21. Math Department of Babylon, the harmonies of Pythagoras, Zeno's Paradoxes, Aristotle's logic, Kepler'sVisions, Newton's clockwork, cantor's infinities, Einstein's railway http://www.saintanns.k12.ny.us/depart/math/mathcourses.html

Math Department ALGEBRA 1 (The Department) This course is an introduction to the principles of elementary algebra. Topics covered include simplification and evaluation of algebraic expressions including algebraic functions, exponential and radical expressions; solution of concrete and linear equations of the first and second degree, of absolute value equations and inequalities, and two-variable systems of equations; graphing of linear and quadratic equations and inequalities; factoring and division of polynomials; function notation; and solution of word problems by algebraic technique. GEOMETRY (The Department) ALGEBRA 2 (The Department) Algebra 2 is devoted to simplifying, problem solving, and graphing nth degree polynomial functions. We review the basic concepts of algebra, including the study of the real number system and the accompanying axioms; solving equations and inequalities with an emphasis on word problems, absolute value, and linear functions. Moving ahead, we study the characteristics of polynomial and rational algebraic expressions. Here too, practical and graphic applications are presented. Irrational and imaginary expressions are explored. Other topics include logarithms and exponential functions, conic sections, and an investigation of algebraic as well as graphical representations of inverse functions. Matrices, determinants and probability may be encountered. Prerequisite: Algebra 1.

22. Ziring Book Review Pages - Current Some of the other topics covered in less depth include Turing machines, Post productionsystems, cantor's infinities, NPcompleteness, Maxwell's demon, neural http://users.erols.com/ziring/bookrev.htm

Welcome to the Ziring Book Pages This page provides two services: links to book review, news, and catalog sources on the web, and capsule reviews of notable books. The reviews are written personally by us, so they are highly subjective; take these reviews with a grain of salt! Current book reviews (New: The Advent of the Algorithm by Dr. David Berlinsk) Book Reviews Master Index Old Review Pages: Summer '96 book reviews Winter '98 book reviews Fall '96 book reviews Spring-Summer '98 book reviews ... My Science Fiction Page (w/ top 100 authors list) For other book news and views, check out the Usenet: rec.arts.books.* newsgroups Ziring Book Reviews for March-June, 2000 Here are some books that you might find interesting. Go Figure! Using Math to Answer Everyday Imponderables by Clint Brookhart ISBN 0-8092-2608-1 Educators and social critics noticed, in the late 1980s, that general familiarity with basic arithmetical concepts and techniques was low among the American population. J.A. Paulos even coined a word to describe it: innumeracy . The real key to improving numeracy, many of its advocates point out, is to make mathematics seem more real and applicable to people's everyday lives.

23. Mathematics Emperor's New Clothes. Not until the late twentieth century did Cohen prove the relationship betweencantor's infinities and the Axiom of Choice. So what's the big deal? http://www.netautopsy.org/jharempr.htm

MATHEMATICS AND THE EMPEROR'S NEW CLOTHES. One of the luxuries of being an amateur mathematician, who does not earn his living at mathematics, is that one may ask questions that might get a salaried mathematician fired from his job. That is, the questions are so outrageous or apparently simple-minded, that one is fired either for blasphemy or for gross incompetence. So what does all this have to do with mathematics? Somewhere late in my graduate school training in biomathematics, it dawned on me that there are about a dozen central ideas in mathematics, all of them basically fairly simple once understood, from which one may derive all the important theories of mathematics. The amazing thing is that such simple things took such a long time to internalize in our culture. For example, why was the Greco-Latin culture so resistant to the idea of ZERO, discovered one thousand years B.C. by the Babylonians (as a place-holder on the abacus)? The idea was banished from ancient Greece, and not really embraced in Europe until the sixteenth century, by merchants not mathematicians, who could do their accounting far more easily with Arabic numerals (with zero) than with Roman numerals (without zero). A few more examples: Pythagoras's proof [Singh]; infinity; infinitesimals (Calculus; Newton/Leibniz; Weierstrass; Robinson's calculus); open/closed sets (Heine-Borel theorem); computational complexity (NP complete problem; why the Sieve of Eratosthenes takes so long to solve); symbolic logic [Boole; Lewis and Langford]; Goedel's proof [Casti and DePauli]; Hilbert's Tenth Problem [Davis]; Fermat's Last Theorem [Singh]; Riemann hypothesis [Davis]; fractals [Lauwerier], etc.

