# Formal Formulations

The and of
Numerical Arithmetics

The 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 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 s

Experiential s of Counting typically involve bending fingers, matching pebbles, and other such primitive "one-to-one" verification systems. This 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 , 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 of reasoning. Just remember that, once upon a time, merely counting to three was very "hi-tech".

Representational s

Representational s of Counting typically involve the use of a symbolic physical to represent a specific numeric . For example, the intellectual concept of the numeric 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 ( ) 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 can be used to extend the meaning of a small number of symbols.

The of Arabic numbers has two very interesting properties.

• Each symbol which represents a digit can be made with a single brush stroke. This allows any numeric substance to be represented with a minimal effort. It is no more difficult to represent eight ( 8 in Arabic, but VIII in Roman numerals ) than it does two ( 2 in Arabic, but II in Roman numerals ).
• Each digital position, within the of an Arabic number, has a value which is ten times greater than the digital position its immediate right. This is a which is infinitely extendible.

Although all of these, and others numbering systems, have equivalent in that they allow Objects to be enumerated, their are very different and impart different levels of utility. This is exactly why The Homeless Mathematician cries out that the and 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:

Arithmetic

Any collection of formal symbols used to represent specific values, or Operands, of , and the specific Operations which represent the interaction of the specific Operands.

Algebra

The formal extension of an Arithmetic to include unspecified Operands of , along with the Operands and Operations of the underlying Arithmetic. Unspecified values are called Formal Variables, when represented symbolically.

Calculus

To quote G. Spencer-Brown in The Laws of Form: Call calculation a procedure by which, as a consequence of steps, a is changed for another, and call a system of constructions and conventions, which allows calculation, a Calculus. In other words, these are the formal symbols which represent the resulting , of the Operations, as well as the specific values.