A mathematician s view on Three gives birth to innumerable things

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1 A mathematician s view on Three gives birth to innumerable things Qinghai Zhang Department of Mathematics, University of Utah, 155 S E., Rm 233, Salt Lake City UT Abstract Philosophy underlies science while science manifests philosophy; The same statement applies to Daoism and mathematics. In this paper the author discusses the connection between general topology and a famous passage in Dao De Jing: Dao gives birth to One; One gives birth to Two; Two gives birth to Three; Three gives birth to innumerable things. It is usually accepted that the two here refers to Yin and Yang, but what is the meaning of three? why does three give birth to all things? The author answers these questions in the language of general topology and show that the cited Daoism paragraph encompasses almost the entire range of modern mathematics. Key words: Yin and Yang; Open and closed sets; Topological spaces; measure; Tai-Chi; Turbulence modeling; 1 Introduction Most doctoral degrees awarded in science and engineering are termed as doctors of philosophy (Ph.D.), instead of doctor of mathematics, doctor of physics, doctor of chemistry, doctor of environmental engineering, and so on. This fact points to a common notion that Philosophy underlies science while science manifests philosophy. Among the scientific disciplines, the role of mathematics to various engineering disciplines is analogous to the role of philosophy to various scientific disciplines. The relations can be very well explained by an everyday example on water: we use water for cooking soup, making coffee, or taking a shower. The act of cooking soup or making coffee is relatively easy compared to distilling and purifying natural water. On the one hand, once water gets into the form of soup or coffee, it loses its generality in that it would be not suitable for a shower (unless one loves such a shower); on the 6th International Conference on Daoist Studies LMU, June

2 other hand, it is due to the needs of soup, coffee, and shower that water is useful. What is mathematics then? Mathematics used to be limited to calculating numbers; but nowadays mathematicians have agreed that mathematics is the science of patterns [1]. To name a few examples, number theory studies the patterns of numbers, geometry studies the patterns of shape, calculus studies the patterns of motion, logic studies the patterns of reasoning, and topology studies the patterns of topological properties. Daoism bibles such as Dao De Jing also concerns patterns, e.g., a famous passage goes as Dao gives birth to One; One gives birth to Two; Two gives birth to Three; Three gives birth to innumerable things; Philosophical patterns like this are not as precise as those in mathematics. If this is considered as a disadvantage, this is also a big advantage in that philosophical patterns can be applied to different disciplines and sometimes inspire scientific work. In this paper, the author connnects the quoted paragraph to basic concepts of general topology[2], a fundamental subject of modern mathematics. In the language of topology, One is the chosen topology of a certain set; Two are the concepts of open and closed; in addition to Two, Three also includes the concept of metric, which produces various metric spaces widely used in modern mathematics. 2 One gives birth to Two As long as we are discussing something, we already fall out of the realm of Dao gives birth to One, one interpretation of which is nothing produces something. So we might as well just start from the second sentence. We are so familar with the concepts of Yin and Yang. After all, we are living in a world full of dichotomy: day and night, man and woman, good and bad, right and wrong. In the realm of topology, Yin and Yang correspond naturally to the concepts of open and closed. Axiom 1 Let X be a non-empty set. A class T of subsets of X is a topology on X iff T satisfies (1) X and belong to T; (2) The union of any number of sets in T belongs to T; 2

3 (3) The intersection of any two sets in T belongs to T ; The members of T are called T -open sets, and the pair (X,T ) is called a topological space. Definition 2 Let X be a topological space. A subset A of X is a closed set iff its complement is an open set. Theorem 3 Let X be a topological space. Then the class of closed subsets of X possesses the following properties: (1) X and are closed sets; (2) The intersection of any closed sets is closed; (3) The union of any two closed sets is closed. Proposition 4 In a topological space X, a subset of A is open iff its complement is closed. One common example of topology is the usual topology in the Euclidean plane. Let an open disc D R 2 be denoted by D = {q : d(q,o) < δ}, where o and δ > 0 are the center and the radius of the disc, d(q,o) is the distance between q and o. Let A be a subset of R 2, a point p A is an interior point of A iff there exists an open disc D p such that p D p A. The set A is open iff each of its points is an interior point. The whole plane is clearly an open set since for any give point, we can find inside the plane an open disc containing the point. It can be easily verified that the set of all open sets in the plane is a topology, i.e., it satisfies all the conditions in Axiom 1, as shown in Fig. 1. This topology, known as the usual topology in the plane, is connected to a famous saying in Daoism, (The Dao is) the smallest that there is no interior to it, and the biggest that there is no exterior to it. Given any non-empty open set, there is always another open set contained in this set by the definition of interior point. The plane does not have an exterior boundary; suppose it does, like the closed disc shown in Fig. 2 (b), the points on its boundary are not interior points and it would not be open. As soon as the topology is chosen, the question of which elements are open and which elements are closed is already determined, in the sense of Definition 2, Theorem 3, and Proposition 4. It is in this sense that we say One gives birth to Two in topology. Naively one might believe that that everything is either yin or yang. This is not true. As far as open and closed sets are concerned, a set might be open, 3

