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Mathematical notation is used in mathematics, and
throughout the physical sciences, engineering, and economics. The complexity of such
notation ranges from relatively simple symbolic representations, such as one and two; to conceptual symbols, such as + and dy/dx; to equations, functions, and variables. See table of mathematical symbols for a systematic
list of the notation.
Counting
It is believed that a mathematical notation was first developed, at least 50,000 years
ago, in order to assist with counting. Early mathematical ideas for counting were represented by collections of rocks, sticks, bone, clay, stone, wood
carvings, and knotted ropes. The tally is a timeless way of counting. Perhaps the oldest known
mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa all used the tally mark method of accounting for numerical
concepts.
Geometry becomes analytic
The mathematical viewpoints in geometry did not lend themselves well to
counting. The natural numbers, their relationship to fractions, and the identification of continuous quantities actually took millennia to take form, much less allow for the development of notation. It
was not until the invention of analytic geometry by Rene Descartes that geometry became more subject to a numerical notation.
However, some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs, for example.
The power and authority of the custom of geometrical style of Theorem and Proof was even followed by the great Isaac Newton's Principia Mathematica,though he did not use geometry to invent his concepts,
but instead blazed a new trail through the invention of calculus to understand the
System of the
World.
Counting is mechanized
After the rise of Boolean algebra and the development of positional notation, it became possible to mechanize simple circuits
for counting, first by mechanical means, such as gears and rods, using rotation and
translation to represent changes of state, then by electical means, using changes in
voltage and current to represent the analogs of quantity. Today, of
course, computers use standard circuits to both store and change quantities,
which represent not only numbers, but pictures, sound, motion, and control.
Computerized notation
The rise of expression evaluators such as calculators and slide rules were only part of what was required to mathematicize
civilization. Today, keyboard-based notations are used for the e-mail of mathematical expressions, the Internet shorthand notation. The wide use of programming languages, which teach their users the need for
rigor in the statement of a mathematical expression (or else the compiler will not accept
the formula) are all contributing toward a more mathematical viewpoint across all walks of life.
The negative side of computerized standards
There is a part of mathematics which is not algebraic, but which seems to use a different faculty of the mind. For those
people with such minds and imaginations, like Isaac Newton's, if they are
to benefit from the wide availability of mathematical devices, then they will need to be served by more graphical, visual, aural,
tactile, and temporal modalities in
notation, as a first step.
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