and
of
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.
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".
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.
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.
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:
,
and the specific Operations which represent the interaction of the specific Operands.
,
along with the Operands and Operations of the underlying Arithmetic. Unspecified values are called Formal Variables, when represented symbolically.
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.
to
comment@formal.com.