Do You Need to Study Mathematics to Understand Philosophy?

When I tell people that I have come back to university after eight years of teaching philosophy, their first reaction is often: “what a radical change!” This has always surprised me, because the reason why I like philosophy is the same reason I like maths, and maths was always one of my favourite subjects in high school. However, in France, philosophy is oddly associated with literature. In the high school curriculum, philosophy is first taught in an optional subject called “Humanities, Literature and Philosophy.” Then, in the last year of high school, philosophy seems to replace the literature module of the previous years. As a result, students generally arrive in the philosophy class thinking that they will continue to study something like literature: reading books, writing essays weighing the pros and cons, and giving their opinion on several topics. Therefore, it is often difficult to make them understand, and sometimes believe, that they are going to think in a way much closer to formal sciences.

This way of studying philosophy is specific to France and some other continental European countries. Most philosophers elsewhere make extensive use of logic when they do philosophy, but it is possible in French universities to study philosophy for five years, and become a teacher, without taking any logic module. Yet, logic is the basis of philosophical reasoning. Unlike the natural and human sciences, which rely on observations and experiments, philosophy progresses through arguments and thought experiments. Some may categorise philosophy as part of the human sciences; but even if I can understand why a few philosophical topics related to existence or politics are strongly linked to human sciences, most philosophical research deals with abstract concepts and formal reasoning, sometimes without any reference to real life.

Returning to my own experience, I had the traditional French curriculum in philosophy: in five years, I only had one semester of logic. I passed the exam to teach in high school, and during my first years, I found it difficult to make students understand what they were supposed to do. I quickly realised what my problem was when I began to study, for my own enjoyment, analytical philosophy and logic : I was trying to explain how to construct a logical argument without any knowledge of logic. That’s why I finally returned to maths : its similarity with philosophy not only helped me better understand my own speciality, but also made me a more efficient teacher.

My aim here will not be to understand why and how there are so many differences in the practice of philosophy between countries. Lots of works in the history of philosophy explain that already, better than I would. We can briefly say that the main division took place during the 20th century. However, at this moment, philosophy was already very different from what it was before, when the “philosopher” was a general scientist, studying every field of human knowledge, from physics to what we now call psychology. I think the best definition of what is current philosophy was given by Bertrand Russell in “The value of philosophy”. As he observed, it is true to say that philosophy doesn’t give any answer to anything; but the reason for this is that each time a philosopher finds a method to get answers, his activity is no longer called “philosophy”. As he writes himself :

If you ask a mathematician, a mineralogist, a historian, or any other person of learning, what definite body of truths has been ascertained by his science, his answer will last as long as you are willing to listen. But if you put the same question to a philosopher, he will, if he is candid, have to confess that his study has not achieved positive results such as have been achieved by other sciences. It is true that this is partly accounted for by the fact that, as soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science. The whole study of the heavens, which now belongs to astronomy, was once included in philosophy; Newton’s great work was called “the mathematical principles of natural philosophy.” Similarly, the study of the human mind, which was, until very lately, a part of philosophy, has now been separated from philosophy and has become the science of psychology. Thus, to a great extent, the uncertainty of philosophy is more apparent than real: those questions which are already capable of definite answers are placed in the sciences, while those only to which, at present, no definite answer can be given, remain to form the residue which is called philosophy.[1]

According to Russell, philosophy concerns the set of questions for which no method of answering yet exists. So how could a discipline which was originally the source of every field we now call “science” have been mainly associated with literature in France and other countries, instead of remaining a scientific subject as it is elsewhere? Despite the interest of this question, it is not our subject here. We can still wonder what is the link between mathematics and philosophy : isn’t mathematics a separate science like physics, biology or psychology? Why should we have to study maths to understand philosophy, rather than the other sciences?

Actually, philosophy is, like the sciences, divided into different fields, and to understand the philosophy of physics, it is necessary to study physics; to understand the philosophy of mind, it is useful to know about neurosciences and psychology… But my aim is more to show that maths remains useful in whichever field of philosophy one chooses to study, not only the philosophy of mathematics.