24. Newsletter On Proof the properties of the prime numbers and the difficulty of finding primes FamousParadoxes Zeno's Paradox and cantor's infinities - The Problem of Points http://www-didactique.imag.fr/preuve/Newsletter/981112.html

Herbst P. G. What works as proof in the mathematics class . Ph.D. Dissertation, The University of Georgia, Athens GA. USA Lopes A. J. Raccah P.-Y. (1998) L'argumentation sans la preuve : prendre son biais dans la langue. Interaction et cognitions . II(1/2) 237-264. Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) A model for analysing the transition to formal proofs in geometry. (Volume 2, pp.24-31) Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) Dragging in Cabri and modalities transition from conjectures to proofs in geometry. (Volume 2, pp. 32-39) Baldino R. (1998) Dialectical proof: Should we teach it to physics students. (Volume 2, pp. 48-55) Furinghetti F. Paola D. (1998) Context influence on mathematical reasoning. (Volume 2, pp. 313-320) Gardiner J. Hudson B. (1998) The evolution of pupils' ideas of construction and proof using hand-held dynamic geometry technology. (Volume 2, pp. 337-344) Garuti R.

25. Physics Forums - The Premier Science And Technology Community infinite sum cantor's infinities refer to the infiniteseries ofa finite number. Ie, Cantor's math refer to a closed-set. However http://www.physicsforums.com/topic.asp?TOPIC_ID=5743&ARCHIVE=

26. Cosmos In Science And Religion 20th Century Science and the new Encounter Mathematics and Physics of the20th Century. cantor's infinities, Godel's theorem. Quantum Mechanics. http://diacentro.physics.auth.gr/eng/eng_topics.html

Topics Cosmogenesis How the world was born. Cosmogenesis in ancient mythologies. Cosmogenesis and ancient greek thought. Genesis and Bible. The birth of Science The ancient Ionians. Logic and beauty in Cosmos. The founding beliefs of Science. The first scientific systems. Democritos, Platonic Timeos, Aristotle. Physics and Metaphysics. Cosmos and Religion Scientific Revolution and the Break The premises of the Scientific Revolution. Descartes, Leibniz, Newton. The new outlook. The supremacy of the subject and the reign of quantity. Classical Mechanics. The conflict with Religion. Creation and Evolution Darwin and the new discoveries in biology. Creation, an instantaneous act, or a continuous process? The human being and the rest of Nature. 20th Century Science and the new Encounter Mathematics and Physics of the 20th Century. Cantor's infinities, Godel's theorem. Quantum Mechanics. Levels of knowledge and triadic structures. The paradigm of Charles Sanders Peirce. The theology of Pavel Florensky. The apophatic tradition. Cosmos as a domain of convergence among Science and Religion.

27. [FOM] As To Strict Definitions Of Potential And Actual Infinities. well as modern 'nonnaive') set theory is based on cantor's theorem on previous message As to strict definitions of potential and actual infinities (see FOM http://www.cs.nyu.edu/pipermail/fom/2003-January/006173.html

