Ask a mathematician

Miscellaneous and Personal Stuff I Must Share
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Perhaps a physicist should answer this one. Physics motivated Newton to invent calculus. (I’m not sure about Leibniz, the other inventor.) Newton actually used it surprisingly sparingly, but only because he was so good at geometry. Nowadays, when you teach introductory physics, calculus is used to reproduce his results.

You could argue that every advance in theoretical physics uses calculus. Even though computers have been used extensively since the 70s or 80s, that which computers compute use calculus. Sure, other areas of mathematics, such as Lie Algebras, are used also, but only in conjunction with calculus. Mechanics (most moving things you can see), Electromagnetics (electricity and magnetism), Thermodynamics (heat), Statistical Mechanics (the basis of Thermodynamics and models of materials, like metals), Relativity, and Quantum Mechanics all use calculus. So, name a technology based on physics and developed after Newton, and it is based on a physical model that uses calculus.

Mathematics uses calculus almost everywhere, also. Pretty much all fields of mathematics encompassing the ideas of “continuous” and “change” use calculus. Modern chemistry is based on Quantum Mechanics, which again requires calculus. Even neglecting QM, chemists still deal with rates of change of compounds and other ideas that require calculus. Physics based fields, such as atmospheric physics, solar physics, meterology, climatology, cosmology, geophysics all use calculus. To a lesser extent, biology also uses it, to model population changes, for example, as discussed before. Another example would be the predictions based on genetic changes for the age of the human race use calculus. I’m not sure there is a science that doesn’t use calculus, and by extension calculus was used to develop almost any technology. (It may not be obvious how biotechnology uses calculus, but, for example, the mathematics behind the testing that shows something is safe uses calculus.)

Calculus is not restricted to just the sciences, either. Let’s call economics a science and ignore it, although it uses calculus. Business also uses calculus. The big investment firms hire “rocket scientists” that use calculus. The finance majors and accounting majors I knew in college all were required to take calculus, also.

122

See, I told you my name was indistinguishable. :slight_smile:

Incidentally, my experience with Numb3rs is precisely the same as yours.

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Thanks guys your examples about calculus really helped me understand.

One more question, though…why did it take until the end of the 17th century to come up with it? What was going on in the world then that demanded a need for such calculations that wasn’t going on in previous years?

In the book I read (Quicksilver, by Neal Stephenson fwiw) the main (fictional) character observed Newton drawing diagrams of shadows cast by the sun and whatever was in his garden, and lots of circles and lines. Not sure if this was while he was coming up with Principia Mathematica or not. Did he come up with calculus to learn more about astronomy maybe?

Also…what is in the Principia Mathematica? Functions upon functions or is it more like some theories to help people make more functions?

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Archimedes was doing things that looked a lot like calculus back in the 200’s B.C. Some say he would have invented calculus if only he had had the algrebraic concepts and notation to do so.

The answer to your question that I’ve often heard is that it took algebraic notation, and analytic geometry, as invented a generation or so before Newton by people like Descartes and Fermat, before calculus as we know it could really be formulated.

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I read Chandrashakhar’s (sp?) translation of Newton’s Principia. I would urge any of the mathematically inclined on this board to do so. I was struck with the sheer simplicity of most of Newton’s proofs. He hardly needed calculus. I read an interview with Chandrashakhar before I bought the book. He makes the comment that he tried every proof first, then read Newton’s. Every time Newton’s was better, in Chandra’s own opinion, and Chandra won the Nobel prize in physics for astrophysical calculations he did on the cruise from England to here!

Newton was preceded by Galileo and Kepler, who were geniuses themselves. IIRC, it was who Galileo proclaimed “the book of nature is written in mathematics”. Perhaps that motivated what I consider Newton’s greatest invention. Before him, no one had ever constructed a mathematical model of the physical world. Equations were used to predict positions, most notably by Kepler, but no one had thought to say something like "the motion of objects can be modeled by these equations, " or “the force of gravity acts at a distance according to this equation”. The former set of equations require calculus. (Aristotle, who from what I’ve read had a pretty sophisticated grasp of experiment and innfinity, never thought of a mathematical model. For example, he did think of performing an experiment to see if the sum of the angles within a triangle is 180 degrees, which would “verify” Euclidean geomtry.) Newton was able to compare his equations to reality by showing he could derive Kepler’s work. (Kepler, btw, probably could not have succeeded without the invention of logarithms, which reduce mutliplication, division and exponentiation to addition and subtraction, by Napier. Even with them, it took Kepler something like 20 years.)

I would say Newton was aided by the concept of cartesian coordinates, the development of algebra. He was aided by the whole philosophical climate that mathematics was the way to describe nature, and experiment was to way to learn about nature (Bacon). He required the work of Kepler and Galileo to verify his models, and they both sprang somewhat naturally from the climate of the time.

Without Kepler and Galileo, there would have been no proof of Newton’s laws. Without the proper philosophical environment, initiated by Bacon, no Kepler, and maybe no Galileo. Could Newton have invented the rest, and succeeded where Archimedes and Aristotle did not? I don’t think Leibniz would have. I wouldn’t bet against Newton, though. The guy was really incredible.