Transformation of Math - 2

Unique Nature of Math

At this point, we need to understand the difference between math and other sciences, which enables “seemingly impossible” questions to be asked.

Sciences like physics & chemistry develop by observing natural phenomena. Then they postulate theories to explain these phenomena. They then conduct experiments to check if their theories are valid. If the experimental results do not align with real observations, then the theory is reviewed.

In math, the results cannot be verified by comparing with reality. The results have to be verified by fellow mathematicians if the internal logic of the topic in question and that of math itself have been rigorously followed.

In that sense, every new idea extends the field of mathematics. It does not replace an older idea. In that sense no previous concept in math is ever proved wrong by a new concept. It just extends its scope. Math spreads like a spider’s web, with new strands attached to older strands as well as new points.

Transformation of Math

Introspections on such questions described above culminated in a different conception of math. A math where the primary focus was no longer on performing calculations or computing answers, but formulating and understanding abstract concepts and relationships. Mathematicians took the cue from Euclid and collectively evolved axioms of arithmetic. The fundamental laws of arithmetic & closure property are part of the axioms. These laws are applied with rigorous logic which again was dictated by these axioms.

We can say that the formulation of axioms of arithmetic was the fourth major change in math.

The rules of operations on integers could be derived by applying these axioms. Similarly, the rules of operations on any newly discovered number could also be discovered.

In fact, a new definition of a number emerged. A number was anything which conformed to the axioms of arithmetic!

The famous rule that the product of two negatives is positive is a result of the axioms. It is not possible to “understand” this rule through our daily experiences. They have to be “understood” only through axioms of arithmetic. All these changes happened in the last 4 centuries. The development of mathematics went along the path of consistency with its internal logic.

Math As Study of Patterns

Math started as the study of patterns in the environment. This new perspective of math gave a new meaning to patterns.

Many real-life events could be seen as consisting of patterns. Except that these patterns came in various forms – physical, visual, biological, digital & behavioural being some of them.

Almost any field can be seen through the perspective of patterns and analysed with tools of math.

This opened the flood gates to the mathematical study of entirely new subjects. Many of these new explorations even necessitated the invention of new tools in math.

The development of math was in turn developing new tools.

A case in point is the invention of calculus to study dynamic motion. Today we are seeing new math techniques being used to study genomes, economics and social relations.

Elementary & Higher Math

Until early 20th century, math was divided into Elementary & Higher Math.

Elementary mathematics was mathematics with addition, subtraction, multiplication, and division, that is, mathematics with four operations.

On the other hand, higher mathematics was mathematics using the limit operation introduced by Augustin Cauchy in addition to the four operations.

This involved grappling with the concept of infinity. Modern mathematics, which accelerated the construction of the modern world, was born entirely from the concept of infinity.

In the 21st century this distinction is also seen to be inadequate to describe the field of math.