What is proof?
Among other things, mathematics deals with numbers, symbols, operations, spatial properties and problem solving. Since earliest times the subject has been developed by mathematicians having an intuitive feeling that some proposition is true and then demonstrating the truth, or otherwise, of this proposition by logical argument. When studying mathematics at secondary school, pupils are mainly concerned with learning techniques but, in reality, pure mathematics is mainly concerned with proof. Theorems and proofs did not occur to their originators in the ‘semi-polished’ form which you find in textbooks. The mathematician has an intuitive, instinctive feeling that some proposition may be true. The essence of proof is to establish whether the result is, indeed, true or whether he has been deceived by such a feeling. In this book we discuss various methods of proof which are commonly used in mathematics. Fundamentally, mathematical proof is based on logical argument, that is, to establish from a hypothesis ‘p is true’ a conclusion ‘q is true’.
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