In this paper, we will explore the link between mathematics and philosophy, considering three approaches: first, a historical one, to show the evolution of the relationships between both disciplines; secondly, we will dwell on philosophical methodology, to understand how having a good practice of maths can help in philosophy; and thirdly, we will present some fields of contemporary philosophy that use mathematics as tools, just as the natural and human sciences do.

A historical approach : from Ancient Greece to analytic philosophy

– Science and philosophy

It is usual, in the French tradition, to trace the origin of philosophy to Socrates and Plato. History of philosophy books may, however, present Thales as the first philosopher. Thales is mostly known for several geometry theorems with which he is credited. In Ancient Greece, a philosopher was a general scientist, interested in knowing the truth about nature, society, or human existence. Thales was part of a school of thought seeking the fundamental component of the universe. They thought the universe was made of one component ; according to Thales, water must be this universal component, as water can be found in many forms. That was the beginning of physics. As Russell said, every scientific field originated from philosophy. However, even at this time, mathematics quickly became kind of a separated science. Indeed, although the Ionian and the Pythagorean schools (respectively Thales’ and Pythagoras’) thought numbers were embedded in reality, Plato, Aristotle and then Euclid understood the abstraction of mathematics. Thus, mathematics was no longer simply part of natural philosophy, but a discipline philosophers needed in order to master abstract reasoning.

It is well known that the sentence “Let no one ignorant of geometry enter” might have been engraved on the door of Plato’s Academy, the philosophy school he founded in Athens. “Geometry” was a more general term then, referring to mathematics in general. According to Plato, if studying maths is necessary before studying philosophy, it is because it is the basis of any abstract and correct reasoning. Before him, mathematical proof could be made using drawings or spatial intuition. For instance, consider the proof of Pythagoras’ theorem based on a construction of two squares and four triangles : it is possible to prove the theorem to anyone, even a child, just by moving around the triangles. Yet, considering mathematics as an abstract field, separate from the real world, we cannot rely on diagrams and spatial perception anymore: proofs have to be based on abstract concepts as well. That is the importance of Euclid’s Elements and the reason it remains such an important work to this day.

Consequently, the study of mathematics was and still is essential for the study of every science. Until the 17th century, natural science was called natural philosophy, because philosophy just meant the search for the truth, whatever the field considered. One important example of a natural philosopher would of course be Aristotle. Not only is he known for his astronomical works (On the Heavens), but also for his biology research (his most famous work must be Parts of Animals). In spite of his mistakes – unavoidable when one writes so much –, he wrote hundreds of pages on nature, the universe, animals, logic, politics, art, and more. This model of the philosopher as a general scholar lasted for several centuries. The Stanford Encyclopedia of Philosophy introduces Leibniz as the last “universal genius”, in the 18th century, due to his important contributions in metaphysics, logic, mathematics, physics and jurisprudence. After Leibniz, sciences became too deep to be accessible by only one person. As a result, the separation of sciences accelerated.

Nevertheless, one similarity remained: mathematics is still the necessary condition for studying science. Another famous quote from a philosopher would be Galileo Galilei’s: “Philosophy is written in this grand book – I mean the Universe – which stands continually open to our gaze (…). It is written in mathematical language, and the letters are triangle, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word”, in The Assayer. This quote is so famous because it foreshadowed the beginning of modern physics, which was going to replace natural philosophy. The reason why mathematics is the right language to understand philosophy is its exactness : mathematical proof is certain, because it relies only on reasoning, and not on the external world.

The historical importance of mathematics for philosophy is the same as the importance of mathematics for every science. It is clear for everyone now that it is impossible to study physics without mathematics. It should not be different for other sciences, including philosophy. Beyond that, mathematics has remained a particular science: it is still possible to study mathematics for its own sake, not merely for the physicist’s. Likewise, although mathematics is a good language to do philosophy, philosophy of mathematics is a separate field of philosophy. In the next section, we will summarise some of the problems we can face in this particular discipline.

– Philosophy of mathematics

Philosophy of mathematics is a large field that we would not be able to present exhaustively here. It has a long history and some parts of it can be considered not as philosophy anymore, but pure mathematics. Some questions about infinity or continuity, for instance, are now well defined in mathematics and may no longer require philosophy. However, it does not prevent mathematicians and philosophers from working together on these questions.

