Continuous mathematics as an approximation of the discrete one.

Transcription

1 Continuous mathematics as an approximation of the discrete one. E.I. Gordon (Eastern Illinois University) November 1, 2015 In this talk I want to discuss a new direction in the foundations of mathematics. The main idea of this direction is that actually all sets are finite and the notion of an infinite set is simply an idealization of a very big finite set. This approach to foundations of mathematics is motivated by a great significance of modern computer methods not only in the science but in mathematics itself. Nowadays we investigate continuous objects and even draw continuous pictures by means of computers, in spite of the fact that computers deal only with finite objects. This situation stimulates more and more investigations of discretization of various continuous objects, e.g. topological and differential geometry structures and arises a lot of foundational problems, which I ll try to explain in this talk. Actually, the contradiction between atomistic point of view on the world and continuous nature of geometry was known already to Ancient Greeks. The first one was formulated by Falles from Milleti and developed by his students Anaxagor, Anaximen and Anaximandr in the 6 century BC and by Democritis in the 4 century BC. The existence of incommensurable segments discovered in Pythagorean school (5 century BC)showed that the atomistic point of view is not applicable for a straight line, which is the central object of geometry (indeed, we can say that the whole mathematics of ancient Greeks was geometry). Zenon from Alexandria (3 century BC) gave many arguments for the both atomistic and continuous points of view. One of his arguments in support atomistic approach was the famous Zenon s paradox. However, the continuous geometry was developed very far by Greeks. Actually, the modern theory of real numbers (Dedekind cuts) was developed by Eudox. The development of Greek s continuous geometry was summarized in the famous Euclid Foundations (2 century BC). However, at that time one had to use discretization to perform various calculation such as areas, lengths, etc. Recall, for example, the Archimedes calculation of the area of a parabolic segment. However, calculations dealing with curved domains were very difficult at that time and not many values were calculated. This situation changed dramatically much later, after creation of differentiation and integration calculus by I. Newton and H. Leibnitz. The great computational power of this calculus allowed to explain many phenomena in mechanics, astronomy, physics and to discover new phenomena confirmed by experiments. The investigations of Cauchy, Weierstrass, Dedekind, Cantor built an irreproachable logical base for calculus. Galileo said The book of nature is written in the language of mathematics. After creation of calculus there was no doubt that more exactly this statement should be understood as The book of nature is written in the language of continuous mathematics. The confidence in the last principle imply requirements to provide the rigorous proofs of the existence of solutions of differential equations, the rigorous proofs of convergence of the numerical solutions to the exact solutions, etc., even for mathematicians whose research areas are connected with applications in 1

