41. Internal Links To Words. Previous The main math page. Up The main page. mathlinks and rings, Awards.The Greek alfabeth. Fields, in Abstract algebra. Field axioms. Field Lines. http://www.torget.se/users/m/mauritz/dict.htm

Index and Glossary. This is an index to my pages of logic and math. It's a glossary of many useful math terms, including Vector Algebra and Analysis, Abstract Algebra and Set Theory terms. Created 971124. Last change 981106. Previous : The main math page . Up : The main page Math-links and rings, Awards The Greek alfabeth The Geek alfabeth ... All the 'elementary' gifs, and an explanation of how to use them. A Addition of Polynomials Algebraic Numbers Alternation , logical. 'Or'. Aleph 0 Alternating groups Alternations or Transpositions of two elements. A kind of permutation. And , logical. Area of a Surface , ff. Arithmetic, The Fundamental Theorem of Argand diagram Arrays and ordinals Associates of units of Integral Domains. Associative law . (ab)c=a(bc)=abc. B Base Vectors Base Vectors, change of Biconditional , logical. Equvivalence. Bijective functions or Inversible functions. Binary operation or composition in Abstract Algebra. C Cancellation laws, right and left Cantors Diagonal Proof Cardinality . and Cardinality Cardinality of the Algebraic numbers Cardinality of N and Z Cardinality of ... Characteristics of rings.

42. Syllabus For Math 511 Syllabus for math 511. is a rigorous course focusing on absolute geometry where eachstep 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.

43. Review Of Chaitin, Limits Of Math In SIAM News Whitehead, before launching into an explanation of Hilbert's plan for deducing allof mathematics from a single finite list of (presumably selfevident) axioms http://www.cs.auckland.ac.nz/CDMTCS/chaitin/siam.html

SIAM News , Volume 34, Number 10, December 2001 Newsletter of the Society for Industrial and Applied Mathematics TABLE OF CONTENTS, p. 2: 7 The Mathematical Implications of Algorithmic Information Theory Reviewer James Case, happening on one of Gregory Chaitin's books on algorithmic information theory, finds that "the author's almost visceral involvement with his subject seems to radiate from every page." Chaitin is "particularly anxious," Case writes, "to interest other theoreticians in using his private version of LISP to explore the capabilities of `self-delimiting' universal Turing machines." FULL REVIEW, pp. 7-9: The Mathematical Implications of Algorithmic Information Theory BOOK REVIEW By James Case The Limits of Mathematics. By Gregory J. Chaitin, Springer-Verlag, Singapore, 1998, 148 + xii pages, $39.95. In perusing the remainder table at my local-and now kaput-megabookstore, I ran across Gregory Chaitin's slender volume on the mathematical implications of algorithmic information theory. The book consists of four chapters and an appendix explaining the rudiments of Chaitin's own personal dialect of LISP. He sees a critical role for such a language in foundational studies, which he deems far from complete. The first three chapters consist of edited transcripts of distinct, self-contained, one-hour lectures given by the author between 1992 and 1996, while the fourth reproduces a 50-page article from the Journal of Universal Computer Science.

44. In A Few Words... author, Abu Ja'far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook Todefine it rigorously we may need a set of axioms, like those proposed by G http://www.cut-the-knot.com/do_you_know/few_words.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site In a few words... While putting together these pages I sometimes feel a need to refer to a term without straying from the topic at hand. Oftentimes it's possible to locate a resource on Internet with a necessary definition but cumbersome to specify the reference. In short, I decided to maintain a page of very short topical descriptions which, if and when a need or inspiration induce me to, I'd be able to expand upon. Many of these have been mentioned on the Did you know... page where, as a group, they, I hope, provided some entertainment. As I had neither immediate need for nor intention to describe them, some terms from that page have been left dangling without any reference or definition. Hence the current page. Absolute value The absolute value is defined for real Algorithm The word algorithm comes from the name of a Persian author, Abu Ja'far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook. The word refers to a precise prescription (given by a step-by-step description) of a solution to a problem. Braids Theory Braids Theory was invented by Emil Artin and is a part of the Knot Theory.