[FOM] As to strict definitions of potential and actual infinities. Alexander Zenkin alexzen at com2com.ru Thu Jan 16 14:19:32 EST 2003 Previous message: [FOM] Question about hard-core independence Next message: [FOM] Question on Equivalence of Theorems Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html ) I have given a quite impressive list of Cantor's opponents as regards the rejection of the actual infinite who, according to W.Hodges' classification, "must be <AZ: considered as> ... dangerously unsound minds" (see his famous paper "An Editor Recalls Some Hopeless Papers." - The Bulletin of Symbolic Logic, Volume 4, Number 1, March 1998. Pp. 1-17, http://www.math.ucla.edu/~asl/bsl/0401-toc.htm ). Now I would like to remind some of appropriate statements of such the "dangerously unsound minds". For example, Solomon Feferman writes (in his recent remarkable book "In the light of logic. - Oxford University Press, 1998."): "[...] there are still a number of thinkers on the subject (AZ: on Cantor's transfinite ideas) who in continuation of Kronecker's attack, object to the panoply of transfinite set theory in mathematics [.] In particular, these opposing <AZ: anti-Cantorian> points of view reject the assumption of the actual infinite (at least in its non-denumarable forms) [...]Put in other terms: the actual infinite is not required for the mathematics of the physical world." The same view as to rejection of the actual infinite is clearly expressed by Ja.Peregrin (see his "Structure and meaning" at:

28. [FOM] As To Strict Definitions Of Potential And Actual Infinities. binary, sequences are = potential in Aristotle's sense then the cantor's theoremon strict definitions of the concepts of the potential and actual infinities. http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html

[FOM] As to strict definitions of potential and actual infinities. Alexander Zenkin alexzen@com2com.ru Tue, 31 Dec 2002 12:45:15 +0300 Previous message: [FOM] Kolmogorov complexity session Next message: [FOM] The Omega Express (Set Theory and Ordinary Language) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... alexzen@com2com.ru URL: http://www.com2com.ru/alexzen/ http://www.w3.org/TR/REC-html40" > <head> <META HTTP-EQUIV=3D"Content-Type" CONTENT=3D"text/html; = charset=3Dkoi8-r"> <meta name=3DProgId content=3DWord.Document> <meta name=3DGenerator content=3D"Microsoft Word 10"> <meta name=3DOriginator content=3D"Microsoft Word 10"> <link rel=3DFile-List href=3D"cid: filelist.xml@01C2B0CA.770035A0 alexzen@com2com.ru <o:p></o:p></span></span></p> <p class=3DMsoNormal style=3D'text-align:justify;tab-stops:14.2pt'><span style=3D'mso-bookmark:OLE_LINK4'><font size=3D3 face=3D"Times New = Roman"><span lang=3DEN-US = style=3D'font-size:12.0pt;mso-ansi-language:EN-US'>URL:<span style=3D'mso-spacerun:yes'>=9A = </span> <p class=3DMsoNormal style=3D'text-align:justify;tab-stops:18.0pt'><span style=3D'mso-bookmark:OLE_LINK4'><span = style=3D'mso-bookmark:OLE_LINK3'><font size=3D3 face=3D"Times New Roman"><span lang=3DEN-US = style=3D'font-size:12.0pt; mso-ansi-language:EN-US'><span style=3D'mso-tab-count:1'>=9A=9A=9A=9A=9A = </span><span class=3DGramE>and</span> the points (P1)-(P3) are the first three axioms = of Peano's system.<o:p></o:p></span></font></span></span></p> <p class=3DMsoNormal style=3D'text-align:justify;tab-stops:18.0pt'><span style=3D'mso-bookmark:OLE_LINK4'><span = style=3D'mso-bookmark:OLE_LINK3'><font size=3D3 face=3D"Times New Roman"><span lang=3DEN-US = style=3D'font-size:12.0pt; mso-ansi-language:EN-US'><o:p>&nbsp;</o:p></span></font></span></span></p= >

29. Cantor's Diagonal Argument The great granddaddy of diagonal arguments is cantor's, which proved thatsome infinities are bigger than others. cantor's Diagonal Argument. http://www.chaos.org.uk/~eddy/math/diagonal.html

