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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.

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

Charming proofs

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

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

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

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

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