41. Question Corner -- Understanding Projective Geometry This definition satisfies all the axioms of projective geometry. Spencer Current Network Coordinator and Contact Person Joel Chan mathnet@math.toronto.edu http://www.math.toronto.edu/mathnet/questionCorner/projective.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Understanding Projective Geometry Asked by Alex Park, Grade 12, Northern Collegiate on September 10, 1996 Okay, I'm just wondering about the applicability of projective and affine geometries to solving problems dealing with collinearity and concurrence. I'd really like to learn more on the topic, but I'm having trouble finding a book that gives the axioms of them both in a way that I can understand it. As far as I can understand it, there are no such things as parallel lines in projective geometry. How does that work? An explanation would be appreciated. There are several different ways to think about geometry in general and projective geometry in particular. 1. The axiomatic approach This approach requires no philosophical definition of what a point or a line actually "is", just a list of properties (axioms) that they satisfy. The theorems of geometry are all statements that can be deduced from these properties. In this approach, the theorems of geometry are guaranteed to be true no matter what concept of "point" or "line" is being used and no matter how they are defined, as long as they satisfy the basic axioms. Euclid wrote down a list of these axioms: five of them (though actually there are some other axioms implicit in Euclid's definitions). He called them

42. Question Corner -- Non-Euclidean Geometry For a description, see our web page http//www.math.toronto.edu/mathnet/questionCorner (Modern mathematicians would tend to call them axioms rather than http://www.math.toronto.edu/mathnet/questionCorner/noneucgeom.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Non-Euclidean Geometry Asked by Brent Potteiger on April 5, 1997 I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Being as curious as I am, I would like to know about non-Euclidean geometry. Thanks!!! All of Euclidean geometry can be deduced from just a few properties (called "axioms") of points and lines. With one exception (which I will describe below), these properties are all very basic and self-evident things like "for every pair of distinct points, there is exactly one line containing both of them". This approach doesn't require you to get into a philosophical definition of what a "point" or a "line" actually is. You could attach those labels to any concepts you like, and as long as those concepts satisfy the axioms, then all of the theorems of geometry are guaranteed to be true (because the theorems are deducible purely from the axioms without requiring any further knowledge of what "point" or "line" means). Although most of the axioms are extremely basic and self-evident, one is less so. It says (roughly) that if you draw two lines each at ninety degrees to a third line, then those two lines are parallel and never intersect. This statement, called

43. [math-ph/0205004] Generalization Of Shannon-Khinchin Axioms To Nonextensive Syst mathematical Physics, abstract mathph/0205004. Generalization of Shannon-Khinchin axioms to Nonextensive Systems and the Uniqueness Theorem. http://arxiv.org/abs/math-ph/0205004

Mathematical Physics, abstract math-ph/0205004 From: Hiroki Suyari [ view email ] Date ( ): Thu, 2 May 2002 07:56:10 GMT (8kb) Date (revised v2): Tue, 28 May 2002 18:09:53 GMT (8kb) Generalization of Shannon-Khinchin Axioms to Nonextensive Systems and the Uniqueness Theorem Authors: Hiroki Suyari Comments: 10 pages Subj-class: Mathematical Physics MSC-class: The Shannon-Khinchin axioms are generalized to nonextensive systems and the uniqueness theorem for the nonextensive entropy is proved rigorously. In the present axioms, Shannon additivity is used as additivity in contrast to pseudoadditivity in Abe's axioms. The results reveal that Tsallis entropy is the simplest among all nonextensive entropies which can be obtained from the generalized Shannon-Khinchin axioms. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math-ph find abs

44. [math/9805104] Axioms For Weak Bialgebras mathematics, abstract math.QA/9805104. From Dr. Florian Nill florian.nill@physik.unimuenchen.de Date Fri, 22 May 1998 173747 GMT (49kb) axioms for Weak http://arxiv.org/abs/math/9805104

Mathematics, abstract math.QA/9805104 From: Dr. Florian Nill [ view email ] Date: Fri, 22 May 1998 17:37:47 GMT (49kb) Axioms for Weak Bialgebras Authors: Florian Nill Comments: 48 pages, Latex Report-no: Preprint SFB 288/11/5/98 Subj-class: Quantum Algebra Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

