81. Zero: The Biography Of A Dangerous Idea -- About Zero calculus, complex numbers, projective geometry, and Georg Cantor s remarkable theory of how some infinities can be shown to be larger than others! http://www.users.cloud9.net/~cgseife/aboutzero.html

Zero The Biography of a Dangerous Idea "From the first page to the last, Seife maintains a level of clarity and infectious enthusiasm that is rare in science writing, and practically unknown among those who dare to explain mathematics. Zero is really something!" Curt Suplee, The Washington Post , 23 January 2000. (Read the whole article Zero: The Biography of a Dangerous Idea describes with good humor and wonder how one digit has bedevilled and fascinated thinkers from ancient Athens to Los Alamos. Charles Seife deftly argues that the concept of nothingness and its show-off twin, infinity, have repeatedly revolutionized the foundations of civilization and philosophical thought. If you're already a fan of mathematics or science, you will enjoy this book; if you're not, you will be by the time you finish it." John Rennie, Editor-in-Chief, Scientific American magazine (more reviews and praise: New York Times Time magazine John Horgan ... The Philadelphia Inquirer A concise and appealing look at the strangest number in the universe and its continuing role as one of the great paradoxes of human thought. The Babylonians invented it, the Greeks banned it, the Hindus worshiped it, and the Church used it to fend off heretics. Now it threatens the foundations of modern physics. For centuries the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. For zero, infinity's twin, is not like other numbers. It is both nothing and everything.

82. Newsletter On Proof Famous Paradoxes Zeno s Paradox and Cantor s infinities The Problem of Points an age-old gambling problem that led to the development of probability http://www.lettredelapreuve.it/Newsletter/981112.html

83. 3. XX Century Paradoxes. The diagonal method allows us to face different infinities, starting from A crucial aspect of Cantor s theory is that it allows to reduce to integer http://www.dm.uniba.it/~psiche/bas3/node4.html

Next: 4. Logic and Up: BEING AND SIGN Previous: 2. Wittgenstein and 3. XX century paradoxes. The same procedure allows Cantor to prove that, for any set S, its cardinality is strictly less than the cardinality of its power set P(S). If now we apply this result to the ''set of all sets'', we have the ''Cantor paradox''. In fact the power set of the ''set of all sets'' is a set of sets and then a subset of ''the set of all sets'', and hence this power-set must have a lesser or equal cardinality than the set, in spite of the above theorem. Maybe the most important modern paradox was the Russell paradox, discovered at the beginning of our century and fatal for the Fregean logic foundation of arithmetic. In such theory it was possible to define S, the ''set of all sets which are not members of themselves''. The problem is: S belonging to S ? If the answer is yes, from the definition of S, it follows that S is not member of itself, and then S does not belong to S. If the answer is not, for the same reason, we have that S belongs to S. The argument can be set in the form of the ''diagonal'' argument, putting member(T,T') = 1 iff T belongs to T', otherwise. Then we define S as build by the sets T such that member(T,T) = 0. It is evident that member(S,S) = == member(S,S) =1, and vice versa member(S,S) = 1 ==

84. Set Theory: Cantor Cantor s last two papers on set theory, Contributions to the foundations of were two kinds of infinities, the consistent ones and the inconsistent ones. http://www.math.uwaterloo.ca/~snburris/htdocs/scav/cantor/cantor.html

Previous: Dedekind Next: Frege Up: Supplementary Text Topics Set Theory: Cantor typing . Another, more popular solution would be introduced by Zermelo. But first let us say a few words about the achievements of Cantor. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather because he initiated the study of infinite sets and numbers which have provided such fascinating material, and difficulties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one-to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and five papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after finishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis. In two papers 1870/1872 Cantor studied when the sequence converges to 0. Riemann had proved in 1867 that if this happened on an interval and the coefficients were Fourier coefficients then the coefficients converge to as well. Consequently a Fourier series converging on an interval must have coefficients converging to 0. Cantor first was able to drop the condition that the coefficients be Fourier coefficients consequently any trigonometric series convergent on an interval had coefficients converging to 0. Then in 1872 he was able to show the same if the trigonometric series converged on

85. SIAM: A Life Of Logic And The Illogic Of Life In Cantor s Paradise (Hilbert s term), completed infinities hobnob with the dancing angels. Set theory contains strange stuff, and when I come to things http://www.siam.org/news/news.php?id=35

86. Infinity infinities actually enacts some of the great paradoxes or thought The textual narrative of Cantor s troubled life is illuminated visually by a http://www.siam.org/siamnews/09-03/infinity.htm

