In mathematics we trust

From Zeno’s paradox to understanding convergent sequences, mathematics has come a long way. However, the popular idea of mathematics oscillates between it being perceived as too abstract or it being perceived as the basis of many applications. Laurent Neveu-Marques ponders if we are right to put our trust in mathematics.

Illustration by Nino

In antiquity, the first philosophers tried to understand our world with mathematics. Parmenides showed us two ways of speaking about reality: the way of the common people’s opinion and the way of truth. This second way is that of strictly using reason. From this point of view and despite appearances, we had to admit that movement does not exist. Being is motionless.

Most people are familiar with the paradoxes of Parmenides’ disciple, Zeno: one of them shows that if Achilles makes a run against a turtle and gives it a head start, he will never overtake this turtle. To overtake the turtle Achilles has to first arrive at its position, but during this time, the turtle goes a little farther, and Achilles now has to overtake this new position again: this will never come to an end because of the infinite divisibility of space. For the same reason, an archer shooting an arrow will never reach the target! Should we not rather avoid such mathematical conceptualization of nature?

Another point is that mathematics is too abstract, too far removed from empirical reality. How then, should we relate these two? Such difficulties arise in Platonism. The main one came from Aristotle whose philosophy dominated science for a long time. He made a distinction between the sublunar world, ours which is under the moon, and the supralunar world which is the universe beyond the moon. In our world, movements are a sign of instability and imperfection of all things subject to generation and corruption. In the supralunar world, there is no movement or only the perfect ones, like the circular movement of planets. This is the reason why use of mathematics was only allowed in astronomy: they are too perfect to be applied to our world. It did not look good for mathematics!

It was only in the sixteenth century that Galileo explained that nature’s laws are everywhere the same. From this time on, physics became mathematical. For Galileo, “the book of nature is written in mathematical language”. What does it mean? Who is the author of this book? What is this book? Is it the book we wrote about nature using mathematics, or is it the nature itself, maybe written by the hand of God? Are we imposing our world of mathematical ideas on top of nature or have we just found mathematical concepts in nature? It’s a long and difficult debate. But in both cases we believe that mathematics is the key code to understand the universe. We believe that the universe can be put into an equation. Since we think that the best theories are the most simple, we postulate that the complexity of phenomena can be reduced to something simple: mathematics is the archetype of this approach. The celebrity status of some formulas like E=mc2 strengthens our belief.

Illustration by Liani Malherbe

At the time of Galileo, Descartes advanced the idea of a method for directing the mind in knowledge of all things. This method is not new: it has already proven itself, but only in mathematics: the rules used in mathematical reasoning (like analysis, inference, etc.) produce results. What is new is the application of this method to all the things we know. Mathematics has become the model of all sciences. This is why Descartes uses the expression mathesis universalis. If the spirit of mathematics prevails, the form will soon follow! In the next century, under the impulse of Leibniz, the logic will undergo a formalization: “let’s not argue anymore, let’s calculate!”. This is the starting point of contemporary logic which rapidly became a logic calculation using mathematical formalism to make demonstrations, as computers will do later. This bolstered the idea that mathematics offers a universal way of understanding the world.

All the reasons we identified here are too abstract for the lay person who does not know much about science or logic. People accept a paradigm (i.e. a certain way of thinking about the world) without understanding or knowing about it. So a distinction had to be made between the reasons that explain this conceptualization of the status of mathematics and the reasons why this conception became a stigma about mathematics. So why do people trust in mathematics?

The answer can be given in a word: applicability. Descartes says that the purpose of modern science is to make men “masters and owners of nature”: this relationship between theory and practice, between science and technique, is possible only if we have a conception of nature as something measurable. As Martin Heidegger said, the requirements of the technical mind made science as we need it to be. The applicability of sciences is an argument for thinking that mathematics gives the truth. Far from the complex debate on the realism of mathematical entities, we think that if mathematics can be applied to reality with success, they are the keys to this reality. And then we trust! The way we learn mathematics cements such beliefs. We learn to count by counting empirical objects, we learn basic operations by solving concrete problems, etc. We know that using the Pythagoras or Thales’s theorems we are able to make the measure we need in our DIY crafts. Then we can easily verify that mathematics is suitable for the daily experience of the world.

In mathematics we trust…but we still can’t believe that Achilles does not win the race against the turtle! Fortunately, modern mathematics can resolve the Zeno paradox: depending on Zeno, the duration of the event “Achilles catches up with the turtle” is calculated by adding all the events of the type “Achilles covers the distance to the current position of the turtle”. The number of these events is infinite and this is why Zeno concluded that Achilles could never overtake the turtle. But this is a mathematical error, for modern analysis shows that an infinite series of strictly positive numbers can converge towards a finite result: so it is possible to calculate the time when Achilles will join the turtle.

Does that mean that we are right to put our trust in mathematics?

Laurent Neveu-Marques

Agrégé in Philosophy, University of Bordeaux

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