The burden of proof	Proof can be elusive as scientists, politicans and the public debate "the facts" of any given situation. Examples include the hunt for weapons of mass destruction in Iraq, the existence (or not) of unidentified flying objects, and the authenticity of religious artifacts such as the shroud of Turin.
Even after all the facts and theorems are in, truth can be a matter of perception
Howard Burton
November 1, 2004

Two years ago, under pressure to join the growing U.S.-led invasion of Iraq, then prime minister Jean Chrétien insisted that the UN weapons inspectors needed more time to find evidence of weapons of mass destruction.

When asked exactly what sort of evidence he would deem sufficient for military intervention, Chrétien famously replied:

"A proof is a proof. What kind of a proof? It's a proof. A proof is a proof. And when you have a good proof, it's because it's proven."

This is the sort of answer that is bound to give mathematicians and philosophers headaches -- not to mention the rest of us. On the surface, it seems like gibberish, but aside from the syntactical confusion and tautological balderdash lies a real issue: just what is a proof anyway?

At first glance, this hardly seems like a profound question: proofs are methods of determining whether something is true. One starts with some accepted premises, axioms or self-evident truths, and then by a series of deductive steps, each of which can be independently demonstrated to be valid, one arrives at a conclusion that is necessarily true. Hence the conclusion is deemed to have been proven.

But dig a little deeper and doubts creep in. What if a "proof" has a mistake in it that nobody catches? Is it still a proof?

What if a theorem is proved using a computer to grind out every possible instance, but no one human being is capable of checking all of the computer's work? Is the statement really proven?

How much is our concept of proof subtly tied to our belief in sociology and faith in the necessary due diligence of our fellow humans?

Mathematicians have been developing rules for proving statements for thousands of years, ever since the Pythagoreans developed the reductio ad absurdum (doubtless they called it something Greek instead): the system whereby one proves a statement by assuming its opposite and deducing a contradiction. A classic example is Euclid's proof that there are infinitely many prime numbers.

To prove that the number of prime numbers is infinite, Euclid assumed the opposite -- namely, that there is a largest prime number; call it n. Now consider the number -- call it z -- created by multiplying all the prime numbers together and then adding 1 (i.e. z = {2 x 3 x 5 x 7 x 11 x 13 x...x n} + 1). Clearly z should be composite (meaning that it is divisible, without remainder, by prime numbers), otherwise it would be a bigger prime number than n, which immediately contradicts our assumption.

But equally clearly, z can't be composite because there is no way to divide z by prime numbers up to n without a remainder (because of the 1 that we added), so any number that does divide z must be greater than n -- an impossibility since n is the largest prime number.

So we are faced with a contradiction: z can't be prime or composite and all numbers must be one or the other, and we must be forced to discard our initial assumption, namely that there is, indeed, a largest prime number. Therefore the number of prime numbers must be infinite.

But reductio ad absurdum arguments aren't the only way to prove things. Despite a general preference for algebraic-style proofs among most mathematicians, the issue of what constitutes a genuine rigorous proof remains highly contested.

Many mathematicians and philosophers believe that pictures and diagrams, while occasionally obscuring the issue, can sometimes be instrumental in the proof process.

Meanwhile, the notion of what is provable became considerably more complicated with the famous incompleteness theorems of the logician Kurt Gödel in the early 20th century.

Among other things, Gödel showed that there will always be true but unprovable statements in any formal system, demonstrating that what is true and what is provable can never be identical after all.

Natural scientists, of course, with their reliance on empirically verifiable facts of the physical world, have understood this all along. It may well be "true" in some sense that green-haired rhinoceroses exist and have green-haired baby rhinoceroses independent of whether anyone has ever actually come across a green-haired rhinoceros. In short, one can only prove the existence of such beasts by observing them.

After all, in the end, a proof is just a fancy way of convincing oneself of the truth of a particular claim. However we try to eliminate subjectivity from the discussion, there will always be people who find certain proofs more compelling then others.

Despite his rather woolly rhetoric, Chrétien's "I'll believe it when I see it" sentiment on Iraq's weapons now seems considerably more persuasive than it might have appeared back in 2002.