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 documented this process in his Elements in 4th century BC. He showed the way of deriving the entire discipline of plane geometry starting with a few axioms whose truth was to be accepted without proof. He said the axioms were “self-evident”.

Axiomatization of geometry was the first major change which happened in math. But 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.

This was one of the first efforts to create an “internal logic” for the development of math. It enabled abstraction of relations within a set with the confidence that it would be valid for any number in the set.

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. 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 Physical Reality

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

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!

When mathematicians asked such questions, it 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.