As we are still in our historical approach, we will begin by introducing Plato’s famous theory about mathematics. Following Pythagoras and his belief that everything in the world is number, Plato recognised that numbers were abstract, insofar as they are not direct properties of things but general ideas. However, according to Plato, mathematics is not constructed by the human mind. It exists in itself in a different world, called by Plato the World of Ideas, which is more real than the sensible world in which we live, because it is in this world that we can find these “Ideas” or “Forms”, entities that are eternal, never change, and structure our own world. Abstract concepts like beauty, justice or good are part of this world, whereas in our world, we only have access to things which share the property of being beautiful, just or good. Mathematical objects are these kinds of forms too.

This view of the nature of mathematics led to an epistemology known as Platonism, or mathematical realism: the belief that mathematics has a separate existence, therefore studying mathematics amounts to discovering what already exists, instead of creating it. One of the philosophical debates about mathematics is to analyse the pros and the cons of each epistemology. Indeed, Platonism is not the only one. Other schools of thought see mathematics as a cultural production, or as arising from nature or human cognition. At the beginning of the 20th century, three philosophical movements about mathematics emerged :

– Logicism aims to reduce mathematics to logic. – Intuitionism considers mathematics as a mental construction. – Formalism sees mathematics as only a formal game, without taking position on its independent existence.

Reflecting on the reasons to adopt one of these positions, the consequences they have and the problems they raise has been a large part of philosophy of mathematics in the last century. Why does a mathematical proof necessarily lead to truth ? What is this “truth” we are talking about ? The answers to these questions will depend on the epistemology we choose to accept.

A wide range of problems in the philosophy of mathematics were addressed at the beginning of the 20th century. Nowadays, the problems also include those related to computer science, particularly regarding new computer proofs. This opens new questions about mathematical understanding.

– Philosophical methods inspired by maths

Let’s return to general philosophy, but still from our historical point of view. I would like to dwell on two specific philosophers who tried to create a new way of doing philosophy, inspired by mathematics. The first one, whom we mentioned earlier, was Leibniz, our last “universal genius”.

Leibniz understood that the strength and certainty of mathematics lay in its precise concepts. If you take the definition of a “group” or a “field” in maths, it always refers to a very specific concept and admits no ambiguity. On the contrary, philosophy is studying concepts from natural language, and the words we use always have different meanings, depending on the context; sometimes just different possible extensions. For instance, talking about “violence” in political philosophy can include psychological violence, structural violence, or be limited only to physical violence. Still in political philosophy, the concept of “Justice” can refer to the human institution or an ideal which should guide our actions. Thus, to construct a philosophical argument, we first need to define it precisely.

Leibniz thought it would be easier to create a language modelled the mathematical one, where every word would have a single and precise meaning. By combining simple elementary symbols, we could construct more complicated ideas. In “Dissertation on the Art of Combinations,” published in 1666, he outlined the project for the construction of a “universal characteristic”, a symbolic universal language in which all human thoughts could be expressed with clarity and precision inside this language. This way, all the problems, including philosophical or legal problems, could be solved by logical calculus. It is obvious that trying to create such a language faces significant limitations. However, with the development of computing science, Leibniz’ project has gained renewed relevance.

The second philosopher we should talk about, also famous as a mathematician (and winner of the Nobel Prize in Literature) is known as the founder of analytic philosophy, which is now the main paradigm in contemporary philosophy. Originally, analytic philosophy was more focused on the logical analysis of the language. It was created at the same time as the reflection about the foundations of mathematics, and it is not a coincidence. Indeed, analytical philosophy makes it a point of honour to use language as clearly as possible. Yet, as we know now, the clearest language is mathematics itself. Philosophical proof has to be as well constructed as a mathematical proof. The more continental philosophy developed, the more it produced neologisms, metaphysical concepts and poetic writings, which were a hindrance to a clear link between the premises and the conclusion. As the opposite movement, analytic philosophy promoted a precise definition of the terms, and a rigorous reasoning. Precise definition of the terms before reasoning is the necessary condition to avoid, as much as possible, the limits of natural language, with its nuances and ambiguities, and resemble mathematical reasoning.