45. Math History - Age Of Liberalism La Habra High School math History Timeline He puts arithmetic on a rigorous foundation giving what were later known as the Peano axioms . http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/mLiberalism.htm

Math History Timeline Age of Liberalism 1848-1914 A.D. Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... 20th Century ... non-Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources Chebyshev publishes On Primary Numbers In his paper On a New Class of Theorems Sylvester first uses the word "matrix". Bolzano's book Paradoxien des Undendlichen (Paradoxes of the Infinite) is published three years after his death. It introduces his ideas about infinite sets. Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers". In particular he gave the example 0.1100010000000000000000010000... where there is a 1 in place n! and elsewhere. Riemann's doctoral thesis contains ideas of exceptional importance, for example "Riemann surfaces" and their properties. Francis Guthrie poses the Four Colour Conjecture to De Morgan.

46. Math History - 20th Century ... La Habra High School math History Timeline any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/m20thCentury.ht

Math History Timeline 20th Century ... 1914-present A.D. Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... 20th Century ... non-Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources Einstein submits a paper giving a definitive version of the general theory of relativity. Sierpinski gives the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written. Hausdorff introduces the notion of "Hausdorff dimension", which is a real number lying between the topological dimension of an object and 3. It is used to study objects such as Koch's curve. Russell publishes Introduction to Mathematical Philosophy which had been largely written while he was in prison for anti-war activities. Fundamenta Mathematica is founded by Sierpinski and Mazurkiewicz. Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy. Emmy Noether publishes Idealtheorie in Ringbereichen which is of fundamental importance in the development of modern abstract algebra.

47. Path News.jmag.net!news.jmas.co.jp!nf9.iij.ad.jp!nr1.iij.ad.jp! See ``axioms of Symmetry Throwing Darts at the Real Line , by Freiling, Journal of Symbolic http//www.jazzie.com/ii/math/ch/ http//www.best.com/ ii/math/ch http://linas.org/mirrors/nntp/sci.math/faq.continuum.html

Path: news.jmag.net news.jmas.co.jp !nf9.iij.ad.jp!nr1.iij.ad.jp! news.iij.ad.jp news.qtnet.ad.jp !news1.mex.ad.jp!news0-mex-ad-jp!nr1.ctc.ne.jp! news.ctc.ne.jp !newsfeed.kddnet.ad.jp!newssvt07.tk!newsfeed.mesh.ad.jp!newsfeed.berkeley.edu!newsfeed.direct.ca!torn!watserv3.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math, news.answers daisy.uwaterloo.ca Summary: Part 25 of 31, New version Originator: alopez-o@neumann.uwaterloo.ca Originator: alopez-o@daisy.uwaterloo.ca Xref: news.jmag.net news.answers sci.answers:152 http://www.jazzie.com/ii/math/ch/ ... http://www.best.com/ ii/math/ch/ Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Last updated: Sat Feb 19 00:01:06 2000

48. Poster Project, What Is Scientific Truth Poster manipulating the axioms according to the rules of logic. It is not to say anything about the real world. This is the big difference between physics and math. http://www.math.sunysb.edu/posterproject/www/materials/truth/truth.html

Visualizing Women in Science, Mathematics and Engineering Home Posters Materials for Study Biographies ... The Design Team What Is Scientific Truth? What do you think of the following "theorems"? Theorem One Proof Suppose a b and neither is zero. Then, a b a b a b Also, a b a ab So, a ab a b a b a a b a b a b a a b a a a a as required! Theorem Two Movement is impossible! Proof This is a proof by contradiction. I will assume that movement is possible and derive a contradiciton, thereby proving movement to be impossible. So suppose movement is possible. In this case, the great Greek warrior Achilles can have a foot race with the very slow tortoise. Now, Achilles is the greatest warrior ever (imagine Rambo and the Terminator combined, and that dude wouldn't last ten seconds against Achilles. Achilles could flatten Goldberg with one arm tied behind his back. In fact, Achilles was so awesome that this Greek guy named Homer wrote a cool book about him called the Iliad .) It isn't really a fair race. I mean Achilles could smoke the tortoise who is slow and old and has to carry her house on her back. In the interests of fairness, we had better give the tortoise a little bit of a head start, let's say twenty feet. Now we're ready to race. There's Achilles (the greatest warrior ever) and twenty feet ahead of him is the old tortoise. Bang! The starter's gun goes off and they're racing.

