Proofs in Mathematics
Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.
Vladimir Arnold 
John Paulos cites the following quotations by Bertrand Russell:
Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Paulos goes on to say
Although the ubiquity of people who neither know what they're talking about nor know whether what they're saying is true may incorrectly suggest that mathematical genius is rampant, the quote does give a succinct, albeit overstated, summary of the formal axiomatic approach to mathematics.
Both opinions are enjoyable and thought provoking. To me, the former just plainly states that proving (that is, deriving from one another) propositions is the essence of mathematics. To a different extent and with various degrees of enjoyment or grief most of us have been exposed to mathematical theorems and their proofs. Even those who are revolted at the memory of overwhelmingly tedious math drills would not deny being occasionally stumped by attempts to establish abstract mathematical truths.
I am not sure it's possible to evict drills altogether from the math classroom. But I hope, in time, more emphasis will be put on the abstract side of mathematics. Drills contain no knowledge. At best, after sweating on multiple variations of the same basic exercise, we may come up with some general notion of what the exercise is about. (At worst, the sweat and effort will be just lost while the fear of math will gain a stronger foothold in our conscience.) Moreover, if it's possible at all for a layman to acquire an appreciation of math, it's only possible through a consistent exposure to the beauty of math which, if anywhere, lies in the abstractedness and universality of mathematical concepts. Nonprofessionals may enjoy and appreciate both music and other arts without being apt to write music or paint a picture. There is no reason why more people couldn't be taught to enjoy and appreciate math beauty.
According to Kant, both feelings of sublime and beautiful arouse enjoyment which, in the case of sublime, are often mixed with horror. By this criterion, most of the people would classify mathematics as sublime much rather than beautiful. On the other hand, Kant also says that the sublime moves while the beautiful charms. I trust math would inspire neither of these in an average person. Trying to make the best of it, I'll seek refuge in a third quote from Kant, "The sublime must always be great; the beautiful can also be small."
Heath Biology, an excellent high school text by
It's important to note that, while proofs and deductive reasoning play an important and practically exclusive role in mathematics, going from a proof to another proof making deductive steps is not how mathematics is done, see, for example, a fascinating article by W. Thorston ON PROOF AND PROGRESS IN MATHEMATICS.
With these preliminaries I want to start a collection of mathematical proofs. I'll distinguish between two broad categories. The first is characterized by simplicity. A proof is defined as a derivation of one proposition from another. A single step derivation will suffice. If need be, axioms may be invented. A finest proof of this kind I discovered in a book by
In the second group the proofs will be selected mainly for their charm. Simplicity being a source of beauty, selection of proofs into the second group is hard and, by necessety, subjective. The first of the collection is due to John Conway which I came across in a book by
Simple proofs
 A Property of Equiangular Polygons
 ab+bc+ca ≤ aa+bb+cc
 An Extension of the AMGM Inequality: A second look
 An integral
 Another simple integral
 A simple integral, III
 Chvatal's Art Gallery Theorem
 Averages of divisors
 Bisecting arcs
 Breaking Chocolate Bars
 Broken Line in Triangle
 Coloring Plane with Three Colors
 Coloring points in the plane
 Gasoline Stations on a Circular Trek
 Gauss and Euler Integrals
 Geometry, Algebra, and Illustrations
 Halving a square
 Heads and Tails
 Integral Is Area
 Intersections of a Circle with the Four Quadrants
 Longest segment
 McDougall's Generalization of Ptolemy's Theorem
 Menelaus Theorem: Proofs Ugly and Elegant  A. Einstein's View
 Number of vowels in a Lewis Carroll game
 Number of X's and O's
 On Gauss' Shoulders
 One Dimensional Ants
 Pigeonhole Principle
 √2 is irrational
 Shapes in a lattice
 Shortest Fence in a QuarterCircle Pasture
 Sine, Cosine, and Ptolemy's Theorem
 Viviani's Theorem
Charming proofs
 4 Travellers Problem
 A Cyclic Inequality in Three Variables XIV
 A Cyclic Inequality in Three Variables with a Variable Hierarchy
 A Proof by Game for a Sum of a Convergent Series
 Areas In Circle
 Assigning Numbers to Points in the Plane
 Averages in a sequence
 BrahmaguptaMahavira Identities
 Clubs in a Vector Space
 Conic sections
 cos(π/7)cos(2π/7)+cos(3π/7) = 1/2
 Countability of Rational Numbers
 Extremal Problem in a Quadrilateral
 Four Pegs That Form a Square
 Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
 Infinitude of Primes
 Infinitude of Primes  A Topological Proof
 Infinitude of Primes Via *Sets
 Infinitude of Primes Via Coprime Pairs
 Infinitude of Primes Via Fermat Numbers
 Infinitude of Primes Via Harmonic Series
 Infinitude of Primes Via Lower Bounds
 Infinitude of Primes Via Euler's Product Formula
 Infinitude of Primes Via Euler's Product Formula for Pi
 Why The Number of Primes Could Not Be Finite?