4 (a) an open disc (b) an closed disc D 1 D 2 (c) intersecting two open discs Fig. 1. Usual topology. closed, or both, or neither. Consider the following example [3]: X = {a,b,c,d,e}; T = { X,,{a},{c,d},{a,c,d},{b,c,d,e} }. (1) X,,{b,c,d,e},{a,b,e},{b,e},{a} are the closed subsets of X; Some subsets such as X and {a} are both open and closed; Some subsets of X such as {a,b} and {a,d} are neither open nor closed. The open set {a,c,d} contains the closed set {a}; the closed set {b,c,d,e} contains the open set {c,d}. This example illustrates two important notions concerning Yin and Yang (I) Yin and Yang are inseparable; (II) Yin contains Yang and Yang contain Yin. The first notion is exemplified by the fact that some sets such as X are both open and closed. The second notion is exemplified by the fact that an open set might contain a closed set and vice versa. In the following, more concepts are defined to demonstrate the notion that (III) Yin and Yang convert to each other. 4

5 For this purpose, we precisely define limit point and closure. Definition 5 A point p X is called an accumulationpoint, or a limitpoint, or a cluster point, or a derived point of a subset A of X iff every open set containing p contains a point of A different from p. G open and p G (G\p) A The set of accumulation points of A is called the derived set of A. Theorem 6 A subset A of a topological space X is closed if and only if A contains each of its accumulation points. Definition 7 Let A be a subset of a topological space X. Then the closure of A is the union of A and its derived set. Axiom 8 (Kuratowski Closure Axiom) The closure operator satsifies (1) cl( ) = ; (2) A cl(a); (3) cl(a B) = cl(a) cl(b); (4) cl(cl(a)) = cl(a). From Definition 5 to Axiom 8, the closure operator converts an open set to a closed set. To convert a closed set to an open set, we need to define precisely the notions of interior, exterior, and boundary. Definition 9 Let A be subset of a topological space X. A point p A is called an interior point of A if p belongs to an open set contained in A. The set of interior points of A is called the interior of A. Proposition 10 The interior of a set A, int(a), is the union of all open subsets of A. Furthermore, (1) int(a) is open; (2) int(a) is the largest open subset of A; (3) A is open iff A = int(a). Definition 11 The exterior of A, ext(a), is the interior of the complement of A. The boundary of A, bry(a), is the set of points that belongs to neither int(a) nor ext(a). To convert a closed set to the largest open subset within it, we can set-minus its boundary from it. Note that the concept of interior and exterior defined as above are more general than their counterparts in the plane. Consider example (1) and the subset A = {b,c,d} of X, int(a) = {c,d} since {c,d} is a T -open set; ext(a) = {a} since the complement of A is {a,e} and {a} is an open set; 5

6 bry(a) = {b,e} since the interior of A s complement is {a}. Definition 12 Let p be a point in a topological space X. If G is an open set containing p, then G is called an open neighborhood of p. A subset N of X is a neighborhood of p if N is a superset of an open set G that contains p. The class of all neighborhoods of p X, denoted by N p, is called the neighborhood system of p. N is a neighborhood of p is the inverse of the relation p is an interior point of N. Here the opposing property of Yin and Yang are reflected functionally. ByDefinition2,aset A X isclosediffx\aisopen,i.e., iffp X\Aimplies p has a neighborhood contained in X \A, or, disjoint from A. Consequently, A is closed iff every neighborhood of x intersects A implies x A. Definition 13 A sequence of points a i, i = 1,2,..., in a topological space X converges to a point b X iff for each open set G containing b there exists a positive integer n 0 such that n > n 0 implies a n G. The above concept of convergence, due to Cauchy, revolutionized the subject of mathematics. It connects to a major piece of philosophy: if the difference between A and B are as small as one wishes, A and B are the same. We can also say the difference of A and B are the smallest. Starting from the primary concept of open sets, we have defined other derived concepts such as interior, exterior, closure, boundary, neighborhood systems. It is possible that we choose the primary concept as neighborhood systems or closure to derive other concepts, as the following two theorems show. This illustrates an important principle that truth of the universe does not depend on the language used for description. Theorem 14 Let X be a non-empty set and let there be assigned to each point p X a class A p of subsets of X satisfying the following axioms: (1) A p is not empty and p belongs to each member of A p ; (2) the intersection of any two members of A p belongs to A p ; (3) every superset of a member of A p belongs to A p ; (4) each member N A p is a superset of a member G A p such that G A g for every g G, then there exists one and only one topology T on X such that A p is the T - neighborhood system of the point p X. Theorem 15 Let X be a non-empty set and let k be an operation which assigns to each subset A of X the subset A k of X, satisfying the following axioms, called the Kuratowski Axioms: 6

