Proof
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[edit] Introduction
In plain English, we say that a thing has been proved when the weight of evidence is such that it would be foolish to doubt it. However, the word proof has technical meanings in logic and the philosophy of science: it is this specialized meaning that we shall discuss in this article.
[edit] Proof in logic and mathematics
In logic and math, proof refers to the deduction of a logical conclusion from a given set of premises. For example, given the premises: "I am an astronaut" and "all astronauts have red hair", I can prove, using the rules of logic, that I have red hair.
Note that this conclusion is only proven in relation to the premises: I do not in fact have red hair. However, the statement: "If I am an astronaut, and if all astronauts have red hair, then I have red hair", is undoubtedly true. It is true whether or not I am an astronaut, whether or not I have red hair, whether or not all astronauts have red hair, whether or not there are such people as astronauts, whether or not I have hair ... it is one of the eternal verities of the universe, precisely because it tells us nothing whatsoever about the universe.
Such exercises may, from this example, seem fairly pointless: using strict logic, we can prove nothing that is not implicit in the premises. However, it becomes useful when the implication is subtle. For example, Newton's laws of motion and gravity imply that planets should orbit in ellipses with the Sun at one focus of the ellipse: this is logically implicit in his laws, but it in not obvious, and we may justifiably feel that we have learned something by proving that the conclusion follows from the premises.
[edit] Proof in science
Philosophers of science tend to reserve the term "proof" for the logical proofs described in the previous section, denying that it is possible to definitively prove any proposition about the state of the world. Instead, they would say that it is only possible to prove propositions such as our example given above: "If I am an astronaut, and if all astronauts have red hair, then I have red hair": propositions which, as a very consequence of being true by force of pure logic, convey no information whatever about the actual state of affairs.
According to this view, even such a seemingly definite proposition about the world as: "I have two legs" cannot be held to be proved absolutely, despite the weight of evidence substantiating it. I might in principle be being deceived by a "Cartesian demon" as described by the philosopher René Descartes:
- I will suppose ... some malicious demon of the utmost power and cunning has employed all his energies in order to deceive me. I shall think that the sky, the air, the earth, colours, shapes, sounds and all external things are merely the delusions of dreams which he has devised to ensnare my judgement.
I might be a one legged man (or a ten-legged sentient lobster) imprisoned in a virtual reality as depicted in the movie The Matrix, being fed artificial data suggesting that I have two legs. I might be a one-legged inmate of a mental hospital with the persistent delusion that I am a two-legged man who is not a mental patient: after all, since there are people in asylums who firmly believe themselves to be the Second Coming of Jesus, mere delusions of bipedalism seem mundane by comparison.
And if all this seems far-fetched, we may note that it is still all possible; and I might, conceivably, at some future date, acquire data convincing me that I do not have two legs.
According to the philosophers of science, then, all I can say is that the data now in my possession are all consistent with the hypothesis that I have two legs, and that I should therefore adopt this as a working theory until such time as I am presented with data to the contrary. But I cannot claim to have proof that I have two legs.
This is all very well for philosophers, but it leads people to make statements along the lines of: "There is no such thing as proof in science". Now, according to the strict philosophical meaning of the word "proof" described above, this is true. But it is so misleading that it would be better if people didn't say it.
In the first place, it is not true in normal English as spoken by non-philosophers. In plain English, we often say that propositions other than purely logical propositions have been proved: we say this when they have been confirmed by such a weight of evidence that it would be foolish to doubt them. And in science, there are many such propositions.
In the second place, it unreasonably singles out science. As we have seen, the philosophical doubt applies to all propositions about the state of the world, as in our example: "I have two legs". There is no reason why it especially applies to that body of knowledge that we call science: it does not.
It would be better, then, to rewrite the statement that "there is no such thing as proof in science" along the following lines:
- According to a definition of "proof" popular amongst philosophers but not employed by anyone else, there is no such thing as "proof" in science, in the same way and for the same reason that there is no "proof" that I have two legs.
This, we feel, would save a great deal of unnecessary confusion.
Still more must we condemn the tendency to describe the lack of "proof" in science by statements such as "all science is tentative". It is true that in principle we can imagine data that would falsify any given scientific proposition, or, indeed, the proposition that I have two legs. But that does not justify the use of the word "tentative", which in ordinary usage implies a high degree of uncertainty and diffidence about a proposition. It does not accurately describe a theoretical anxiety that perhaps some malicious demon is using magic to deceive us; and while some concepts in science are genuinely tentative, others are so thoroughly proved (in the normal sense of the word) that we should have to conjure up some such far-fetched hypothesis in order to call them into question.
