vol 3: Development
chapter 2: Model
page 12: Mathematics
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Is the transfinite network isomorphic with mathematics?
Here we accept the formalist view of mathematics introduced by Hilbert. Richard Zach Hilbert's approach formally distinguishes mathematics from the study of reality and sees it as like a game played with certain symbols and certain rules. The symbols cannot move themselves to implement the rules: all the action comes from the mathematicians (and their computers) manipulating immobile symbols according to the rules of their current game.
The mathematics industry, like any other, thus comprises workers (mathematicians) who 'do' the mathematics and their mathematical communications which may be conversations, lectures, papers, books, models or any other means of mathematical communication.
Although mathematicians are very prone to talk about continuous and infinite entities like lines and real numbers, all their communication is symbolic, that is finite and quantized. Everything that a mathematician needs can therefore be represented by finite strings of symbols manipulated according to finite sets of rules. This situation is reflected in the mathematical literature, which is finite even if when it talks about infinite entities like the set of natural numbers.
The transfinite network provides us with a model of the whole mathematical process. Mathematicians themselves are represented in network by nodes, which are themselves networks of cells. molecules and so on. These networks have a certain cardinality, the cardinal of humanity. Mathematicians communicate with one another using mathematical protocols which enable them to encode and decode ideas like 'complex numbers', 'Hilbert space', 'differential and integral operators' and so on.
The transfinite network is a formal model of the universe, able to formally model its own dynamics, including the mathematics industry. We use this insight to found the claim that the transfinite network is in effect isomorphic to mathematics. Like mathematics, it has no boundaries except formal consistency and the rate at which its operators can work.
In the chapters which follow this, we will explore the construction of the universe of from its simplest state, the initial singularity, to its present state (which includes mathematicians) and beyond.
Revised 27 November 2007
Further reading
Books
| Brouder, Felix, Mathematical Developments Arising from Hilbert's Problems (Proceedings of Symposia in Pure Mathematics volume 28, American Mathematical Society 1983 Amazon book description: 'In May 1974, the American Mathematical Society sponsored a special symposium on the mathematical consequences of the Hilbert problems, held at Northern Illinois University, DeKalb, Illinois. The central concern of the symposium was to focus upon areas of importance in contemporary mathematical research which can be seen as descended in some way from the ideas and tendencies put forward by Hilbert in his speech at the International Congress of Mathematicians in Paris in 1900.' Amazon back |
| 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 |
| Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). 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 |
| Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. ... The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' Amazon back |
| Gellert, Walter, and et al (eds), The VNR Concise Encyclopedia of Mathematics , Van Nostrand Reinhold 1994 Preface: '... there is a wide demand for a survey of the results of mathematics ... Our task was to describe mathematical interrelations as briefly and precisely as possible. ... Colours are used extensively to help the reader. ... Ample examples help to make general statements understandable. ... A systematic subdivision of the material, many brief section headings, and tables are meant to provide the reader with quick and reliable orientation. The detailed index to the book gives easy access to specific questions. ...' The Editors and Publishers Amazon back |
| Hodges, Andrew, Alan Turing: The Enigma, Burnett 1983 Author's note: '... modern papers often employ the usage turing machine. Sinking without a capital letter into the collective mathematical consciousness (as with the abelian group, or the riemannian manifold) is probably the best that science can offer in the way of canonisation.' (530) Amazon back |
| Kramer, Edna Ernestine, The Nature and Growth of Modern Mathematics, Princeton UP 1982 Preface: '... traces the development of the most important mathematical concepts from their inception to their present formulation. ... It provides a guide to what is still important in classical mathematics, as well as an introduction to many significant recent developments. (vii) Amazon back |
| Smith, David Eugene, A Source Book in Mathematics, Dover 1984 Jacket: 'This book presents, in English translation, the great discoveries in mathematics from the Renaissance to the end of the 19th century. You are able to read the writings of Newton, Leibniz, Pascal, Riemann, Bernoulli, and others, exactly as the world saw them for the first time. Succinct selections from 125 different treatises and articles, most of them unavailable elsewhere in English, offer a vivid first-hand story of the growth of mathematics. ... ' Amazon back |
| Stewart, Ian, Life's Other Secret: The new mathematics of the living world, Allen Lane 1998 Preface: 'There is more to life than genes. ... Life operates within the rich texture of the physical universe and its deep laws, patterns, forms, structures, processes and systems. ... Genes nudge the physical universe in specific directions ... . The mathematical control of the growing organism is the other secret ... . Without it we will never solve the deeper mysteries of the living world - for life is a partnership between genes and mathematics, and we must take proper account of the role of both partners.' (xi) Amazon back |
| Tymoczko, Thomas, and (Editor), New Directions in the Philosophy of Mathematics: An Anthology, Princeton University Press 1998 Jacket: 'The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. ... By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, TT reveals an evolving effort to account for the nature of mathematics in relation to other hman activities.' Amazon back |
Papers
| Calder, Alan, "Constructive Mathematics", Scientific American, 241, 4, October 1979, page 134-143. 'This approach is based on the belief that mathematics can have real meaning only if its concepts can be constructed by the human mind, an issue that has divided mathematicians for over a century.'. back |
| Wigner, Eugene P, "The unreasonable effectiveness of mathematics in the natural sciences", Communications in Pure and Applied Mathematics, 13, 1, February 1960, page . 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.'. back |
Links
| Eugene Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back |
| Guillermo Ferreyra The Mathematics Behind the 1997 Nobel Prize in Economics 'On October 14 the Royal Swedish Academy of Sciences announced the winners of the 1997 Nobel Prize in Economics. The winners were Professor Robert C. Merton, of Harvard University, Cambridge, USA and Professor Myron S. Scholes, of Stanford University, Stanford, USA, for the discovery of "a new method to determine the value of derivatives". ' back |
| Richard Zach Hilbert's Program (Stanford Encyclopedia of Philosophy) 'In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called "finitary" methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics.' back |
| University of Tennessee, Knoxville Mathematics Archives WWW Server The goal of the Mathematics Archives is to provide organized Internet access to a wide variety of mathematical resources. The primary emphasis is on materials which are used in the teaching of mathematics. Currently the Archives is particularly strong in its collection of educational software. . . . A second strength of the Archives is its extensive collection of links to other sites that are of interest to mathematicians. back |
| Wolfram Research Mathworld: The World's Most Extensive Mathematics Resou7rce 'MathWorld has been assembled over more than a decade by Eric W. Weisstein with assistance from thousands of contributors. Since its contents first appeared online in 1995, MathWorld has emerged as a nexus of mathematical information in both the mathematics and educational communities. It not only reaches millions of readers from all continents of the globe, but also serves as a clearinghouse for new mathematical discoveries that are routinely contributed by researchers. Its entries are extensively referenced in journals and books spanning all educational levels, including those read by researchers, elementary school students and teachers, engineers, and hobbyists.' back |
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