volume II: Synopsis
Part II: A brief history of dynamics
page 15: Continuity
The observable world is a collection of discrete objects and events like atoms and people. On the other hand changes in the world, like growth or the movement of an object from here to there appear to be continuous. The relationship between discreteness and continuity is an age old mathematical and philosophical problem.
Mathematically, the study of continuous metric and topological spaces is called analysis. The mathematical study of motion has been problematic for a long time. Parmenides of Elea (fl circa 510 bce) postulated that 'real' reality is static and eternal. His follower Zeno of Elea (490-430 bc) may have set out to support Parmenides' position by developing arguments to show that motion is impossible and so illusory. Mathematical analysis - Wikipedia, John Palmer: Zeno of Elea
The modern mathematical treatment of motion began with Newton's invention and application of calculus. Newton's work prompted a careful study of continuity. Like Zeno, mathematicians consider a line to be a set of points. It has been known since antiquity that there are too many points in a line for them to be placed into correspondence with the natural numbers. It is simply proved that the square root of two (representing the length of the diagonal of a unit square) is not a rational number. Square root of 2 - Wikipedia
This led Georg Cantor to seek an expression for the number of points in a continuous line, the cardinal of the continuum. We turn to Cantor on the next page, but here we want to discuss whether it is really possible to represent a continuous line by a set of discrete points, since continuity and discreteness seem to be incompatible with one another.
The modern treatment of continuity in point sets depends upon limiting processes. We define the differential coefficient of a function y = f(x) at the point x by the expression dy/dx = f '(x) = Limit δx → 0 [f(x + δx) - f(x)] /δx . This assumes that there are no surprises in the process of making δx smaller and smaller.
We use a similar approach to define the continuity of a function, by which we mean that small changes in the independent variable are accompanied by small changes in the dependent variable. This idea is represented in Weierstrass's 'ε δ' definition of continuity.
Continuous function - Wikipedia, Karl Weierstrass - Wikipedia, (ε, δ)-definition of limit - Wikipedia
It seems that most mathematicians are comfortable with this pointwise approach to continuity, that is they consider the arguments presented by Weierstrass and others to be logically sound. Even though there is difficulty reconciling discrete points and continuous lines, the discussion is effectively about the distances between the points (measured, for instance by ε and δ) rather than the points themselves which maintain their classical property of having no size. What is important is that ε and δ are here assumed to be themselves continuous so that that can shrink to as near to zero as we wish. This makes the argument look a little circular.
As in all other mathematical arguments, what matters here is the 'logical continuity' of the arguments for geometrical or metrical continuity. By logical continuity, we mean the correct application of the rules of logical inference so that the conclusion of a proof is 'logically bound' to the axioms or assumptions upon which the proof is built. Rules of inference - Wikipedia
Here we consider logical continuity to be prior to metric continuity, and in fact to all the other mathematical conclusions that we reach from axioms via logical argument. Such a logical argument, once discovered, is a mechanical process which can be carried out by a suitably programmed computing machine. Much of the progress of mathematics depends on the creative discovery of logical connections between the established conclusions of mathematics and new propositions which begin their lives as inspired guesses and (if they are correct) gradually work their way toward inclusion in the set of results considered proven.
We see this process in operation in the work inspired by Hilbert's problems formulated at the beginning of last century, and in the more recent Millennium Prize problems formulated by the Clay Institute. In each case, the problem is (or will be) considered solved, when someone finds a continuous logical path from established mathematics to a solution to the problem in question. Of course, our everyday mathematical operations seek a similar logical path between the statements and solutions of simpler problems, like doing accounts or computing prices. Hilbert's problems - Wikipedia, Millennium Prize Problems - Wikipedia
Logical continuity is the key definition of continuity for this site. As we can see from Weierstrass's definition, the essence of geometric continuity is that nothing very exciting happens to a function as we move through the domain of a function: small changes in the domain are accompanied by small changes in the range. A logically continuous process on the other had, is not so constrained. The function not for instance, completely reverses the truth value of a statement.