45. Multiplication Of Numbers Associativity and commutativity of both addition and multiplication follow fromthe Peano axioms. math problem no answer in sight Posted by Dmom 4 messages 10 http://www.cut-the-knot.com/do_you_know/mul_num.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Multiplication of Numbers On a page on addition of numbers , I assumed that we know how to add, subtract and multiply whole numbers. I mentioned Peano axioms as the foundation on which these operations are defined and their properties established. Let's see how it can be done. First of all we assume that there is something to talk about: there exists an entity known as the set of natural numbers N whose properties (explicitly or implicitly) are given by the following Peano axioms 1 is a natural number. This says that the set N is not empty. There is at least one natural number. This number is denoted by the symbol 1 (pronounced one or unit For every x in N there exists a number x' known as the successor of x. Since x = y means that x and y are one and the same number, x = y implies x' = y'. x' 1. In other words, 1 is not a successor of any natural number. x' = y' implies x = y. Different numbers have different successors. Axiom of Induction ). Let M be a (sub)set of natural numbers with the following properties:

46. Math 300 - Fall 2002 - Collins math 300 Introduction to Abstract mathematics - Fall 2002. Handouts,Resources, Etc 29, W Oct 30, The Field axioms (Ch. 5), No Homework! http://www.math.utk.edu/~ccollins/M300/

Math 300 - Introduction to Abstract Mathematics - Fall 2002 Handouts, Resources, Etc: Class Syllabus (Postscript) (PDF) Corrections to Text Grade Status: I'm using Online@UT for the grade book. You should be able to logon there and check your grades. Links to Other Resources Class Schedule: (Homework marked with a is to be turned in the next class period) Class Number Date Material Assignment W Aug 21 Class Intro, What is a Proof?, Example with Changelings Read Chapter 1 (twice) Change FOUR to FIVE, ONE to TWO, BLACK to WHITE F Aug 23 A Bit of Logic, Intro to Sets Logic Handout (PS) (PDF) Sets Worksheet (PS) (PDF) (Due Monday) M Aug 26 Problems from Section 1.1, 1.2 (Intervals, Reflections) W Aug 28 Proof by Contradiction; If and Only If Sets Worksheet Solution (PS) (PDF) F Aug 30 Quantifiers 7a*, 7b*, 9, 11*, 15a-h M Sep 2 Labor Day Holiday W Sep 4 Negation of Quantifiers; Contrapositives and Converses 12a*, 12c*, 13 (for closed intervals), 19 Extra Credit Homework (PS) (PDF) F Sep 6 Negation; Counterexamples 15a-h, give negation*, 22, 23*, 24*

47. Math Game David Hilbert's math Game . F must contain formulas of the form P and ¬P. A, theset of axioms, must be a decidable set of formulas defined by a finite set of http://cs.wwc.edu/~aabyan/CII/MathGame.html

David Hilbert's "Math Game" Mathematics is a game played according to certain simple rules with meaningless marks on paper.- David Hilbert The structure of the game A language The rules of deduction (S) = S and D n+1 (S) = D(D n (S)). Constraints F, the set of formulas , must be a decidable set of strings of finite length composed of symbols in L and defined by a finite set of rules. F must contain formulas of the form P and ¬P. A, the set of axioms , must be a decidable set of formulas defined by a finite set of rules. D, the set of rules of deduction , must be a decidable set of functions defined by a finite set of rules. The rules must satisfy the properties of the logical connectives. T, the set of theorems , must be a decidable set of formulas constructed from the axioms using rules of deduction i.e., P=D n (A). Consistency : For any formula p in F, T cannot contain both p and ¬p. The goal of the game is to: Make A as small as possible. Make D as small as possible. completeness ) or that P is as large as possible. The rules Basic Setup Select a language L.