Diagonal arguments Various arguments prove extremely strong results by phrasing the result in such a way that some `two-parameter' thing can be used to supply a `diagonal', the special case where the two-parameter thing has both parameters equal, analysis of which suffices to prove some property of the two-parameter thing which suffices to imply the desired strong result. The great grand-daddy of diagonal arguments is Cantor's, which proved that some infinities are bigger than others. Cantor's Diagonal Argument can can enumerate any set of subsets of the naturals as long as there's an upper (finite) bound on the size of the subsets. One may use each natural in turn as choice of upper bound and construct an enumeration of all the subsets of the naturals smaller than this. It might seem that `in the limit at infinity' this process should get us at least an enumeration of all finite sub-sets of the naturals; and one might hope that some analogous trickery might enable one to get all sub-sets of the naturals. In any case the diagonal argument shows only one The diagonal argument can be read as an algorithm for converting a function, from a set to its power set, into a function from the set's successor to

30. Re: Infinities There is a greater immediacy of completed infinities in cantor's application forthe uncountability of the real numbers; but natural number arithmeticas, for http://hhobel.phl.univie.ac.at/phlo/199707/msg00192.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Infinities To phil-logic@bucknell.edu Subject : Re: Infinities From n_nelson@ix.netcom.com Date : Thu, 31 Jul 1997 09:48:50 -0500 (CDT) Prev by Date: Re: Infinities Next by Date: Re: Infinities Prev by thread: Re: Infinities Next by thread: Re: Infinities Index(es): Date Thread

31. Re: "completed Infinite" You then responded, complicating the issue by saying that cantor's diagonal isn't isalgorithmic and the use of completed infinities introduces ambiguities. http://hhobel.phl.univie.ac.at/phlo/200009/msg00054.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: "completed infinite" To phil-logic@bucknell.edu Subject : Re: "completed infinite" From : "Drake O'Brien" < dobrien@lightspeed.bc.ca Date : Wed, 6 Sep 2000 07:26:42 -0700 References 000801c016c8$686391c0$1c778ad1@default 39B41E13.18EABF5F@pacbell.net 000901c016c1$9a7daa80$4718fea9@desynre 39B43654.42A9A7F4@pacbell.net ... 39B63898.A1FC84FB@pacbell.net Reply-To phil-logic@bucknell.edu Sender owner-phil-logic@bucknell.edu Follow-Ups Re: "completed infinite" From: References Re: diagonalization From: "charles silver" <silver_1@mindspring.com> Re: diagonalization From: Re: diagonalization From: "Drake O'Brien" <dobrien@lightspeed.bc.ca> Re: diagonalization From: Re: diagonalization From: "Drake O'Brien" <dobrien@lightspeed.bc.ca> Re: diagonalization From: Re: diagonalization From: "Drake O'Brien" <dobrien@lightspeed.bc.ca> Re: diagonalization From: Re: diagonalization From: "Drake O'Brien" <dobrien@lightspeed.bc.ca> "completed infinite"

32. Deterministic Chaos: Mandelbrot's Fractal Dimensions Are Not Special Species Of next 10 cantor's schizophrenic infinities A reminder of the wayfor generating numbers according to set theory. A reminder of http://pro.wanadoo.fr/quatuor/mathematics.htm

home french version contents Mathematics where is the order of deterministic chaos hiding From the dimensions with coordinates to the dimensions of deformation start what is a dimension ? The evolution of the notion of dimension, from antiquity to fractal dimensions of Mandelbrot. [next] the measurement of the deformation of a contrast How we can measure a phenomenon without any notion of coordinates on an axis. The fundamental difference between coordinate dimensions and deformation dimensions. [next] in theory the dimension of a contrast can be self-similar The problem of dimensions which vary with the scale of the measurement. [next] the trap in the vectorial representation of the forces Usually a force is summarized by a vector which applies on a point. This representation leads to a serious abnormality when we want to calculate the interference of several forces. We suggest another type of representation that correctly uses the effects of their interference in all directions of space: this representation requires the measurement of infinity of vectors in every point. The objet of the following text is precisely to show how it can be easy to measure infinity of vectors with a single number.

