What is math for and why is it so unreasonably effective

What is math for? Ever since the Pythagoreans and Plato based reality on the very existence of geometry, postulating that it is the very essence of things, and Galileo declared that the great book of nature is written using numbers, many thinkers have wondered why its most powerful description is the mathematical one, and on why, in the end, reality, once its independence from the observer is admitted, can be mathematised. In 1921, this view led Einstein to ask: “How can it be that mathematics, which is after all a product of human thought independent of experience, is so admirably appropriate for describing objects of reality?” Subsequently, in 1959, Wigner spoke of “the unreasonable effectiveness of mathematics”, precisely to underline this unexpected property of what we can grasp of the universe around us.

Without answering these kinds of questions, it is perhaps difficult to justify why to best describe the physical world we need mathematics and not some completely independent and different system – a claim, the latter, shared by many proponents of all sorts of vitalistic pseudosciences.

Here I would like to summarize some points which, a century after Einstein’s question, are among those I find most convincing. First of all, instead of asking ourselves if and why the structure of the universe is mathematical, that is, if Plato wasn’t right after all, we can ask ourselves whether for an observer the use of a mathematical description of physical reality does not present some special advantage, compared for example the use of poetry or other purely intuitive and qualitative means.

So let’s imagine a poet who wants to describe to the world the color of the rose he intends to give to his beloved, and tell us about the intense red that symbolizes his love. Many readers will be carried away by the words and the image, but among these there may be a color-blind reader; much to the poet’s horror, she may well be her lover, although it is rare for women to be in such a condition. How could the poet explain himself, after having declared his love in a more shrewd way, if the woman he loved was really curious to understand what she meant? He could show her that the light reflected from the rose he gave her has a wavelength between 620 and 760 nanometers, like the blood that gives her life.

Compared to the use of the word “red”, the advantage of using a measure to describe a particular property of physical objects is evident: there is no ambiguity between two distinct observers about the objects showing that property. Therefore, if it is necessary to create a shared worldview, the best way we have to disambiguate the description we create is to start from the accurate measurement of what interests us. Because we are a social species, we live by constantly communicating something of the physical world to others; the reduction of ambiguity in this communication is therefore essential, but why do we have to go so far as to use numbers, rather than more ambiguous but more comfortable words? For example, we can very well warn a child that touching a flame is dangerous, because it burns; we do not need to tell him its temperature to make him understand the danger, and instead a description of the world in which the flames burn is enough for that child to know enough to be relatively safe when near a gas stove. Of course, there are cases in which we need to count, and in those cases the advantage of making a measurement is obvious; but why is it better to describe the entire universe through mathematics, rather than relying on an alternative system?

Here we come to a really important point: “mathematizing” a problem does not mean measuring and calculating, but revealing a hidden skeleton of conceptual relationships linking the observable properties of the universe. Mathematization consists precisely in formulating the idea underlying the connection between apparently different physical properties in an abstract mathematical language. Even these relationships, in order to be graspable by everyone and to be able to become a shared heritage, must be expressed in an unambiguous way; but what matters above all is that the axiomatic-deductive system of mathematics is the most powerful we have, when we want to describe and investigate the existing relationships between measurable properties of reality, i.e. expressible by numbers. Our ordinary language is absolutely insufficient for the task of describing even simply the variety of relationships existing between the physical properties of the world around us, while the expressions obtainable through mathematization have the expressiveness needed, do not introduce ambiguity and, in many cases , allow us to compact and make the description of what we have learned much, much more efficient.

Our very varied experience of the physical world can thus be traced back to the discovery and study of the relationships between measurable variables, which well describe the properties of each physical object that we can encounter; these relationships allow us not only to describe what we perceive, but also to make predictions about the state of what interests us in the past, in the future or in places very distant from us, because some variable that we are able to measure can be mathematically related with the state of the physical system that we intend to study even if very distant in space and time. The process of mathematization of what we experience works so well, that our entire scientific knowledge of the world today rests on the determined value of a few dozen fundamental constants, from which, through the relations we have discovered, we are able to derive all the description of any phenomenon that science has hitherto explained. This description is not complete, and it is said that it never will be, for reasons that may even be intrinsic to the very process we use to build it; at least for now, however, we are equipped with nothing better to explore our universe than a brain capable of mathematizing it and the collective mind that we have created with our global society.

And here we finally get to answer the initial question of why this tool is so unreasonably effective. Perhaps, after all, it is simply wrong to judge its reasonableness from the breadth of its success, just as it is wrong to marvel at the perfection of the eye and then to invoke divine intervention for its existence: given some link between the properties of the parts that make up the physical world, starting from the quantification of these properties by computation (a process that also occurs at the level of individual cells) a way of expressing those links has evolved as quantitative relationships between the measurements made. This process underlies the logical-mathematical processes that we have subsequently formalized and condensed into what we call mathematics.

It is not the universe that is unreasonably mathematical, but our reason that is mathematical, and therefore universal, due to the evolutionary advantage that this entails in a social species, where it better allows for the sharing and collective exploration of a description of the cosmos. And those who do mathematics discover relationships that could always find application to describe and make us understand some new piece of reality, which escapes us today; it is the mathematical description of those relationships that will allow us new and more fruitful collective tests of understanding the universe, and it is for this reason that the exercise of mental exploration in the green mathematical pastures is among the most important activities of our intellect.