Euclid & Logical Thinking

< 28.11 Coordinate Geometry | Topic Index | 29.2 Logic &amp; Proof – 1 >

What is logical thinking?

In this book, we have mentioned in several places that math is a subject studied mainly to develop the skill of logical thinking. Let us understand the idea of logical thinking.

In a formal sense the development of geometrical theorems & properties of whole numbers is supposed to have begun with Thales around 600 BC. He is supposed to have tried to prove that the sum of the angles of a triangle is equal to 2 right angles and that the sides of 2 similar triangles are proportional. Pythagoras who is famous for his theorem lived around 400 BC. He is also credited with the discovery of irrational numbers.

By the time of Euclid, around 300 BC, many such theorems had been stated and proved. What Euclid did was to collate all discoveries in geometry and properties of numbers of the 3 centuries before him into a logical structure. He organized all geometrical statements (or propositions) in a sequential way that each proposition (called a theorem) was proved using theorems which had been proved earlier.

So, each theorem followed logically from the previous theorems. "Logically" means, if you accept the result of a particular theorem, then you have "no alternative other than" to accept the theorem which follows from it. Euclid also established the idea of "proof" of a theorem.

The Need for Axioms

This system raised the following question. If each theorem depended on the previous theorems, how was the first theorem proved? Euclid started with certain assumptions which he said were self-evident in that they were obvious and did not need any proof. His axioms were "self evident" as they were rooted in the physical reality which we experience all the time.

These assumptions were called axioms. The first theorems followed from the original set of axioms.

Euclid is also considered the father of the axiomatic method, where a few basic rules or axioms or assumptions are defined and then by the application of logic, various verifiable results are arrived at.

Axiomatic Method

This method of starting with certain axioms and then proving a series of theorems which follow logically either from the original assumptions or from the earlier theorems is called the "Axiomatic Method". He documented his entire method and the theorems in an encyclopedic work called "The Elements". The geometry was called plane geometry as it exclusively dealt with figures which could be drawn on a plane surface.

The Elements

Elements consisted of 13 books covering plane geometry, solid geometry, properties of numbers and ratios. Five of these books deal with plane geometry and start with 10 assumptions, 5 of them related to geometry (called as postulates) and 5 being general notions. He also provides 23 definitions & descriptions of geometrical terms.

His work was so comprehensive and clear that it became the standard textbook for studying plane geometry. Euclid's contribution to logical thinking is so unique, that it dominated it for almost 2000 years. Next to the Bible it is the book which was translated into the maximum number of languages.

Axiomatic Method in Other Disciplines

The Axiomatic Method has been used to develop several scientific disciplines. In the physical & biological sciences, the idea of logic had to be in consonance with the cause and effect of phenomena in the real world. For example, the logic of bodies moving in space was dictated by the effect of gravity. It has become one of the backbones of the scientific method.

It was also used in many other areas of knowledge like philosophy &amp; politics. The preamble to the American Declaration of Independence is a good example.

Axiomatic Method In Math

But the logic of math was not dependent on the realities of the physical world. It was dictated by the original axioms one had accepted and the application of logic to extract propositions which followed from the axioms. It became the predominant method or the "Gold Standard" by which ideas in math were developed.

Hence even though initial ideas in math were extracted from observations in the real world, the internal logic of math, took it to areas where they were no more related to the real world around us. This is the strength of math as a discipline for developing logical thinking, as well as its weaknesses when it comes to teaching & learning it in the classrooms.

This aspect of math is best illustrated by the history of the centuries-old confusion over Euclid's Fifth Postulate. We will discuss this in a subsequent chapter 29.5.

Let us now see two examples of proof of a theorem; one from arithmetic and the other from geometry.

< 28.11 Coordinate Geometry | Topic Index | 29.2 Logic &amp; Proof – 1 >