33. Ecclectica - Aleph However, Cantor reasoned that Kronecker would only read the title and abstract,looking for some mention of cantor's objectionable infinities. http://www.ecclectica.ca/issues/2002/1/williams.asp

Aleph Aleph by Jeff Williams Where there is the Infinite there is joy. There is no joy in the finite. - The Chandogya Upanishad No one shall expel us from the paradise which Cantor has created for us. - David Hilbert The book, or movie, A Beautiful Mind, has recently entranced us all with a view into the troubled thoughts of Princeton mathematician and Nobel Laureate John Nash. He is often compared with the painter, Vincent Van Gogh, another unhappy spirit who spent much time in and out of mental institutions but never wavered from his quest. Many of us have a weakness, even envy, for these people, driven to madness by their single-minded search for truth and their determination to declare it. Georg Cantor is another example. He was born in Russia in 1845, but lived most of his life in Germany. Although he trained with some of the foremost mathematicians of the day, and obtained his doctorate in 1869, he was unable to secure a position at any of the prestigious research universities. Disappointed, he accepted a post at Friedrich's University in the small industrial town of Halle, almost midway between the two great university cities of Gottingen and Berlin. Time and again, he applied for positions in both of these places, always to be refused. Cantor's work with mathematical set theory, which would eventually lead to a revolution in our understanding of the infinite, began with the question: How do I count the number of elements (members) in a set? The answer, he concluded, was to associate each element in turn with the so called

34. 10 Fatal Mistakes Of The Cantor's Proof 2) The cantor's proof itself is not a reductio ad absurdum proof, but it metamathematiciansand symbolic logicians to talk about that infinities are different http://www.com2com.ru/alexzen/papers/Cantor/10_mistakes.html

10 Fatal Mistakes of the Cantor's Proof of the Real Numbers Uncountability. (a fragment of my Letter to the Editor-in-Chief of the Bulletin of Symbolic Logic) prove the following (here B = "the enumeration (*) contains all real numbers of X"). contradiction B not-B of Cantor's proof has no relation to classical Aristotle's Logic at all. quasi -logical, i.e., pathological, version of the well-known counter-example method where, however, (in contrast to classical mathematics) a counter-example itself (the Cantor anti -diagonal number which is different from every number of the enumeration (*)) is deduced (!) logically and algorithmically from the non-authentic statement ( assumption B which then that counter-example itself "must" disprove. meta -mathematics (as every philosopher knows well, all meta -mathematical proofs of famous Geodel's, Tarski's, Church's, etc. theorems are based just upon the Cantor diagonal method [9]) and the only ground, allowing meta -mathematicians and symbolic logicians to talk about that infinities are different in their cardinalities, does not distinguish infinite sets from finite sets just by their cardinalities.

35. Guestbook understanding of the fundamental distinction between countable and uncountable infinities ,but not from a critical logical analysis of the cantor's proof . http://www.com2com.ru/alexzen/guestbook/guestbook.html

GUEST BOOK Add message 04 December, 2001. AZ's ANSWER TO Dr. R.N. Deepak's message on Tuesday, November 06, 2001 at 08:44:16 AZ: unnecessary and tiresome " IFF one has in mind the millennial, speculative, quasi-philosophical, fruitless discussions as to whether "the actual infinity exists" which were going on from Aristotle's time to the middle of theXIX Century (sometimes - even hitherto). But just Cantor's proof of theorem on the uncountability changed the situation cardinally: that proof at the first time USED the actuality not at a vague, verbal level, but algorithmically in (allegedly) mathematical argument. The sense of the "actuality" becomes sitrictly defined by the Cantor's diagonal rule: "for ANY i, if i-th diagonal digit is then 1, and vice-versa". generates a new INDIVIDUAL mathematical object (Cantor's anti-diagonal real number differing from ANY real of (1)), then the INFINITE sequence (1) is ACTUAL. http://www.com2com.ru/alexzen/papers/ ), but I would like to emphasize once more the obvious (after Cantor) truth: the ACTUALITY of the set X of reals and of the sequence (1) is a NECESSARY condition of Cantor's proof. If someone does not wish to leave the mathematics ground, he/she has no logical right to state that a NECESSARY condition of a proof (in particular, the Cantor's proof) is " unnecessary and tiresome ", and then that the term itself "actual infinity" is "naive", and "consequently" can be simply omitted and excluded from the logical analysis. As they say, "if there is not term, then there is not problem".