2 science. As a rule, physicists are not so concerned in rigorous proofs. They are confident in results of experiments and don t see any need in some additional formal logical proofs of phenomena discovered experimentally. However, there are some reasons for physicists to value mathematical proofs. One of these reasons was observed by the Russian physicist Andronov, who called it The miracle of a small parameter. Andronov investigated dynamical systems by the Poincare method of a small parameter. He noticed that if the convergence of this method is rigorously proved for a very small interval of values of the small parameter µ, then the method is applicable for the physically reasonable values of µ even if they do not belong to the interval of proved convergence. A similar situation occurs very often for the proofs of convergence. Another reason for physicists to value mathematical proofs is that the failure to prove rigorously, say, the existence of a solution of a differential equation in the case, when this solution is interpreted as some physical process, the existence of which is clear by physical reasons, may mean that the differential equation itself is not a physically adequate model for this phenomenon. On the other hand many physicists and nowadays even some applied mathematicians, who appreciate calculus as a powerful computational tool do not consider formal logical foundations of calculus as physically adequate. There are many serious reasons for such attitude. V.I. Arnold (in the book What is Mathematics ) retells his conversation with the prominent Russian physicist Zeldovich, who explained to Arnold his understanding of the derivative: Actually, we are interested in the ratio of two very small values, but is very hard to compute such ratio and, thus, we have to use derivatives for approximate computation. Indeed, physicists may have a strong reservation about the definition of a derivative as the limit of y x as x becomes arbitrary small. If, for example, we use derivatives in some problems of classical mechanics, then it is senseless, to consider arbitrary small x, since for very small distances the laws of classical mechanics do not work. The laws of quantum mechanics are applicable for such distances. However, it is impossible to indicate the greatest lower bound for the distances. We see here that two basic postulate of reals come to contradiction with the physical reality: 1. Every property of real numbers defines a set, that consists of those reals that satisfy thus property. 2. Every bounded from below set of reals has the greatest lower bound. The first axiom holds not only for the set of reals R but for an arbitrary set X: Every property defines a subset of X that consists of all those elements of X, that satisfy this property. This axiom is called the Axiom of Separation in axiomatic set theory. Actually, every physical property of real numbers fails to have exact bounds for the reals that satisfy this property. Thus, logically rigorous mathematical analysis as an idealization that sometimes gives a good enough approximation to reality and sometimes does not. The history of the discovery of the Max Planck law of the black body radiation gives an impressive example of the situation, when the law formulated in terms of integrals gives a bad approximation of reality under some conditions. A blackbody is an object that is perfect absorber of radiation. In ideal case it absorbs all light that falls on it no light is reflected by it and no light passes through it. However, being heated it can radiate light. The first problem, where classical approach failed is the problem of explaining the thermal radiation of a blackbody. The spectrum of the radiation emitted by a blackbody under the temperature T is the distribution of energy by frequencies, which is measured by the density ρ(ν, T ). So, the energy of radiation 2

3 of the unit of volume in the interval of frequencies [ν, ν + dν] for an infinitesimal dν is equal to ρ(ν, T )dν. The following Reyleigh-Jeans formula was obtained by some theoretical considerations based on statistical physics and the Maxwell theory of electricity. ρ(ν, T ) = 8πν2 kt, (1) c3 which immediately leads to a contradiction, since it follows from (1), that the energy of the radiation emitted by the unit of volume of a blackbody is infinite. Actually, the Reyleigh-Jeans formula works only for small frequencies. W. Wien found experimentally the following formula for ρ(ν, T ) for big frequencies: ρ(ν, T ) = Aν 3 e λνt (2) for certain parameters A and λ. Trying to find the general law (that works for all frequencies), M. Planck first of all found a formula, that is asymptotically equal to the Reyleigh-Jeans formula for small frequencies an to the Wien s formula for big frequencies. This is the formula ρ(ν, T ) = Aν3 e. λν T 1 Trying to explain this formula Planck noticed that the term kt in the Reyleigh-Jeans formula actually was obtained from the equality: kt = 0 0 Ee E kt de e E kt de M. Planck suggested to replace integrals in this formula by discrete sums with the width ε = hν. Thus, he replaced the term kt in (1) by the expression: (3) Finally he obtained the formula: n=0 nhνe nhν kt n=0 e nhν kt (4) ρ(ν, T ) = 8πh kt c 3 ν 3 e hν kt 1, (5) which completely agrees with the experiments. The variable E in (4) assumes the values of energy of oscillators involved in the blackbody. There are so many oscillators that it was quite natural to consider their energies as continuously distributed. However, we saw that this assumption gives a wrong result. The situation her is much more complicated. Actually, this result already based on the laws of quantum mechanics, which has absolutely another formalism than classical physics. Indeed, Planck 3

