*Laurent Neveu-Marques delves into the historical relationship between philosophy and mathematics and argues for continuous metaphysical reflection.*

There is a lot to say about the relationships between mathematics and philosophy, but it seems to be recapitulated by an inscription at the entrance of Plato’s Academy: “No one enters here if he is not a geometer”. Clearly it means there is a link between these two disciplines, but this could be understood in different ways. Commonly we understand that mathematics is a qualifying program, which provides the foundation to do philosophy because of its high level of abstraction. However, this is the visible part of the iceberg and even the good reasons why they could be seen as a propaedeutic, show us a more complex relation.

In his seminal work *The Republic*, at the end of book six (509d-511e), just before the famous allegory of the cave, Plato makes a hierarchy of types of knowledge, figured by a line which is divided into two sections: one regards the visible realm (all we can know by our sense), and the second one regards the intelligible realm (all we can know only by intellect). These two sections are subdivided in two according to the same proportion. Thus we have four sections that map types of objects with types of knowledges.

Firstly, representation: this is knowledge of images, like reflects or shadows. Secondly, belief: this is knowledge by the perception of physical objects. Thirdly, thinking: this is knowledge of mathematical objects like numbers or figures. Fourthly, intellection: this is knowledge by direct contemplation of what Plato called Ideas or Forms like Beauty, Justice, etc.

We can easily understand that we need to follow this line to attain the last section and therefore we first need to be geometers before becoming philosophers. And why does Plato write “geometer” rather than “mathematician” if the objects of the third section are not only figures but also numbers?

It is because geometry, which is a part of mathematics with algebra, shows with more evidence that objects of mathematics are not so abstract as ideas which are objects of philosophy, because they suppose a kind of possible representation. We can have a representation of a triangle, for example, but we do not have the same about justice, beauty, and goodness. As we say, “geometry is the art of making correct reasoning with false figures”: geometry maintains a relation with a kind of representation. In the first two sections, knowledge is better if objects are perceived directly rather than their images. This is the same between the third and the fourth sections. But, if geometers use images, they know they are images and not reality (and therefore they can make correct reasoning despite these images): this is a crucial step toward philosophy which does not use image anymore.

At the time of Plato, mathematical objects were simpler than those of recent times: numbers and figures can easily be put in relation with representation. For this reason, we can easily make an objection based on the very high level of abstraction of modern mathematics. Therefore, we must consider one more difference between mathematics and philosophy: mathematics is hypothetico-deductive. Principles of mathematics are hypothesis (like axioms) from where thinking is possible. According to Plato, philosophy (dialectic) is a knowledge by direct intellection of her objects. This is the reason why philosophy is the kingly science: “comprehensive mind is always the dialectical”. But the inscription at the entrance of Plato’s Academy does not put philosophy and mathematics in competition: it would be an error to read the hierarchic difference between a comprehensive mind and a hypothetico-deductive system in this way. The global knowledge is a line. There is therefore a continuity and complementarity between them.

According to Plato, our world and all objects it contains are copies of forms or ideas located elsewhere, outside the famous cave where we are living. In the Timeus, Plato explains the relation between these two places by using a cosmological myth (a common practice at his time); but for the first time ever, he uses mathematics in order to express the way our world is made. For example, the four elements (air, water, fire and earth) are based on different regular polyhedrons and this explains their different qualities. All we can see is reducible to two types of triangles and rectangles, and the world’s soul has the structure of an armillary sphere. This is enough to see that mathematics and philosophy are complementary. And this is still the case!

The history of Science shows us that a lot of philosophers were mathematicians: Thales, Pascal, Descartes, Leibniz, etc. They explained our world not only by using paradigmatic methods of mathematics (like Descartes or Spinoza) but also mathematical tools and concepts, as we just saw that Plato had done. At all times we are surprised by the world’s harmony and by the fact that this harmony can be expressed by mathematics: is the book of the world written in mathematical language as Galileo said? When a mathematician has this kind of question about mathematical realism, when s/he wonders about the basis of mathematics, s/he is no longer simply a mathematician but a philosopher too. The famous Gödel’s theorem of incompleteness shows us a structural limit of mathematics. How powerfully they could be, they always call for metaphysical reflection to try, again and again (may be ever and ever), to see outside the cave!

### Laurent Neveu-Marques

Agrege in Philosophy,

UNIVERSITY OF BORDEAUX