Transcript Organic Mathematics

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Organic Mathematics
New approach to Hilbert’s 6th problem
Introduction Presentation
Moshe Klein , Doron Shadmi
17 June 2009
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Introduction
“…An old French mathematician said:
A mathematical theory is not to be considered
complete until you have made it so clear that you can
explain it to the first man whom you meet on the
street...“
David Hilbert , in his lecture at ICM1900
Distinction is a very important part of our life. Similarly to
Hilbert’s analogy about the completeness of a mathematical
theory, Organic Mathematics claims that any fundamental
mathematical theory is incomplete if it does not deal with
Distinction as first-order property of it.
This presentation is focused on the structure of Whole Numbers.
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The Partition function as a motivation
of the notion of Distinction (part 1)
We are all familiar with the Partition Function Pr(n).
Pr(n) returns the number of possible Partitions of a
given number n.
The order of the partitions has no significance.
Pr(5) =7 as follows:
5=5
5=4+1
5=3+2
5=3+1+1
5=2+2+1
5=2+1+1+1
5=1+1+1+1+1
Total = 7.
n
Pr(n)
1
2
3
1
2
3
4
5
6
5
7
11
7
8
15
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The Partition function as a motivation
of the notion of Distinction (part 2)
But here is the Catch!
Every partition of n gives us some way
of looking at the Whole number n.
We go one step further and analyze
every partition:
We define (this will take some time) for
every given partition a of a given
number n , the number of distinctions
it has.
We call that number D(a) .
We denote the sum of the D(a) ‘s of all
partitions a of a given number n ,
Or(n).
Or(n) will be called the Organic
Number of number n .
We begin by understanding the way we
got the first 4 organic numbers.
All the rest of them are easily calculated
by using the principals we will
introduce now.
Partition: a
D(a)
5
0
4+1
9
3+2
6
3+1+1
3
2+2+1
3
2+1+1+1
2
1+1+1+1+1
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Total = Or(5)
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Representation of Distinction as points
on a line (part 1)
Let us consider n=2.
Situation A and Situation B describe two different states of
Distinction that stand at the basis of partition 1+1:
Situation A is where we cannot distinguish between a and b.
Situation B is where we can distinguish between a and b.
D(1+1) =2.
So Or(2) =2.
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Representation of Distinction as points
on a line (part 2)
3=1+1+1:
Situation A’ is the case where one cannot distinguish between a,b,c points.
3=2+1:
In B’ – one can distinguish one of the points – say a – but cannot
distinguish between – say – b and c.
C’ – is quivalent to situation B of n=2; where in this case we can
distinguish between the three points.
Situations B’ and C’ are a recursion of n=2 within n=3.
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The Organic Sequence for n>3
So far we have seen that Or(n)=n, for n=1,2,3.
However, this is changed if n>3.
For example: Or(4) =9.
In the next few slides we will look at Or(n) for n>3.
The first 12 values of Or(n) are :
1, 2, 3, 9, 24, 76, 236, 785, 2634, 9106, 31870, 113371 . . .
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The case of 4
For n=4, things become more complex.
4=1+1+1+1
4=2+1+1
4=2+2
4=3+1
(4=4)
Let us explain n=4 in details, in the next few slides.
In order to do that we introduce new notation:
#
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1=1
The point’s identity is clearly known
a
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2=(1)+(1)
The identity of the points is in a superposition.
b
a
b
a
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2=(1)+1
The identity of the points is not in a superposition.
a
b
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Representation of n=2
Or(2) = 2.
b
a
b
a
a
b
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Representation of n=3
Or(3) = 3.
c
b
a
c
b
a
c
b
a
c
b
c
b
a
c
b
a
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4=(1)+(1)+(1)+(1)
Superposition of identities:
dddd
cccc
bbbb
aaaa
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4=(2)+(1)+(1)
Recursion of n=2 within n=4
A
c c
B
d d
cc
bbb b
aaa a
dd
bb
aa aa
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4=(2)+(2) - order has no significance
D(2+2) =4 -1=3 because order has no significance.
