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

statement	утверждение
turn for	обращаться за
instead of	вместо
to reason	рассуждать
to obtain =	to get
to conclude - conclusion
advantage over	преимущество над
approach	подход к проблеме
to derive from	выводить из
to insist on	настаивать на
to establish	устанавливать
to rely on	полагаться на
to deduce	выводить, делать вывод
feature	особенность, черта
to justify	оправдывать, подтверждать

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.

Unit 2. Geometry

Text. 7. Geometry

Geometry (from the Greek geometria, the Earth's measure) has its roots in the ancient world, where people used basic techniques to solve everyday problems involving measurement and spatial relationships. The Indus Valley Civilisation, for example, had an advanced level of geometrical knowledge- they had weights in definite geometrical shapes and they made carvings concentric and intersecting circles and triangles. Gradually, over the centuries, geometrical concepts became more generalised and people began to use geometry to solve more difficult, abstract problems.

However, even though people in those times knew that certain relationships existed between things, they did not have a scientific means of proving how or why. That changed during the Classical Period of the ancient Greek civilisation (490 ВС-323 BC). Because the ancient Greeks were interested in philosophy and wanted to understand the world around them, they developed a system of logical thinking (or deduction) to help them discover the truth. This methodology resulted in the discovery of many important geometrical theorems and principles and in the proving of other geometrical principles that had been known by earlier civilisations. For example, the Greek mathematician Pythagoras was the first person that we know of to have proved the theorem

Some of the most significant Greek contributions occurred later, during the Hellenistic Period (323 BC-31 BC), Euclid, a Greek living in I: Elements, in which, among other things, he defined basic geometrical terms and stated five basic axioms which could be deduced by logical reasoning. These axioms or postulates were: Two points determine a straight line. A line segment extended infinitely in both directions produces a straight line. A circle is determined by a centre and distance. All right angles are equal to one another. If a straight line intersecting two straight lines forms interior angles on the same side and those angles combined are less than 180 degrees, the two straight lines if continued, will intersect each other on that side. This is also referred to as the parallel postulate. The type of geometry based on his ideas is called Euclidean geometry, a type that we still know, use and study today. With the decline of Greek civilisation, there was little interest in geometry until the 7^th century AD, when Islamic mathematicians were active in the field. Ibrahim ibn Sinan and Abu Sahl al-Quhi continued the work of the Greeks, while others used geometry to solve problems in other fields, such as optics, astronomy, timekeeping and map-making. Omar Khayyam's comments on problems in Euclid's work eventually led to the development of non-Euclidean geometry in the 19 ^th century.

During the 17^th and 18^th centuries, Europeans once again began to take an interest in geometry. They studied Greek and Islamic texts which had been forgotten about, and this led to important developments. Rene Descartes and Pierre de Fermat, each working alone, created analytic geometry, which made it possible to measure curved lines. Girard Desargues created projective geometry, a system used by artists to plan the perspective of a painting. In the 19^th century, Carl Friedrich Gauss, Janos Bolyai and Nikolai Ivanovich Lobachevsky, each working alone, created non-Euclidean geometry. Their work influenced later researchers, including Albert Einstein.

Vocabulary

measurement	мера, измерение
spatial	пространственный
relationship	отношение, связь
advanced	передовой
weight	вес
carving	резьба
gradually	постепенно
generalise	обобщать
even though	даже если
means	способ
result in	приводить к
occur = happen = take place
determine	определять
straight	прямой
extend	простирать(ся), тянуться

1. Ask each other questions about the text. Work in pairs.

2. Answer the questions:

1) What is geometry, how did it come into being?

2) How did ancient people understand it?

3) What is Euclidean geometry?

4) Who were the followers of Euclidean geometry?

5) How was analytic geometry created?

6) What other geometries do you know?

3. Use the words in the box in the sentences below:

relationships	timekeeping
thinking	take an interest in
resulted	created
advanced

1) Certain ___ exist between things.

2) They developed a system of logical ___.

3) This methodology ___ in the discovery of theorems.

4) Islamic mathematicians used geometry to solve problems in other fields such as optics, ___ and map-making.

5) Europeans began to ___ geometry.

6) N.I. Lobachevsky ___ non-Euclidean geometry.

7) Some civilizations had an ___ level of geometrical knowledge.

4. Translate from Russian into English.

1) Он первым доказал эту теорему.

2) У них не было научных способов доказательств.

3) Отрезок линии, простирающийся до бесконечности в обоих направлениях, образует прямую линию.

4) Упадок греческой цивилизации привел к потере интереса к геометрии.

5) С течением веков геометрические понятия стали все более обобщаться и геометрию использовали для решения более трудных, абстрактных задач.