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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 J. E. McLaren and L. Rotundo, talking about experimental sciences, has the following to say about proofs: "Notice also that scientists generally avoid the use of the word proof. Evidence can support a hypothesis or a theory, but it cannot prove a theory to be true. It is always possible that in the future a new idea will provide a better explanation of the evidence." Thus we see that proofs are a peculiar attribute of mathematical theories. The proofs may only exist in formal systems as described by B.Russell.

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 I. Stewart. Most of the proofs I think of should be accessible to a middle grade school student.

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 R. Honsberger. Many a mathematician would insist that math objects (even the most abstract) have existence of their own like physical objects. Mathematicians may only discover them and study their properties. Look into the proof. Think of those powers of the golden ratio. Has Conway invented them, or have they been filling the grid all along?

A Checker - Jumping Problem

Simple proofs

1. A Property of Equiangular Polygons
2. ab+bc+ca ≤ aa+bb+cc
3. An Extension of the AM-GM Inequality: A second look
4. An integral
5. Another simple integral
6. A simple integral, III
7. Chvatal's Art Gallery Theorem
8. Averages of divisors
9. Bisecting arcs
10. Breaking Chocolate Bars
11. Broken Line in Triangle
12. Coloring Plane with Three Colors
13. Coloring points in the plane
14. Gasoline Stations on a Circular Trek
15. Gauss and Euler Integrals
16. Geometry, Algebra, and Illustrations
17. Halving a square
18. Heads and Tails
19. Integral Is Area
20. Intersections of a Circle with the Four Quadrants
21. Longest segment
22. McDougall's Generalization of Ptolemy's Theorem
23. Menelaus Theorem: Proofs Ugly and Elegant - A. Einstein's View
24. Number of vowels in a Lewis Carroll game
25. Number of X's and O's
26. On Gauss' Shoulders
27. One Dimensional Ants
28. Pigeonhole Principle
29. √ 2 is irrational
30. Shapes in a lattice
31. Shortest Fence in a Quarter-Circle Pasture
32. Sine, Cosine, and Ptolemy's Theorem
33. Viviani's Theorem

Charming proofs

1. 4 Travellers Problem
2. A Cyclic Inequality in Three Variables XIV
3. A Cyclic Inequality in Three Variables with a Variable Hierarchy
4. A Proof by Game for a Sum of a Convergent Series
5. Areas In Circle
6. Assigning Numbers to Points in the Plane
7. Averages in a sequence
8. Brahmagupta-Mahavira Identities
9. Clubs in a Vector Space
10. Conic sections
11. cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
12. Countability of Rational Numbers
13. Extremal Problem in a Quadrilateral
14. Four Pegs That Form a Square
15. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
16. 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?
17. Integers and Rectangles
18. Lucas' Theorem
19. Maxwell's Theorem
20. Menelaus from 3D
21. Negative Coconuts
22. Number of Regions N Lines Divide Plane
23. Property of the Line IO: a proof from the Book
24. Ptolemy by Inversion
25. Rectangle on a Chessboard
26. Partitioning 3-Space with Circles
27. Point in a square
28. Property of the Line IO
29. Seven Concyclic Points in Equilateral Bumps
30. Splitting piles
31. Symmetries in a Triangle
32. Three circles
33. Three Circles and Common Chords
34. Three Circles and Common Tangents
35. Two-Sided Inequality - One Provenance
36. 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

1. An Old Japanese Theorem
2. A Property of aⁿ
3. About a Line and a Triangle
4. An Inequality of the Areas of Triangles Formed by Circumcenter And Orthocenter
5. Arbelos' Morsels
6. Beatty Sequences
7. Butterfly Theorem
8. Carnot's Theorem
9. Cevians Through the Circumcenter
10. Curious Irrationality in Square
11. Dan Sitaru's Cyclic Inequality In Many Variables
12. Dimensionless Inequality in the Euclidean Plane
13. Dots and Fractions
14. Exponential Inequalities for Means
15. Ford's touching circles
16. Function in the Plane That Vanishes
17. Geometric Mean In Trapezoid
18. Haruki's Theorem
19. How Do Angle Trisectors Divide the Area?
20. Intersecting Chords Theorem
21. More On Inscribed Angles and Pivot Theorem
22. Morley's Miracle
23. Napoleon's Theorem
24. Orthocenters
25. Pentagon And Decagon, Both Regular
26. Points Generated by the Nine Points
27. Proizvolov's Identity
28. Properties of Circle Through the Incenter
29. Ptolemy's Theorem
30. Salinon: From Archimedes' Book of Lemmas
31. Shifting Digits and a Point of View
32. Squares in Semicircle and Circle
33. Property of Semicircles
34. The Shoemaker's Knife
35. Yet Another of Euler's Formulas
36. The Size is in the Eyes of the Beholder
37. Three Concurrent Chords at 60 Degrees Angles
38. Volumes of Two Pyramids
39. 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. 1 = 0
2. 1 = 2
3. 1=2 via Continued Fractions
4. 1/2 = 1
5. A Circle With Two Centers
6. A Faulty Dissection
7. All Integers Are Equal to 1
8. All Integers Are Even
9. All Powers of x are Constant
10. All Powers of 2 Are Equal to 1
11. All Triangles Are Isosceles
12. Curry's Paradox
13. Delian Problem Solved
14. $\pi ^e$ is rational
15. Every Parallelogram Is a Rectangle
16. Four Weighings Suffice
17. Galton's Paradox
18. In Calculus too 1 = 0
19. Langman's Paradox
20. Rabbits Reproduce; Integers Don't
21. Rouse Ball's Fallacy
22. SSA
23. Sam Loyd's Son's Dissection
24. Sum of All Natural Numbers
25. 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 Leviathan, ch. 46 Penguin Classics, 1982

1. Ancient Problem = Ancient Solution
2. Calculus Proof of the Pythagorean Theorem
3. Delian Problem
4. Equilic Quadrilateral I
5. Eyeball Theorem, proof #5
6. Exterior Angle Theorem
7. Faulty Symmetry
8. Fermat's Last Theorem
9. However You Solve It ... A Wonderful Equation
10. Is Every Trapezoid Parallelogram?
11. Is the Triangle Inequality Necessary?
12. Morley's Theorem: A Proof That Needs Fixing
13. Pythagorean Theorem: Some False Proofs
14. SSS
15. When A Quadrilateral Is Inscriptible?
16. An Inequality from Marocco, with a Proof, or Is It?

References

1. R. Honsberger, Mathematical Gems II, MAA, 1976
2. I. Kant, Observations on the Feeling of the Beautiful and Sublime, University of California Press, 1991
3. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
4. S. Savchev, T. Andreescu, Mathematical Miniatures, MAA, 2003
5. Ian Stewart, Nature's Numbers, BasicBooks, 1995

MANIFESTO

• Manifesto. Interactive Mathematics Miscellany and Puzzles
• Is anything wrong with math education?
  • Is There Always One Right Answer?
  • Free vs. Pedantic Thinking
• Can I do anything?
• Math was the most difficult subject I ever...
• Is Mathematics all around us?
  • How Fast Does One's Shadow Grow?
• 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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