vol 3: Development
2 Model
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Immensity: the symmetric universe
Mathematics
Traditionally, God is infinite. Aquinas To talk about God, therefore, we need a language capable of dealing precisely with infinity. To get such a language, we augment natural language with mathematics..
Mathematics begins from natural languages like, as here, English. Nowak et al note that
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'. (Nowak et al)
Mathematics goes further making transfinite use of infinite means. The infinite means are the natural numbers, 0, 1, 2, 3 ... . The properties of these numbers are summarized by the Peano axioms. Peano axioms - Wikipedia. Beginning with zero, each new number is generated by adding one to its predecessor. There is no last natural number. Even though every natural number is finite, the set of all the natural numbers is infinite, in the language of set theory 'countably infinite'.
Set theory was founded by Georg Cantor toward the end of the nineteenth century. Cantor Because of its clarity and simplicity, set theory has become an important way to represent mathematical ideas. Although it has been formalized and vastly extended since Cantor's time, we will follow Cantor's 'naive' approach. Jech.
Cantor defined a set (or 'aggregate') as 'any collection into a whole S of definite and separate objects s of our intuition or our thought. These objects are called the elements of S. Symbolically, we write
(Cantor, 85)
So we write the set of natural numbers N = {0, 1, 2, 3, ...}.
Cantor defines the 'power' or 'cardinal number' of a set as 'the general concept, which, by means of our active faculty of thought, arises from the set S when we make abstraction of the nature of the various elements s and the order in which they are given.' (Cantor, 86) That is we consider the set as simply a collection of units of no specific kind.
Since there are an infinite number of natural numbers, the cardinal number of N cannot be represented by natural number. Instead Cantor uses the first letter of the Hebrew alphabet, aleph, and we write card(N) = aleph(0). (Cantor, page 103)
Order
The natural numbers described by the Peano Axioms have a natural order which arises from the process of their generation, so that we may call N an ordered set. From an ordered set we derive the notion 'ordinal type' 'which is itself an ordered set whose elements are units which have the same order of precedence amongst one another as the corresponding elements of S, from which they are derived by abstraction.' (112)
Cantor notes that
'the concept of ordinal type developed here, when it is transferred in like manner to multiply ordered aggregates embraces, in conjunction with the concept of cardinal number or power . . . everything capable of being numbered (Anzahlmässige ) that is thinkable, and in this sense cannot be further generalized. (Cantor page 117)
Following the lead of natural language, we can make larger numbers out of the natural numbers by combinations and permutations. Given the set {1, 2, 3} we can produce the following six permutations: <1, 2, 3>, <1, 3, 2>, <2, 1, 3>, <2, 3, 1>, <3,1,2>, <3,2,1>. A set of cardinal number four can be arranged in 24 different permutations, and in general, a set whose cardinal is n can be ordered in n ! (n factorial) = n x (n-1) x ... x 2 x 1 different ways. The size of n ! grows very quickly with n, as Stirling's formula shows.
The permutations of the natural numbers N may be considered the elements of a set with an even greater transfinite cardinal, aleph(1). Since these permutations have a natural ('dictionary') ordering, we may consider the permutations of these permutations to arrive at a set with cardinal number aleph(2). This construction of larger and larger ordered sets may continue without end. Like the natural numbers, there is no last transfinite number.
We may call the Cantor universe envisaged as a recursive process of permutation the symmetric universe, by analogy with the symmetric groups constructed by permutation.
The transfinite tree
We may visualize the structure described above using a tree. We begin from the binary tree, a very common denizen of the computer world.
A transfinite tree may grow much faster than a binary tree.
We will use this transfinite structure as the foundation for a model of God.
The systematic application of this model to the world begins in the section on physics. Here we note two correspondences. First, the natural numbers provide a model for counting and arithmetic that applies to all unique objects in the universe, dollars, sheep, ticks of a clock, etc etc.
Second, we notice that in some respects the changing world behaves very much like a permutation process: things are swapping their positions all the time: for instance when a leaf falls the position occupied by the leaf an the tree is replaced by air, and some air near the ground is replaced by a leaf.
We now examine some details of the structure of the formal, symbolic universe here described.
(revised 19 february 2008)
Further reading
Books
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 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.' Amazon back |
| 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.' Amazon back |
| Engeler, Erwin, and others , The Combinatory Programme, Birkhauser 1995 Jacket: 'The programme in combinatory logic, developed at the ETH in Zurich, takes its philosophical basis primarily from the works of two mathematicians of widely separate eras. The first was Richard Dedekind, who was probably the first to base the development of mathematics on pure thought. The second was Haskell Curry who, as a young man, took on the task of creating a formal basis for the foundation of all mathematics. Probably neither of them would have forseen the extension of their ideas to a profound influence on a discipline that did not even exist in their time. For the purpose of the programme is no less than to rework the mathematical foundations of computer science on such a theory of pure thought. It begins from the idea that, if logic is to be the science of correctly dealing with thought-objects, the underlying theory must be in some sense a part of, or at least preliminary to its structure: i.e., a protologic. From this idea, a combinatory algebra is constructed, using a programmatic mixture of the classical axiomatic and set-theoretic approaches.' Amazon back |
| 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. Amazon back |
| 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.' Amazon back |
| Zemanian, Armen H, Graphs and Networks: Transfinite and Nonstandard, Birkhäuser 2004 Amazon editorial review :'For about thirty years Zemanian has been developing a theory of infinite electrical networks. This book is the latest in a series of books...on the subject. The subject is necessarily abstract and sophisticated because infinite objects are the main objects of discourse.... The first few chapters are important not only to remind the reader of the terms, but also to give an improved or alternate treatment of some earlier results. There does not yet seem to be a large following of researchers in this area, but it seems very attractive and ripe for investigation. Its intriguing to see the connections between set theory and electrical network problems.... To understand these concepts fully the reader must consult the book under review. The reviewer highly recommends devoting the effort needed to understand these original and surprising concepts.' SIAM Review Amazon back |
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 |
| Dauben, Joseph W, "Georg Cantor and the origins of transfinite set theory", Scientific American, 248, 6, June 1983, page 112-121. 'How large is an infinite set? Cantor demonstrated that thee is a hierarchy of infities each one 'larger' than the preceding one. His set theory is one of the cornerstones of mathematics.'. back |
Links
| Aquinas 35 Summa: I 7 1 Is God infinite? 'Since therefore the divine being is not a being received in anything, but He is His own subsistent being as was shown above back |
| Peano axioms - Wikipedia Peano axioms - Wikipedia - the free encyclopedia 'In mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory' back |
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