The character that Plato gives us is truly wonderful. I too have a fondness for him but....

Tell us truthfully, Nawbut. Are you sure your affection is not for the example he offers of what happens to people who decline professional legal representation?

"Well...you could defend yourself but let me tell you what happened to..."


I doubt somehow that he lived up to the myth Plato created. But he must have been singular enough to have inspired that myth. A magical blend of two minds?

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Outside of a dog, a book is man's best friend. Inside of a dog it's too dark to read.
Marx (G.)

Actually, calculus was developed in India centuries before either man was born.

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"Some people complain about the system. The system is not good, so they can't do anything. It's an excuse. Freedom is in your heart." (Jin Xing)

Einstein because he studied all those before him.

I steered this discussion slightly off topic in discussing those great minds from history for whom I have a 'fondness'. IRL, ICantBreathe and others were properly discussing the 'cleverness' of the individuals, and perhaps we should get back to that.

Though a narrowed or temporally-confined personality, for me, would be indicative of a certain want-of-cleverness'....in the greatest.

Einstein, Socr-ato, John Harrison, Heisenberg, Heidegger(?), Shakespeare....

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"A man's work is nothing but this slow trek to rediscover, through the detours of art, those two or three great and simple images in whose presence his heart first opened."

They were able to solve a few isolated functions, but they did not have a full scope understanding of the calculus principles. Rather, they understood a couple of aspects of it, but did not recognize the larger set of guiding principles (evident in the fact that they never developed proofs). But they were not unique in this. Ancient Greeks and Romans realized a few points of calculus also, more than 1000 years prior to the Indian school in question (in fact the word calculus comes from the Romans).

It's not like Leibniz and Newton just completely developed the complete set of principles from scratch though. They had a lot of previous knowledge to work from.

That is true of the Ancient Greeks (such as Archimedes), but Bhaskara II (1114-1185) clearly understood the basic principles of calculus long before Newton, Leibniz or even the Kerala School.

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"Some people complain about the system. The system is not good, so they can't do anything. It's an excuse. Freedom is in your heart." (Jin Xing)

There are two subsets of calculus, differental and integral. Differential calculus is the set of principles for describing rates of change - this is fairly easy and more or less common sense. Integral calculus is the set of principles for relating the rate of change to other quantities. It's not nearly so easy or straight forward as differential calculus. A comparison can be made by saying that the derivative is like the tree, whereas the integral is like the forest. Sort of.

So anyway, Bhaskara and later individuals of the Kerala School, figured out ways to apply what are principles from differential calculus, but they didn't have any understanding whatsoever of integral calculus. Had they understood calculus fully, I can assure you, the world we know would be a very, very different place.

What I'm trying to say is that saying ancient Indians developed calculus is like saying that the first person who created a spoke-like object invented the wheel.

DUPLICATE POST

The rate of change is always related to some other quantity (technically). It would be more accurate to state, therefore, that Bhaskara II is to calculus what the Wright Brothers were to the airplane. Today's airplanes are far, far more advanced than what the Wright Brothers flew in 1903, but they are based on the same principles. It is worth noting that "Rolle's" theorem was actually discovered by Bhaskara, and while I know next to nothing about calculus, it is rather obvious to me that integral calculus is a logical extension of that and similar theorems.

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"Some people complain about the system. The system is not good, so they can't do anything. It's an excuse. Freedom is in your heart." (Jin Xing)

Yes, that's pretty much it, but that's not the type of relationship I was referring to when I was describing what integral calculus is. I just couldn't think of a good way to put it into non-technical terms.

It would be more like designing a rudder and understanding what it does without understanding how it would be used in conjunction with many other mechanisms to achieve flight. The 'calculus' of Bhaskara and others couldn't 'fly'.

Rolle's Theorem is absolutely common sense once you understand the concept of derivation. And while Rolle's Theorem is necessary to make the step to integral calculus, the progression isn't that straight forward (otherwise it wouldn't have taken over 1000 years for somebody to make it).