49. Mathematics 340: Axiomatic Geometry Hilbert s axioms for Euclidean geometry, projective geometry, history of parallel axiom, hyperbolic geometry, elliptic geometry. Outline...... Course http://math.ndsu.nodak.edu/courses/340.html

Mathematics 340: Axiomatic Geometry General Information: Title: Mathematics 340: Axiomatic Geometry Credits: 3 credits Prerequesites: Mathematics 270 Required for: Suggested Text: Course Description: Hilbert's axioms for Euclidean geometry, projective geometry, history of parallel axiom, hyperbolic geometry, elliptic geometry. Outline: Introduction Relations, functions, real number system, fundamentals of axiomatic method. Plane Euclidean Geometry Existence and incidence axioms, order and congruence axioms, Parallels axiom for Euclidean Geometry, axioms of continuity and completeness. Solid Euclidean Geometry Axioms of solid Euclidean Geometry, parallels in space. Projective Geometry Axioms of the projective plane, duality. Axioms of projective space, Theorem of esargues (*), harmonic sequences (*), and Projective transformations. The Parallel Postulate History of the parallel postulate, alternatives to the parallel postulate of Euclid. Hyperbolic Geometry Axioms of hyperbolic geometry and hyperbolic parallels axiom, Saccheri quadrilaterals, consequences of hyperbolic parallel axiom, angle sum theorem, the are problem. Elliptic Geometry Parallels axiom of elliptic geometry, finititude of a line. Incidence and seperation axioms, axioms of congruence, consequences of elliptic parallels axiom, angle sum theorem and the area problem.

50. Math 504 312) 9963069 e-mail marker@math.uic.edu course webpage http//www.math.uic.edu be formalized inside of set theory and that a simple set of axioms could be http://www.math.uic.edu/~marker/504.html

Math 504 Set Theory I Spring 2002 Instructor: David Marker Class: MWF 303 Adams Hall, 11-11:50 Office: 411 SEO Office Hours: M,F 9:00-10:30 and by appointment phone: (312) 996-3069 e-mail: marker@math.uic.edu course webpage: http://www.math.uic.edu/~marker/504.html Description In the first half of the century it was shown that most of mathematics can be formalized inside of set theory and that a simple set of axioms could be given so that every acceptable proof followed formally from these axioms. Godel's Incompleteness Theorem implies that there are mathematical truths not settled by these axioms. The most famous is the Continuum Hypothesis (CH) that asserts that there are no infinite sets of cardinality greater than the natural numbers but less than the real numbers. This course will start by introducting the axioms for set theory and developing the basic theory of cardinals and ordinals. We will then begin looking at models of set theory and prove that the Continuum Hypothesis is neither provable nor refutable. The topics covered will include: the Zermelo-Frankel axioms for set theory the axiom of choice ordinals and cardinals models of set theory Godel's constructible universe and the consistency of CH Cohen's method of forcing and the independence of CH Texts K. Kunen

51. Methodology Of Social Science Is empty math useless in economics? No. You may impose some axioms on top of it and then use it as a model, but it is a logic system by itself. http://mywebpages.comcast.net/ylding/method/method.html

Home My View on Social Science Research Although this looks like a manifesto, it was nothing more than a post to a public forum as a part of discussion on the methodology of economics and finance. Minimal edit was applied after mark up. Logic and Content I think that exposure to the history of economic thoughts and some thinking about philosophy of science are important in order to stay awake in a tour of economics. I'm a strong believer of Carnap's logic positivism, which essentially claims that any theory trying to say something about nature can be decomposed into two parts: pure logic and the contents applied to the logic. Math by itself is just a logic system. Based on some assumptions, you can prove something else that's not a theory about nature. Here comes the first argument against naive application of the positivist approach populated by Milton Friedman. If you build an internally consistent logic/mathematical system, you haven't done anything yet about the economic science. To put the flesh on the singleton of a logic system, you start adding contents. The starting point is so-called axiom. An axiom is an assumption based nothing other than belief. When you say that the sun rises from the east, you can't prove it you assume that I observe the same thing as you do within the same space. If you study the history of science, you'll notice that axioms in different paradigms are not necessarily consistent. It's not a matter of right or wrong both are right and both are wrong. Take Newton mechanics and theory of relativity for example. The axioms in two systems are obviously inconsistent, but they are both useful in their own areas. The key is to realize that the purpose of axioms is to help you start the exploration with some reasonable approximation without obvious deviation from the "truth" because no one knows the "truth".