Join Renew Contact SIAM SIAM Journals Online WWW From SIAM News, Volume 36, Number 7, September 2003 Hilbert's Hotel, Other Paradoxes, Come to Life in New "Math Play" By Kirsten Shepherd-Barr "Mathematics provides a new language for the theatre," says Luca Ronconi, director of John D. Barrow's exciting play Infinities , which finished a second successful run in Milan in May. Barrow's play does for mathematics what Michael Frayn's Copenhagen did for physics. Infinities actually enacts some of the great paradoxes or "thought experiments" about infinity: the Hotel Infinity in all its vastness, the notion of time-travel, the idea of living forever, Borges's Library of Babel with its endless corridors of books. Watching Infinities brings such concepts to life in a stunning combination of mathematics, philosophy, science, and theatre. For those who felt that David Auburn's play Proof was only tangentially and incidentally about mathematics, here is a play that truly engages the subject, for specialists and general audiences alike. Barrow provided the text, a mixture of original passages and selections from essays on infinity of his own and other people, ranging from Nietzsche to Borges to Hawking. The acclaimed Italian director Ronconi developed the staging in conjunction with the Teatro Piccolo and Sigma Tau Foundation in Italy. The result is a play that demonstrates the very concepts it deals with, and takes the genre of "science plays"-so popular in recent years-to a new level.

87. Infinity And Infinities Infinity and infinities. It was not until the 19th Century that mathematicians One of the amazing consequences of Cantor s work is that it proves the http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L4.html

MT2002 Analysis Previous page (Functions) Contents Next page (Axioms for the Real numbers) Infinity and infinities It was not until the 19th Century that mathematicians discovered that infinity comes in different sizes. Georg Cantor (1845 to 1918) defined the following. Definition Any set which can be put into one-one correspondence with N is called countable Remarks Given such a set, one can count off its elements: 1st, 2nd, 3rd, .. and will eventually reach any element of the set. We will see later that many infinite sets are countable but that some are not. Some versions of the above definition include finite sets among the countable ones, but we will (mostly) not do so. Examples of some countable sets The set Z of positive, zero and negative integers is countable. Proof Here is a counting. That is, we list the elements N Z You can (if you wish) write down a formula: n if n is even, n - 1) if n is odd. The set N N is countable. Proof Count the points with integer coefficients in the positive quadrant as shown. (The formula is now rather tricky to write down.) The set Q of all rationals is countable.

88. Atomism And Infinite Divisibility - Chapter 4 - Aristotle On Infinite Divisibili As a result of Cantor s work, we differentiate among infinities of different cardinality (size). The first such division is between the size of the natural http://www.xenodochy.org/rekphd/chapter4.html

Atomism and Infinite Divisibility prev CHAPTER IV next ARISTOTLE (384-322 BC) AND INFINITE DIVISIBILITY Before we can reasonably examine Aristotle's views on the subject, we need to briefly outline the events and conditions that transpired between Zeno and Aristotle. It was during this period that true Atomism was born. The Birth of Atomism Proper The birth of atomism in its modern form can be traced to a reinterpretation of Melissos's arguments to support Eleatic monism. Melissos re-presented Parmenides' arguments in Ionic prose, but he deviated from Parmenides' teachings. Parmenides claimed the real was a sphere, which suggests that the real was finite. Melissos claimed that the real was infinite. The real, he said, could only be limited by empty space, and there is no empty space. Melissos also presented a reductio argument against pluralism. If there were many things, they would have to be of the same description as I say the One is. While Parmenides had earlier advocated the spherical nature of the one, it was Melissos's assertion that there was no empty space that suggests the next development. Combining Parmenides sphere with the denial of both Melissos's assertions, that the real is infinite and that there is no empty space, yields a spherical non-infinite "real" in existing empty space. The denial of monism multiplies these non-infinite reals and produces atoms. That task fell to Leukippos. Leukippos (450-420 bc) It is certain that Aristotle and Theophrastos both regarded [Leukippos] as the real author of the atomic theory.

