volume II: Synopsis
Part II: A brief history of dynamics
page 16: Georg Cantor
(1845-1918)
The natural numbers, 1, 2, 3 . .. are infinite, since we can always add another one. It was known in antiquity that Pythagoras' theorem implies that there are quantities that cannot be measured by the natural numbers or by the rational numbers, ratios (eg 3/5) of natural numbers. To measure such quantities, we have invented the 'irrational' or real numbers. Cantor showed that the step from natural to real numbers is not unique, but the first of an endless series of steps to even bigger number spaces, which he called the transfinite numbers. Following Cantor, our natural theology is based on the hypothesis that the transfinite numbers can help us to understand God. Incommensurable magnitudes - Wikipedia, Michael Hallett: Cantorian Set Theory and Limitation of Size
One might argue that the oldest scientific problem in the world is the relationship between discrete and continuous systems. A continuous system is something like a wheel, which moves smoothly without jumps or gaps. A discrete system is the exact opposite, more like walking, step by step. Each unit of the system is separate from the others, and if one is to move, one must cross the gaps between the elements of the system. Language is a discrete system. Different words and sentences have gaps between them.
Aristotle, and many others, felt that the existence of motion meant that space must be continuous. The idea of continuity is closely connected to the idea of infinity: one may think of a continuous line as an infinity of points; alternatively one may think of motion as indeterminate or infinite in the sense of not being definite. In the first point of view, the points are imagined to be dense, having no gaps between them. This idea seems to contradict the idea of a point, which something distinct, having, one feels, some sort of boundary between itself and the next thing. John L Bell (Stanford Encyclopedia of Philosophy): Continuity and Infinitesimals
Cantor set out to find the 'cardinal of the continuum', that is the number of point it takes to make a continuous line. Cohen later showed his project was beyond the power of an axiomatic set theory, but during his search Cantor invented the original theory of sets, which has developed to become a foundation of mathematics. Cardinality of the continuum - Wikipedia, Set theory - Wikipedia, Paul J Cohen: Set Theory and the Continuum Hypothesis
Language is discrete, but it is also infinite. There is no limit to the number of new sentences that the speakers of a language can produce and understand. This infinity arises through the combination and permutation of elements of the language in accordance with the rules (if any) of its grammar. Essential to the notions of combination and permutation are the ideas of subset and order. Cantor used the notion of ordered set to construct a representation of the transfinite space of mathematical language. Nowak et. al.: The evolution of syntactic communication
The basic set of mathematical symbols is the set of natural numbers, {0, 1, 2, 3, . . . }. These numbers are represented in natural languages by very different words and symbols, eg one, two, three; bir, iki, uc; etc, but we all know what they mean. Because the natural numbers are infinite, they provide us with an infinite vocabulary, and so an infinite domain of potential meanings. By meaning here we can understand a mapping between one symbol and another. Cantor imagined a set or class containing the natural numbers and called the number of numbers in this set ℵ0, the first transfinite number. Georg Cantor: Contributions to the Founding of the Theory of Transfinite Numbers
He then went on to show that the infinite set of natural numbers may be assembled into a set which has a higher degree of infinity than the infinity of natural numbers. A standard proof of this result assumes the existence of the power set P(S), the set of all subsets of the set S, and shows that the number of elements in P(S) is strictly greater than the number of elements in S. This next transfinite number Cantor called ℵ1. This, he thought, might be the cardinal of the continuum. Beyond this, we may use the same approach to generate sets of even greater infinity, yielding sets with cardinal ℵ2, ℵ3 and so on. The generation of the transfinite numbers depends upon ordering, analogous to the ordering of words to form sentences in a language or of the elements of the Universe to give complex entities, like ourselves. On this site we take the transfinite numbers to be a foundation on which to build a language big enough to begin talking about God.