 Integers and Rectangles
 Lucas' Theorem
 Maxwell's Theorem
 Menelaus from 3D
 Negative Coconuts
 Number of Regions N Lines Divide Plane
 Property of the Line IO: a proof from the Book
 Ptolemy by Inversion
 Rectangle on a Chessboard
 Partitioning 3Space with Circles
 Point in a square
 Property of the Line IO
 Seven Concyclic Points in Equilateral Bumps
 Splitting piles
 Symmetries in a Triangle
 Three circles
 Three Circles and Common Chords
 Three Circles and Common Tangents
 TwoSided Inequality  One Provenance
 Uncountability of the Reals  via a Game
There are also facts, mathematical statements that seem to hold some secret, being counterintuitive to most or suprising. Often their proofs are either straightforward or insignificant in themselves, which suggests an additional list of
Attractive facts
 An Old Japanese Theorem
 A Property of a^{n}
 About a Line and a Triangle
 An Inequality of the Areas of Triangles Formed by Circumcenter And Orthocenter
 Arbelos' Morsels
 Beatty Sequences
 Butterfly Theorem
 Carnot's Theorem
 Cevians Through the Circumcenter
 Curious Irrationality in Square
 Dan Sitaru's Cyclic Inequality In Many Variables
 Dimensionless Inequality in the Euclidean Plane
 Dots and Fractions
 Exponential Inequalities for Means
 Ford's touching circles
 Function in the Plane That Vanishes
 Geometric Mean In Trapezoid
 Haruki's Theorem
 How Do Angle Trisectors Divide the Area?
 Intersecting Chords Theorem
 More On Inscribed Angles and Pivot Theorem
 Morley's Miracle
 Napoleon's Theorem
 Orthocenters
 Pentagon And Decagon, Both Regular
 Points Generated by the Nine Points
 Proizvolov's Identity
 Properties of Circle Through the Incenter
 Ptolemy's Theorem
 Salinon: From Archimedes' Book of Lemmas
 Shifting Digits and a Point of View
 Squares in Semicircle and Circle
 Property of Semicircles
 The Shoemaker's Knife
 Yet Another of Euler's Formulas
 The Size is in the Eyes of the Beholder
 Three Concurrent Chords at 60 Degrees Angles
 Volumes of Two Pyramids
 Wonderful Inequality on Unit Circle
To prove means to convince. More strictly, proof is a sequence of deductions of facts from either axioms or previously established facts. A deduction that follows the rules of logic is tacitly assumed to be sufficiently convincing. Sometimes, however, by mistake or oversight, an error crops into a proof. The proof then may present a convincing argument of the correctness of a fact that, in itself, may be true or false. If a proof presents a convincing argument of the validity of an incorrect statement it's called fallacious or a fallacy. Sometimes, an incorrect deduction leads to a correct statement. Such crippled deductions that lead to correct results I shall designate simply as false, wrong or invalid proofs each of which should be judged an oxymoron.
Fallacies
 1 = 0
 1 = 2
 1=2 via Continued Fractions
 1/2 = 1
 A Circle With Two Centers
 A Faulty Dissection
 All Integers Are Equal to 1
 All Integers Are Even
 All Powers of x are Constant
 All Powers of 2 Are Equal to 1
 All Triangles Are Isosceles
 Curry's Paradox
 Delian Problem Solved
 $\pi ^e$ is rational
 Every Parallelogram Is a Rectangle
 Four Weighings Suffice
 Galton's Paradox
 In Calculus too 1 = 0
 Langman's Paradox
 Rabbits Reproduce; Integers Don't
 Rouse Ball's Fallacy
 SSA
 Sam Loyd's Son's Dissection
 Sum of All Natural Numbers
 Two Perpendiculars From a Point to a Line
Invalid Proofs
By philosophy is understood the knowledge acquired by reasoning, from the manner of generation of anything, to the properties; ... Nor are we therefore to give that name to any false conclusions; for he that reasoneth aright in words he understandeth can never conclude an error. 
Thomas Hobbes 
 Ancient Problem = Ancient Solution
 Calculus Proof of the Pythagorean Theorem
 Delian Problem
 Equilic Quadrilateral I
 Eyeball Theorem, proof #5
 Exterior Angle Theorem
 Faulty Symmetry
 Fermat's Last Theorem
 However You Solve It ... A Wonderful Equation
 Is Every Trapezoid Parallelogram?
 Is the Triangle Inequality Necessary?
 Morley's Theorem: A Proof That Needs Fixing
 Pythagorean Theorem: Some False Proofs
 SSS
 When A Quadrilateral Is Inscriptible?
 An Inequality from Marocco, with a Proof, or Is It?
References
 R. Honsberger, Mathematical Gems II, MAA, 1976
 I. Kant, Observations on the Feeling of the Beautiful and Sublime, University of California Press, 1991
 J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
 S. Savchev, T. Andreescu, Mathematical Miniatures, MAA, 2003
 Ian Stewart, Nature's Numbers, BasicBooks, 1995
MANIFESTO

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