89. The Spiritual Function Of Mathematics, By Thomas J McFarlane 58 Here Wolff refers to Cantor s mathematical theory of transfinite numbers. These infinities are just two of the smallest infinities. http://www.integralscience.org/sacredscience/SS_spiritual.html

The Spiritual Function of Mathematics and the Philosophy of Franklin Merrell-Wolff www.integralscience.org If one searches the historical record for evidence of rational and logical thought, one finds among the most highly developed intellects the spiritual philosophers such as Shankara, Nagarjuna, and Plato, for whom the primary function of the intellect is to serve the ends of spiritual realization. Moreover, mathematics, perhaps the most subtle and rigorous form of thought, traces its origins back to Pythagoras and Plato, for whom mathematics is first and foremost a spiritual activity. Although today the spiritual function of mathematics, and rational thought in general, has been largely forgotten, yet there are a small number among us who remember; perhaps the most notable to live in our century is Franklin Merrell-Wolff. This essay presents Merrell-Wolff's writings on the spiritual function of mathematics, selected from his three major works, Pathways Through To Space The Philosophy of Consciousness Without An Object , and Introceptualism . Mathematics, according to Wolff, functions as a bridge between the relative and transcendent states of consciousness. It serves, on the one hand, as a vehicle for crossing from the transcendent to the relative by providing a highly subtle and precise language for expressing the immediate contents of transcendent states with minimal distortion. On the other hand, it also serves as a vehicle for crossing from the relative to the transcendent by providing highly abstract and universal symbols for generating insights through contemplation. Wolff emphasizes, however, that although the structure of this mathematical bridge is provided by the highly subtle forms of thought, an actual crossing of the bridge requires the motivating power of love and devotion.

90. [Two Different Models Of Infinity] - A New Kind Of Science The Peirce s intuition in the matter right around Cantor s time seems to me still Continuum infinities involve the idea of distinguishability breakdown, http://forum.wolframscience.com/archive/topic/747-1.html

A New Kind of Science: The NKS Forum Pages: Two different models of infinity (Click here to view the original thread with full colors/images) Posted by: Doron Shadmi Important: This topic is based on proofs without words ( http://mathworld.wolfram.com/ProofwithoutWords.html A one rotation of the Archimedean Spiral is exactly 1/3 of the circlefs area ( http://www.calstatela.edu/faculty/h...html#Prop.%2022 http://www.geocities.com/complementarytheory/ARCHCON.jpg If this area is made of infinitely many triangles (as can be seen in the picture below) , it cannot reach 1/3 exactly as 0.33333... cannot reach 1/3: http://www.geocities.com/complementarytheory/ARCHDIS.jpg http://www.geocities.com/complementarytheory/MULTIKOCH.jpg http://members.cox.net/fractalenc/fr6g6s.577m2.html In any arbitrary level that we choose, the outer boundary of this multi-fractal has sharp edges. 0.333... = 1/3 only if the outer boundary has no sharp edges. Now we can understand that a one rotation of the Archimedean Spiral is exactly 1/3 of the circle's area only if we are no longer in a model of infinitely many elements, but in a model that is based on smooth and non-composed elements (and in this case the elements are a one rotation of a smooth and non-composed Archimedean Spiral and a one smooth and non-composed circle). A model of infinitely many elements and a model of a non-composed element have a XOR connective between them.

91. Www.phil.uni-passau.de/dlwg/ws02/11-2-94.txt nth power is greater than n, that there are infinities of different sizes. Indeed Cantor s Paradise proves to be a truly nonexistent platonic heaven, http://www.phil.uni-passau.de/dlwg/ws02/11-2-94.txt

92. Slater 20 Tiles thinks that Cantor s proofs of nondenumerability, nevertheless, still require us to discriminate between infinities. She goes on http://ejap.louisiana.edu/EJAP/2002/Slater.html

Namely-Riders: An Update B. H. Slater The Department of Philosophy, The University of Western Australia http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html I here recall Ryle's analysis of Heterologicality, but broaden the discussion to comparable analyses not only of Heterologicality but also other puzzles about self-reference. Such matters have a crucial bearing on the debate between representational and non-representational theories of mind, as will be explained. [1] The most remarkable thing about the so-called Heterologicality Paradox is that a straightforward logical analysis of the case shows there is no proper paradox. The appropriate analysis, moreover, has been published more than once, yet even its author did not see that there was no problem. [2] If the reader will take the trouble to consult, for instance, the 4th edition of Irving Copi's Symbolic Logic (Copi 1973), then, in the chapter about the theory of types, he will find the definition of heterologicality, and then a supposed proof of the contradiction: Het'het' iff -Het'het'.

