Power and Limits of Math

Power of Mathematics 

There have been many instances of branches of math which were developed by following an internal logic, but were found to explain real life phenomena sometimes many decades later.

Many of the ideas in math were discovered much before any use or application was found for them. Imaginary numbers were discovered much before a discovery that they could be used in equations related to electric currents.

One hundred years after the invention of Boolean Algebra, it was found to be useful in designing electronic circuits and the flow of information inside computers.

This has led to a speculation that since mathematicians are looking for patterns, some of them also have a seventh sense in sensing future developments and inventing the mathematics necessary to work on these future developments.

It is almost as if math provides an internal telescope to understand future developments and invent the mathematics necessary to work on these future developments.

We are also witnessing the use of math to identify patterns in any topic and understand it by using tools of mathematics.

Mathematics has become a general tool for understanding almost any topic, including the universe in which we live.

The same idea was proposed by scientist Eugene Wigner in his seminal 1960 paper “"The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

One reason seems to be that the universe itself is mathematical in structure and evolution seems to have built in math structures in our brain. So by extending these mathematical ideas humans are able to “understand” the universe through the language of math.

Limits of Mathematics

At the same time mathematicians and philosophers understood a serious limitation of math.

In the light of rapid developments, some mathematicians tried to understand the very root of what mathematics is really about.

Since math is essentially logic, they asked if the entire field of mathematics can be logically derived from first principles.

The answer, from philosopher-mathematician Kurt Godel, was a surprising no.

He proved that any field of mathematics has to be based on certain assumptions which themselves cannot be proved. They have to be just accepted. To that extent, math was subjective.

Godel showed the difference between Truth & Proof. Some truths cannot be proved. They have to be accepted.

Euclid seems to have anticipated all these developments 2300 years back, by defining a set of axioms which have to be accepted without any proof.

In a way math has turned a full circle!