From this point of view, Russell (along with his German contemporary Frege), agreed with Leibniz’s conclusion about the problem of natural language, and it is not a surprise that founders of analytical philosophy have been mathematicians and logicians as well as philosophers.

The reader will not be surprised, at the point of the paper, to learn this, because they already know that in the early history of philosophy, all philosophers were scientists. However, we must remember that between Leibniz, the last “universal genius”, and Russell, two centuries had passed. Philosophers were already losing their general knowledge and most of the philosophers in the 19th century were interested in metaphysics, losing their interest in natural sciences – actually, scientists began to separate from philosophers, as we have seen before. Analytic philosophy was created at the beginning of the 20th century. However, I see it as simply a return to what philosophy was at its origin. I hope this first historical approach has helped the reader to understand why.

The philosophical method: reasoning and arguments

At the end of our previous section, we saw that contemporary philosophy, in the vast majority of the world, is based on logical rules and reasoning, just as mathematics is. Contrary to natural sciences, to check the validity of a philosophical theory, we cannot use experiments, or compare with reality. Natural sciences are constrained by the rules of nature, whereas philosophy is constrained by the rules of logic. A philosophical argument rests on a set of propositions, called premises. From these propositions, we deduce other propositions : that is the way we can prove an assertion in philosophy. To refute an idea in philosophy, we can either refute one of the premises, or highlight a contradiction in the reasoning – that means to find a mistake in the reasoning from a logical point of view.

As with the rest, this is not an invention of analytic philosophy. In Aristotle’s books we find the definition of simple logical structures in three parts, called syllogism. It can take this form :

All A are B All C are A Therefore, all C are B

Of course, current proofs in philosophy are much more complicated than this one. However, the basis of philosophical reasoning in a logical way was already present in ancient Greece. The idea remains the same : every conclusion must be deduced from premises. Actually, this is a particular type of argument called “deduction”, and if it is valid, the conclusion is necessarily true if the premises are. This is a very strong proof because, there is no exception where the premises are true and the conclusion is not. This property of necessary truth preservation led to considering the deductive argument as the paradigm of valid reasoning.

The opposite type of argument is the “induction”. It can be seen as opposite in various ways:

– First, instead of going from general premises to general or particular conclusion, induction goes from particular fact to a general conclusion. – Therefore, on the contrary to deduction, induction is not a valid argument. It is, however, a type of reasoning we vastly use in our everyday life : seeing the same event keep repeating, we make the conclusion that it will happen again under the same circumstances. – Finally, deduction is associated with philosophy and mathematics : the formal sciences which are based on a set of axioms and conclude what must follow from them. Induction is more related to natural and human sciences, which rely on observation.

This last point leads to the most important question in philosophy of science, called the “problem of induction”. Because natural sciences rely on an invalid kind of argument, the question of their truth needs to be asked. However, mathematics and philosophy does not work with induction, but deduction. The difference between psychology and philosophy of mind is that psychology observes phenomena in the world and tries to create general rules about the way people think, feel and act, whereas philosophy of mind seeks to define coherently what mind, consciousness or will are. By giving answers to how conscious people and the brain work, psychological and neurological knowledge is crucial for philosophers, but their methodologies are opposite. In contrast, the common point between philosophy and mathematics is that they do not begin with observation, but abstract postulates.

This leads to another kind of problem, specific to deduction, called the “premises problem”: if premises have to be true to guarantee the truth of the conclusion, how can we be sure that they are indeed true ? This question led to the fundamental crises of mathematics at the end of the 19th century, and the search for a solid basis for mathematics. In philosophy, the premises are exactly the points that are interesting. As it is not possible to question the conclusion of a right deduction, the philosophical debate focuses on the premises – with exception, sometimes, about the logical structure.

Consider the following argument, known as the sceptical argument (scepticism is a philosophical movement according to which any knowledge is certain, therefore possible, provided that a knowledge is a true justified belief. The Global scepticism even consider that the existence of the external world itself cannot be proven), where “S” is a sceptical hypothesis and P any proposition about the external world:

1) If I cannot refute the possibility of S, then I am not justified in believing P 2) I cannot refute the possibility of S. 3) Therefore, I am not justified in believing in P.