7 (1) k = ; (2) A A k ; (3) (A B) k = A k B k ; (4) (A k ) k = A k, then there exists one and only one topology T on X such that A k will be the T -closure of the subset A of X. 3 Two gives birth to Three Yin and Yang represent the two ends of a dichotomy, such as big and small, reckless and timid, substantial and insubstantial, etc. However, frequently we have to ask the question: how small is small? how timid is timid? how substantial is substantial? The full spectrum of being timid or reckless can be divided into three parts: the timid, the brave/cautious, and the reckless. More often than not one wants to be brave, but neither timid nor reckless. In this sense, a dichotomy often give birth to a third concept naturally. However, the above interpretation are not satisfactory in two facets. Firstly, there are discrete dichotomies that do not extend to a continuous interval, e.g., the open-closed sets elaborated in the last section. Secondly, even for dichotomies that do have a continuous interval, it only defers the question to the next level rather than answers it, since one can ask again: what is the fine line between the timid and the brave? Hence to answer the question that how brave is brave, we need a criteria to judge how timid is timid and how brave is brave. It is this criteria that should be interpreted as the third one, which in mathematics follows from quantification, and leads to the concept of metric in general topology. Definition 16 Let X be a non-empty set. A real-valued function d defined on X X, i.e., ordered pairs of elements in X is called a metric or distance function on X iff it satisfies the following axioms: (1) d(a,b) 0 and d(a,a) = 0; (2) d(a,b) = d(b,a); (3) d(a,c) d(a,b)+d(b,c); (4) If a b, then d(a,b) > 0. The real number d(a,b) is called the distance from a to b. The most familiar metric to us is the Euclidean metric, or simply the distance between two points as the length of the straight-line segment connecting them. Definition 16 generalizes the Euclidean metric by capturing the four essential 7

8 (a) Metric (2a) 1 (b) Metric (2b) 1 (c) Metric (2c) properties of it. Fig. 2. Different shapes of the open sphere for different metric. 4 Three gives birth to innumerable things Metric together with open and closed sets give birth to a wide range of mathematical objects. Definition 17 Let d be a metric on a set X. For any point p X and any real number δ > 0, let S d (p,δ) denote the set of points within a distance of δ from p: S d (p,δ) = {x : d(p,x) < δ}, which is called the open sphere, or simply sphere with center p and radius δ. It is also called a spherical neighborhood or ball. Three examples of the metric in the Euclidean plane are d 1 (p,q) = (x p x q ) 2 +(y p y q ) 2 (2a) d 2 (p,q) = max( x p x q, y p y q ) d 3 (p,q) = x p x q + y p y q (2b) (2c) Consider the point p = (0,0) in the Euclidean plane and the real number δ = 1. The three metrics in (2) are the subsets of R 2 illustrated in Fig. 2. By changing the metric function to different forms, we obtain different shapes of the open sphere in Euclidean geometry. In Riemann geometry, there exists interesting and useful metrics that lead to Einstein s general relativity theory. The three objects, metric, open, and closed, are united in the concept of metric space. Definition 18 Let d be a metric on a non-empty set X. The topology T on X generated by the class of open spheres in X is called the metric topology 8

9 (or, the topology induced by the metric d). Furthermore, the set X together with the topology T induced by the metric d is called a metric space. Metric space can be defined in various ways, depending on the associated set of objects and the metric. One famous example is Hilbert space. Definition 19 Let R deonte the class of all infinite real sequences that converges. Let p = p n, q = q n denote two sequences that belong to R. The function d(p,q) = p n q n 2 n=1 is a metric and called the l 2 -metric on R. The metric space consisting of R with the l 2 metric is called Hibert space. Other well-known metric spaces include Sobolev space and Banach space; the study of these metric spaces is one of the main topics in the analysis branch of modern mathematics [4 6]. 5 Conclusion The concepts of open and closed sets in general topology furnish a precise and rigorous example of Yin and Yang since their definitions satisfy major principles of the Yin and Yang theory. The metric space that unites the three concepts of open, closed, and metric has innumerable applications in modern mathematics. Acknowledgements TheauthorhaswouldliketothankMasterYuanmingZhangforhisteachingof Daoism physical exercises such as breathing techniques and internal alchemy. References [1] K. J. Devlin. Sets, functions, and logic: an introduction to abstract mathematics. Chapman & Hall/CRC, third edition, ISBN: [2] J. L. Kelley. General Topology. Springer-Verlag, ISBN: [3] S. Lipschutz. General Topology. Schaum s Outline of Theory and Problems. McGraw-Hill, ISBN:

10 [4] H. L. Royden. Real Analysis. Prentice Hall, third edition, ISBN: [5] W. Rudin. Real and complex analysis. McGraw-Hill Science, third edition, ISBN: [6] W. Rudin. Functional analysis. McGraw-Hill Science, second edition, ISBN:

Topology notes. Basic Definitions and Properties.

Topology notes. Basic Definitions and Properties. Intuitively, a topological space consists of a set of points and a collection of special sets called open sets that provide information on how these points