Proof

Proof, in its general sense, is evidence that is sufficient to justify belief in a claim. With the exception of purely logical proofs, such as those in mathematics, proof is largely subjective, requiring that it meet the criteria of the person(s) weighing evidence. Therefore, what constitutes sufficient evidence for one person may not be the same for others.

Common usage
Proof is commonly taken to mean that the claim is absolutely true. For general and unremarkable claims this can be considered correct, although the philosophy of pyrrhonism states that nothing can ever be proven in this way &mdash; even in the unremarkable daily lives of people. In the "real world", proof might consist of a printed source stating what you claim it states (e.g., "The Times really did say that the earth is just 6000 years old, and here is the text as proof") or a demonstration (e.g., "my new car can do 0 to 60 in 6 seconds, watch!").

Scientific proof
It is often said that proof is for alcohol and mathematics, as these are really the only areas where proof has any real meaning. "Proof" is something that the opponents of science are always clamouring for, yet is not actually something that science suggests it will give &mdash; specifically in the common sense definition of "proof" that suggests that a claim has been proven 100%.

Surprisingly to some, science does not deal in proof, in spite of the word being associated with science a whole lot more than perhaps it should be. Scientists will throw around phrases like "we've just proven our theory" when they should know better, but this can be excusable human nature. With some theories it's very difficult not to think of them as proven (e.g., Newton's laws of motion, for instance, which always have and always will describe the behavior of objects within the limits of those theories.) Science as a method, however, deals not in proof but in evidence, and perhaps disproof when the evidence contradicts a hypothesis.

For example, finding a fingerprint at the scene of a crime may be construed by most people as "proof" that the owner of the fingerprint was, at some time, at that location. Science would say that the fingerprint is "evidence" that somehow a means of leaving that fingerprint was at that location. More evidence can be collected that shows a person was there at the right time, or perhaps some evidence is shown that they weren't there. It is up to someone working in the world of science to put together a hypothesis which is consistent with this evidence when taken together. A scientist hasn't strictly proven anything by putting together this hypothesis, but they have constructed a realistic scenario based on overwhelming evidence. Anti-evolutionists often ask for proof of evolution, and act triumphant when none is forthcoming. This is perfectly reasonable for science to do — after all, no scientist can even be certain of what evidence will come up in the future that may contradict the current theory. Yet those of the anti-science movement who cling to this detail usually fail to hear the words put in place of "proof" — overwhelming evidence. In a way they know they are being disingenuous, for they understand — with some exceptions — the way the legal system uses the word proof in that overwhelming evidence is considered to prove a case, yet reject that same usage for when science talks of the overwhelming evidence for evolution.

Alcohol
In the USA measurement of alcohol, "proof spirit" is an equal mixture of alcohol and water. The alcoholic content of a spirit is sometimes expressed as a percentage of proof. A whiskey that is 80 proof is 40% alcohol by volume, and a rum of 120 proof is 60% ABV. The term actually comes from "proving a spirit," i.e. proving it had not been watered down excessively. In the 18th century, sailors tested the strength of their rum rations by dousing a small amount of gunpowder in a small amount of rum. If the powder would still ignite, this demonstrated that the liquor was proof strength or above.

Until 1980, the UK had a different scale whereby 100 degrees proof was equal to 4/7 alcohol by volume. Thus 100% alcohol was 175% proof. Nowadays a straight percentage is used.

Maths
In math or logic, a mathematical/logical proof is a statement based on a set of rules which necessarily leads inevitably to a particular conclusion. This is seen most simply in a list of mathematical equations which are rearranged sequentially to show that one thing leads to another. Providing all the mathematical rearrangements are correct, these proofs are considered quite correct. There are many ways of going about this, directly, inductively or by showing contradictions.

Legal
Proof also has a legal meaning which is much fuzzier than the mathematical or scientific meanings.

Political
Former Prime Minister of Canada, Jean Chrétien, "explained" the concept of proof as:

Music
"Proof" is the fourth track on Paul Simon's 1990 album The Rhythm Of The Saints: