[FOM] FOM - reply to Haney and others
ehlert-abler at sbcglobal.net
Fri Jan 27 15:15:27 EST 2006
Persistent questioning, especially from Richard Haney, emphasizes that the
mathematical membership of [FOM] are interested in basic mathematical
questions, but are not exactly ruthless in their pursuit of foundations.
The question of foundations is not so much, "What is basic in mathematics?"
as, "What causes mathematics in the first place?"
Two main answers are commonly available.
1) Mathematics is wholly arbitrary, a product of the mind. It has its
ultimate source in human biology and sociobiology.
2) Mathematics is not arbitrary, but is more found than made.
For several reasons, the first answer is wrong. We can't arbitrarily
decide, for example, that "2+2=5", or that "pi=3". More profoundly, if
mathematics is an indirect product of human evolution, which is driven by
selective advantage, then there is an equation that explains why people can
do equations, i.e., because selective advantage is expressed by an equation.
Solution 1 is circular.
The second solution looks like closet Platonism: If mathematics is found,
where are the pieces hidden that we are trying to find? The right answer,
then, must be a non-Platonic form of solution 2. Such a solution may be
found by reducing the problem to one that has already been solved. The
orbits of planets, for example, are ellipses, and have to be, yet no one
imagines that the orbits are material objects that were floating in space
until they captured the planets, or the planets captured them.
The orbits are a non-material, or abstract consequence of the dimensional
properties of matter; and I suggest that mathematics has a similar
foundation. A more complete treatment of this and related questions will be
found in my book,
Abler, William L. 2005.
Structure Of Matter, Structure Of Mind.
Philadelphia: Bainbridge; Sofia: Pensoft,
which I respectfully submit to the [FOM] readership.
At least some founding properties of mathematics are demonstrably not
arbitrary. The discreteness property of the symbols, whether spoken or
written, is a necessity because blending symbols would lose information to
their neighbors through information-averaging, and would eventually lose
their identities. The property of discreteness is more famously a necessity
for the gene.
Equations are based on the property of symmetry, i.e., symmetry about the
"equals". When we say that "2+2=4" is "true", and that "2+2=5" is "false",
we are observing that 2+2=4 is symmetrical, while 2+2=5 is not. Thus the
concept of truth in mathematics has its beginnings, at least, in the
property of symmetry. Since asymmetry is not self-regulating, and is thus
unstable, symmetry is also a necessity for mathematics, at least as far as
equations are concerned.
William L. Abler
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