52. Cornell Mathematics- Robert Connelly- Math 452 Home Page Telephone (607) 2554301 (voice mail). math 452, Classical Geometries. One stated one s axioms. But the idea was to minimize such unproved statements. http://www.math.cornell.edu/~connelly/452.stuff.html

Classical Geometries Math 452 Instructor: Bob Connelly Office: 455 Malott Hall Telephone: (607) 255-4301 (voice mail) Math 452, Classical Geometries This is an introduction to geometry and how it has been moved by a combination of classical Greek influences as well as a more modern desire simply to understand the world as it is seen. The influence of Euclid's Elements was pervasive in western thought, especially mathematics. The idea is that you should explicitly state what is assumed. One stated one's axioms . But the idea was to minimize such unproved statements. Euclid created a small list of such axioms, but he delayed using the fifth until he was really forced. Did he really need that fifth axiom after all? Could it be deduced it from his other axioms? Meanwhile, during the Renaissance, artists began to seriously ask just how should one draw a picture that accurately shows what we see? The realization of just how this should be done came almost as a revelation. This led to the development of perspective drawing, and that led to projective geometry. The difference between what is drawn in "correct" perspective and what is not, is striking. The principles perspective are simple, but some of the consequences are not what one might expect. In a way, projective geometry is an example where one can apply the insight obtained from a simple set of axioms, unlike the situation of Euclid, where there were a very large number of hidden, subtle, complicated, axioms. Projective geometry has just three simple axioms, two are just mirror duals of each other, and the third does not really count. We will use these to show how they provide a great perspective into the nature of geometry, even Euclidean geometry. Projective geometry provides a toy axiomatic system, without a lot of fuss or mess, yet still delivering what is needed.

53. Syllabus For Math 511 Syllabus for math 511. is a rigorous course focusing on absolute geometry where each step of the argument is justified by previously proven axioms and theorems. http://www.uncp.edu/home/truman/mat511/511syl.htm

Syllabus for Math 511 Advanced Topics in Geometry Textbook: College Geometry a Discovery Approach by David C. Kay Narrative: The graduate course of Advanced Topics in Geometry differs from the undergraduate course called College Geometry not so much in the content but the philosophy of each course. The undergraduate course is all about writing formal proofs. It is a rigorous course focusing on absolute geometry where each step of the argument is justified by previously proven axioms and theorems. The graduate course also deals with formal proofs; however the focus of this course is to make justification of theorems as visual as possible. The formal proof is the end product of a process of exercises where the student discovers fundamental properties of geometry. In the undergraduate course, a model of the theorem is not required; in some cases a model is undesirable since students tend to rely on the model as justification of the theorem. In the graduate course, we deal with how to select a model to use to illustrate the theorem to be proven. The graduate course deals with properties of good models and how we can encourage students to discover the many principles and phenomena of geometry for themselves. In the graduate course, students, who usually teach geometry in high school, are taught the transition of proof-writing, how to start with proofs that are largely intuitive and go through the process of developing a more formal proof. In general, the graduate course assumes the student has mastered the proof-writing procedures covered in the undergraduate course in geometry. The purpose of the graduate course is how to go from observation or discovery of a principle to a formal justification.

