Mathematics, discovered or invented?

In nature, everything exists in proper weight and measure. To know something then, is to measure it. Any insight devoid of numbers, equations and/or formal proof is merely conjecture. The systematic study of natural phenomena is incomplete without mathematics. In fact, there is no science without mathematics. Is mathematical truth a fundamental constituent of the reality, or is it merely a tool that we have invented to understand how the world works? This question has confounded many modern logicians, thinkers, scientists and the philosophers of science. In this short essay, I will expound the subject in detail with my own opinion on the matter.

Some might argue that the mathematical truths are embedded in the structure of the universe, waiting to be discovered. The order underpinning the clockwork of the world is the same reflected in the mathematical structures. For example, let us consider the Pythagorean Theorem. It is a mathematical statement that is true irrespective of whether we know it or not. Furthermore, it applies to every 3-sided right-triangular shape. The school of thought arising from this line of reasoning is called the Platonist position. It considers mathematics as a fundamental constituent of the reality. If the universe were to disappear tomorrow, the mathematical truths would continue to exist in that ideal, abstract realm of Platonist space. The natural world, then, is merely a subset of the Platonist hyperspace. It is from this hyperspace that some truths or descriptions may apply to and explain our reality.   

The non-Platonist position maintains that mathematics is a product of human mind and its power is limited to the approximate descriptions of the structures existing in nature. The universe, from this perspective, is not inherently composed of mathematical truths; rather, mathematics exists as a way to develop approximate models of reality. Unlike the mathematical Platonism, in a non-Platonists' world, the mathematical truths vanish with the non-existence of the universe like every other contrivance of human thought does.

Our mathematical descriptions of natural phenomena are subject to change and revision. This implies that our mathematical understanding of the natural world is ad hoc, provisional and borne out of necessity. Moreover, in principle, it is possible to create two different sets of self-consistent logical rules to explain the same phenomenon or in a general sense arrive at the same conclusion. This argument follows from the perspective of anti-realism, which asserts that our mathematical theories of nature invoke entities that do not exist in the literal sense of the word. A number of completely disjoint formalisms of different, ideal mathematical entities can be used to describe a particular physical phenomenon with more or less the same predictive power. The lack of robustness is another important issue which is evident in a variety of theories in modern physics. For example, consider the Standard Model of physics, the model that unifies three of the fundamental forces, has about 18 free parameters that are not predicted by the theory. That is quite cumbersome! Such mathematical treatment lacks the predictive power and cripples the theory. Imagine fitting a higher order polynomial to a set of data points. Higher the order of the fit, more tight the fit is. However, it is interesting to note that a higher order polynomial with more free parameters lacks the power of description and prediction. It can not explain another set of slightly different observations originating from the same data source. A famous quote by John von Neumann comes to mind,

Source: https://libquotes.com/john-von-neumann/quote/lbe6z3h

To illustrate this, consider a typical tank emptying problem. We know that the time '\(t\)' of emptying a cylindrical tank of diameter '\( d \)'  and fixed height is,

\[t = C d^2\]

where \(C\) is a calibrated constant. 

Model predictions M1 and M2 for tank emptying problem.

Suppose two students Student1 and Student2 are provided with the same set of data recorded by the experimenter. Student1 models the problem using a polynomial fit of degree 2 while the Student2 uses a higher order polynomial of degree 18 to achieve a tighter fit. It is easily discerned that Student1's model \(M1\) is robust while Student2's model \(M2\) is not. If they are provided data recorded at a different time by a different experimenter with a different set of cylindrical vessels of varying diameter, \(M2\) would be grossly in error, while \(M1\) will continue to give precise results and a reasonable prediction. This is illustrated in the attached figure.

The provisional nature, incompleteness, anti-realism/instrumentalism coupled with inadequacy and the added need for experimental verification of the mathematical models makes a strong case for the non-Platonist position, and I subscribe to that. Attaching an absolute truth value to the figments of our imagination, contrivances and ideas, scientific or philosophical - no matter how effective they may be in describing the world around us - is far fetched. Mathematics is not a portal to access the absolute truth of the universe, rather it is about what we can say and predict about the world. Mathematics, as applied to physical sciences, thus is invented, not discovered.   

Cite as

Bader, Shujaut H., “Mathematics, discovered or invented?” Backscatter, August 16, 2020, https://backscatterblog.blogspot.com/2020/08/mathematics-discovered-or-invented.html.