That trivialism ought to be rejected is almost universally held. I argue that the rejection of trivialism should be held in suspicion and that there are good reasons for thinking that trivialism is true. After outlining in chapter 1 the place of trivialism in the history of philosophy, I begin in chapter 2 an outline and defense of the various arguments in favor of the truth of trivialism. I defend four such arguments: an argument from the Curry Paradox; and argument from the Characterization Principle; an argument from the Principle of Sufficient Reason; and an argument from the truth of possibilism. / In chapter 3 I build a case for thinking that the denial of trivialism is impossible. I begin by arguing that the denial of some view is the assertion of an alternative view. I show that there is no such view as the alternative to trivialism and so the denial of trivialism is impossible. I then examine an alternative view of the nature of denial – that denial is not reducible to an assertion but is a sui generis speech act. It follows given such an account of denial that the denial of trivialism is possible. I respond to this in two ways. First, I give reason for thinking that this is not a plausible account of denial. Secondly, I show that even if it is successful, the denial of trivialism is still unassertable, unbelievable, and severely limited in its rationality. / In chapter 4 I examine two important arguments that purport to show that it is impossible to believe in trivialism: one from Aristotle and a more recent one from Graham Priest. According to Aristotle, it is not possible to believe in trivialism because such a belief is incompatible with being able to act in a discriminating manner. According to Priest, belief in trivialism is incompatible with being able to act with a purpose. I show that there are a number of ways to respond to such arguments, and so it is far from obvious that it is impossible to believe in trivialism. / In chapter 5 I reply to one of the few sustained arguments against the truth of trivialism. According to this argument, trivialism cannot be true because it entails that every observable state of affairs is contradictory - which is clearly not the case. After raising a number of objections to this line of reasoning, I argue that a contradictory state of affairs will necessarily appear consistent. As such, that the world appears consistent is not a good reason for thinking that it fails to be contradictory. / In chapter 6 I defend the claim that the observable world is indeed contradictory in the way that trivialism implies. I show that a dialetheic solution to Zeno’s paradox of the arrow requires one to postulate that a body in motion is located at every point of the path of its journey at every instant of the journey.

Ce travail vise à identifier et définir une nouvelle tendance du platonisme mathématique que l'on propose d'appeler « platonisme sobre ». Comme le platonisme mathématiques classique, le platonisme sobre admet la fiabilité de la connaissance mathématique et l'existence d'objets mathématiques. Contrairement au platonisme mathématique classique, son engagement ontologique aux objets mathématiques est atténué par des arguments démontrant qu'un monde sans objets mathématiques ne serait pas cohérent. Quand bien même il le serait, on ne pourrait pas accepter de rejeter les mathématiques pour des raisons philosophiques. Le platonisme sobre suggère donc de concilier l'enquête philosophique avec la pratique mathématique. Dans le premier chapitre, on analyse le platonisme mathématique classique. Le deuxième, troisième, quatrième et cinquième chapitre sont respectivement dévoués à l'examen du platonisme pur-sang, du structuralisme ante rem, de la théorie de l'objet abstrait du trivialisme. Cette théories sont explicitement platoniciennes, mais seulement sobrement engagées dans l'existence d'objets mathématiques. Elles traitent l'existence d'objets mathématiques, la possibilité d'accéder à la connaissance mathématique, le sens des énoncés mathématiques et la référence de leur termes en tant que questions philosophiquement pertinentes. Cependant, elles sont dévouées à l'élaboration d'une description précise des mathématiques en tant que telles. Dans le dernier chapitre, le platonisme sobre est défini comme une description méthodologique de la façon dont les mathématiques sont réalisées, plutôt que comme une prescription normative de la façon dont les mathématiques doivent être réalisées. / This work aims at identifying and defining a new trend in mathematical platonism I propose to call “Sober Platonism”. As classical mathematical platonism, Sober Platonism acknowledges the reliability of mathematical knowledge and the existence of mathematical objects. But, contrary to classical mathematical Platonism, its ontological commitment with mathematical objects is softened by several arguments that demonstrate the claim that a world without mathematical abjects wouldn't be consistent. And even if it would be, rejecting mathematics for philosophical reasons wouldn't be acceptable. As a result, Sober Platonism suggests to lined up philosophical inquiry with mathematics as practiced. In the first chapter, I analyzed classical mathematical Platonism. The second, third, fourth and fifth chapters are devoted to the examination of full-blooded Platonism, ante rem Structuralism, Object Theory and Trivialism respectively. This theories are explicitly platonist, but only soberly committed with the existence of mathematical abjects. They take into account the existence of mathematical abjects, the possibility to access to mathematical knowledge, the meaning of mathematical statements and the reference of their terms as philosophically relevant questions. But they are firstly focused on providing an accurate description of mathematics by its own. In the last chapter, Sober Platonism is defined as a methodological description of how mathematics is performed, rather than as a normative prescription of how mathematics should be performed. In conclusion, Sober Platonism admittedly achieves the goal of providing both philosophy and mathematics with a proper domain of inquiry.