48. Math Game Alfred Tarski's math Game . The structure of the game. A, the set of axioms,must be a decidable set of formulas defined by a finite set of rules. http://cs.wwc.edu/~aabyan/CII/MGTarski.html

Alfred Tarski's "Math Game" The structure of the game A language The rules of deduction (S) = S and D n+1 (S) = D(D n (S)). Let S be a relational structure valuation function Constraints F, the set of formulas , must be a decidable set of strings of finite length composed of symbols in L and defined by a finite set of rules. F must contain formulas of the form P and ¬P. At is the set of atomic formulas A, the set of axioms , must be a decidable set of formulas defined by a finite set of rules. D, the set of rules of deduction , must be a decidable set of functions defined by a finite set of rules. The rules must satisfy the properties of the logical connectives. T, the set of theorems , must be a decidable set of formulas constructed from the axioms using rules of deduction i.e., T=D n (A). Correspondence Consistency : For any formula f in T, both f and ¬f are not in T. The goal of the game is to: Make A as small as possible. Make D as small as possible. completeness ) or that T is as large as possible. The rules Basic Setup Select a structure S.

49. Sci.math Topic The math Forum. sci.math, Search. Date, Topic, Author. 03 Feb 03, ReScience is axiom system progressing most when axioms 03 Feb 03, http://mathquest.com/discuss/sci.math/t/479479

The Math Forum sci.math Search All Discussions sci.math Topic Date Topic Author 03 Feb 03 Re: Science is axiom system progressing most when axioms ... 03 Feb 03 Re: Science is axiom system progressing most when axioms... Piet Holbrouck 03 Feb 03 Re: Science is axiom system progressing most when axiom... 08 Feb 03 Re: Science is axiom system progressing most when axiom... Y.Porat 08 Feb 03 Re: Science is axiom system progressing most when axio... Piet Holbrouck 03 Feb 03 Re: Science is axiom system progressing most when axioms... 03 Feb 03 Re: Science is axiom system progressing most when axiom... Piet Holbrouck 03 Feb 03 Re: Science is axiom system progressing most when axiom... 03 Feb 03 Re: Science is axiom system progressing most when axio... Piet Holbrouck 05 Feb 03 Re: Science is axiom system progressing most when axi... Gregory L. Hansen 05 Feb 03 Re: Science is axiom system progressing most when ax... Piet Holbrouck 05 Feb 03 Re: Science is axiom system progressing most when a... Gregory L. Hansen 05 Feb 03 Re: Science is axiom system progressing most when ...

50. Sci.math Topic The math Forum. sci.math, Search. Date, Topic, Author. 03 Jan 01, axioms ofmodern mathematics, Ahimog. 03 Jan 01, Re axioms of modern mathematics, Virgil. http://mathquest.com/discuss/sci.math/a/t/311564

The Math Forum sci.math Search All Discussions sci.math Archive Topic Date Topic Author 03 Jan 01 axioms of modern mathematics Ahimog 03 Jan 01 Re: axioms of modern mathematics Virgil 03 Jan 01 Re: axioms of modern mathematics Peter L. Montgomery 03 Jan 01 Re: axioms of modern mathematics Joe 04 Jan 01 Re: axioms of modern mathematics Herman Rubin 04 Jan 01 Re: axioms of modern mathematics Jesse F. Hughes 04 Jan 01 Re: axioms of modern mathematics glenn 04 Jan 01 Re: axioms of modern mathematics Jesse F. Hughes 05 Jan 01 Re: axioms of modern mathematics Mike Oliver 05 Jan 01 Re: axioms of modern mathematics Jesse F. Hughes 05 Jan 01 Re: axioms of modern mathematics G. A. Edgar 05 Jan 01 Re: axioms of modern mathematics Mike Oliver 06 Jan 01 Re: axioms of modern mathematics Herman Rubin 10 Jan 01 Re: axioms of modern mathematics Norman D. Megill 03 Jan 01 Re: axioms of modern mathematics 03 Jan 01 Re: axioms of modern mathematics Dave Seaman 04 Jan 01 Re: axioms of modern mathematics Herman Rubin 03 Jan 01 Re: axioms of modern mathematics Ronald Bruck 03 Jan 01 Re: axioms of modern mathematics Bob Silverman 05 Jan 01 Re: axioms of modern mathematics 05 Jan 01 Re: axioms of modern mathematics glenn 05 Jan 01 Re: axioms of modern mathematics Jesse F. Hughes

