Transformation of Math - 1

We have talked in brief in earlier chapters that math as a discipline itself has been changing. Let us look at this transformation in some detail.

Math developed from observing patterns in us & our surroundings. Early numbers, shapes, operations on them and their relations were firmly rooted in our observations of the physical reality.

Axiomatization of Geometry

The study of shapes led to the discipline of geometry and the development of logical thinking.

Euclid first accepted a few axioms & postulates about shapes which could be drawn on a plane surface as true without any proof. He said these axioms & postulates were “self-evident”. They included, among others, the idea of a point & a line.

Starting from these axioms & postulates, he derived, using just logical arguments & results proved earlier, a series of "theorems or geometrical truths" about the properties of shapes which could be drawn on a plane surface. These truths could be verified by actual construction of these shapes & measuring their properties.

These theorems constituted the entire discipline of plane geometry. This process of starting a discipline with a few unproved axioms and proving other results by use of logical arguments was called Axiomatization. Euclid documented this process in his book 'Elements' in 4th century BC.

Axiomatization of geometry was the first major change which happened in math. Its true impact was understood only in the 18th century when mathematicians tried to understand the “true nature” of arithmetic.

The Number System

Numbers easily modelled our social & commercial transactions. We can say that the invention of the Decimal Place Value system was the second major change in mathematics. This was in the early centuries of the CE.

The Closure Property

When different numbers were invented, mathematicians also imposed on the subject, a limitation that all the numbers invented thus far should form a closed group with respect to arithmetic operations. This was called the Closure Property.

It enabled abstraction of relations within a set with the confidence that it would be valid for any number in the set.

The set of Natural numbers was 1, 2, 3 etc. Additions with natural numbers yielded natural numbers. But subtraction like 3 - 3 or 3 - 5 yielded numbers that were not natural numbers. Hence first 0 was included with natural numbers and called whole numbers. Then negative numbers were included with whole numbers and called Integers.

Now the set of integers yielded only integers under addition & subtraction. Hence generalisations could be thought about addition & subtraction operations with the set of integers. There was no fear that other kinds of numbers would emerge.

This was one of the first efforts to create an “internal logic” for the development of math.

This property led to the invention of new kinds of numbers. This process ultimately culminated in the invention of the set of complex numbers.

Textual Math to Symbolic Math

Until the invention of the printing press, math documents & books were copied by hand. To avoid any confusion, unfamiliar symbols were avoided and all the math was written out in text. The only symbols which were used were numerals & the four operating symbols.

This can be called the era of “rhetoric” math.

Mathematicians found it convenient to use more & more symbols to convey abstract ideas. As interactions among mathematicians increased, more symbols were invented to represent ideas.

The invention of the printing press enabled multiple copies of a book to be printed and circulated. It made math knowledge easily available to common people. Hence symbols became more commonly known & got standardized.

Symbols helped concise expressions of ideas. They also abstracted events & reality into ideas.

The statement 2 + 3 = 5 holds within it a million different real life situations expressed in concise form. It could be a story of a person have 2 fruits and then buying 3 fruits. It could also be 2 families comparing the number of their children, saying that we have 3 children more than your 2 children!

Thus “rhetoric” math slowly became “symbolic” math. This happened in the 16th century.

Symbolic math can be called the third major change in math.

Freedom From the Constraints of Physical Reality

Symbolic math also encouraged mathematicians to ask questions which could never be asked if math was only related to real life.

Development of Negative & Imaginary Numbers

If 5 – 2 indicates that 2 cows have been sold out of a stock of 5 cows, we can never ask if 5 cows can be sold out of a stock of 2 cows!

But if the problem is just stated, without any context as 5 – 2 =3, some maverick mathematician could ask “what then is 2 – 5?”

Obviously, someone asked this question and mathematicians invented negative numbers!

Another related question was “we know that 2X3=6. What then is -2 X -6?”

Yet another such question was “If √4 = ± 2, what then is √(-4) ?”. This led to the invention of imaginary numbers!

But negative & imaginary numbers do not have any relation to physical reality!

Another mathematician asked the question whether logical relations can be mathematized? The result was Boolean Algebra which is the mathematics of logical values (True & False).

Though Euclid's book, Elements, was widely studied, there was always a doubt about the "Fifth Postulate".

Development of Non-Euclidean Geometries

The questions about this postulate were settled in the 19th century when it was shown that the Fifth Postulate was true only on a plane.

Assuming that the Fifth Postulate was not true led to the invention of Spherical & Hyperbolic geometries which are called non-Euclidean geometries.

Freedom to Ask Any Question

When mathematicians asked such questions, they led to a lot of discussions & introspections. Most other mathematicians never said that these questions were meaningless or irrelevant. They were only interested in the consequences of following these questions.

So the question “what happens if?” is an important question for mathematicians which laid the seeds which transformed math.