Introduction

Philosophy was born in Greece because in that country there were some totally new ideas about mathematics, art, politics and emotions. Then, Badiou suggests that philosophy really only develops in this system of 4 "truths": science, art, politics and love

I. Mathematics must be saved

Mathematics requires to become a Subject whose freedom, far from being opposed to discipline, demands it. When you work on a mathematical problem, the discovery of the solution is not some sort of blind wandering but rather the determination of a path that is always lined by the obligations of overall consistency and demonstrative rules.

Lacan said that "desire and law are not opposites, but dialectically identical". Mathematics, for its part, combines intuition and proof in a unique way, in which philosophy has a lot to say.

Sartre was the one who revealed philosophy to Badiou, but there was an incompatibility with mathematics: "Science is zilch; morality's an asshole".

Majority of mathematicians elitist relationships with the discipline. There is a need for philosophy as well as a democratization of both subjects. They should be taught a lot sooner.

So the divergence between mathematics and philosophy also stems from the fact that philosophy, the reactionary figure of “the new philosopher,” has undergone an incredible trivialization of its status. In mathematics, this is not the case.

II. Philosophy and Mathematics, or the Story of an Old Couple

Mathematics is part of democratic thought, which moreover appeared in Greece at the same time. There is a close link between mathematics, democracy and philosophy.

Spinoza said that "If mathematics hadn't existed, man would have remained in ignorance, in particular, because he would have continued to explain everything by final causes, mythologies, or the influence of supernatural powers"

There are three types of knowledge according to Spinoza:

• Sensory and imaginative representation (ordinary ignorance)
• Methodical conceptual knowledge (mathematical proofs)
• Intuitive knowledge of God, philosophical knowledge

Kant said that "mathematics is necessary in order for philosophy to exist"

Kant's conception of mathematics is an a prioric conception. This means that the organization of mathematical thinking does not originate in concrete experience but is prior to it. According to Kant, if everyone is in agreement about a mathematical proof, it's not because it refers to anything that touches the thing in itself or the real world; it's because the structure of the human mind obeys a single paradigm, such that what will be a proof for one person will be a proof for another. Badiou thinks this point of view as a sophisticated version of the formalist thesis

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There has been a break between mathematics and philosophy. Philosophical romanticism, from Hegel to Sartrian existentialism, moved away from analytical and demonstrative rationality. There were also institutional reasons, mainly the separation between disciplines.

III. What is Mathematics About

Badiou points out that the question of the definition of mathematics is not a mathematical question, it is a philosophical one.

There are two main orientations:

• Realist or platonic. Holds that mathematics is an ontology. Mathematics is part of the thinking of what there is. There are structures that recur in everything that exists. This even explains why physics could not exist without mathematics. This path was taken by mathematicians like Gödel and Erdős.

• Formalist. Holds that mathematics is merely a language game whose rigor cannot claim to have an ongoing relationship with empirical reality. This path was taken by some analytical philosophers such as Wittgenstein.

Badiou chooses the realist current, holding that mathematics structures "exist" in a certain way. The position of Badiou is that mathematics is simply the science of being qua being, or what philosophers traditionally call ontology. What it means to think only in the being is to think only in the structure, the abstraction of the characteristics. This ontology can be developed for its own sake.

If we switch back to philosophy and think about the Hegelian system, we encounter that the negation of negation is not at all identical to the original affirmation. Its logic is therefore nonclassical. We cannot, then, link this philosophy with the axioms of mathematics everyone of us agrees with.

IV. An attempt at a Mathematics-based Metaphysics

Badiou calls trues (in plural) particular creations with universal value: works of art, scientific theories, politics of emancipation, passionate loves...

• Scientific theories are truths about being itself (mathematics) or the natural laws of the worlds we can have experimental knowledge.

• Political truths concern the organization of societies, laws of collective life and universal principles such as freedom and equality.

• Artistic truths are related to formal consistency of finished works that sublimate what our senses perceive.

• Truth of love relates to the dialectical power contained in the experiencing of the world, not from the point of view of the One, but from the point of view of the Two, and hence with a radical acceptance of the other person.

The two properties these categories fulfill are absolute and eternal.

Mathematics offers what contemporary relativism doesn't: the possibility of an absolute ontology. An absolute ontology is the existence of a universe of reference, a site for the thinking of being qua being, having four characteristics:

• Motionless or changeless. Any rational thought is itself foreign to that category

• Completely intelligible in its being on the basis of nothing. Non-atomic.

• Can only be described in terms of axioms. No experience or construction. Non-empirical

• Principle of maximality: any intellectual entity whose existence can be inferred without contradiction from the axioms that prescribe it exists by this very fact.

Badious' thesis: "Being is multiplicity. THh rational theory of the different possible forms of the multiple is set theory. A truth, like everything that exists, is also multiple."

V. Does Mathematics Bring Happiness?

Mathematics is wrapped in a sort of mystery, but it's ultimately a mystery in broad daylight. So it is true that there is the experience of a strange pleasure. If you have understood and grasped something it is because you have seen something you had never seen before, and it is this ineffable pleasure that will remain.

The real roots of happiness lie in subjective commitment to a truth procedure (remember the categories art, science, politics and love).

Badiou thinks that philosophy has no other objective than this: to help anyone understand, in the sphere of his or her own life experience, what a happy direction in life is. By doing philosophy you are freeing your Subject from the passive or empty life.

What is this happiness that's greater than the petty pleasures available in stores? Is there happiness greater than those pleasures? That's the big question of philosophy. Our societies, domesticated by Capital and commodity fetishism, say that there isn't. But philosophy says that we can open much larger windows onto this profit-driven outside. We can, as Plato's famous allegory has it, get out of the cave.

Conclusion

How can we get people to discover (or rediscover) mathematics? How can get them to love it? The notion of a problem solved plays an important role in the task. It encourages the children to face a problem and get interested in it.

First of all, an important key to motivation is to present problems of history of mathematics. For example, the question Socrates raised to a farmer: "Given you a square of land, how do you construct another square that double its area?". After a trial and error discussion finally the farmer was able to solve the puzzle.

The second point is to be armed with philosophy. What is interesting about mathematics is to wonder what mathematics is. It's known that three-year-old children are far better metaphysicians than eighteen-year-old ones, because they wonder about all the questions of metaphysics.

Philosophy is still an endangered discipline in the final years of school and mathematics a deplorable operator of social selection.