36. Archimedes Plutonium Thus in this fashion any and every Real becomes a flipped over adic. Each correspondoneto-one. There, cantor's orders of infinities disappear forever. http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/File118.html

Cantor's transfinites are fakery by Archimedes Plutonium Naturals = Adics have same cardinality as Reals this is a return to website location http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/

37. Infinity And Infinities Infinity and infinities. One of the amazing consequences of cantor's work is thatit proves the existence of a class of real numbers which previously had been http://www.gap-system.org/~john/analysis/Lectures/L4.html

38. Proof Of Infinities Proof That Not All infinities Are The Same Size. cantor's first proof is complicated,but his second is much nicer and is the standard proof today. http://math.bu.edu/INDIVIDUAL/jeffs/cantor-proof.html

Proof That Not All Infinities Are The Same Size The proof is as follows: we count by matching the natural numbers to some set. For example, the set: has three elements in it; we match each bullet with a natural number: and the last number is 3. In infinite sets, we do the same thing. For example, the number of squares and the number of natural numbers is the same. To show this rather odd result, consider the following matching: You can continue this matching; as you can see, every natural number gets a unique square, so the number of squares and the number of natural numbers is the same. (Another way of putting this is that every natural number has a square, and every square is associated with a natural number). How about the rational numbers? It might seem that the set of all rationals, like 3/8 and 5/9 and 12321/98732 would be much larger than the set of natural numbers. However, you can match them up. The proof is fairly simple, but difficult to format in html. But here's a variant, which introduces an important idea: matching each number with a natural number is equivalent to writing an itemized list. Let's write our list of rationals as follows: and so on. Notice that first we list all the fractions whose numerator and denominator add to 1, then those that add to 2, then those that add to three, etc. Every fraction is somewhere on this list (and a little elementary arithmetic sequence calculation can tell you

39. [math-ph/9909033] Infinities In Physics And Transfinite Numbers In Mathematics Upon examining these examples in the context of infinities from cantor's theoryof transfinite numbers, the only known mathematical theory of infinities, we http://arxiv.org/abs/math-ph/9909033

Mathematical Physics, abstract math-ph/9909033 Infinities in Physics and Transfinite numbers in Mathematics Authors: P. Narayana Swamy Comments: 16 pages, Latex Subj-class: Mathematical Physics; Classical Analysis and ODEs Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known mathematical theory of infinities, we conclude that apparent inconsistencies in physics are a result of unfamiliar and unusual rules obeyed by mathematical infinities. We show that a re-examination of some familiar limiting results in physics leads to surprising and unfamiliar conclusions. It is not the purpose of this work to resolve the problem of infinities but the intent of this analysis is to point out that the study of real infinities in mathematics may be the first step towards delineating and understanding the problem of infinities in physics. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Links to: arXiv math-ph find abs

40. Mediev-L: Re: Different Infinities Re Different infinities. the real numbers) cannot be put into oneto-one correspondencewith the natural numbers, so there are, in cantor's sense, more real http://www.ku.edu/~medieval/melcher/matthias/t98/0060.html

Re: Different infinities Gordon Fisher ( gfisher@SHENTEL.NET Wed, 5 Nov 1997 14:06:34 LCL

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