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.

Take for instance the classical example of imaginary & complex numbers. They were related in the 16th century just to contribute to the fundamental theorem of algebra by allowing every polynomial equation to have a solution. They did not show any empirical interpretation and did not express any magnitude.

However, in the later centuries, they came to be essential for various theories in physics and technology, such as fluid dynamics, quantum mechanics, relativity, electromagnetism, control theory, signal analysis and many others.

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.

AI software have analysed games like Go & Chess and learnt to play by playing against themselves! Today chess enthusiasts are sad that the game is changing from a creative one to that of memorizing many moves from past games.

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

Mathematics has perpetually fed on the problems of sciences, has created new concepts and theories just for solving these problems, and thus has contributed to the advances of those sciences; mathematics has also developed itself and strengthened its ability to solve future problems.

Understanding the brain is the ultimate challenge for humans. Brain scientists are trying to solve one of neuroscience’s greatest mysteries: where does the brain ‘store’ memories?

They are also realising that the ordinary mathematics of networks cannot explain the brain. Hence they are trying more abstract math like algebraic topology to understand the structure and working of the brain.

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.

In his famous article On Formally Undecidable Propositions of ‘Principia Mathematica’ and Related Systems, 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.

This completely demolished attempts by mathematician David Hilbert's effort to prove that a complete and consistent axiomatic theory can be constructed to explain the entire field of mathematics.

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!