FF Logo

Formal Formulations


The Form and Substance of
Numerical Arithmetics

The Substance of a Numerical Arithmetic is the purely intellectual concept of the "Enumeration of Objects", whether the Objects are real or imagined.

The algorithms for Counting may have a Form which is either Experiential or Representational.

A brief discussion to clarify the distinction among an Arithmetic, an Algebra, and a Calculus is also presented. It may surprise the kind reader to understand that these three mathematical concepts merely correspond to the abilities of counting, generalizing, and calculating, respectively.


Experiential Forms

Experiential Forms of Counting typically involve bending fingers, matching pebbles, and other such primitive "one-to-one" verification systems. This Form of Counting is visible and directly observable. The primary advantage of such systems is that they provide immediate, absolute, and non-controvertible proof of the integrity of the algorithm. For example, we either display three fingers, or we don't. The fact of the matter, the fact of the numerical Substance , is immediately real, clear, and evident.

For the primitive analogical consciousness of early humans, the absolute reality and integrity of the experiential counting algorithms was a strong counterpoint to the uncertainty of their other perceptions. It is precisely this recognition of certainty that allowed analytical consciousness it validate itself as a viable Form of reasoning. Just remember that, once upon a time, merely counting to three was very "hi-tech".


Representational Forms

Representational Forms of Counting typically involve the use of a symbolic physical Form to represent a specific numeric Quantity. For example, the intellectual concept of the numeric Quantity called "nine" is represented, variously, as:


Interestingly, the Greeks and Romans never formalized the concept of zero, as the Mayan and Arab scholars did.

The Romans attempted to extend their system of symbols to allow for any quantity to be represented. This was fine for merely counting objects, but made any formal calculations impossible. With Roman numerals, the value of any symbol depends on the symbols to the immediate left and right of the particular symbol. They never discovered the Fundamental Theorem of Arithmetic, namely

         Any Natural Number, N, can be represented as:

                             N = n * B + r
         where:  n, B, and r are Natural Numbers such that 0 <= r < B,
                 and B is called the Number Base.

The Mayans made use of the concept of zero ( Maya Zero ) to count in what we now call a "base 20" number system. By placing a dot above their symbol for a value between zero and nineteen, they essentially multiplied the value by a factor of 20. Although they never delved into the world of fractions, their system did respect and exploit the concept of how Form can be used to extend the meaning of a small number of symbols.

The Form of Arabic numbers has two very interesting properties.

Although all of these, and others numbering systems, have equivalent Substance in that they allow Objects to be enumerated, their Form are very different and impart different levels of utility. This is exactly why The Homeless Mathematician cries out that the Form and Substance must be respected with equal dignity. Imagine building a computer, or doing your income taxes, using Roman Numerals.


Arithmetic, Algebra, and Calculus

And here, kind reader, The Homeless Mathematician must draw an unusual, but extremely useful, distinction. This distinction distinguishes a Calculus of Symbols from a mere Collection of Symbols. Any set of scribbles, distinct or otherwise, may be grouped to form a Collection of Symbols. However, a Collection of Symbols must have certain formal properties before it may be called a Calculus of Symbols, or more simply, a Calculus. The formal progression is as follows:

Copyright © 1996. Formal Formulations. ~ ~ ~ Send e-Mail to comment@formal.com.
###### hits since 95/11/28. Updated 96/02/01.