Gottfried Wilhelm Leibniz

[lpetrich]

 Gottfried Wilhelm Leibniz (1646 - 1716) has a Google Doodle, as I write this.

He did a *lot* of work in mathematics.

Systems of linear equations

He expressed systems of linear equations in matrix form: A.x = b for matrix a, vector x to be found, and vector b.

He also worked on determinants of square matrices, a sort of sum of products of the matrix's elements. It turns out to be useful for solving systems of linear equations, and Leibniz discovered Cramer's rule: each component of x has this value:

det(A with that component's location replaced by b) / det(A)

where "det" means calculate a determinant. Though Cramer's rule is impractical for all but the smallest systems of linear equations, it is nevertheless good theoretically. If det(A) = 0, then matrix A is "singular", and a system of equations with it either has no solutions or some infinite set of solutions.

Leibniz worked out how to calculate determinants recursively, in terms of determinants of subsets of one's matrix. Though theoretically correct, for size n, this algorithm takes O(n!) operations. Leibniz also did some Gaussian elimination for systems of linear equations, a method that requires only O(n^3) operations. It can also be used on determinants, with that same overall runtime.

Functions

He had a general idea of function, finding some value by doing some operations on some input value. He used the idea in geometry: a point (x,y) on a curve has y being a function of x, or else both x and y being functions of some parameter t.

Calculus

This is the mathematics of slopes of curves (differentiation) and areas underneath curves (integration). He invented it independently of Sir Isaac Newton, and he got involved in a nasty fight over who invented it. For integration, he used a big S, for "sum", and for differentials of quantities, he used the letter d. His notation became much more widely used than Newton's, though Newton's is still used in some cases, like a dot over something for time derivatives.

He used infinitesimals, quantities that are smaller than any nonzero number but are nevertheless nonzero. For instance:
df(x)/dx = (f(x+dx) - f(x))/(dx)
where dx is an infinitesimal. In the nineteenth century, Karl Weierstrass developed the notion of limits, thus making infinitesimals unnecessary.

With calculus, I'm sure, he derived this formula for pi:
(pi/4) = 1 - 1/3 + 1/5 - 1/7 - 1/9 + ...

This is from arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

How to derive it. Consider integral of 1/(1+x^2) by x. Set x = tan, a trigonometric function, and thus y = arctan(x) its inverse. Then the integral becomes
integral of 1/(1 + tan^2) * d(tan)/dy by y = integral of 1 by y = y

Returning to the original integral, 1/(1+x^2) = 1 - x^2 + x^4 - x^6 + ... Integrating over x gives x - x^3/3 + x^5/5 - x^7/7 + ...

Thus, what we wanted to derive.

 

[lpetrich]

Leibniz did a lot of other work.

Theology

He argued in his book Theodicy that this is the best of all possible worlds that God could have created, something that Voltaire satirized in Candide.

Pangloss was professor of metaphysico-theologico-cosmolo-nigology. He proved admirably that there is no effect without a cause, and that, in this best of all possible worlds, the Baron's castle was the most magnificent of castles, and his lady the best of all possible Baronesses.

"It is demonstrable," said he, "that things cannot be otherwise than as they are; for all being created for an end, all is necessarily for the best end. Observe, that the nose has been formed to bear spectacles—thus we have spectacles. Legs are visibly designed for stockings—and we have stockings. Stones were made to be hewn, and to construct castles—therefore my lord has a magnificent castle; for the greatest baron in the province ought to be the best lodged. Pigs were made to be eaten—therefore we eat pork all the year round. Consequently they who assert that all is well have said a foolish thing, they should have said all is for the best."

Symbolic thought

The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right.

He did some early work on symbolic logic to make possible his "calculus ratiocinator".

He also worked on binary numbers, the subject of that Google Doodle: someone writing binary numbers with a quill pen. He was not the first, and he noticed a binary-number sequence in a version of China's I Ching. But he simplified them and associated them with various properties of sets and operations on them.

He also designed a mechanical adding machine, one that could do all four arithmetic operations. He also thought of giving such a machine a memory in the form of presence or absence of marbles.

He was evidently groping toward a unified way of implementing reasoning and computation, one that would be pursued with greater success in the mid 19th century by Charles Babbage and Ada Lovelace, and in the early 20th century by Alan Turing and others. It laid the theoretical groundwork for computer design: one could do essentially any operation as a combination of some simple ones on lists of ones and zeros. Sometimes a very big combination, and sometimes very big lists, but still a combination on lists.

An especially nice thing about this work is that it is independent of its operations' substrate. As long as one can implement its operations, it does not matter how they are implemented. Thus, we have gone through numerous computing technologies from purely mechanical clockwork to hundreds of millions of transistors on a single computer chip.

Leibniz did a lot of other things, but I'll leave off here.

 

[Wiploc]

Il Monstro. He was called the monster. Not because he was evil, but because he was just insanely smart.

He was untutored, untaught, but he got into his father's library and taught himself to read--in several languages.

Quicksilver, which is wonderful for a number of reasons, gives a sympathetic portrait of him.

 

[Speakpigeon]

Leibniz did a lot of other work.
(...)
He did some early work on symbolic logic to make possible his "calculus ratiocinator".
(...)

Yeah, and apparently he couldn't finish.

And seems nobody did.

Well, I guess someone will have to eventually.
EB

 

[steve_bank]

The standardized notation set the stage for all the ensuing science and technology.

The princciples of integration and differentiation preceded Newton and Liebnitz. Same with Newtons laws of motion.

 

[lpetrich]

The standardized notation set the stage for all the ensuing science and technology.

Or at least the more mathematical parts. Also, I don't know if it is as much "standardized" as it is "more convenient".

The princciples of integration and differentiation preceded Newton and Liebnitz.

Please point out where the two were anticipated.

Same with Newtons laws of motion.

Please point out where he was anticipated.

 

[steve_bank]

I used to have a history of math. Egyptians had techiques for calculating areas of odd shapes. Tangets to a curve, the derivative, was around well before the Europeans.

An ancient scroll shows Chinese were doing practical linear algebra.

Newton and Leibniz did not create in a vacuum. Neither did Maxwell or Einstein.

I watched a show a few yeras back the movement of science and math through history. It follows the money. As Arabs and Persians declined and European developed excess wealth, it moved with the money. Newton used Persian observation data for his work. The first comprehensive algebra text came from the Arabs. It was not as complete as Newton, the Arabs had the basic principles of motion. They had texts on optics.