4 conclusion that the energy of an oscillator in atoms may assume only values nhν, n = 1, 2... was not correct. It is shown in quantum mechanics that this energy assumes values (h )hν. This it not our topic. I discussed it here simply to show that already M.Planck understood the continuous values as some idealization of discrete ones, and, thus, the replacement of the integral by infinite sum more adequate for this problem was very natural. The point of view on continuous mathematics became very common among applied mathematicians after the invention and wide expansion of computers in scientific research. I will cite here the well-known specialist in discrete mathematics D. Zeilberger from Rutgers. Continuous analysis and geometry are just degenerate approximations to the discrete world... While discrete analysis is conceptually simpler... than continuous analysis, technically it is usually much more difficult. Granted, real geometry and analysis were necessary simplifications to enable humans to make progress in science and mathematics.... In what follows, D.Zeilberger expresses his confidence in decrease of the role of continuous mathematics now due to the increase of the computers power. Certainly, this point of view is disputable. In classical continuous mathematics we have to distinguish constructible proofs, that provide, e.g. good estimates. The value of these proofs is indisputable independently of what point of view on relation between discrete and continuous mathematics is accepted. However, the constructive proofs are much more difficult and could be obtained very rarely. Pure existence or convergence proofs in continuous mathematics enables us to understand the qualitative picture of investigated problem, which often is very important (see the miracle of a small parameter mentioned above). However, the above discussion shows that the value of these proves maybe not so significant by the reason of the very strong idealizations of real world accepted in the classical theory of real numbers. If we restrict ourselves only by computer mathematics, we do not have an appropriate language to formulate qualitative problems, while constructive proofs are even more difficult than in continuous mathematics. So, to obtain necessary estimates we still have to use classical proofs based on continuous mathematics. However the correlation between continuous statements and their computer analogs is not so obvious and sometimes an existence of even constructive proof of convergence of a certain numerical method is not enough for this method to work in computers. Let me discuss a couple of examples. The first of them is taken from one of the old manuals in FORTRAN (McCracken D.D., Dorn W.S. (1964) Numerical Methods and Fortran Programming. John Wiley, New York, London, Sydney.) Although the Taylor series for sin x converges for all x, the approximate computation of sin x for large x based on its Taylor expansion gives an incorrect answer in a floating-point system. For large x, the first few terms in a partial sum of this series are also very large. Due to the fixed number of digits in the floating-point representation of real numbers, the addition of terms in a partial sum of the series should be done with the terms taken in ascending order, to avoid roundoff error; this is explained in the chapter 2 of this book. However, calculation of the k th term of the Taylor series for sin x produces exponent overflow for large x and k. A second example we elaborated together with Dr. Duane Broline and students Jessica Murray and Brad Heller from EIU. Consider the system of three linear equations with three unknowns and two parameters 4

5 The determinants are of the following forms: 5x 7y + 8z = b 3x ay + 4z = 5 ax + 4y bz = 2 = 8a 2 + 5ab 28a 21b + 16 x = ab a 51b y = 4ab + 3b 2 40a 25b + 8 z = a 2 b 45a + 12b 58 (6) (7) We are interested in the case, when this system has infinitely many solutions. This happens if the parameters a and b satisfy the following system of equations: = x = y = z = 0. (8) Certainly, one of the equations (but not = 0) can be skipped. The solution of this system has the following form: a = b b2 19 b, (9) 232 while b is the root of the polynomial f(t) = 3t 4 25t t t (10) The resulting general solution of the system (6) is of the form x = 10 b + u[ b b b3 ] y = u z = b + u[ b b3 ], (11) where b satisfies f(b) = 0 and fis the polynomial (10). It can be shown that the polynomial (10) has only two real solutions and both of them are irrational. The values of a obtained by formula (9) for these values of b are irrational as well. Suppose that we try to solve the system (6) on a computer numerically with the parameters a and b, satisfying (8) and wish to find the general solution. In this case we may deal only with rational approximations of a and b. Substituting these rational approximations ã and b in the system (6) we obtain a system that has a only unique solution. The question is how to find approximately the general solution (11). More general question is the following one. Given arbitrary rational parameters a and b how can one determine whether these parameters are close two solutions of system (8) and, thus, to the case of infinitely many solutions, or two the case of non-existence of solutions. In other words: both of these cases for the system (6) belongs to continuous mathematics. How can they be reflected in computer mathematics? I will answer this question later. To investigate the inter-relation of the continuous mathematics and the finite one P.Andreev and myself introduced a new axiomatic set theory, which we called The Theory of Hyperfinite Sets, the main feature of which I am going to discuss now. First of all let me recall the main features of classical axiomatic set theory. I ll have to explain both variants (ZF and NBG). 5

