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Math is everywhere. Everything can be assigned a number that has some meaning.
The rock you just picked up might have a mass of 40 grams; with a volume of 15
cubic millimeters. Your child might be 6 years old and in grade 1. Your bank
account has a many digit number and has a certain number of dollars and
a number of cents at any given time. At the last Star Wars movie you sat in the
5th seat in the 10th row from the screen on its left
side. So, whatever you might say exists has one or more numbers associated with
it. In fact, not much thought is really made about these numbers yet we use
them day in and day out, we acquire this ability right along with our spoken
language as soon as we can speak. We do this in terms of answering the
questions how much and how many.
Mathematics is based on numbers; the characteristic logical thought patterns
found in the study of mathematics stem from the relationships amongst these
numbers, and in general we wish to know if such relationships are true in
general, never true, or true only under certain circumstances. So, to fully
appreciate mathematics, start with understanding these numbers and how real
world problems tended to direct our understanding in certain ways that led to
what we call the field of Mathematics today.
To answer "how many", or derive a mathematical representation, we use
numbers such as 1, 2, 3, etc. This mathematical notation wasn’t always used.
The ancient Romans used capital letters, I for 1, II for 2, III for 3, IV for 4
(one less than 5), V for 5, VI for 6 (one more than 6), X for 10, L for 50, C
for 100, etc. (see
Roman Numerals). So 56 would be LVI, that is 50 + 5
+ 1; 54 would be LIV. Although it’s possible, addition with this style of
notation is tedious. What was more important at the time, perhaps, was the
cardinality of the number. The number represented quantity. You are lucky you
weren't a Roman mathematician. (see
Cardinality) So, what is XCVII + LXIV ?
The ancient Greeks, the Pythagoreans in particular,
considered numbers to be reality itself. Starting with, one, each number in
turn had a very specific meaning and purpose in their world view. You probably
recall something about the Pythagorean Theorem, or perhaps were introduced to
geometry using a text based on Euclid’s Elements. In any case, the study of
numbers, that is, the study of mathematics will start here for us.
The Pythagoreans used pictographs for representing
numbers and mathematical meaning; one can imagine using pebbles placed in the
sand. One was a single pebble, all by itself, a single point in the sand. Two
was represented with two pebbles, three with three pebbles, etc. However, these
pebbles weren’t laid out in just any old way. The Pythagoreans thought about
numbers in terms of geometry, shapes they experienced in every day experience.
Geometry = Math to a Pythagorean. Thinking mathematically, a single pebble
is a point. Two pebbles defined a line. Three pebbles (arranged as an
equilateral triangle, two on the bottom and one vertex above the two, that is,
a line as a base and a point as the vertex) defined a triangle, four pebbles a
square, etc.
Numbers and reality where one and the same to Greeks.
For example, one meant unity, not divisible, the generator of all numbers, and
more abstractly, reason itself. Monad (meaning unity in Greek) was their term
for one. Two represented opinion, diversity, and represented the first female
number. Dyad was the word they used for two. Three was the first possible sum
of the numbers so far, monad plus dyad, but for the Greeks, unity combined with
diversity meant harmony and they called three the first male number. (Of course
this line of thought is quite debatable today, however, it does give insight
about the Greeks and how they approached mathematics.) Four was represented as
4 pebbles arranged in a square, a shape with four equal sides, and with all
being equal, four represented fairness and justice in particular. Phrases such
as “I was treated squarely by your people…” still ring true to this original
Greek meaning.
Another good
Math history reference.
...Russ Lewis
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