51. Re: Madden, Explain Math's Flaw Please In Reply to Re Madden, explain math's flaw please posted by MaddenL3 on December butequivalent formulations of the same theory, the number of axioms is not http://superstringtheory.com/forum/topboard/messages4/57.html

String Theory Discussion Forum String Theory Home Forum Index Re: Madden, explain math's flaw please Follow Ups Post Followup Topology IV FAQ Posted by DickT on December 25, 2002 at 10:39:02: In Reply to: Re: Madden, explain math's flaw please posted by on December 24, 2002 at 18:31:27: Lee, The general rule on axiom systems is the same as Einstein's "Everything must be made as simple as possible, but no simpler". Of course if any axiom in a set is a consequence of any of the others, it should be dropped. That is, the axiom must be independent, or in another terminology "a minimal spanning set". In different but equivalent formulations of the same theory, the number of axioms is not invariant; one formulation may require fewer axioms than another. In such a case it might be tempting to just adopt the formulation with the fewer axioms, but this might be unwise. It may be that the other formulation gives you a better angle on new discoveries. In any case, removing an axiom that has independent content (such as existence of irrationals) is not justifiable simply on grounds of parsimony, The set of axioms should be minimal, but it should also span. Peace to you on Christmas Day

52. Math(s) Described Area (Visit) Site Exits Links for math etc (Recommended) math HOWTOs Algebracontinues with axioms that say in general when different calculations or http://whyslopes.com/freeAccess/mathTopics.html

Appetizers and Lessons for Mathematics and Reason Site Areas: Volume 1, Elements of Reason Volume 1A, Pattern Based Reason Volume 1B, Mathematics Curriculum Notes Volume 2, Three Skills For Algebra Volume 3, Why Slopes and More Math 4 Lecons (Mathematiques et Logique) Complex Numbers Revisited Help Your Child Learn LaTeX2HotEqn Automation Order above Volumes via DoubleHook Book Store Order Volumes via PayPal (Credit Card) Order Volumes via OrderForm (Check/Money Order) Order Volumes (or contact author) via Email Key Pages: Feedback form] [ Study Tips Site Entrance ] [Member Area ( Visit Site Exits: Links for math etc (Recommended Math HOW-TOs Website Reviews Lessons Essays Etc Subject or Topic Descriptions Numbers by themselves may say how many or how much, or they may give position or coordinates in time or place. The first source of skill and confidence in mathematics comes from the use of numbers to count or give position. Results are supposedly repeatable and reproducible independent of the person observing position or counting or measuring. Arithmetic or figuring Arithmetic done well

53. MATH 360: Foundations Of Geometry math 360 Foundations of Geometry. course serves as an example of how the disciplineof mathematics works, illuminating the roles of axioms, definitions, logic http://people.hws.edu/mitchell/math360f02.html

Math 360: Foundations of Geometry Offered: Fall 2002 Instructor: Kevin J. Mitchell Office: Lansing 305 Phone: (315)781-3619 Fax: (315)781-3860 E-mail: mitchell@hws.edu Information Available: About the course Outline of Weekly Readings Assessment Office Hours ... Additional Sources on Reserve About the Course This course is about geometry and, in particular, the discovery (creation) of non-Euclidean geometry about 200 years ago. At the same time, the course serves as an example of how the discipline of mathematics works, illuminating the roles of axioms, definitions, logic, and proof. In this sense, the course is about the process of doing mathematics. The course provides a rare opportunity to see how and why mathematicians struggled with key ideas-sometimes getting things wrong, other times having great insights (though occasionally they did not recognize this fact). History is important to this subject; this course should convince you that mathematics is a very human endeavor. The course focuses on Euclid's Parallel Axiom: "For any line l and any point P not on l, there is a unique line through P parallel to l." In particular, could this axiom be deduced as a consequence of the earlier and more intuitive axioms that Euclid had laid out for his geometry? Mathematicians struggled with this question for 2000 years before successfully answering it. The answer had a profound philosophical effect on all later mathematics, as we will see.

