vol 4: Glossary
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Continuity
[Hazewinkel, continuity] [Borowski, continuity]
The notion of continuity is closely related to our ideas about continuous motion. We imagine that in moving from A to B an object must occupy every spatial point between A and B. This idea lay at the foundation of Zeno's arguments against the possibility of motion. Zeno was a disciple of Parmenides, who held that motion was illusory and that true being was one and immobile, an idea which carried over into Christian notions of God.
Aristotle, perhaps thinking of the continuity of a chain, saw continuity as a spatial relationship related to contact and contiguity. Things are said to be "in contact" whose extremes are together in a place. [Metaphysics XI, xii]. Aristotle
The invention of calculus, which we might see as the mathematics of motion, ushered in a new era in the understanding of continuity. Cantor's discovery of transfinite set theory and the infinite structure of the line led to a slow redefinition of the notion of continuity. Naive set theory led to contradiction because it placed no limit on the size of sets. This led to formal theories, deep investigations into the foundations of logic and mathematics and the theorems discovered by Goedel, Turing and Shannon.
Through this period, the notion of continuity underwent a radical change. In the beginning, we had continuity according to physical contact, touching or interlinking. Now we see continuity more abstractly in terms of logic and dynamics.
From a logical point of view, a sequence propositions may be considered continuous if each proposition is derived from its predecessors according to a set of agreed logical rules. The opposite of logical continuity is logical inconsistency or contradiction, something which is outside the rules of inference.
From a dynamic point of view, continuity means that one entity influences or communicates with another. With our definition of entity, two communicating entities are in fact one entity, that is dynamically continuous. In this way we see the universe to be both many discrete entities, and, insofar as these entities communicate with one another, one.
Further reading
Books
| Aristotle, and (translated by P H Wickstead and F M Cornford), Physics books I-IV, Harvard University Press, William Heinemann 1980 Introduction: 'The title "Physics" is misleading. .. "Lectures on Nature" the alternative title found in editions of the Greek text, is more enlightening. ... The realm of Nature, for Aristotle, includes all things that move and change ... . Thus the ultimate "matter" which, according to Aristotle, underlies all the elementary substances must be studied, in its changes at least, by the Natural Philosopher. And so must the eternal heavenly spheres of the Aristotelean philosophy, insofar as they themselves move of are the cause of motion in the sublunary world.' Amazon back |
| Aristotle (translated by H Tredennick), Metaphysics I-IX, Harvard University Press, William Heinemann 1980 Introduction: "[Aristotle] felt that there must be a regular system of sciences, each concerned with a different aspect of reality. At the same time it was only reasonable to suppose that there was a supreme science which was more ultimate, more exact, more truly Wisdom than the others. The discussion of ths science - Wisdom, Primary Philosophy or Theology, as it is variously called - and of its scope, forms the subject of the Metaphysics' page xxv. http://www.amazon.com/exec/obidos/ASIN/0674992997/tnrp">Amazon back |
| Aristotle (translated by H Tredennick and G Cyril Armstrong), Metaphysics X-XIV, Oeconomica and Magna Moralia, Harvard University Press, ; William Heinemann Ltd. 1977 Jacket: 'Aristotle, great Greek philosopher, researcher, reasoner and writer, born at Stagirus in 384 B.C., was the son of a medical Doctor, Nichomachus of Phaestis. He studied under Plato at Athens and taught there 367-347; spent three years at the court of former pupil Hermeias in Asia Minor and married Pythias a relation of his; after some time in Mitylene, in 343-2 he was appointed by King Philip of Macedon to be the tutor of his teen-aged son Alexander, and had other pupils. After Philip's death in 336, Aristotle became head of his own school (of 'Peripatetics'), the Lyceum at Athens. Because of anti-Macedonian feeling there after Alexander's death in 323, he withdrew to Chalcis in Euboea and died there in 322.' Amazon back |
| Aristotle (translated by P H Wickstead and F M Cornford), Physics books V-VIII, Harvard University Press,William Heinemann 1980 Introduction: 'Simplicius tells us that Books I - IV of the Physics were referred to as the books Concerning the Principles, while Books V - VIII were called On Movement. The earlier books have, in fact, defined the things which are subject to movement (the contents of the physical world) and analyzed certain concepts - Time, Place and so forth - which are involved in the occurrence of movement.' Book V is a further intoduction to the detailed analysis in Books VI - VIII. Book VI deals with continuity, Book VII is an introductory study for Book VIII, which brings us to the conclusion that all change and motionin the unvierse are ultimately caused by a Prime Mover which is itself unchanging and unmoved and which has neither magnitude nor parts, but is spiritual and not in space. Amazon back |
| Borowski, Ephraim J, and Jonathan M Borwein, HarperCollins Dictionary of Mathematics, Harper Collins 1991 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership.' 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 |
| Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
Links
| John Burnet Early Greek Philosophy: Parmenides of Elea: The Poem back |
| mathacademy.com Zeno's paradox of the Tortoise and Achilles 'Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles.' back |
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