93. October 97 Stammtisch Notes: Minddance Chronicles Comments (This part extends on Cantor s infinities a bit.) This is true in maths because different number systems (scales) deal with integers, fractions and all that http://www.ideatree.net/stammtsh/oct97cmt.htm

The Idea Tree Member I.C.I.E.R. A Virtual Learning Environment Updated: 2001 May 13 Z It's A Sin To Waste A Good Idea!! NOTICE: This material may be copied for personal use only provided that the entire header remains intact. Absolutely no "for-profit" use of this material is allowed without full written permission from The Idea Tree and October 97 Stammtisch Notes: Minddance Chronicles Comments TABLE OF CONTENTS Chaos and Determinism (31 Oct 97) (30 Oct 97) Holographic Realities (28 Oct 97) Zones, Bio-Matter and Organisations (26 Oct 97) Quantum Bees (26 Oct 97) CACs and the Will to Power (26 Oct 97) Structural Persistence (26 Oct 97) All-Scales Thinking (26 Oct 97) Can Spiders "Think"? (25 Oct 97) Volunteer Teamwork (25 Oct 97) Networks, Feedbacks, and Process (25 Oct 97) Revisiting Persistent Command-and-Control (CAC) (25 Oct 97) Engineering and Complexity (23 Oct 97) Decision by Committee? (20 Oct 97) "Zones" & Persistence (19 Oct 97) Cycles and Scales (18 Oct 97) (18 Oct 97) Conjecture Concerning Life Systems (18 Oct 97) The Costs of Credit Assignment Problems (16 Oct 97) Credit Assignment, Revisited

94. The Cardinal I Ching the cognitive processes behind Cantor s derivation of different infinities with transfinite cardinal numbers; thus Cantor s alephs reflect qualitative http://members.ozemail.com.au/~ddiamond/cardinal.html

The Cardinal I Ching A fundamental distinction made by the human brain is that of ordinality from cardinality. Ordinality deals with sequence, cardinality with size. These distinctions are reflected in the I Ching through modes of interpretation of hexagrams where ordinality refers to a path through a hexagram where the only difference is in the position of a line - before/after - , whereas cardinality refers to a more qualitative emphasis with the distinctions of 'raw' and 'refined' and so an emphasis on exageration, distortion. (this gets into topolgy etc) Consider the hexagram as a whole and that whole can vary in colour expression or chord expression, sometimes it is loud and other times soft etc etc Note there is no need for order here, no explicit causality. When we combine the two distinctions, cardinality and ordinality, we deal with complex expressions such as the development PATH from raw (base line of a hexagram) to refined (top line of a hexagram); we combine the 'horizontal' with the 'vertical'; orthogonally-derived map making. The particular path from raw to refined has a cardinality emphasis within an ordinality context but the ordinality is WITHIN the hexagram, the emphasis is on the qualitative expressions of the hexagram in question; mathematically we are dealing with a histogram and to do this we have to suspend time.

95. Number As Archetype But the continuum hypothesis says that there are no infinities that lie between As early as 1905 René Baire suggested that Cantor s continuum hypothesis http://www.goertzel.org/dynapsyc/1996/num.html

DynaPsych Table of Contents NUMBER AS ARCHETYPE Robin Robertson This paper has been adapted from the final chapter of Jungian Archetypes: Jung, Godel and the History of Archetypes The sequence of natural numbers turns out to be unexpectedly more than a mere stringing together of identical units: it contains the whole of mathematics and everything yet to be discovered in this field. Carl Jung It has turned out that (under the assumption that modern mathematics is consistent) the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic; i.e., the domain of the kind of elementary indisputable evidence that may be most fittingly compared with sense perception. Kurt Godel Mathematics followed a similar path. When it came out of its slumber early in the 17th century, the first product was analytic geometry, which demonstrated the equivalence of geometry and arithmetic. And with no clearly defined problem, there was no mathematical Kant to resolve the problem. Instead Euler was the chief representative of the era, and Euler was more concerned with building a great edifice based on mathematics, than with examining its foundations. Though still consciously unresolved, the issue continued to evolve in the unconscious. By the 2nd half of the 19th century, psychology finally emerged from the unconscious, wearing the twin faces of experimental and clinical psychology. It took someone like Freud, who bridged both camps, to discover the unconscious. It took someone like C. G.

96. 4.06: PHYSICS AND MATHEMATICS -- Logic And Computational Theory The problem of large infinities in mathematics from a vedic perspective One of the most successful attempts used Cantor s original conceptual framework http://www.imprint-academic.demon.co.uk/SPECIAL/04_06.html

Classified Abstracts PHYSICS AND MATHEMATICS 4.6 Logic and computational theory A recursive theory of self-conscious machines M. This work is motivated by the desire to model the degree to which one must consciously attend to a problem to solve it. This `degree' of conscious attention is not the time or space (or any of the usual resources) needed to solve the problem, as is the case in complexity theory. Rather, it is based on the degree to which a problem solver can monitor and control himself. A problem will be called more complex here (or `deeper') if one must have a higher degree of consciousness to solve it. `Problems' will be modelled by recursive functions from the natural numbers to the natural numbers, `problem solvers' by Turing Machines, and `degrees' of consciousness by constructible ordinals. For any constructible ordinal a , an a -self-monitoring machine, or a -machine, (as they will be called) behaves as follows: Before seeing the input, it places the initial ordinal a into an ordinal clock. This is the degree of consciousness by which it must compute the function on all f g such that For all x g x f x sane machines are defined which are well-behaved in the sense that any two a -machines will behave similarly. The class of sane functions that are