A A
b bbb
a aaa
B
A
bb
ab aa
A B
bb
aa a b
B
B
ab ab
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4=(3)+1
Recursion of n=3 within n=4.
A’
ccc
bbb
aaad
B’
C’
bb
aacd
abcd
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Representation of 4
There are nine different distinctions in 4. Or(4) =9.
4=(1)+(1)+(1)+(1)
4=(2)+(1)+(1)
4=(2)+(2)
4=(3)+1
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OR(n) algorithm
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OR(5) detailed representation
Unclear
ID
Clear
ID
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Ramanujan’s way of thinking
Ramanujan published 3900 formulae,
Without being able to prove them!
We think that Ramanujan’s way of
thinking is different than the way most
mathematicians think.
We call it “Parallel”, as opposed to
“Serial”.
Organic Mathematics looks at lines and
points as different atoms that are not
derived of each other.
Organic Numbers are the result of
“Parallel” (line-like) AND “Serial”
(point-like) observations of the concept
of Number.
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Dialog in Mathematics
Young children apply an “Organic way
of Thinking” based on imagination,
intuition, feelings and logic.
While working with children, we have
developed a serial way of thinking
(points) as well as a parallel way of
thinking (lines).
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The Interaction Between Parallel and
Serial Thinking
1)Do you see “Necker cube” from outside or inside ?
2) In which direction the Pyramid is turning?
If you see it turning in clockwise you are using the right
side of your brain. If you see it turning on the other
way, you are using the left side of your brain.
Some people see both directions, but most people see
only one direction. See if you can change directions
by shifting the brain's current perception.
BOTH DIRECTIONS CAN BE SEEN!
This can explain the way Ramanujan got his discoveries
in Number theory, engaging simultaneously both
sides of his brain.
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Paradigm’s change: Distinction
Organic numbers are based on a new philosophy, which says
that a point and a line are two abstract observations that if
associated, enables to define things mathematically, where
Distinction is their first-order property.
The line represents a Parallel Thinking Style where things are
understood at-once, without using a step-by-step analysis.
The point represents a Serial Thinking Style where things are
understood by using a step-by-step analysis.
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New approach to Hilbert’s 6th problem
In 1935 the EPR thought experiment introduced Non-locality into
Physics.
The 6th’ problem of David Hilbert is about mathematical treatment
of the axioms of physics:
“The investigation on the foundation of geometry suggests the
problem : to treat in the same manner, by means of axioms, those
physical sciences in which mathematics plays an important part ; in
the first rank are the theory of probabilities and mechanics“.
Organic Numbers bring Distinction to the front of the
Mathematical research, by associate the non-local (line-like
observation) with the local (point-like observation).
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Historical Background to Organic
Mathematics
1. Introduction of Non-Euclidean Geometry, Bolyai and
Lobachevsky 1823.
A new observation of Geometry.
2. The 23 problems of Hilbert and the Organic Vision in
ICM 1900.
The Organic Vision: From Hilbert’s Lecture (last page): ”Mathematical science is
in my opinion an indivisible whole, an organism whose vitality is conditioned
upon the connection of its parts[…] The organic unity of Mathematics is
inherent in the nature of its science.”
3. The Sixth Problem of Hilbert - Mathematical Axioms
for Physics.
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Continue..
4. Foundations of Probability Theory by Kolmogorov
1933.
5. Non-locality in Quantum Theory - the EPR thought
experiment 1935.
Or(n) uses non-locality as one of its building-blocks.
6. Remarks on the foundation of Mathematics Seminar in
Cambridge by Wittgenstein 1939.
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Summary
1) Since Euclid's “Elements” 13 Books, for 2,500 years,
Mathematics did not develop “parallel” observation
methods.
2) We believe that by using both non-local (parallel) and local
(serial step-by-step) observations of the mathematical
science, fundamental mathematical concepts are changed
by a paradigm-shift (presented in a more advanced
presentation)
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Thank you for listening..
Moshe Klein ,Doron Shadmi
Gan Adam L.T.D
[email protected]
ISRAEL
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