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MATHEMATICS PROOF

### Proof - the deductive method of mathematics

Induction versus Deduction

Arguments (or reasonings) divide into two classes: inductive arguments and deductive arguments. An inductive argument draws a general conclusion from observation of particular cases. A deductive argument draws a particular conclusion from general laws.

The premises of Inductive Arguments claim to provide incomplete or partial reasons in support of the conclusion. Arguments in the sciences are of this type. There the acceptance of the premises would make acceptance of the conclusion more reasonable than not. In contrast, the premises of Deductive Arguments claim to provide conclusive reasons for the conclusion. A deductive argument is characterized by the claim that its conclusion follows with strict necessity from the premises. A mathematics proof is a deductive argument.

Although induction and deduction are processes that proceed in mutually opposite directions, they are closely related. One could say, induction is the mother of deduction. Why? First, the aim of deduction is to establish a claim or conjecture. But coming up with a claim or a conjecture involves induction. Because, based on plausibility or partial evidence, one makes the claim or conjecture. This making is induction. Second, the construction of the proof itself involves induction. For when seeking the deductive argument one looks at examples, facts, and other proven results. These encourage one to choose a line of argument. It is as though one is observing many particular cases as in induction to bring out the hidden general law.

Mathematicians are proud that their deductive proofs are irrefutable. Assuming that the proof is correct, this is true. However, note that this is in spite of the mathematician. It is in the nature of the deductive method - from the general to the particular. When proceeding from the general to the particular, one often has more information than needed to arrive at the conclusion. Yet, in most first attempts at the proof of a specific statement, one is at a loss to produce that chain of information to produce the conclusion. But all the information is there, acting as conditions that allow only the sought-after conclusion.

A cautionary note to the novice: The induction discussed above is not the mathematics proof technique using the Principle of Mathematical Induction which is often referred to as Mathematical Induction. What we discussed above is the inductive method of science.

Deduction

An argument is deductively valid when the conclusion is correctly deduced from the premises, irrespective of whether the premises are true or false. How do we ensure that a conclusion is correctly deduced?

A conclusion is said to be correctly deduced if the argument correctly employs the appropriate inference rules of Propositional and Predicate Logic - two basic branches of Mathematical Logic. That is, the conclusion of the theorem being proved must be derived from hypotheses, axioms, definitions, and proven theorems using inference rules. In actual proofs presented by mathematicicans short-cuts are taken using already proven theorems, using multiple inference rules in one step without explicitly mentioning them individually, omitting "obvious" proof steps, and so on.

A proof in mathematics is then a deductively valid argument establishing a theorem. Thus, a proof is done in a meta language - here meta means 'beyond' as in metagalaxy - which combines the language one is working in, like English, with the inference rules of Mathematical Logic. The proof process may be compared to a game like chess. Just as the valid chess moves are governed by rules of the game, the valid steps in a proof are governed by the rules of inference.

In order to discuss Rules of Inference used in proofs, we need to have a rough idea of Propositional Logic and Predicate Logic. However, we will not treat Predicate Logic for two reasons: (1) We need to keep our discussion simple and (2) Propositional Logic contains enough inference rules to enable us to discuss proof methods.

Propositional Logic

G.R.T.

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