Text. 6. Mathematical Proof
A proof is a demonstration that some statement is true. Maths involves proofs and it is even doubted by some people whether “proof’ in the precise and rigorous sense which the ancient Greek mathematicians gave to this word, is to be found outside maths. We may say that this sense did not change because what constituted a proof for Euclid is still a proof for us. It is to the Greeks that modem mathematicians turn again for models of proof. The Greeks were the first to apply the deductive procedures developed by the Greek philosophers in maths. They are credited with the use of deductive methods of proof in geometry instead of intuition, experiment and trial-and-error methods of the Egyptians. Philosophers and mathematicians do not reason and prove as do scientists on the basis of personally conducted experiments. Rather their reasoning centres about abstract concepts and broad generalization. Deduction as a method of obtaining conclusion has many advantages over reasoning by induction and analogy. Some historians claim that it was the discovery of the incommensurable line segments that forced the Pythagoreans to accept the axiomatic and synthetic approach in math proofs (i.e., an approach without using numbers) and led to the method of deriving theorems from axioms. The Greeks insisted that all math conclusions should be established by deductive reasoning only. Math proof, thus, demands a specific kind of reasoning. In a formal math proof the mathematician cannot rely on his intuition, insight and imagination. He must reason logically and start with (1) the definitions of basic concepts for the theory involved, (2) axioms (or postulates) and (3) deduce a conclusion without making further assumptions. By analysis of the mechanism and structure of proofs we can see that the main feature of formal math proofs is that every statement in the proof must be justified by referring to (a) definition; (b) axioms (or postulate); (c) chain substitution; (d) the theorem already proved. An important property of the equality is that of substitution, e.g., if a=b and b=c, then a=c (a,b,c are natural numbers). We can express this in words by saying that “natural numbers equal to the same natural number are equal to each other” (axiom). (1893) Vocabulary
1. Answer the following questions: 1) What is a proof in mathematics and what does it involve? 2) How did ancient Greek regard the problem of proof? 3) What procedures did they apply? 4) What are they credited with? 5) What does their reasoning centre around? 6) What forced the Pythagoreans to accept the axiomatic and synthetic approach in proofs? 7) What kind of reasoning does math proof demand?
2. Find English equivalents for the following Russian word combinations: важное свойство равенства, главная особенность доказательства, ссылаться на определение, доказать теорему, вместо (взамен) интуиции, несоизмеримые отрезки, принять подход к доказательству, получить вывод (заключение), проводить опыты, приписывать кому-либо использование методов, широкое обобщение, заставить принять метод, обращаться за моделями доказательства, полагаться на интуицию и воображение, определение основных понятий. 3. Translate into English. 1) Основная черта математического доказательства должна подтверждаться ссылками на а) определения, b) аксиомы, c) цепочку замен и на уже доказанные теоремы. 2) Важное свойство равенства – это замена (подстановка). 3) Мы можем выразить это следующими словами. 4) Дедукция – один из методов получения заключения. 5) Именно открытие несоизмеримых линейных отрезков заставило пифагорийцев принять новый подход к математическому доказательству. 6) То, что было доказательством для Евклида, остается им и для нас.
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