6 Soon after the set theory was invented by G. Cantor some logical contradictions in it called paradoxes were discovered by various mathematicians. The most known is the paradox of B. Russell, who showed that the set M = {x x / x} can not exists: both assumptions M M and M / M immediately lead to a contradiction. In G. Cantor s set theory it was assumed by default that every formulated property defines the set of all objects, that satisfy this property (possibly this is the empty set ). This assumption was the reason of all discovered paradoxes. To avoid these paradoxes it was necessary to introduce some restrictions on properties which define sets. This was done by E. Zermelo and in another way by B. Russel. We concentrate on Zermelo s system. The main assumption is that all objects are sets, and some sets are elements of other sets (x y). Thus all properties can be formulated in term of basic properties x y, x = y using logical connectives and quantifiers. 0.1 Axiomatic of Zermelo Theory. Z1 (Extensionality). x = y z(z x z y). Z2 (Pair). For every sets a and b there exists the set {a, b}. Z3 (Separation) If F (t) is an arbitrary property of sets (F may include parameters) and x is a set, then there exists a set {y x F (y)}. Z4 (Power set). For every set x there exists a set {y y x}. Z5 (Union). For every set x there exists the set y Z6 (Choice.) Z7 (Infinity). x( x y x(y {y} x)). Z8 Regularity Let F (t) is an arbitrary property such that tf (t), then y(f (y) z y F (z)). It is easy to prove using axiom of infinity and separation the existence of minimal set N such that N n N(n {n} N). This set satisfies all axioms for the set of natural numbers if to assume n = n {n}. Then n < m n m and all other operation are defined by recursion in a usual way. The first few terms of N 0, 1, 2, 3,... looks as follows: y x, { }, {, { }}, {, { }, {, { }}},... One more basic notion, the notion of ordered pair is formalized in set theory in the following way: a, b = {{a}, {a, b}}. The basic property of ordered pairs a, b = a, b a = a b = b 6

7 can be proved easily. Having ordered pairs, we can prove the existence of x y - the Cartesian product of x and y, and, thus, of the sets Z, Q and R, as the set of Dedekind cuts of Q. It is almost obvious that all mathematical proofs can be formalized in this theory. 0.2 Axiomatic of von Neumann, Bernays, Gödel. The equivalent axiomatic was suggested by J. von Neumann, P. Bernays and K. Gödel, which is more relevant for the Theory of Hyperfinite Sets. So I have to expose it briefly. This axiomatic deals with classes. The sets are classes that are elements of other classes: Set(x) := X(x X). Usually we use the lower case letters to denote sets. The classes that are not sets are called the proper classes. Intuitively, the proper classes are properties of sets, which in Zermelo s axiomatic were described by formulas. Here they occur as variables (denoted by capital letters). Axioms NBG1, NBG2, NBG4, NBG5, NBG6, NBG7, are exactly the same as axioms Z1, Z2, Z4, Z5, Z6, Z7 respectively. Axiom NBG3 (Separation): X, x Set(X x). Axiom NBG8 (Regularity): X = y X( z y z / X) We need one more axiom - the axiom of existence of classes. We say that a formula F (t) is predicative if only quantifiers over set variables are involved in F. A predicative formula F (t) may contain some set and class free variables. Axiom NBG9 (Comprehension) For every predicative formula F (t), there exists the class {x F (x)}. The last Axiom actually consists of infinitely many axioms. They can be replaced by finitely many axioms: Axiom NBG9(1). There exist classes E = { x, y x y} and V = {x Set(x)}. Axiom NBG9(2). For every classes X and Y there exist classes X Y, X Y and X = V \ X. Axiom NBG9(3). For every class X there exists the class Y = {x y x, y } The class Y defined in the Axiom NBG(3) is said to be the domain of X and denote dom(x). It is easy to see that axioms NBG9(1), NBG9(2) and NBG9(3) are equivalent to the axiom NBG Theory of hyperfinite sets. As it was mentioned above, we want to develop the theory of continuous objects basing on the hypothesis that all objects are discrete, but some of them are so big that some of their parts are indiscernible. Visible images of such objects are continuous objects. From the point of view of set theory discrete objects are finite sets. So we have to define what are very big sets. Obviously, it is 7