54. Paul Howard Department Of Mathematics Eastern Michigan University KinnaWagner selection principles, axioms of choice and multiple choice, by Paul Howard, Arthur L. Rubin, and Jean E. Rubin, Monoshafte f\ ur math. http://www.math.purdue.edu/~jer/Papers/papers.html

Paul Howard Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 Phone: 734-487-1292 e-mail: phoward@emunix.emich.edu Consequences of the Axiom of Choice by Paul Howard and Jean E. Rubin The book has been published by the American Math Society. (Math. Surv. and Monographs, vol 58, 1998.) The introduction and the web page for the book is available below. . Introduction(.dvi) Introduction(.tex) Consequences web page If you have comments or questions about our project or if we have missed some form that you think should be included, please contact either Paul Howard (phoward@emunix.emich.edu) or Jean E. Rubin (jer@math.purdue.edu). PAPERS RELATED TO THE AC PROJECT The axiom of choice for well ordered families and for families of well orderable sets, by Paul Howard and Jean E. Rubin, Journal of Symbolic Logic, vol 60, (1995) pp 1115-1117 WO-choice.dvi WO-choice.tex WO-choice.pdf Elementary Abelian p-groups revisited, by Paul Howard and Jean E. Rubin, Bulletin, Austral. Math. Soc., vol 52, (1995) pp 373-376. p-groups.dvi

55. Math In The Media 1003 math in the Media. math in the Media Archive. Klarreich continues with the history of the hypothesis, and of its relation to the standard axioms of set theory. http://www.ams.org/new-in-math/10-2003-media.html

Math in the Media Math in the Media Archive October 2003 Infinite Wisdom Wolfram in Washington African Institute Paul Wolfowitz, "former mathematician" Infinite Wisdom , a piece by Erica Klarreich in the August 30 2003 Science News , surveys some recent work on the continuum hypothesis. Klarreich starts with a review of Cantor's proof that the set R of real numbers is strictly larger, in a precise sense, than the set Z of integers. In this connection she shows Helaman Ferguson's clever visualization of Cantor's diagonal argument: Cantor's Flickering Diagonal. The left half of this stereo pair shows the beginning of an enumeration of the real numbers between and 1. The top line represents the start of the binary expansion of the first number on the list (white=0, black=1). The next line corresponds to the second number on the list, and so on. The right half is identical, except that each diagonal element (the first digit of the first number, the second digit of the second number, and so on) has been reversed: changed from white to black or from black to white. When the two images are fused, the reversed diagonal flickers in and out. The reversed diagonal is the binary expansion of a real number that cannot occur on the original list. Since this will happen for any list, the construction shows that there is no way of listing the real numbers between and 1. Click here for the stereo image of a larger array. Image courtesy Helaman Ferguson, used with permisssion.

56. Math In The Media 0203 math in the Media Archive. rich to do elementary arithmetic, there will be some statements that are true but cannot be proved (from the axioms). He presents http://www.ams.org/new-in-math/02-2003-media.html

Math in the Media Math in the Media Archive February 2003 Splits at Clay Science "Math's Wild and Crazy Guy" Figure skating ... Democracy and despotism Reproduced with permission from Nature Splits at Clay The recent administrative changes at the Clay Mathematics Institute were the subject of a piece ("Resignations rock mathematics institute") by Geoff Brumfiel in the January 23 2003 Nature Science. Science "Math's Wild and Crazy Guy" is how the January 6 2003 Washington Post describes Maryland's James A. Yorke, the recipient, two weeks before, of the Japan Prize for his work in chaos theory. He shared the prize with Benoit Mandelbrot, "another major mojo in the chaos biz." Peter Carlson, the author of the piece, takes us on a visit to Yorke's lab where we him talk about current projects. The Rat Genome: "We're not the official guys doing it, but we hope our results are better than theirs." An improved computer model for weather forcasting, in collaboration with the National Weather Service. An epidemiological study of AIDS. Yorke shows us a double pendulum: "You see the motion gets pretty complicated. ... This is what chaos is. It's predictable in the short run but not in the long run. Chaos is about lack of predictability, basically. Obviously, the spin of the pendulum is determined by physical laws, but it's very hard to predict because very small changes in the spin cause very big changes in the output." And then, of course, chaos intrudes in the lab. There's asbestos work going on, so they have to keep moving the computers from room to room. And Yorke's graduate student's motherboard just fried. The article is available

57. Math 113-3 BC math 11 (or equivalent) with a grade of at least C or permission of the Line axioms, angle axioms, angle/line axioms, the axiom of congruent triangles. http://www.math.sfu.ca/math/courses/03-2/MATH/MATH_113.html