This argument is valid, but its conclusion can seem absurd. There are many things we feel justified to believe. To refute the conclusion, we need to refute one of the two premises. The more fragile one seems to be the second, as the first is based on definitions of knowledge and scepticism : is it true that I cannot refute S ? Philosopher G. E. Moore suggests the following, with P = “I have hands”:

1) If I cannot refute the possibility of S, then I am not justified in believing P. 2) I know P. 3) Therefore, I can refute the possibility of S.

Moore’s argument structure is called modus tollens and relies on a contraposition: the first premise says that “If X, then Y”, which is logically equivalent to “If not-Y, then not-X”. The proof by contraposition is often used in maths as well as in philosophy.

Other kinds of arguments exist, and philosophy is far from being reduced to the two we just presented. Similarly, mathematics does not only use deductive arguments: mathematical induction exists as well, proof by contradiction, proof by exhaustion – all of which have great importance in philosophy too. By studying mathematics, and learning to use each of these proofs efficiently, we tend to develop our philosophical skills, and vice versa.

Current fields of philosophy using maths

In this last section, we will stop talking about the methodology to return to the content of mathematics and ask our final question: are there forms of calculation that are required for specific fields of philosophy ? Of course, a deep understanding of philosophy of mathematics requires a good understanding of mathematics themselves, but in this section, we will not be talking about philosophy of mathematics, as our purpose is to show that mathematics is useful in many fields of philosophy.

We can first broaden our topic by considering philosophy of science in general, not only mathematics. The situation is nevertheless the same : most of the natural sciences use mathematics to study their objects and to understand these sciences; therefore, to practice rigorous philosophy of science, some basics in maths are unavoidable. It is particularly true in philosophy of physics, where the link between mathematics and reality is the most studied.

However, mathematics is also essential in fields of philosophy not related to science. I usually avoid speaking too much about fields I do not master, so I will focus on one particularly field: ethics. Ethics is the reflection on good and evil, and at first sight, it seems far removed from the study of mathematics. Ethics is divided in several subcategories:

– Descriptive ethics studies the way people act and think about morality. – Normative ethics tries to construct a coherent and complete moral system, to help us determine how we should behave; this part of ethics is sometimes compared to the research in the foundation of mathematics. – Applied ethics studies real cases like war, medical situations, or the condition of animals. – Metaethics is a more fundamental field, studying the ontological, semantic or epistemological presuppositions of morality.

Particularly in descriptive and applied ethics, the use of statistics and probability is common in studying and analysing human behaviour. Yet, the misuse of statistics is common. Therefore, to avoid mistakes and carry out precise philosophical studies, it is essential to have a sound knowledge and mastery of these fields. Finally, many ethical questions can now be approached through the angle of game theory, another area of mathematics.

Conclusion:

Overall, even if the history of philosophy and most of the continental tradition of philosophy do not need to study mathematics to be understood, the practice of philosophy requires at least mastery of logic, often related with mathematical proof. In any case, studying mathematics before philosophy would be helpful, especially for methodology: from the philosophers of ancient Greece to the founders of analytic philosophy, the ability to think mathematically has been considered essential for a clear philosophical thought. From my own experience, I can say how much the study of mathematics, even in its basic form, has been useful for my reasoning, teaching and understanding of classical as well as contemporary texts. Recognising this connection should remind us that philosophy, at its best, is more than an exercise in opinion or rhetoric. It is a disciplined search for truth, conducted through reasoning that must withstand the same standards of coherence and validity that govern mathematics. In this way, mathematics does not just support philosophy: it reveals what philosophy aspires to be.

Bibliography :

To learn more about argumentation:

– Stanford encyclopedia of philosophy, “Argument and Argumentation”, Argument and Argumentation (Stanford Encyclopedia of Philosophy)

To learn more about philosophy of mathematics:

– Introduction à l’épistémologie des mathématiques, Olivier Fouet, Ellipses, 2025 – Stanford encyclopedia of philosophy, “Philosophy of mathematics”, Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) – Introduction to mathematical philosophy, Bertrand Russell, Taylor Francis Ltd, 2023

[1] Russell, « The value of philosophy », in Problems of philosophy, Oxford university press, 2001