About Foundation of Math

The science of number and the science of form are called – mathematics. The science of number is covered by arithmetic and its big children, algebra and calculus. The science of form (triangles, circles, cubes, and such) is covered by geometry, and all its offspring.

Zero is the Father of Mathematics

The introduction of the symbol “o” is the greatest step ever taken in the history of arithmetical science, and one that completely escaped the Greeks and Romans. The zero was surely a happy thought. It passed from the Hindus of India to the Arabs of Arabia, Egypt, and Coney Island. The figures we now use came in the same package from the same people. These figures took the place after awhile of the letters used by the ancients, I, V, X, L, for example. Our present figures, however, are deformed descendents of letters.

It was the zero (which means empty space) that brought about in some way the idea that a figure represents ten times as much in any place as in the succeeding place. That is, the zero gave birth to the position system in number.

The old Hindu (the great mathematician Aryabhatt) who first thought of this scale of ten (units, tens, hundreds, thousands, tens of thousands, etc.) got his notion probably from Nature's own counting machine – the ten fingers of man.

Columbus came around us, somewhere, in 1492. Well, in 1478 the first arithmetic came out. Four years later another arithmetic appeared. These pioneer books explained the use of the zero system. They caused the shape of the figures to be accepted as fixtures, and thus put a stop to the changes in their shape that had been going on.

Basic Difference Between Arithmetic, Algebra and Calculus

A B C of arithmetic, algebra, and calculus, is something like this: Say the side of a square is two inches, then the square itself is 2x2 = 4 inches. So we express this fact in arithmetic thus:

4=2²

In algebra we may use the letters a and b instead of the figures 4 and 2, thus:

A=b²

In arithmetic, you see, we are dealing with a ‘specific’ square and no other. In algebra we are dealing with a square of ‘any’ size; that is, with all squares.

Now in calculus they say the size of the square ‘depends on’ the length of its side. That is, the square increases or diminishes as the side changes in length. That is, the square increases or diminishes as the side changes in length. You may say that this is an absurdly simple principle. So it is, but it is a very powerful idea in mathematics, just as that other simple thing, gravity, is powerful in physics. So we will now take x and y instead of a and b or 4 and 2, and say the size of x depends on the size of y, or in technical language, x is a "function" of y. The way that notion is expressed in calculus is:

x = f(y)

You need not bother your head about why this artifice is more powerful in problems than arithmetic or algebra, because it is not used except in complex questions, such as finding out how things change, or rather at what rate they change.

Thus, roughly speaking, in arithmetic we have known quantities, in algebra unknown quantities, in calculus changing quantities.