MATHEMATICS 113-3 EUCLIDEAN GEOMETRY Summer 2003 DAY COURSE Instructor: Dr. A. LACHLAN (K 10502) Prerequisite: BC Math 11 (or equivalent) with a grade of at least C or permission of the department or the non-credit course, Basic Algebra. Textbook Geometry for College Students, by I. Martin Isaacs, published by Brooks/Cole. Course Description: Plane Euclidean geometry, congruence and similarity. Theory of parallels. Polygonal areas. Pythagorean Theorem. Geometrical constructions. Outline: GOAL: The course will provide an introduction to plane Euclidean geometry from a modern viewpoint, and also serve as an introduction to the axiomatic method. LIST OF TOPICS: Preliminaries. Sets, ordered pairs, partitions, relations, linear orderings, equivalence relations, measuring angles, the real numbers, the axiomatic method. The Euclidean plane ‹ the basics. Points, lines, the metric, angle measure, the two directions of a line. Line axioms, angle axioms, angle/line axioms, the axiom of congruent triangles. The two sides of a line. The two sides of a line, partition of the plane by a triangle, the inside and outside of simple -gons, convexity.

58. Dr. King - Math 333 - Key #2 math 333 Key to Homework 2 October 5, 1998. 1. Plane dual to Fano s geometry axioms. (5 pts) a. There exists at least one point. http://spruce.flint.umich.edu/~lmk/Math333key2.html

Math 333 Key to Homework # 2 October 5, 1998 1. Plane dual to Fano's geometry axioms. (5 pts) a. There exists at least one point. b. There are exactly three lines on every point. c. Not all lines are on the same point. d. There is exactly one point on any two lines. e. There is at least one line on any two distinct points. Representation of the model and justification. (5 pts). Fano's geometry has the property of being self-dual; i.e., the plane dual axioms of Fano's geometry are all true in Fano's geometry. Therefore, a model for the plane dual axioms is any model of Fano's geometry. 2. Show that each axiom of Fano's Geometry is independent. (2 pts for each axiom). a. Axiom 1: a single point and no lines. The model "no points and no lines" would work vacuously for any set of axioms. Therefore, I gave credit for it but I really didn't like it. e. Axiom 5: A model for Young's geometry works. Let the points be A B, C, D, E, F, G, H, I, and the lines be the columns in the following table. (Note: There are nine points and twelve lines.) A D A A B B B C C D G H B E D E E D F F E H H F C F G I H I G I G C I A 3. Prove: Fano's geometry consists of exactly seven lines. (5 pts for proving at least seven, and another 5 pts for proving there can be no more than seven lines.)

59. Information Outpost's Web Site Directory: Category Listings For Math of Algorithms Notes to Dr. Paul E. Dunne s math history lecture Rate this link! Translate If a set of points (a plane) satisfies all the above axioms, it is http://www.informationoutpost.com/directory_home.cfm?newcat=32&categorypath=3

60. The Utility Of Mathematics One could believe that the Euclidean axioms constituted a kind of perfect apriori knowledge about geometry in the phenomenal world. http://www.catb.org/~esr/writings/utility-of-math/

The Utility of Mathematics Abstract Does mathematics have some sort of deep metaphysical connection with reality, and if not, why is it that mathematical abstractions seem so often to be so powerfully predictive in the real world? Originally written 14 May 1993 for the Extropians mailing list, and re-published it with minor changes in 2001 at the urging of a list member. any of which could be modeled in the phenomenal universe , called the whole relationship between mathematics and physical theory into question. deduce the correct choice from first principles, producing a-priori descriptions of reality to be confirmed, as an afterthought, by empirical check. Principia Mathematica The majority of mathematicians quickly became "Formalists", holding that pure mathematics could not be philosophically considered more than a sort of elaborate game played with marks on paper (this is the theory behind Robert Heinlein's pithy characterization of mathematics as "a zero-content system"). The old-fashioned "Platonist" belief in the noumenal reality of mathematical objects seemed headed for the dustbin, despite the fact that mathematicians continued to feel like Platonists during the process of mathematical discovery.

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