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Tue, Oct 09 2007 100 Heisenberg and Prokofiev Each sees their own image of reality.
One sees facts and mathematical proofs
that show that the reality of the other is elsewhere.
The other sees emotions and perceptions.

And what kind of bird are You - said duck - if you can't swim.
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Fri, Jul 13 2007 140 The Origin of Racism A true disbeleiving Jew can mathematically prove the Protocols of Zion from The Origin of Species.

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David Bowie
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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,
through mathematics.


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
NP complete.
These are defined by means of a concept called Reduction.

A reduction is a logical transofrmation of one problem
to another.
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.

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Fri, Jul 13 2007 400 Functions The blending function is that which takes an apple
and turns it into apple juice.
You can also use the blending function on strawberries
and get strawberry juice.
But you can't blend oranges.

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For clarity, we will call this function "blended()".

We use parenthesis, (brackets),
to show that a function acted on something
by writing like this:

blended(apple)

and then we can write down the result:

blended(apple) = appleJuice

and

blended(x) = xJuice

but only if x is in the universe where blended() operates.

the function:
double(x) = 2x
can be drawn easily on paper:

Make a horizontal line from one side of the paper to the other.
Divide it into 21 parts by drawing short vertical lines
across the line, and number them from -10 to 10.
Lets call this drawing "The X axis".
The Y axis is vertical, and is otherwise the same.
Once you draw it, we call the drawing, "The Axis System".

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If X is 5, than double(x) is 10,
so we draw a dot above the 5 on the X axis, (where x=5),
at the height where y=10.

double(-6) = -12 and is not on the paper,
so we draw all the dots from x equals -5 to 5,
and connect the dots with a ruler.

Together with the axes, we call this drawing now:
"The graph of y = 2x".


What would you blend to get absolutely nothing?

Well, if you put nothing in, you'll get nothing back,
that's for sure.

But is it that obvious that you cannot put a few ingredients
that would cancel each other out and you'll still get nothing.
For example if you put in the blender
matter and anti matter in just the right quantities?

The root of a function is the value of X where the value of Y is zero.
So at least one root of blended() is also zero.

Finding the root of a function is like solving an equation:

blended(x) = nothing.
What is x?

y = double(x)
what is X if y is zero?
y = 2x
0 = 2x
2x = 0
x = 0

But with the function:
y = 2x - 2
the root is 1.

So what is the root of y = x*x?
Zero.

of:
y = x*x -4 ?
2, right.

Well if we look carefully, -2 is also a root of this function.

The equation:
square(x) - 4 = 0
Is the equation to find the root of the function:
y = square(x) - 4

If in an function definition x is squared,
but no higher powers are there,
we call the function a Quadratic function,
and the equation a Quadratic equation.

I you draw the line of the function, the roots of the equation
are those where y=0, or when the line crosses the X axis.

We also use letters for the numbers in the definition itself,
and not to be confused with the numbers from the axes,
we take the letters A,B and C for the Constants in the equation,
and X,Y for the variable numbers from the axes
we originally marked X and Y in our drawing of the graph.

y = a*sqaure(x) + b*x + c

The quadratic equation to find the roots of
the quadratic function is:

a*sqaure(x) + b*x + c = 0

If all this down to earth mathematics
was a bit stressful,
remember at least that quadratic is not Quadruped,
nor Aquatic, which is more on the Oceanic side,
that math can predict the future,
that the suns will rise tomorrow,
in a galaxy, far, far, away....


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Keep Up the Good Work.
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