The article deals with the concept of intuition (ϑεωρία, intuitio, Anschauung, Wesensschau) and the method of doubt in the philosophy of mathematics from Antiquity to the present day. In the classical philosophy of mathematics, there was no doubt about the attainability of reliable mathematical knowledge. Doubt was always present, but did not claim to deconstruct contemplation. The objects of contemplation are either the higher world of “ideas” (Plato, Neoplatonists), or innate ideas (René Descartes, Gottfried Leibniz), or a priori forms (Immanuel Kant), or something mysterious, perhaps the subconscious (Henri Poincaré). Edmund Husserl was the last classical philosopher to require intuition, and Kurt Gödel was the last mathematician. Modern mathematics is characterized by a departure from intuition, and therefore the philosophy of mathematics has fundamentally changed: now intuition is being questioned. The article provides a brief overview of the history of doubt, from Plato to the triumph of doubt in the second half of the 19th and early 20th centuries. The overthrow of intuition occurs in the works of Hans Hahn, Ludwig Wittgenstein, David Bloor and modern nominalists (Hartry Field). Often, intuition is replaced by a formal conclusion, logic, as was the case with Poincaré and David Hilbert. However, the deconstruction of intuition has deeper implications for the philosophy of mathematics than replacing it with logic. Logic, in turn, also needs to be justified. Why are modus ponens and the substitution rule considered reliable formal operations? Here logical evidence comes into play: Husserl showed that it is also a kind of intuition. However nothing should be obvious, including the simplest formal operations should be considered as purely conventional. What philosophy of mathematics awaits us in the future? The article makes the assumption that mathematics will be considered as a kind of game according to arbitrary rules. This is the current trend of its development. However, this trend is contradicted by the problem of applications of mathematics and Eugene Wigner’s riddle of “the unreasonable effectiveness of mathematics in the natural sciences.”