuları/fikirleri hatırlamak için  

3. Sınav öncesi etkili bir çalışma yöntemi olarak kullanabilirsiniz. Bu yöntemi kullanarak bir fikri uzun yıllar hatırınızdan çıkmayacak şekilde, kısa sürede derinlemesine kavrayabileceksiniz.

Feynman Tekniğine şimdi bir göz atalım:

1. Adım: Konuyu Belirleyin 

Boş bir kağıt alın. Öğrenmek istediğiniz konunun başlığını kağıdın en üstüne yazın.

2. Adım: Konuyu Bilmeyen Birine Anlatır gibi Anlatın  

Kağıdın geri kalanına konuyu hiç bilmeyen birine anlatıyormuşçasına, mümkün olduğunca karmaşık ifadeler kullanmaktan kaçınarak öğrendiklerinizi yazın. Bir çocuğun bile anlayabileceği kadar basit bir dil kullandığınızda kendinizi de konuyu daha derin bir seviyede anlamaya ve konular arasındaki ilişki ve bağlantıları basitleştirmeye zorlamış olursunuz. Aynı zamanda yazdığınızı sesli olarak tekrar etmek çok daha etkili olacaktır.

3. Adım: Takıldığınız Noktada, Kaynağa Geri Dönün 

2. adımda hatırlamakta ya da anlatmakta zorlandığınız yerler olduğunu fark ettiğinizde konu hakkında çalıştığınız kaynaklara geri dönün. Öğrendiklerinizi kağıda aktarabilecek hâle gelinceye kadar tekrar tekrar okuyun ve çalışın. Sözgelimi biyolojiden yazılınız var ve evrimi basit cümlelerle açıklamakta zorlanıyorsunuz. Biyoloji kitabınızı açın ve evrimle ilgili kısmı yeniden okuyun. Şimdi kitabı kapatın ve yeni bir boş kağıt alarak öğrenmiş olduklarınızı yazın. Bu aşamayı sorunsuzca hâlletiyseniz, asıl çalışma kağıdınıza dönerek çalışmaya devam edebilirsiniz.

4. Adım: Basitleştirin ve Benzerlikler Kurun  

Artık kağıda döktüklerimizi gözden geçirebiliriz. Einstein’ın “Bir şeyi 6 yaşında bir çocuğa anlatamıyorsanız, siz de anlamamışsınız demektir” sözünden de anlayabileceğimiz gibi karmaşık bir jargon kullanıp kafa karıştırıcı açıklamalar yapmak yerine, dilimizi basitleştirmek ve benzerlikler kurmak anlamayı kolaylaştıracaktır. Opsiyonel olarak: 6 yaşında birini bulup, öğrendiklerinizi ona anlatmayı deneyin. Sorunsuz bir şekilde anlıyorsa, siz de gerçekten anlamışsınız demektir.

Bu harika yöntem yalnızca öğrenmeyi ve hatırlamayı kolaylaştırmakla kalmıyor, aynı zamanda farklı düşünme şekillerine pencere açarak fikirleri baştan aşağı yeniden inşa etmemizi sağlıyor. Fikir ve konuları daha derinden anlamamızı kolaylaştırıyor. Hepsinden önemlisi, sorunlara bu şekilde yaklaşarak, ne konuştukları hakkında en küçük bir fikri bile olmayanları anlamamızı sağlıyor.

Kaynak: Matematiksel

Richard Feynman: The Difference Between Knowing the Name of Something and Knowing Something

Richard Feynman, who believed that “the world is much more interesting than any one discipline,” was no ordinary genius.

His explanations — on why questions, why trains stay on the tracks as they go around a curve, how we look for new laws of science, how rubber bands work, — are simple and powerful. Even his letter writing moves you. His love letter to his wife sixteen months after her death still stirs my soul.

In this short clip (below), Feynman articulates the difference between knowing the name of something and understanding it.

See that bird? It’s a brown-throated thrush, but in Germany it’s called a halzenfugel, and in Chinese they call it a chung ling and even if you know all those names for it, you still know nothing about the bird. You only know something about people; what they call the bird. Now that thrush sings, and teaches its young to fly, and flies so many miles away during the summer across the country, and nobody knows how it finds its way.

Knowing the name of something doesn’t mean you understand it. We talk in fact-deficient, obfuscating generalities to cover up our lack of understanding.

How then should we go about learning? On this Feynman echoes Einstein, and proposes that we take things apart:

In order to talk to each other, we have to have words, and that’s all right. It’s a good idea to try to see the difference, and it’s a good idea to know when we are teaching the tools of science, such as words, and when we are teaching science itself.

[…]

There is a first grade science book which, in the first lesson of the first grade, begins in an unfortunate manner to teach science, because it starts off with the wrong idea of what science is. There is a picture of a dog–a windable toy dog–and a hand comes to the winder, and then the dog is able to move. Under the last picture, it says “What makes it move?” Later on, there is a picture of a real dog and the question, “What makes it move?” Then there is a picture of a motorbike and the question, “What makes it move?” and so on.

I thought at first they were getting ready to tell what science was going to be about–physics, biology, chemistry–but that wasn’t it. The answer was in the teacher’s edition of the book: the answer I was trying to learn is that “energy makes it move.”

Now, energy is a very subtle concept. It is very, very difficult to get right. What I mean is that it is not easy to understand energy well enough to use it right, so that you can deduce something correctly using the energy idea–it is beyond the first grade. It would be equally well to say that “God makes it move,” or “spirit makes it move,” or “movability makes it move.” (In fact, one could equally well say “energy makes it stop.”)

Look at it this way: that’s only the definition of energy; it should be reversed. We might say when something can move that it has energy in it, but not what makes it move is energy. This is a very subtle difference. It’s the same with this inertia proposition.

Perhaps I can make the difference a little clearer this way: If you ask a child what makes the toy dog move, you should think about what an ordinary human being would answer. The answer is that you wound up the spring; it tries to unwind and pushes the gear around.

What a good way to begin a science course! Take apart the toy; see how it works. See the cleverness of the gears; see the ratchets. Learn something about the toy, the way the toy is put together, the ingenuity of people devising the ratchets and other things. That’s good. The question is fine. The answer is a little unfortunate, because what they were trying to do is teach a definition of what is energy. But nothing whatever is learned.

[…]

I think for lesson number one, to learn a mystic formula for answering questions is very bad.

There is a way to test whether you understand the idea or only know the definition. It’s called the Feynman Technique and it looks like this:

Test it this way: you say, “Without using the new word which you have just learned, try to rephrase what you have just learned in your own language.” Without using the word “energy,” tell me what you know now about the dog’s motion.” You cannot. So you learned nothing about science. That may be all right. You may not want to learn something about science right away. You have to learn definitions. But for the very first lesson, is that not possibly destructive?

I think this is what Montaigne was hinting at in his Essays when he wrote:

We take other men’s knowledge and opinions upon trust; which is an idle and superficial learning. We must make them our own. We are just like a man who, needing fire, went to a neighbor’s house to fetch it, and finding a very good one there, sat down to warm himself without remembering to carry any back home. What good does it do us to have our belly full of meat if it is not digested, if it is not transformed into us, if it does not nourish and support us?

https://www.farnamstreetblog.com/2015/01/richard-feynman-knowing-something/