Tag Archive: proof

Nothing can be proved

The critical attitude, the tradition of free discussion of theories with the aim of discovering their weak spots so that they may be improved upon, is the attitude of reasonableness, of rationality. It makes far-reaching use of both verbal argu­ment and observation—of observation in the interest of argument, however. The Greeks’ discovery of the critical method gave rise at first to the mistaken hope that it would lead to the solution of all the great old problems; that it would establish certainty; that it would help to prove our theories, to justify them. But this hope was a residue of the dogmatic way of thinking; in fact nothing can be justified or proved (outside of mathematics and logic). The demand for rational proofs in science indicates a failure to keep distinct the broad realm of rationality and the narrow realm of rational certainty: it is an untenable, an unreasonable demand. [67]

The relativity of proof

Every proof must proceed from premises; the proof as such, that is to say, the derivation from the premises, can there­fore never finally settle the truth of any conclusion, but only show that the conclusion must be true provided the prem­ises are true. [ch. 11, 260]

Any objective question is subject to proof

Questions of ultimate ends are not amenable to direct proof. Whatever can be proved to be good, must be so by being shown to be a means to something admitted to be good without proof. The medical art is proved to be good by its conducing to health; but how is it possible to prove that health is good? … There is a larger meaning of the word proof, in which this question is as amenable to it as any other of the disputed questions of philosophy. The subject is within the cognisance of the rational faculty; and neither does that faculty deal with it solely in the way of intuition. Considerations may be presented capable of determining the intellect either to give or withhold its assent to the doctrine; and this is equivalent to proof. [ch. I, 157-8]

Critical control over our scientific debates

Deductive arguments force us to choose betweeen the truth of their conclusions and the falsity of (one or more) of their premises. Inductive arguments do not. This, in and of itself, does not show that anything is true or false. But if an argument is deductively valid, then we simply cannot, without contradicting ourselves, deny its conclusion unless we also deny (one or more of) its premises. In this way, deductive arguments enable us to exercise critical control over our scientific debates. [105]

Proof theory is computer science

So, a computation or a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties. It is our window on the abstract. It works because we use such entities only in situations where we have good explanations saying that the relevant physical variables in those objects do indeed instantiate those abstract properties.

Consequently, the reliability of our knowledge of mathematics remains for ever subsidiary to that of our knowledge of physical reality. Every mathematical proof depends absolutely for its validity on our being right about the rules that govern the behaviour of some physical objects, like computers, or ink and paper, or brains. So, contrary to what Hilbert thought, and contrary to what most mathematicians since antiquity have believed and believe to this day, proof theory can never be made into a branch of mathematics. Proof theory is a science: specifically, it is computer science.

The whole motivation for seeking a perfectly secure foundation for mathematics was mistaken. It was a form of justifi­cationism. Mathematics is characterized by its use of proofs in the same way that science is characterized by its use of experimental testing; in neither case is that the object of the exercise. The object of mathematics is to understand – to explain – abstract entities. Proof is primarily a means of ruling out false explanations; and sometimes it also provides mathematical truths that need to be explained. But, like all fields in which progress is possible, mathematics seeks not random truths but good explanations. [188-9]

The subject-matter of mathematics

Abstract entities that are complex and autonomous exist objectively and are part of the fabric of reality. There exist logically necessary truths about these entities, and these comprise the subject-matter of mathematics. However, such truths cannot be known with certainty. Proofs do not confer certainty upon their conclusions. The validity of a particular form of proof depends on the truth of our theories of the behaviour of the objects with which we perform the proof. Therefore mathematical knowledge is inherently derivative, depending entirely on our knowledge of physics. [256-7]

No, not even maths is certain

Thanks to Gödel, we know that there will never be a fixed method of determining whether a mathematical proposition is true, any more than there is a fixed way of determining whether a scientific theory is true. Nor will there ever be a fixed way of generating new mathematical knowledge. Therefore progress in mathematics will always depend on the exer­cise of creativity. It will always be possible, and necessary, for mathematicians to invent new types of proof. They will validate them by new arguments and by new modes of explanation depending on their ever improving understanding of the abstract entities involved. Gödel’s own theorems were a case in point: to prove them, he had to invent a new method of proof. I said the method was based on the ‘diagonal argument’, but Gödel extended that argument in a new way. Nothing had ever been proved in this way before; no rules of inference laid down by someone who had never seen Gödel’s method could possibly have been prescient enough to designate it as valid. Yet it is self-evidently valid. Where did this self-evidentness come from? It came from Gödel’s understanding of the nature of proof. Gödel’s proofs are as compelling as any in mathematics, but only if one first understands the explanation that accompanies them.

So explanation does, after all, play the same paramount role in pure mathematics as it does in science. Explaining and understanding the world – the physical world and the world of mathematical abstractions – is in both cases the object of the exercise. Proof and observation are merely means by which we check our explanations. [235-6]

Science: more or less likely

It is not unscientific to make a guess, although many people who are not in science think it is. Some years ago I had a conversation with a layman about flying saucers—because I am scientific I know all about flying saucers! I said “I don’t think there are flying saucers”. So my antagonist said, “Is it impossible that there are flying saucers? Can you prove that it’s impossible?” “No”, I said, “I can’t prove it’s impossible. It’s just very unlikely”. At that he said, “You are very un­scientific. If you can’t prove it impossible then how can you say that it’s unlikely?” But that is the way that is scientific. It is scientific only to say what is more likely and what less likely, and not to be proving all the time the possible and im­possible. [165-6]

The power of logic

Most people think that the purpose of an argument is to justify its conclusion — and to thereby establish its certainty — and that the problem with inductive arguments is that they fail to establish their conclusions with objective certainty since their conclusions may be false even if all of their premises are true. But this entirely confuses the issue, and it has even enabled inductivists to take the high ground in the debate, arguing that objective certainty is an impossible dream, and that inductive arguments are not, as a consequence, at fault for failing to achieve it.

But if the uncertainty of their conclusions were the problem with inductive arguments, then we would also have a similar problem with deductive arguments. For the premises of a deductive argument may be false. And the conclusion of a deductive argument may be false as well.

Contrary to what most people think, logical arguments cannot establish the truth, let alone the certainty, of their con­clusions. And so contrary to what most people think, the problem with inductive arguments has nothing to do with the uncertainty of their conclusions.

The best that a logical argument can do is test the truth of a statement. But it can do this only by showing that its falsity is inconsistent with the truth of other statements that can only be tested and never proved. Our so-called ‘proof’ methods are really techniques for testing consistency. And the demonstration that a ‘conclusion’ follows from a ‘premise’ shows only that the falsity of the ‘conclusion’ is inconsistent with the truth of the ‘premise.’

That is all that is involved.

But so long as we regard contradictions as unacceptable, it is really quite a lot.

The inconsistency that marks a valid deductive argument — the inconsistency, that is, between the truth of is premises and the falsity of its conclusion — cannot force us to accept the truth of any belief. But it can force us, if we want to avoid contradicting ourselves, to reexamine our beliefs, and to choose between the truth of some beliefs and the falsity of others — because the falsity of the conclusion of a valid argument is inconsistent with the truth of its premises. …

And this is just another way of saying that what we call a proof actually presents us with the choice between accepting its conclusion and rejecting its premises. …

We construct logical arguments in order to persuade others of our beliefs. But the best we can do is to clarify a choice that they have to make. Inductive arguments, however, cannot even do this. [86-7]