8 impossible to formalize the notion of a very big set in the framework of the continuous mathematics, since it is impossible to determine the minimal very big number. On the other hand every set of natural numbers has a minimal number and every property, in particular, the property to be very big. This is the same situation as in the well-known paradox of Eubulidies (4th century BC): 1 grain of sand is not a heap, if n grains of send do not form a heap, then n + 1 grains do not form a heap as well. When, then, a heap starts? To overcome this logical contradiction we should accept the existence of not well defined parts of sets, that cannot be considered as sets. After understanding these we have to assume that classical mathematics deals only with well defined aggregates that are sets. This is may be the main idealization of classical mathematics, that, as we discussed before, is not relevant to real world. The question is how to include these non well defined aggregates in the logically consistent way. We use the same approach that was used in the theory NBG. Our theory THS is the a theory of classes. The sets are those classes that are elements of other classes. The proper classes were considered as aggregates that are too big to be sets. This was reflected in the axiom of separation. The crucial point is that we sacrifice the Separation axiom and replace the axiom of infinity by its negation. Now we can define the notion of a very big set. We prefer to use the term a hyperfinite set A set x is hyperfinite if there exists a proper subclass X x In what follows proper subclasses of sets are said to be semisets. The sets that are not hyperfinite are simply finite. Hereditarily finite sets (i.e. finite sets, all whose elements, elements of elements, etc., are finite) are said to be standard sets. We are going to keep all classical properties of finite sets (i.e. those true statements, in which proper classes are not involved. The axioms THS1, THS2, THS3, THS4 are the same as axioms NBG1, NBG2, NBG4 and NBG5 respectively, i.e. axioms of extensionality, pair, power set and union respectively. THS5 (Axiom of set formation) Set( ) x(set(x) = Set(x {x}). THS6 (Axiom of induction and regularity.) Let F (t) be a formula that contains only set variables. Then F ( ) x(f (x) = F (x {x}) = y F (y). Axiom THS6 is equivalent to Z7 Z8 (see E. Zermelo s Axiomatic). The axioms THS1 THS4 and THS6 represent the classical axiomatic of the theory of finite sets ZF fin. This axiomatic is equivalent to formal arithmetic. The equivalence is given by the Accerman s function ac that maps bijectively the family of all hyperfinite sets onto the set of natural numbers N. This function is defined by induction: ac( ) = 0; ac(x) = a x ac(a). The Separation axiom of ZF fin follows from THS6. Since we deal only with finite sets in THS, this makes it possible to formulate a more general Comprehension axiom, than the one in the theory NBG. THS7. (Axiom of Comprehension) Let F (t) be an arbitrary formula, maybe, containing set parameters or class parameters (compare with Axiom NBG9), then there exists the class {x F (x)}. This axiom implies the existence of the following classes: 8

9 1. H = {x x = x} the class of all sets; 2. E = { x, y x y}, 3. F = {x X x Set(x)} - class of all finite sets; 4. S = {x S ([ S a, b S(a {b} S] = x S)} - the class of all standard sets (it is an easy exercise to show that S consists of all hereditarily finite sets); 5. X Y, X Y, H \ X, dom(x) for arbitrary classes X and Y. 9