Order of Friars Minor Franciscans of Canada

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The Unreasonable Effectiveness of Mathematics in the Natural Sciences


The physicist Eugene Wigner asked this question in a lecture given in 1959: Why is mathematics so effective in explaining the world of the natural sciences? This has been shown over and over again: when a mathematical theory explains a physical phenomenon, then it ends up explaining more than what is known. It makes predictions about things entirely unexpected and not observed until we were told by the mathematics what to look for. Positrons were discovered when the theory explaining electrons predicted that there was another such particle, but positively rather than negatively charged. Other elementary particles were discovered in a similar way. This has been going on for some time now. In the 19th century, the planet Neptune was discovered by careful analysis of the movement of known planets. Newton’s physics predicted how the planets would move, and if the observations deviated from these predictions, then these deviations would show were to look for the unknown planet that has caused them. The history of physics is full of examples of theories leading to the discovery of things that had not previously known to exist.

It seems that mathematical theories describe the world better than the list of observations that let to them, as they not only organize what we already know, but they also tell us what to look for next. This experience explains the strength of the conviction held by many, if not all, physicists, that mathematics goes beyond making accurate predictions about measurements, but it tells us something about the basic fabric of reality.

While I shared this conviction when I began my studies as a Franciscan, I eventually saw that some skepticism is in order due to the tentative and preliminary nature of physical theories. Since their beginnings of mathematical physics in the 17th century, each theory has always been a stepping stone to another theory. Since none of them was a perfect match with reality, they could not be true in the strict sense. Currently, our best theories—quantum mechanics and relativity—are incompatible with each other, meaning that they cannot both be true. Either one or both require serious modifications. Maybe this is impossible, and their reconciliation and a final theory will always remain out of reach. However, a friend of mine who is an expert on this topic recently took the time to correct me on this. In his and, as he assured me, in the opinion of most of his colleagues, a comprehensive theory of all physical phenomena is possible, and we may well live to see its discovery. There are signs that the journey of discovery that has been going on for 400 years may one day come to its conclusion.

Therefore, the question cannot be dismissed. Why is mathematics so powerful, explaining not only what we perceive directly with our senses but reaching much more deeply, and possibly all the way down to the bottom of reality? There is no simple answer. Eugene Wigner himself concludes that it is a “wonderful gift which we neither understand nor deserve. We should be grateful for it.” It gives us much depth of understanding, much knowledge with useful consequences, and it places considerable power into our hands.

In spite of my friend’s assurance, I am not quite convinced that physics can accomplish the goal of finding a “theory of everything,” but I think that it is worth thinking about what it would mean if it were accomplished. Mathematical physics seems the quintessential modern accomplishment. I wonder whether it shows that classical philosophical thought, and especially philosophy grounded in the Aristotelian tradition, needs to be subordinated to philosophies that begin with the being of persons. If I wanted to express in Aristotelian language what has been accomplished by modern physics, then I would say that the universe as a whole has a form. Just as the form of a rabbit shapes unformed matter into what looks and acts like a rabbit, the form of the universe shapes unformed matter into the stuff that now makes up the universe, ourselves and all else that is in it. What we call mathematical physics is the rational understanding of this form. Just as we know that the rabbit is an herbivore, we know that the universe has features, such as gravity, and these features govern changes in the “life” of the universe.

The trouble is that higher forms rule, and that the lower forms of the beings included in them are no longer, as the “Angelic Doctor” would put it, actually but only virtually present. The forms of flowers and rabbits and the moon and the stars were subsumed under the one, comprehensive form of the universe. One can deduce what powers they had were there taken out of the universe and observed by themselves, but this makes them a figment of the imagination, as they could not possibly be taken out of the universe and observed by themselves. In other words, the objects of our perception are not actual objects in the universe. This is how I would, in Aristotelian terms, argue that mathematical physics, if it is a true and full description of physical reality, does not leave room for any distinct beings within reality in as much as it is physical. It is a closed system that operates as a whole, with no place for any distinct real beings. There would be no room for us in this universe. Since any action within the material world that is not according to its laws is a violation of its laws, human actions, or actions in which we are truly first causes of the actions that we initiate in the physical world, are impossible in such a universe. And without acting that is truly our own in the physical world, we could hardly be persons made of matter. We might be spirits impotently caught up in matter, but this is hardly the Christian view of the human person.

Clearly, these are absurd conclusion, which means that we have a problem. One can reconcile Aristotelian philosophy either with personal being, which is what the scholastic philosophers did, or one can reconcile Aristotelian philosophy with modern physics, which was done by more recent philosophers in the Aristotelian-Thomistic tradition. But it seems to me that one cannot do both without contradiction. Therefore, I am back to where I was 10 years ago, when I started my studies. Aristotelianism just does not work when I try to make sense of both my faith and my understanding of the physical world.

Why did it fail, though, after having survived 2,000 years of scrutiny by scholars coming from very different cultural backgrounds? There was the grand synthesis of human knowledge in the scholastic philosophy of the 13th century, but it developed in two different directions. One is the development towards modern physics, with its comprehensive view of reality in as much as it is accessible to observation. There is another one, though, and this is the more important one. It is the development that leads to the importance of the individual human person with personal dignity, rights, and obligations. William of Ockham serves as the best example for both developments as they begin to diverge, as he made important contributions to both in his natural philosophy and in his political philosophy.

Now, at the conclusion of this development, we are left with two incompatible insights: the all-encompassing unity of the physical universe that includes all, including you and me, and the unity and distinctness of each individual person that is always its own whole, and never just a subordinate part of a whole.

When I consider the branch leading to personalism more important than the branch leading to physicalism, I do so for a very simple reason. The concept of distinct entities, separate things in the world, such as persons, other living beings, and just material things, is an insight that precedes the insight of physics that all these things are governed by overarching laws that form a larger unity. My own personal being, and the personal being of those people who I came to know in my life, is more real than the being of the physical entities that I learned about in mathematical physics.

It seems that we must accept that being can come by a matter of degree, with persons being more real than the material parts of which they are made. Which seems to take me back to Plato. Somehow, philosophy never seems to make progress, but it certainly provides a framework to express what we know at any time in the life of humanity. And while mathematical physics has given us much power, the concomitant discovery of the superior importance of personal being provides us with the ethical framework to use this power.