54. Archimedes Plutonium Subject Re Correcting and Overhauling the axioms of mathematics Date 1 Nov 1998002713 GMT Organization physics overhauls the axioms of math Lines 16 http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/File102.html

Physics corrects and overhauls the axioms of mathematics by Archimedes Plutonium this is a return to website location http://www.newphys.se/elektromagnum/physics/LudwigPlutonium/ - From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) Newsgroups: sci.logic,sci.physics Subject: If one major physics parameter is infinite, then all are Date: 12 Jan 1999 08:15:53 GMT Organization: infinities in parameters of physics Lines: 26 Distribution: world Message-ID: In article In article I wrote: > (1) there is no valid concept of "finite" but that all numbers have a > component of infinity. Looking at the 3 types of Numbers, (1) infinite string right, Reals, (2) infinite string to the left, p-adic (3) doubly infinite strings which of those three fits the old Peano axiom system the best? The answer is that p-adic strings fit the Peano axiom system the best although there are predecessors to and mathematical induction does not hold. But at least with the p-adic integers, an overhauled Peano axiom system can build the p-adics. With Whole Reals, remember, you must first build the entire set of Reals before you can have the Whole Reals.

55. Godel And Godel's Theorem: Math Overview of Hofstadter's explanation of Gödel's Theorem.Category Computers Artificial Intelligence...... of math, in terms of TNT, and in terms of Gödelized TNT. Zero equals zero istrue. The string 0=0 is a valid TNT theorem (ie can be derived from axioms). http://www2.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/godel.html

by Kenny Felder In the nineteenth and early twentieth centuries, one of the big mathematical goals was to reduce all of number theory to a formal axiomatic system. Like Euclid's Geometry, such a system would start off with a few simple axioms that are almost indisputable, and would provide a mechanical way of deriving theorems from those axioms. It was a very lofty goal. The idea was that this system would represent every statement you could possibly make about natural numbers. So if you made the statement "every even number greater than two is the sum of two primes," you would be able to prove strictly and mechanically, from the axioms, that it is either true or false. For real, die-hard mathematicians, the words "true" and "false" would become shorthand for "provable" or "disprovable" within the system. Russell and Whitehead's Principia Mathematica was the most famous attempt to find such a system, and seemed for a while to be the pinnacle of mathematical rigor. there is always a statement about natural numbers which is true, but which cannot be proven in the system. In other words, mathematics will always have a little fuzziness around the edges: it will never be the rigorous unshakable system that mathematicians dreamed of for millennia.

56. Math 123 General Course Outline math 123 General Course Outline. Catalog description. 123. Foundations of GeometryLecture three hours; discussion - one hour. Requisite course 115A. axioms http://www.math.ucla.edu/undergrad/courses/math123/outline.html

UCLA Department of Mathematics Math 123: General Course Outline Catalog description 123. Foundations of Geometry: Textbook Greenberg, M., Euclidean and Non-Euclidean Geometries , Third Edition, W.H. Freeman and Co. Reviews and Exams The following schedule, with textbook sections and topics, is based on 24 lectures. The remaining five classroom meetings (only four in Winter Quarter) are for leeway, reviews and midterm exams. These are scheduled by the individual instructor. Schedule of Lectures Lecture Section Topic Chapt. 1 Euclid's geometry; Euclid's Elements, common notions, Euclid's five axioms, Propositions 1, 2 and 4 from Book 1. Instructor should consult Book 1 of Euclid's Elements.* Chapt. 2 Logic; incidence geometry, affine and projective planes, finite geometries. Chapt. 3 Hilbert's axioms. Chapt. 4 Neutral geometry. Chapt. 5 History of the parallel postulate; Wallis' postulate, Clairaut's axiom. Chapt. 6 (pp. 177-191) "Discovery" of non-Euclidean geometry, AAA criterion in hyperbolic geometry. Chapt. 7

57. Stephen Wolfram: A New Kind Of Science -- Index T-z and discreteness of space, 1027 and geons, 1054 Whitehead, Alfred N. (England/USA,18611947) and axioms for logic, 1151 and foundations of math, 1149 and math http://www.wolframscience.com/nks/index/names/t-z.html?SearchIndex=t-z

