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|Fri, Jul 13 2007||155||Gauss||
His true genious was to connect mathematics and reality,
by prooving and classifying from reality,
the true axioms of mathematics.
In math, we first take axioms for granted to be true.
If we use a system of logic where the axioms are not true,
then the system can only be used for teaching logic:
What is true then, in an axiom, if we can not proove it is true,
to make it different from the axioms that lead to
no Aristotelian logic constructs that we will ever use?
This is in fact the difference. They pertain to reality,
and Gauss has shown us that,
In complexity theory -
a field in computer science theory that deals with
the complexisty of computer programs -
there is this set of very complex problems called
These are defined by means of a concept called Reduction.
A reduction is a logical transofrmation of one problem
If one can reduce problem A to problem B,
then B must be a harder problem or equal,
and if B were soleved,
then A is considered solved,
because it is reducible to a solved problem.
Computer science theorist have collected a large set
of such problems, showing reductions in both directions,
thereby establishing equivalence of the problems,
without ever solving any.
They are too complex, probably, to be actually solved.
But if one is ever solved in the future,
then automatically all others are solved too.
From a theoritical standpoint,
this concept is very powerful.
A world of logic consturcts with many reductions
after long research, has been built,
atop a foundation yet to be discovered.
Reduction is a mathematical construct, not just a concept.
It is a methemtical description of what logically
constitutes what is termed by this, and proof
of why it is logically correct.
Gauss connects math to reality. It is implied
by looking at his conclusions from the mathematical point
of view of reductions, and follow aristotelian logic.
The primary concept is obvious:
Reality can be counted,
and pure mathematical deductions can be drawn.
But if a certain demographic of reality can be
prooven statistically to be correct,
whereas we know this fact also to be correct
from other observations of reality,
then we can take the statistical conlusion from Gauss
to be inhetrently correct, because we know it to be true
from other sources.
This being the case we start a process of reduction
that will in the end up saying the follwing:
The axioms are prooven to be true, (in as far as reality is true,)
in all cases where the conclusions regarding reality,
which we otherwise know to be true somehow,
where the axioms are used in mathematically
arriving at the same known conclusion.
The details of the reduction process itself are not important
in this case, but it is worth noting
that the same can probably be prooven
by other means, even though mathematical proofs
in general do not necesserily work if you reverse
the order of deduction.
"A prooves B" does not necessarily imply "B prooves A",
and so a reversal may or may not be logically correct.
In other words, Gauss has classified the axioms
of mathematics that were used for prooving the central
limit theorem as correct.
The axiom, and the thorem are equivalent,
to the extent that conclusions from the theorem,
can be shown from other means to be correct.
For example, if election poles match statistical predictions,
then it is proof that A+B always equals B+A.