posted on Sep, 4 2004 @ 12:58 AM
"Isn't the only pure reality mathematical? Everything else is perception? I suppose a personal conception of math is really a series of concepts,
but outside of everything, there exists numbers."
I’ll try to respond to you, Ktprktpr. Although my degree was in math, I make no claims as an authority. Forgive me if this seems like a dissertation,
but your questions are not easily answered with only a few words. Perhaps this should be a thread all on it’s own for those who might be interested.
As it relates to “reality”, though, I’ll continue on.
The content of "pure" mathematics is ultimately derived from the material world. The idea that “mathematical truths” are the result of some special
kind of knowledge, or divine inspiration, simply doesn’t bear out under serious examination. The notion that “pure” mathematics is absolute thought is
far from the truth. In the final analysis, mathematics deals with the quantitative relationships observed in the “real world”. For instance, the
decimal system is not the result of logical deduction or "free will," but instead came about because we have ten fingers. The word "digital" is a
derivative of “fingers” in Latin. To this day, we often count our “material” fingers beneath a “material” desk to arrive at the answer to an
“abstract” mathematical problem. In doing this, we unconsciously are demonstrating the same behavior used by early humans when they learned to count.
If humans had evolved with 12 fingers instead of ten, then we would no doubt have a duo-decimal number system, based on 12 digits, instead of 10. A
base 12 number system is perfectly valid. In fact, computers commonly implement logic using base 2, base 8 and base 16 number systems. And so, from
the very beginning, the development, and very foundation, of mathematics is based upon human observations of the material world; our 10 fingers to be
Mathematics has always been riddled with inconsistencies, contradictions and paradoxes. For example, Galileo pointed out that every integer (whole
number) has only one perfect square, and every perfect square is the square of only one positive integer. Thus, in a sense, there are just as many
perfect squares as there are positive integers. This may seem to make sense, but upon further consideration it immediately leads to a logical
contradiction. It contradicts the axiom that the whole is greater than any of its parts, in that not all positive integers are perfect squares, and
all perfect squares form only a subset of all positive integers. I hope that made sense.
At any rate, that’s only one of many paradoxes which have plagued mathematics ever since the Renaissance.
Mathematical systems developed in the 19th century forced mathematicians to realize, reluctantly, that mathematics proper and the mathematical “laws”
were indeed not “truths”. It was discovered that several different geometries apparently fit spatial experience equally well. So, the question arose,
“Which geometry was the truth?”. They couldn’t all be the truth. It would seem that mathematical design, in and of itself, was not inherent in nature,
or if it was, the mathematics created by human beings was not necessarily of that same design. Therefore, the key to “reality” was lost. Mathematics,
in other words, has no objective basis, but is purely the product of the human mind!
In recent years, the limitations of mathematical models to represent “reality” has come under intense scrutiny. Differential equations, for example,
one of the jewels in the bag of Isaac Newton’s astonishing discoveries, represents “reality” as a continuum, where changes in time and position occur
smoothly and without discontinuity. It offers no room for sudden breaks or interruptions. In nature, however, discontinuity takes place all the time.
Chaos theory “attempts” to address the issues involving breaks in continuity. Euclidean geometry reduces “reality” to planes, spheres, lines, cones
and points. However, the shapes defined in Euclidean geometry are totally inadequate to express the complexity and irregularity of surfaces found in
nature. Clouds are not spheres, mountains are not cones and lightning does not travel in a straight line.
Despite the advances of the 20th century, it’s noteable that such a large number of what seems to be simple phenomena are still not clearly understood
and do not lend themselves to expression through mathematical formulation. And so, despite exaggerated claims, mathematics remains only a rough
approximation to the real world. Even the most advanced mathematical system cannot describe “reality”, and has no validity outside certain bounds.
Frankly, I doubt seriously if human beings will ever be in a position to understand “reality” on an objective level. I’m not even sure there is such a
thing as an objective level or an objective "reality". In the meantime, though, it can’t hurt to try …
PS: Sorry that was so long-winded. It was the only way I could think of to answer the questions. I hope it at least made a small bit of sense, though.