58. Lecture Notes 2 - Math 3210 axioms for the Finite Geometry of Pappus. There exists at least one line. axiomsfor the Finite Geometry of Desargues. There exists at least one point. http://www-math.cudenver.edu/~wcherowi/courses/m3210/hg3lc2.html

Lecture Notes 2 Pappus of Alexandria (340 A.D.) Pappus' Theorem: If points A,B and C are on one line and A', B' and C' are on another line then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and CB' lie on a common line called the Pappus line of the configuration. Axioms for the Finite Geometry of Pappus There exists at least one line. Every line has exactly three points. Not all lines are on the same point. [N.B. Change from the text] If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line. If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'. With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them. Theorem 1.10 Each point in the geometry of Pappus lies on exactly three lines. Pf . Let X be any point. By corrected axiom 3, there is a line not containing X. This line contains points A,B,C [Axiom 2]. X lies on lines meeting two of these points, say B and C [Axiom 5]. There is exactly one line through X parallel to BC [Axiom 4]. There can be no other line through X since by Axiom 4 it would have to meet BC at a point other than A, B or C [Axioms 6 and 5], and this would contradict Axiom 2. Pappus geometry has 9 points and 9 lines.

59. Lecture Notes 1 - Math 4220 (This nondegeneracy condition can be replaced with There exist three pointsnot on the same line.) An axiom system is consistent if the axioms are not self http://www-math.cudenver.edu/~wcherowi/courses/m4220/hg2lec1.html

Lecture Notes 1: Affine Planes Defn : An affine plane is an ordered pair ( ) where is a nonempty set of elements called points and a nonempty collection of subsets of called lines which have the following properties: If P and Q are distinct points, there is a unique line l such that P l and Q l . [This line is denoted l P,Q If P is a point not contained in the line l , there is a unique line m such that P m and m l . (When l m l is said to be parallel to m , written l m There are at least two points on each line; there are at least two lines. (This non-degeneracy condition can be replaced with : There exist three points not on the same line.) An axiom system is consistent if the axioms are not self-contradictory, that is, the assumption of the truth of the axioms will not lead to a contradiction. If an axiom system has a model (i.e., an example) in which all the axioms hold, then it is consistent. The following examples each show that the axioms for an affine plane are consistent. Examples: The Real Coordinate Plane and l in iff l , a + b (This last condition is just a fancy way to say that both of a and b can not be simultaneously) The Rational Affine Plane and l iff l , a + b The Smallest Affine Plane and One can also view this example as a coordinate plane over the finite field Z (= GF(2)) by setting A = (0,0), B = (1,0), C = (0,1) and D = (1,1).

60. Me? A Math Geek? You Must Be Joking! She won't tell you that her math books are wrong. is proven or true , a betterway to read that is that it is a logical consequence of a set of axioms . http://www.ihoz.com/math.html

On the newsgroup alt.society.generation-x a discussion ensued as to the correct way to teach math. In the conversation, someone said that Algebra was not all proof, but it was what was done with proof that was important. Assuming that Algebra meant the subject about groups and rings, not the one with trains meeting each other, I saw a springboard to something that I wanted to write- an essay on the nature of mathematics. Enjoy Professor. It isn't just my experience that is like this. In their book The Mathematical Experience , practising mathematicians Phillip J. Davis and Reuben Hersh wrote an essay on "The Ideal Mathematican" (ideal as in the platonic mathematican, not as in best). The purpose of this essay was to show mathematicans what they tend to say and believe and how it comes across to the outsiders. In their essay, a public information officer interviews our ideal mathematican: PIO: Perhaps I'm asking the wrong questions. Can you tell me about the applications of your research? IM: Applications? PIO: Yes, applications. IM: I've been told that some attempts have been made to use non-Riemannian hypersquares as models for elementary particles in nuclear physics. I don't know if any progress was made. Later on in the interview: PIO: Do you see any way that the work in your area could lead to anything that would be understandable to the ordinary citizen of this country? IM: No. PIO: How about engineers or scientists? IM: I doubt it very much.

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