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1 Organic Mathematics New approach to Hilbert’s 6th problem Introduction Presentation Moshe Klein , Doron Shadmi 17 June 2009 2 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. 3 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 22 4 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 1 Total = Or(5) 24 5 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. 6 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. 7 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 . . . 8 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: # 9 1=1 The point’s identity is clearly known a 10 2=(1)+(1) The identity of the points is in a superposition. b a b a 11 2=(1)+1 The identity of the points is not in a superposition. a b 12 Representation of n=2 Or(2) = 2. b a b a a b 13 Representation of n=3 Or(3) = 3. c b a c b a c b a c b c b a c b a 14 4=(1)+(1)+(1)+(1) Superposition of identities: dddd cccc bbbb aaaa 15 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 16 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 17 4=(3)+1 Recursion of n=3 within n=4. A’ ccc bbb aaad B’ C’ bb aacd abcd 18 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 19 OR(n) algorithm 20 OR(5) detailed representation Unclear ID Clear ID 21 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. 22 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). 23 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. 24 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. 25 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). 26 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. 27 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. 28 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) 29 Thank you for listening.. Moshe Klein ,Doron Shadmi Gan Adam L.T.D Gan_adam@netvision.net.il ISRAEL 30