vol 3: Development
cap 1: Epistemology
page 2: Abstraction
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Abstraction
Epistemology is quality control for knowledge, separating the true and trustworthy from the dodgy and unreliable. Before we can talk about quality control, however, we need a theory of knowledge. This we base around the age old idea of abstraction. When I come to know something, like a bag of flour, I have something new inside me, not the bag of flour, but an abstract representation of the bag of flour.
On this site, we affirm the hypothesis proposed by Landauer, that all information is physically represented. Landauer
It is this abstract representation of the bag of flour, encoded physically within me in some complex set of symbols , which, enables me to imagine all sorts of things about that bag of flour: how to approach it, pick it up and load it into a truck; how to sample its quality; estimate how much to pay for it; foresee the pleasure of eating it, and so on.
Mathematics
We have described scientific method as a cycle of imagination and testing. Popper Modern science began its career when Galileo and others began to use mathematics to expand their imagined models of the world.
Here we think of mathematics as the exploration of pure abstraction. Mathematicians can communicate anything that can be written down. There is simply not enough matter in the earth to construct a physical infinite set. Yet abstraction allows us to imagine, talk, and write about infinite sets. The practical criterion for mathematics is that its creations be consistent, useful and beautiful
Mathematical language is an aid to consistency and communication. It is an extension of natural language. Mathematics uses numbers stretching to infinity to enable us to deal with huge sets of objects like all the points in a space, and complex relationships between them.
We see mathematics stretching back to the beginning of recorded history, so we can only speculate about its origins. Kramer Here we use naming as a starting point to explore the nature of mathematics. Naming establishes a correspondence between two things.
In natural languages, names are seen as very different from things. My name is a word; I am a massive and complex physical object. Mathematics deals only with names, so that its correspondences exist between names only. It completely ignores the physical embodiment of names that Landauer suggests is necessary for their realization.
Arithmetic and geometry
Traditionally, mathematics is divided into two areas, arithmetic and geometry. Arithmetic is concerned with numbers, geometry with shapes, forms, pictures and spaces. Applied arithmetic uses numbers to model the relationships of distinct, countable objects like sheep and monetary units. Applied geometry deals with the measurement and calculation of continuous objects like land and buildings.
The interface between arithmetic and geometry is a fertile source of mathematics. It was here that people realized that integers were not enough to describe continuous geometric objects. So it become necessary to invent fractions (rational numbers). Then it was discovered that the diagonal of a unit square cannot be represented by a rational number. Heath This pointed to the new world of real numbers which both includes the rational numbers, and fills the spaces between them.
In addition to numbers, arithmetic is built on a set of operations on numbers, addition, subtraction, multiplication and division. Arithmetic was put on a firm foundation with the invention of set theory by Georg Cantor late in the nineteenth century. Dauben Using the imagery of sets, we can explain clearly to ourselves how things like addition and multiplication work.
Order and correspondence
Two important ideas in set theory are order and correspondence. The discovery of algebra led to the development of the complex numbers so that every algebraic equation would have a numerical solution. A complex number is an ordered pair of numbers with special rules for their arithmetic operations. Set theory does not restrict itself to ordered pairs, but may consider infinite ordered sets and operations that operate on such sets. This mathematical formalism has proved very useful in physics.
Set theory put mathematics on such a firm foundation that Cantor discovered a new realm of numbers. Cantor's work was inspired by the need for an arithmetic treatment of the geometrical continuum.
He was able to reveal an enormously complex structure within the continuum which can be represented by transfinite numbers. Cantor The transfinite cardinal and ordinal numbers that Cantor invented form the mathematical backbone of this site.
Cantor imagined (and it seems true) that anything thinkable can be represented in the space of transfinite numbers. Following his lead, we hope to exploit transfinite numbers to study god.
Consistency, completeness and computability
As with most great inventions, set theory raised more questions than it solved. In the early part of the twentieth century, people were inclined to think that mathematics was a logical linguistic structure in which all questions could be answered. In other words, mathematics, although infinite, was in some way bounded.
Kurt Goedel and Alan Turing, building on Cantor's work, showed that this was not the case. Goedel, Hodges Goedel found that if mathematics is consistent, it is not complete. Turing found that if mathematics is consistent, it is not computable.
These two ideas are of great importance for theology, since they suggest that any consistent god cannot be of a fixed infinity, but must be an ever growing, living and evolving entity. Such an abstract system could possibly describe our growing, living and evolving universe, and our infinitely imaginative minds.
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 |
| 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 |
| 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 |
| Goedel, Kurt, and Solomon Feferman et al (eds), Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' Amazon back |
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' 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 |
| Kneebone, G T , Mathematical Logic and the Foundations of Mathematics, van Nostrand 1975 Preface: 'The present book ... is designed to serve in the first instance, when supplemented by reference to original sources, as a comprehensive introduction to the earlier phases of the historical development of the philosophy of mathematics. p vi.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 |
| Popper, Karl Raimund, Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge and Kegan Paul 1972 Preface: 'The way in which knowledge progresses, and expecially our scientific knowledge, is by unjustified (and unjustifiable) anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism; that is, by attempted refutations, which include severely critical tests.' [p viii] 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 |
| Stewart, Ian, Why Beauty is Truth: A History of Symmetry, Basic Books/Perseus 2007 Jacket: ' ... Symmetry has been a key idea for artists, architects and musicians for centuries but within mathematics it remained, until very recently ,an arcane pursuit. In the twentieth century, however, symmetry emerged as central to the most fundamental ideas in physics and cosmology. Why beauty is truth tells its history, from ancient Babylon to twenty-first century physics.' Amazon back |
Papers
| Landauer, Rolf, "Information is a physical entity", Physica A, 263, 1-4, 1 February 1999, page 63-7. 'This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that, on the ultimate nature of the laws of physics are included.'. back |
| Landauer, Rolf, "The Physical Nature of Information", Physica A, 217, 4-5, 15 July 1996, page 188-93. 'Information is inevitably tied to a physical representation and therefore to restrictions and possibilities related to the laws of physics and the parts available in the universe. Quantum mechanical superpositions of information bearing states can be used, and the real utility of that needs to be understood. Quantum parallelism in computation is one possibility and will be assessed pessimistically. The energy dissipation requirements of computation, of measurement and of the communications link are discussed. The insights gained from the analysis of computation has caused a reappraisal of the perceived wisdom in the other two fields. A concluding section speculates about the nature of the laws of physics, which are algorithms for the handling of information, and must be executable in our real physical universe.'. back |
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