Doron Shadmi Posted October 14, 2005 Report Posted October 14, 2005 Dear Prof. Alain Connes, Last Thursday my friend Moshe Klein called your secretary, asking to arrange a meeting between us. Your secretary suggested sending an email to you, and I am glad to do so. I will be staying in Paris with my wife. Four years ago, Moshe Klein heard your lecture about Non-commutative Geometry, which was given in "100 to Hilbert" U.C.L.A conference. You closed your lecture with these words: " ...We need a new understanding in Mathematics which is based on geometry more then on logic". For the past 20 years I have developed a mathematical framework, which uses geometrical notions, in order to research the most fundamental mathematical concepts, (like set, number, point, segment, function, continuum, discreteness, finite, non-finite,... etc.) according to this orientation. I think that some of my results may be useful to your own research, and can contribute to the language of Mathematics and its logical reasoning, in general. Euclid in his book of elements, started by the definition of a point: Definition 1: A point is that which has no part. In other words, a point is an atomic element (indivisible, non-partitioned, non-composed, etc.) Euclid did not define a line as an atomic element, and the result is that a line can be defined in terms of points, whereas only a point is considered an atomic element. During my research about the point and the segment concepts, I have found that if we understand a segment in terms of an atomic element, we invent/discover new mathematical universe, which enables us: a) A more precise understanding of the current mathematical paradigm. b) To create a paradigm-shift in the most fundamental notions of the language of Mathematics and its logical reasoning. In order to demonstrate this ability, let us examine, for example, the Set concept. But now, both a point and a segment are atomic elements, or in other words, none of them is defined by the other. A set can be empty {} XOR non-empty {x}. Let x be a XOR b, where a is a single point . and b is a single segment _ If x = a then x can be in{x} XOR out{}x of some set, and it cannot be simultaneously in{x} AND out{}x of some set. If x = b then x can be simultaneously {x} AND {}x some set. For example: The single point case cannot be but: in{.} XOR out{}. The single segment case can be: in{_}, out{}_ and also in AND out{ _}_ In AND out{ _}_ is a property that can be found only in a single segment, and no collection of many elements have the property of being simultaneously in{x} AND out{}x of some set. Some claim, for example, that the set of all even numbers is "in" and "out" of the set of prime numbers (with intersection = {2}). Both consist of many elements, which contradicts my argument about the inability of a collection to be both "in" AND "out" of some set. But as I explained, at this first-order level I research the simplest case, where a single point and a single segment are compared to one another. In this case {2} intersection cannot be in{x} AND out{}x of the intersection. Furthermore, at this first-order level a segment or a point must not be understood as geometrical objects, but as two atomic information forms, which are fundamentally different from one another. Conclusion: A segment cannot be defined in terms of points. Furthermore, a segment cannot be defined in terms of a collection/sequence of segments or sub-segments, because one and only one segment has the property of being simultaneously in AND out{ _}_ some set and no collection/sequence of segments or sub-segments has this property, for example: in{_._._} XOR out{}_._._ , and never {_._._} in AND out{}_._._ Strictly speaking, we define a fruitful universe that exists between continuous realm (represented by a single ___} and a discrete realm (represented by collections/sequences of ...(points) _ _ _ (segments) or any arbitrary mixing of them (_.._ _.)) . Moshe Klein calls _._.._. the fundamental "Morse key" of our own cognition,where 0 is . and 1 is _ Monadic Mathematics takes the next step by developing the Organic Natural Numbers, which are based on the associations between a continuous segment and a finite/non-finite collection of points/segments. A preface of my number system can be found in: http://www.geocities.com/complementarytheory/TAP.pdf An overview of my axiomatic system can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf A new point of view about the natural numbers can be found in: http://www.geocities.com/complementarytheory/ONN1.pdf http://www.geocities.com/complementarytheory/ONN2.pdf http://www.geocities.com/complementarytheory/ONN3.pdf Most of these papers have a non-standard format, because most of the subjects need new representation techniques, in order to correctly express the new notions. A lot of work has to be done, and I hope that you will find my work interesting enough for a possible meeting between us. Sincerely yours, Doron Shadmi Quote
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