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. Non-professionals 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 AM-GM 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 Quarter-Circle 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
- Brahmagupta-Mahavira 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 3-Space 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
- Two-Sided 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
- Manifesto. Interactive Mathematics Miscellany and Puzzles
- Is anything wrong with math education?
- Can I do anything?
- Math was the most difficult subject I ever...
- Is Mathematics all around us?
- Is math beautiful?
- Do we need Mathematics?
- Proofs in Mathematics
- Artisans vs Mathematicians
- Pleasant math morsels
- Mathematics and Puzzles
- Learn Mathematics for Its Value
- Need Mathematics Taught in School Be Relevant?
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