Return to my Mathematics pages
Go to my home page

Proof

© Copyright 1999, Jim Loy

The average person does not have a firm grasp of the meaning of the word "proof." He/she may hear that someone saw a UFO, and consider that a proof that we are being visited by creatures from outer space. They may see a blinding light, just before dying (and then recover) and consider that proof of heaven. Instead, these are examples of evidence (and flimsy evidence at that), and not proof. But these people live in a world in which nothing is certain. In such a world, "proof" does not have much meaning.

Mathematicians (and scientists to a large extent) live in a world in which some things are certain. Mathematicians have Euclid's (and other people's) axioms, postulates, and theorems. Physicists have Newton's (and other people's) laws. These are ideas that are so basic that it would be silly to deny them (under normal circumstances). And mathematicians and scientists use these postulates, theorems, and laws to deduce other theorems and laws. This is proof.

There are several kinds of proof:

• Direct proof: Show that the intended theorem can be deduced from basic truths. Start with the basic principles, end with what you are trying to prove.
• Proof by contradiction: Assume the intended theorem is false and show that this leads to a contradiction. This is often among the easiest kinds of proof. [Note: Once you have shown a contradiction, your proof is done. State what you have just disproved. Continuing on with other deductions only complicates, unnecessarily]
• Mathematical induction: Show that the intended theorem is true for the first case. Then show that if it is true for any given case it is true for the next case. This shows that it is true for all positive integers (case 1, case 2, case 3...).
• Impossibility proof: Show that something is impossible to do, by showing that it cannot be done in certain ways, and then show that there are no other ways. This is often the hardest kind of proof.
• Existence proof: Usually, to show that some things with certain properties exist, just show an example. To prove there is an even prime number, just mention 2. If an example cannot be found, this can be a very difficult kind of proof indeed.
• Proof by contraposition: If we have a statement of the form A implies B (If A then B), then the contrapositive is: not B implies not A. A statement is always equivalent to its contrapositive. If you have proved one, you have proved the other.
• Backward proof: Assume the intended theorem is true. See that it leads to basic truths. This is a flawed method; it is not a proof. It may help you discover a proof, however. See if you can make the backward proof go forward.

In my WWW pages are a few proofs (some of which are informal):

Direct proofs:

• The Monty Hall Trap. Proof that you should change your choice.
• 0! Proof that 0!=1.
• Two Equals One? Find the flaw in this famous proof.
• Quadratic Formula. I derive this formula.
• Volumes Of Cones & Pyramids. I derive the formula for these.
• Regular Solids. Proof that there are only five regular solids.
• Hero's Formula. I derive this formula.
• The Pythagorean Theorem. Several proofs of this.
• Isosceles Triangles. Proof that the base angles of an iscosceles triangle are congruent.
• Trisection Of An Angle. Proof that one of the illegal methods (by Archimedes) does indeed trisect an angle.
• Unbreakable Ciphers. Proof that such ciphers are possible.

Proof by contradiction:

• Irrational Numbers. Pythagoras' proof that the square root 2 is not a rational number.
• Congruence Of Triangles, Part II. Four proofs by contradiction.
• The Infinitude Of Primes. Euclid's proof that there are infinitely many primes.

Proof by mathematical induction:

• Dividing The Plane. I guess at a formula, and then prove it.
• Casting Out Nines. This proof is stated informally.

Impossibility proof:

• Trisection Of An Angle. I talk about such a proof here. But I don't prove that trisection of an angle is impossible.

Existence proof:

• Four Color Theorem Intro. I show that at least 4 colors are necessary for some map or maps. It is a trivial proof.
• Unbreakable Ciphers. I show that there are unbreakable ciphers.

The backward proof is sometimes diabolical. At worst, you can assume something that is false and end up with some obvious truth, and you think you proved your false statement. At best, you assume something that is true, and end up mistakenly thinking you have proved it. The following erroneous proof is too obvious to fool anyone: Assume 3=2. The by the commutative law, 2=3. Add the two equations, and we get 5=5. Subtract 5 from both sides, and we get 0=0, which I happen to know is true. Therefore 3=2. Not! To actually try to prove that 3=2, we would have to start with 0=0 and end with 3=2. And that won't work.

There is also proof by overwhelming evidence. This is not a mathematical proof. But, it is often the best there is in Science, and can be very convincing. My article All Numbers Are Less Than A Million is a joke based on this idea. My computer simulation, in The Monty Hall Trap, is a serious use of this idea.

There is also proof by waving your arms. This is when you pretend to prove something. You give some arguments and then leave the proof hanging. This may be satisfactory, under the circumstances. You just may not want to get into it at the time.

There is also proof by appeal to authority. Many of my science and pseudoscience articles use this method. It is, of course, no proof at all. I am just informing you that other people have done the proving. You may have to go elsewhere to find the evidence or proof. When the authority that you appeal to is "the scientific community" or "biologists" or "physicists," that is not proof. They could all be wrong. They have been wrong in the past. But I think that they are probably not all being unreasonable or stupid (See Phlogiston Theory). They have sound logical arguments for what they say. An appeal to the authority of the opinions of average people, or to some government organization, or to "scientists" quoted in the supermarket tabloids should be less convincing.

And we have proof beyond a reasonable doubt. This is the standard for proof in our courts, with juries. Ideally, this sounds pretty good, and it seems to work well in most cases. Both sides present their arguments, and the jury judges, based on the facts as presented. And they cannot railroad someone, because they must be sure of guilt to judge a person guilty. But there are problems. For many reasons, jury decisions may not reflect the truth. They may have never been presented with the truth. And so, such "proof" is less reliable than any of the above kinds of proof.

My mathematical induction joke (from my Jokes page):

You may have seen the ad that says "Penn Tennis Balls, you've seen one, you've seen them all." Supposedly, they are all the same. Here's my improved ad:

1. Weaker statement: Penn Tennis Balls, you've seen n, you've seen n+1.
2. My ad shows you one Penn tennis ball.
3. Then, by mathematical induction: Penn Tennis Balls, you've seen one, you've seen them all.

Addendum:

There is a book called The Brendan Voyage. There are legends of an Irish monk (Saint Brendan) who explored the northern Atlantic, long before Columbus. Some of the legends sound as if America were discovered by Brendan. This book is subtitled, "Sailing to America in a leather boat to prove the legend of the Irish saints." This re-enactment proves that such a voyage is possible. But it falls far short of proving the legend. This misuse of the words "prove" and "proof" is very common.

Another good example of a backward proof is this: Prove that 1=-1. Square both sides, getting 1=1, which is true. Therefore 1=-1 (not).

Return to my Mathematics pages
Go to my home page