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Arithmetic as a Mathematical Model

In elementary school, mathematics is often taught as a set of rules for counting and computation. Students learn that there is only one right answer and that the teacher knows it. There is no room for judgment or making assumptions. Students are taught that Arithmetic is the way it is because it's the truth, plain and simple. Often students go on to become trapped in this view of the universe. As fairy tales fade from the imagination, so is mathematical creativity lost.

There is evidence that Mathematics and Arithmetic existed over 3000 years ago, but only the very well educated leisure class had access to it. The rules for simple computation only were developed recently, so much of the computation of sums and products was much more complicated. Imagine adding and multiplying Roman Numerals for example. Because of this difficulty, computations were laid out only to solve very specific practical problems.

Although mathematics was mainly limited to solving practical problems, people in eastern countries and South America developed methods for solving systems of linear equations, the concept of zero, and negative and irrational numbers. Without the place holder zero, modern arithmetic computations would not be possible.

The Greek contribution was the use of axioms and the first mathematical modeling. The Greeks viewed mathematical models as abstract idealizations of the real world. Numbers were represented as the lengths of line segments. Drawn line segments were viewed as imperfect inexact representations of abstract lines. This was the beginning of mathematical modeling.

The Greeks created the concept of commonly accepted assumptions which were the basis of their mathematical modeling. The assumptions were called axioms and postulates and formed the basis for a model. Then general minor conclusions (called lemmas), major conclusions (called theorems), and consequences (called corollaries) were proved using logic.

Now, for the first time, the link between mathematics and reality was broken forever. A line had no width or thickness and a plane had no thickness. These were abstract assumptions to which nothing in the universe corresponded. The only requirement for the assumptions which were the basis for a mathematical model was that they did not contradict one another.

Then nothing much happened until the 14th or 15th century except that eastern countries developed the concept of place value, the modern base-ten number system and the basis for Algebra. In addition, the beginnings of modern computational methods were discovered. The computational methods were known as "the method of the Indians".

It wasn't until the 19th century that the assumptions of arithmetic were investigated thoroughly. Giuseppe Peano, a mathematician in the 1800's, formulated the standard axioms that today serve as the basis for standard arithmetic of the natural numbers. Those axioms can be found here: http://en.wikipedia.org/wiki/Peano_axioms. Thus the natural numbers and operations on the natural numbers became a mathematical model in the spirit of the Greeks. Today every number system in use is a mathematical model based on axioms, lemmas, theorems, and corollaries.

The rise of the concept of a mathematical model took Mathematics far away from the other sciences because now mathematical systems could be expressions of pure imagination, completely divorced from any reality. Mathematicians could now explore the unknown and let their imagination take them to another universe with different assumptions. And this was the bases for the major mathematics discoveries of the 20th century.

One interesting, though minor, consequence of the concept of mathematical models was the development of the Calculus of Infinitesimals. Basically when calculus was being developed, many mathematicians assumed the existence of infinitely small quantities called infinitesimals. This use of infinitely small quantities allowed the concepts and theorems of Calculus to be developed without the use of limits at all. Later mathematicians took fault with this intuitive approach and re-developed the same concepts using limits. It was taught that the concept of limits was required for rigorous proofs to take place in Calculus.

But the concept of infinitesimals was vindicated using the concept of a mathematical model. It was shown that the approach using infinitesimals was internally consistent and could be developed from basic assumptions - the concept of the infinitesimal did not contradict anything in the standard models.

The concept of Mathematical Model has allowed the export of arithmetic to sets of objects completely unrelated to numbers on the real number line. Mathematical systems called Fields, Groups, Rings, and others are systems that satisfy some, but not all, of the assumptions used in standard arithmetical computations, and the objects you add, subtract, or multiply in these systems are conceptually far removed from real numbers.

Just like the parallel universes of science fiction, by changing a few assumptions you can create your own mathematics. To understand mathematics today one needs to understand the power of mathematical modeling.
 
 
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Comments

You have opened my mind to the wonder and enjoyment of math thru your eloquent description of the subject. Thank you for presenting mathematical modeling as the foundation for understanding the math
we use today. This is very helpful to folks like me, who have always struggled with this subject.
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