Aristotle and Aquinas model God as pure act, actus purus. God is alive, that is an entity which moves itself, or, as mathematicians might say, maps itself onto itself. It can be proven that any complete formal system that maps onto itself has fixed points. The idea to be developed here is that Cantor's transfinite numbers are a representation of the fixed points in the pure dynamism of God. Aquinas, Summa I, 18, 3: Is life properly attributed to God?, Fixed point theorem - Wikipedia
Cantor had a similar idea. He felt that his set theory was divinely inspired by God and that the transfinite numbers existed in the mind of God. If for 'mind of God' we read 'stationary points in the divine dynamism', we come to the same idea. Joseph Warren Dauben: Georg Cantor: His Mathematics and Philosophy of the Infinite, pp 271 sqq
(revised 5 April 2013)
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Further readingBooks
Bernays, Paul, Axiomatic Set Theory, Dover Publications 1991 Jacket: 'Since the beginning of the 20th century, set theory . . . has become increasingly important in almost all areas of mathematics and logic. In Part I of this excellent monograph, A A Fraenkl presents an introduction to the original Zermelo-Fraenkel form of set-theoretical axiomatics and a history of its subsequent development. In Part II Paul Bernays offers an independent presentation of a formal system of axiomatic set theory, covering such topics as the frame of logic and class theory, general set theory, transfinite recursion, completing axioms, cardinal arithmetic, and strengthening of the axiom system.'
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1895, 1897, 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'
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Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'
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Dauben, Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press 1990 Jacket: 'One of the greatest revolutions in mathematics occurred when Georg Cantor (1843-1918) promulgated his theory of transfinite sets. . . . Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradox in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.'
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Hallett, Michael, Cantorian Set Theory and Limitation of Size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson.
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Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. . . . It can be used as a textbook for a graduate course in set theory and can serve as a reference book.'
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Papers
Nowak, Martin A, Joshua B Plotkin and Vincent A A Jansen, "The evolution of syntactic communication", Nature, 404, 6777, 30 March 2000, page 495-498. Letters to Nature: 'Animal communication is typically non-syntactic, which means that signals refer to whole situations. Human language is syntactic, and signals consist of discrete components that have their own meaning. Syntax is requisite for taking advantage of combinatorics, that is 'making infinite use of finite means'. ... Here we present a model for the population dynamics of language evolution, define the basic reproductive ratio of words and calculate the maximum size of a lexicon.'. back |
Piatelli-Palmerini, Massimo, "The barest essentials", Nature, 416, 6877, 14 March 2002, page 129. Grammar: It is useful to consider the concept of grammar as primary and that of language as derived. . back |
Links
Aquinas, Summa I, 18, 3, Is life properly attributed to God?, Life is in the highest degree properly in God. In proof of which it must be considered that since a thing is said to live in so far as it operates of itself and not as moved by another, the more perfectly this power is found in anything, the more perfect is the life of that thing. ' back |
Cardinality of the continuum - Wikipedia, Cardinality of the continuum - Wikipedia, the free encyclopedia, 'In mathematics, the cardinality of the continuum (sometimes also called the power of the continuum) is the cardinal number of the set of real numbers R (sometimes called the continuum). This cardinal number is often denoted by c, so c = R.' back |
Fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.
The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' back |
Georg Cantor - Wikipedia, Georg Cantor - Wikipedia, the free encyclopedia, Georg Ferdinand Ludwig Philipp Cantor (March 3 [O.S. February 19] 1845[1] – January 6, 1918) was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware' back |
Incommensurable magnitudes - Wikipedia, Incommensurable magnitudes - Wikipedia, the free encyclopedia, 'The Greek discovery of incommensurable magnitudes changed the face of mathematics. At its most basic level it shed light on a glaring contradiction within the then current Greek conception of mathematical thought, which eventually resulted in a reformulation of both the methods and practice of mathematics in general. These reformulations brought about a new era in mathematics, and were the first stepping stones of some of our most important modern day conceptions, such as calculus.' back |
John L Bell, Continuity and Infinitesimals - Stanford Encyclopedia of Philosophy, 'The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity—a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.' back |
John L Bell (Stanford Encyclopedia of Philosophy), Continuity and Infinitesimals, 'The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity—a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.' back |
Set theory - Wikipedia, Set theory - Wikipedia, the free encyclopedia, 'Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.' back |
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