Mathematics and reality

In 1960, physicist Eugene Wigner wrote his famous essay “Unreasonable Effectiveness of Mathematics in the Natural Sciences”, wherein he popularised the idea that many mathematical concepts that had been generated in a purely abstract way, in an unexpectedly manner, came out to be the very tools required to explain phenomenon observed in nature in later years. Quoting Wigner, “Max Born noticed that some rules of computation, given by Heisenberg (to explain quantum mechanics), were formally identical with the rules of computation with matrices, established a long time before by mathematicians.” As if the mind had created a conceptual framework that would be suitable for explaining a phenomenon in nature, long before it was discovered.

Wigner made this situation akin to a position of a man who had a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial, making him skeptical about the uniqueness of the coordination between the keys and doors.

Wigner had opened a question upon the foundational basis of mathematics, as to how ‘mathematics’, which is a purely thought-thing, a language created in the mind by giving symbols and number-names to generalised properties of the things we observe in nature; and a game of building sets of axioms upon these symbols — most of the times turned out to be so ‘uniquely appropriate’ in explaining a phenomenon under observation and even in explaining phenomena yet to be observed. To understand Wigner’s perplexity, perhaps it would be useful to understand the basis of mathematics first.

We know that in math we assign ‘numbers’ which are our imagined symbols to ‘properties’ of things. So, when we say ‘two apples’, the number tells us nothing intrinsic about the apple rather only that we have two of them. Then we further generalise it by saying 2x, which is like saying ‘two of anything’ we want to call x. Later, we go on to create ‘sets of axioms’ that make up mathematical frameworks or ‘systems’ that will yield certain kinds of results. For instance, logarithms are a mathematical field based on axioms that define the relationship between a number and its exponent. This field is like a ‘game’ that can be played under this particular ‘sets of rules’ and that will yield only this range of results. While logarithms were introduced by John Napier in 1614, later logarithms were found to be working in astronomy, psychology, probability, chaos theory, music, etc. and it was found that the shells of the marine mollusk Nautilus increase in size logarithmically.

The question is: how do patterns created in an abstract mental field, correspond so appropriately to real systems in nature? Rather the surprise in the first place is, how and why nature and the human thought share the same language – that of mathematics? After all, if we and everything around us are all the resultant of millions of chance-events occurring in billions of years, then what was the chance of the becoming of a thinking matter that was to be not only consciously aware of intricate phenomenon surrounding it, but simultaneously possessing an abstract tool, the language of mathematics, that could measure its effects and devise formulations depicting these phenomena and the interrelations between these phenomena.

How is it that nature and its laws, that are grounded in its material basis, in the world of matter, which scientists tend to deem as ‘reality’, correspond in ‘uniquely appropriate’ ways with the patterns of ‘thought’ or ‘conscience’, which, according to positivist neuroscientists, is an ‘illusion’, a ‘confusion’ or ‘by-product’ of the neural system — even when, all our sciences and physics and chemistry are by-products of this so-called by-product — even our mathematics.

Moreover, the effectiveness of mathematics in understanding reality becomes more astonishing when we consider the fact that these axiomatic setups that constitute the foundations of all calculations are always ‘inconsistent’ and ‘incomplete’; Gödel proved in 1930 that in all axiomatic systems there can be true statements such that their opposites/false statements can also be proven right in the same system; and that no axiomatic system can completely prove all true theorems regarding the properties in that system. Rendering all mathematical certainties tending towards uncertainty!

This ‘uncertainty’ is not new, it has haunted mathematicians as early as the Greeks, who discovered that the hypotenuse of a right-triangle with unit sides, was a never ending, unsolvable fraction! How is a never-ending entity, tending to infinity, real? How is the circumference of every circle, something also tending to infinity because it contains the infinite fraction ‘π’, making it something unimaginable, or something like a broken circle, because the line of the circumference never ends at any given point, real then? So, is our math a weird, unreal abstraction, or is our real world spooky, non-discrete and abstract in its core? Does mathematics, the very instrument that all positivists hold dearly to their hearts, and consider anything that fits in a mathematical casting to be ‘real’, deceive their surety with constantly recurring infinitely deep loopholes?

Or does it at least prove the continuous nature of reality, wherein one reality overlaps the other or one phenomenon blends in the other, literally smudging the boundaries between discrete things.

In the 80s, Hartry Field, realising this, introduced mathematical ‘fictionalism’, the doctrine that all mathematical statements are merely useful fictions, and should not be taken to be literally true. Yes, mathematics is useful, it is beautiful, in fact it is magical, and it is as close to truth as can be. Yet perhaps it cannot touch truth like a real touching, it always has to regress away from real complete truth, perhaps that real truth, that the soul perceives of, yet does not find a tool to touch, is really a set of aggregates that fit gracefully in the large mural of nature, but as we come closer and closer to the mural, the nature of things starts changing face.

Perhaps the real beauty of reality is how it dances all around us, touches our soul subliminally, yet always remains undefinable, intangible. Perhaps reality is as abstract as mathematics is!