If you have a good imagination, picture the following: An
eight-by-eight board (a chess or checker board will do) and a set of
dominos. If not, perhaps you can lay your hands on the real things to
augment your imagination. Now, lay (or imagine laying) one domino on
two adjacent squares of the board. Since the board has 64 squares it
should take 32 dominoes to do the job.
The fact that this is possible is obvious. Anyone should get it
right on the first try. Now, try to lay out the dominoes such that the
lower left and upper right corner squares are left uncovered. This
covers two fewer squares, so it should take one less domino. Try it.
If you succeed, you will have proved that it’s possible. If you fail,
can you prove that it’s impossible?
There are at least two ways to complete the proof that this task is
impossible. One way is to program a computer to check every possible
arrangement of dominos. This is the “brute force” method of proof.
The other way is more elegant. It consists of noticing one of the
features of a chess board: The fact that adjacent squares are of
opposite colors, and that squares of the same color are always
diagonal. Combine this with the fact that a domino must always cover
adjacent squares and never diagonals and you have a proof. Notice that
the opposite diagonal squares are the same color. Notice that each
domino must always cover two squares of opposite colors. You simply
can’t cover two differently colored squares with each domino and have
two squares of the same color left over, if the board has the same
number of squares of each color.
What does this illustrate in a larger sense? For one thing, it
illustrates the varied nature of proofs. Statements may affirm
something. Statements may be provably true or false, they may be
suspected to be true or false, or their status may simply be unknown.
A person’s “belief structure” is the set of statements that person
would accept as true. Most people accept statements as true that they
can’t themselves prove to be true, but they believe that others can.
Many people accept statements as true that they know can’t be proven,
or they accept inadequate proofs, or they are simply not concerned with
the criteria for proof (or the truth, which to them may not even be the
A proof of existence consists of demonstrating a single example of
something. On the other hand, you can’t ever prove that something
(like God) does not exist. Some statements assert a relationship or
predict an outcome. These are more tricky than statements about
existence. These statements may involve single instances, categories
with any number of instances, or even sets with an infinite number of
instances. A proof of truth has to consider every instance. A proof
that a statement is false only has to find a single counter-example.
Sometimes an assumption is made that you can prove a statement is false
by proving that its opposite is true. Likewise, you can prove a
statement is true by proving its opposite is false. But these methods
of proof are subject to the weakness of the assumption that there is no
middle ground between the true and the false. This type of proof
implies that everything is provably true or false, and that itself, is
an assertion that has been proven false. It is Goedel’s Incompleteness
Theorem. It states, in essence, that statements in any adequate
language cannot always be proven either true or false. There are
always three possibilities, not just two: A statement is provably
true, a statement is provably false, or a statement is not known to be
either. In this latter category there are, again, three types of
statements: Those that will later be proven true, those that will
later be proven false, and those that remain, either suspected to be
true or false, or proven to be undecidable.
An interesting conjecture is the statement that every even number is
the sum of two primes. A prime number, remember, is a number not
evenly divisible by any number other than itself and one. Since every
prime number must be odd, and two odd numbers always add up to an even
number, it appears that the conjecture could be true. However, this
conjecture has so far defied proof. It appears that each even number,
and there are an infinite number of even numbers, is a separate
problem. No one has yet found an even number that was not the sum of
two primes. But, there appears not to be a pattern or algorithm that
tells us how to get the two primes that will add up to any given even
number. Without such a procedure, we are left with an infinite number
of separate problems, and an infinite number of problems cannot be
solved. Thus, if a procedure for this does not exist, we are left with
a simple statement that is probably true, but can never be proved.
Statements that are undecidable fall into at least three categories:
Those which generalize about an infinite set of things, those which are
self-referential, and those involving things about which we have
insufficient evidence. I like the statement “The Barber of Seville
shaves every man who does not shave himself.” The question is, who
shaves the Barber? Since the first statement ought to answer that
question, but contradicts itself when doing so, it makes the truth of
the statement undecidable in a sense. My own answer to the paradox is
that the Barber of Seville is obviously a woman, but that’s a cheap
solution, and the general point remains. The statement in the
preceding paragraph, “every even number is the sum of two primes,” is
an example of a generalization about an infinite set of things. And,
statements that lack sufficient proof are why we have courtrooms,
juries, judges, and lawyers.
Spend some time Numinating about proofs (if you were unconvinced by
“What are the Odds?” you might try constructing your own proofs one way
or the other, and test them against my assertions, but be careful that
you don’t create a slightly altered version, rather than an exact
equivalent of what I asserted, or we will be in dispute over apples and
oranges). Next time we’ll kick it up a notch, and Numinate once again
about Truth itself.
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