97. The Power Set Using Cantor s cross section method, it is simple to create an enumeration of the power set of The power set, however, has infinities within infinities. http://descmath.com/diag/power.html

The Power Set The goal of transfinite theory is to create a dichotomy between the rational and real numbers and to use this dichotomy as the foundation for a definition of the continuous line. Transfinite theorist claim to have accomplished this goal with the diagonal method . However, I have found the diagonal proof lacking. For example, using the diagonal method, I am able to show that the set of namable numbers is both denumerable and non-denumerable. It appears that the diagonal method itself is not sufficient to establish the claimed distinction between rational and reals, and I am unable to even begin a study of transfinite theory. Transfinite theorists claim that the dichotomy that exists between the rational and the real numbers also exists between the set of ordered pairs and the power set of the integer. Exploring these sets might give me better answers to the nature of the transfinite. n elements. The distribution of the elements follow the binomial pattern (Pascal's Triangle): Using Cantor's cross section method, it is simple to create an enumeration of the power set of an arbitrarily large set. This exercise, should give a clearer understanding of the issue addressed by the theory.

98. NOVA | Infinite Secrets | Working With Infinity | PBS NOVA How did Georg Cantor s set theory refine mathematicians thinking about Do you allow in infinities beyond the first, most simple infinity or not? http://www.pbs.org/wgbh/nova/archimedes/infinity.html

Working with Infinity: A Mathematical Perspective Infinite Secrets homepage For mathematicians, infinity means something completely different than for philosophers. It's not something vague and unapproachable, but rather something with a precise definition that lies at the core of modern mathematics. To explain how this eminently practical form of infinity evolved over time, and to translate for the layman how mathematicians think of it, we approached Stanford University classics historian Reviel Netz. A scholar who discovered that, contrary to belief, the ancient Greeks, through the work of Archimedes, had actually toyed with infinitely large sets, Netz knows a thing or two about mathematical infinity. For a philosophical take on the subject, see Contemplating Infinity Defining infinity NOVA: How do mathematicians define infinity? Netz: Something which is equal to some of its parts. That's really the technical definition. NOVA: Netz: That's the curious thing. Infinity became a really clear and well-defined quantity mathematically in the late 19th century, which it wasn't before and which makes it rather different from what we ordinarily talk about when we talk about infinity, namely, about something very, very big. In mathematics nowadays, when we think about infinity, we think about a set whose properties are different from those of ordinary sets. NOVA: Can you explain?

99. The Repressed Content-Requirements Of Mathematics Cantor s interpacking of infinite decimals to disprove the invariance of Wittgenstein s arguments against Cantor s infinitiesin turn considered http://www.henryflynt.org/studies_sci/reqmath.html

Back to H.F. Philosophy contents The Repressed Content-Requirements of Mathematics Henry Flynt [started c. 1987; this draft 1994] (c) 1994 Henry A. Flynt, Jr. A. Mathematics, as it is conceived in the twentieth century, has presuppositions about perception, and about the comprehension of lived experience relative to the apprehension of apparitions, which are repressed in professional doctrine. It also has presuppositions of a supra-terrestrial import which are repressed. The latter concern abstractions whose reality-character is an incoherent composite of features of sensuous-concrete phenomena. In the twentieth century, mathematicians have been taught to say by rote, "We are beyond all that now. We are beyond psychology, and we are beyond independently subsisting abstractions." This recitation is a case of denial. Indeed, if this recitation were true, it would allow mathematics nowhere to live. Certainly social conventionsbeloved by the Vienna Circleare too unreliable to found the truths which mathematicians claim to possess. (Truths about the decimal value of [pi], or about different sizes of infinity, for example.) To uncover the repressed presuppositions, a combination of approaches is required. (One anthropologist has written about "the locus of mathematical reality"but, being an academic, he merely reproduces a stock answer outside his field, namely that the shape of mathematics is dictated by the physiology of the brain.)

100. Untitled Document The nineteenthcentury mathematical genius Georg Cantor s answer to this question not Cantor s counterintuitive discovery of a progression of larger and http://www.wwnorton.com/catalog/fall03/000338.htm

Page 5     81-100 of 100    Back | 1  | 2  | 3  | 4  | 5