(2007-01-09)
Only "negatives" are worth proving.
"You can't prove a negative" is a proverb about tests, not proofs...
To a mathematician, "proofs" are not restricted to mere "tests".
Arguably, in the above adage, the very word "negative" lacks clear meaning as well
(since most statements could be cast in either "positive"
or "negative" forms).
To play along, we should dub the first sentence below "positive"
and the second one "negative":
- A square can be the sum of two nonzero squares.
- A cube cannot be the sum of two nonzero cubes.
Both of those statements happen to be true in the realm of integers.
The first one can be "proved" merely by giving an example
(the most popular of many
is 25 = 16+9).
On the other hand, the second statement tells that counterexamples do not exist...
That affirmation can only be supported by a piece of reasoning,
since a lack of solutions can never be demonstrated by many failed attempts.
To whomever came up with the above infamous proverb,
such a thing was clearly inconceivable.
Yet, it's precisely what mathematicians call a "proof".
Once you've seen and understood just one such proof, you shall know better.
A proof that a cube can't
be the sum of two cubes can be given using the so-called
method of infinite descent.
An example of that method is
a clever two-line proof (which repays study) that
there's no rational whose square is 2 :
Since 1 < Ö2 < 2,
if a positive integer n was making
nÖ2
an integer, the smaller positive integer
m = (Ö2-1) n
would make mÖ2 an integer also!
Another proof
invokes the concept of divisibility.
It may be easier and more intuitive, but it's less "elementary"
(i.e., it relies on more previous knowledge).
(2007-01-16)
Stochastic proofs are wrong with
vanishing probability.
Sometimes, an elusive truth is reinforced by many failures to attack it.
One celebrated example is the iterated
Rabin-Miller test
which tells (beyond the shadow of a doubt) whether a large number
is prime or not, without actually proving anything
when that number happens to be prime...
For a composite number,
each iteration stands a substantial chance (over 75%) of proving it's
not prime.
Thus, if several iterations fail to provide such a proof,
we may be very confident that the number is indeed prime
(the probability of error decreases
exponentially with the number of iterations).
Another example consists in determining
whether a (large) finite group
is cyclic (knowing the factorization into primes of its
order).
A finite group is cyclic if and only if it
has a primitive root.
It turns out that a random element of a cyclic group is
primitive with a fairly large probability
(and it can be proved to be primitive very efficently
if the prime factors of the group's order are known).
Thus, if many random elements are not primitive,
then the group is "almost surely" not cyclic.
(2007-01-16)
Heuristic Arguments
Establishing the likelihood of a conjecture with an approximative proof.
For example, I argue (against the dominant opinion)
that there are probably
infinitely many Wieferich primes,
although only two of them are known (in spite of
great efforts to find a third).
A proper heuristic argument is not a hasty generalization.
It's actually a strict mathematical proof about a modified
problem, where part of the original mathematical structure is
substituted with a probabilistic model.
Quantitative conclusions from such a model can be enlightening
while an exact solution to the original problem remains elusive.
This may be construed as "relaxing" some mathematical
constraints while retaining the problem's essential aspects.
A good heuristic argument must be supported with convincing
justifications
of the probabilistic assumptions underlaying the model.
A heuristical argument is never foolfproof (or else it would be a proper
mathematical proof) but it should be nearly so...
The qualifier "heuristic" shouldn't be an excuse for sloppiness !
(2009-06-21)
Ruling out proofs
Facts which can't be established by some or all
types of proofs.
Generations of mathematicians have attempted to
prove Euclid's fifth
postulate of plane geometry from the other four axioms of Euclidean
geometry.
It is now known that such a proof is not possible.
The reason why this is so is rather subtle:
If Euclidean geometry
(including the fifth postulate about parallel lines)
is at all consistent, then it can serve as a framework to
describe other surfaces besides a plane.
One such surface is the sphere...
The geometry of the surface of a sphere provides one example
where the first four axioms of Euclid are verified with
suitable redefinitions of the concepts involved
("points" are actually pairs of diametrically opposite locations
and "lines" are great circles). Yet, the
fifth postulate is not verified, as all "lines" intersect
(there is no such thing as two "parallel" great circles).
Therefore, the fifth postulate cannot
be a consequence of the other four.
Note that this conclusion can be reached without settling
the question of whether the Euclidean postulates
are consistent or not. We just note that if
they are consistent, then a consistent "model" can be
constructed (spherical geometry)
where the fifth one is false.
Hence, that fifth postulate is truly an independant
axiom which may be assumed to be true or false.
In examples of lesser historical significance,
similar arguments can be used to rule out some types
of proofs for a given statement.
For example, "Fermat's last theorem" can be shown to be false
within certain "models"
(involving beasts like p-adic integers).
This shows that it is a so-called global
statement whose proof must involve some peculiar
property of the rational integers besides
ordinary algebra, ordering
and divisibility by finitely many prime numbers.
The proof must involve something very specific to the integers,
like the validity of Fermat's own
infinite descent method...