The perfect example of a logical continuum is a halting Turing machine which proceeds from some initial state to a final state by a series of logically determined steps. One feature of the Turing machine is that it can stop after every operation and restart again without losing its place in the computation. Logical continuity is independent of continuity in space and time, so we can imagine the very early universe having logical process before it has processes in spacetime. Nevertheless, in practical computing we are often quite concerned with how fast our machines go, since other things (eg the design of the software) being equal, a faster machine delivers an answer more quickly, which is important when time is, for some reason, of the essence. Turing machine - Wikipedia
(revised 4 April 2020)
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Further readingBooks
Berberian, Sterling K, A First Course in Real Analysis, Springer Verlag 1994 Jacket: 'This book offers an initiation into mathematical reasoning and into the mathematicians's mind-set and reflexes. Specifically, the fundamental operations of calculus - differentiation and integration of functions, and the summation of infinite series - are built, with logical continuity (i.e., "rigor"), starting from the real number system. The first chapter sets down the axioms for the real number system, from which all else is derived using the logical tools summarized in the Appendix. The discussion of the "fundamental theorem of calculus", the focal point of the book, is especially thorough. The concluding chapter establishes a significant beachhead in the theory of the Lebesgue integral by elementary means.'
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Gaukroger, Stephen, Descartes: An Intellectual Biography, Clarendon Press 1995 Jacket: 'Rene Descartes (1596-1650) is the father of modern philosophy and one of the greatest of all thinkers. This is the first intellectual biography of Descartes in English; it offers a fundamental reassessment of all aspects of his life and work. . . . Descartes' early work in mathematics and science produced ground-breaking theories, methods and tools still in use today. This book gives the first full acount of how this work informed and influenced the later phisosophical studies for which, above all, Descartes is renowned. . . . [It] offers for the first time a full understanding of how Descartes developed his revolutionary ideas. It will be a landmark publication, welcomed by all readers interested in the origins of modern thought.'
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Kolmogorov, A N, and S V Fomin, Elements of the Theory of Functions and Functional Analysis Volumes 1 and 2, (Two volumes bound as one), Dover 1999 Jacket: Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces. The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces - a topic seldom discussed in textbooks. . . . Part two focusses on an exposition of measure theory, the Lebesgue interval and Hilbert space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions and theorems.'
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Kreyszig, Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik
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Links
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The “computable” numbers may be described briefly as the real numbers whose expression: s as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back |
Continuous function - Wikipedia, Continuous function - Wikipedia, the free encyclopedia, 'IIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".' back |
(ε, δ)-definition of limit - Wikipedia, (ε, δ)-definition of limit - Wikipedia, the free encyclopedia, 'In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. It was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy never gave an (ε, δ) definition of limit in his Cours d'Analyse, but occasionally used ε, δ arguments in proofs. The definitive modern statement was ultimately provided by Karl Weierstrass.' back |
Hilbert's problems - Wikipedia, Hilbert's problems - Wikipedia, the free encyclopedia, 'Hilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the International Congress of Mathematicians, speaking on 8 August in the Sorbonne. The complete list of 23 problems was later published, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.' back |
John Palmer (Stanford Encyclopedia of Philosophy), Zeno of Elea, 'Zeno of Elea, 5th c. B.C. thinker, is known exclusively for propounding a number of ingenious paradoxes. The most famous of these purport to show that motion is impossible by bringing to light apparent or latent contradictions in ordinary assumptions regarding its occurrence. Zeno also argued against the commonsense assumption that there are many things by showing in various ways how it, too, leads to contradiction.' back |
Karl Weierstrass - Wikipedia, Karl Weierstrass - Wikipedia. the free encyclopedia, ' Karl Theodor Wilhelm Weierstrass (German: Weierstraß 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany and gymnastics.
Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.' back |
Mathematical analysis - Wikipedia, Mathematical analysis - Wikipedia, the free encyclopedia, Mathematical analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry. However, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).' back |
Millennium Prize Problems - Wikipedia, Millennium Prize Problems - Wikipedia, the free encyclopedia, 'The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of May 2013, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. The Poincaré conjecture, the only Millennium Prize Problem to be solved so far, was solved by Grigori Perelman, but he declined the award in 2010.' back |
Planck constant - Wikipedia, Planck constant - Wikipedia, the free encyclopedia, ' Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. . . . For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.626 070 150 × 10-34 J⋅s exactly. ' back |
Rules of inference - Wikipedia, Rules of inference - Wikipedia, the free encyclopedia, 'In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference modus ponens takes two premises, one in the form of "If p then q" and another in the form of "p" and returns the conclusion "q".' back |
Square root of 2 - Wikipedia, Square root of 2 - Wikipedia, the free encyclopedia, 'Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.' back |
Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer.
The "machine" was invented in 1936 by Alan Turingwho called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.' back |
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