By looking at three significant examples in analysis, geometry and dynamical systems, I propose the possibility of having two levels of realism in mathematics: the upper one, the one of entities; and a subordinated ground one, the one of objects. The upper level (entities) is more the one of ‘operations’, of mathematics in action, of the dynamics of mathematics, whereas the ground floor (objects) is more dedicated to culturally well-defined objects inherited from our perception of the physical or real world. I will show that the upper level is wider than the ground level, therefore foregrounding the possibility of having in mathematics entities without underlying objects. In the three examples treated in this article, this splitting of levels of reality is created directly by the willingness to preserve different symmetries, which take the form of identities or equivalences. Finally, it is proposed that mathematical Platonism is – in fine – a true branch of mathematics in order for mathematicians to avoid the temptation of falling into the Platonist alternative ‘everything is real’/‘nothing is real’.

通过分析分析，几何和动力学系统中的三个重要示例，我提出了在数学中具有两个层次的现实主义的可能性：上层是实体，一个是实体；第二层是实体。以及从属地面之一，即对象之一。上层（实体）更多是“操作”，数学在行动中，数学动力学中的一种，而底层（对象）则更专注于文化定义明确的对象，这些对象是从我们对物理或物理的感知中继承而来的真实世界。我将说明高层比地面宽，因此提出了在没有底层对象的情况下具有数学实体的可能性。在本文讨论的三个示例中，现实水平的这种分裂是由保持不同对称性的意愿直接造成的，采用身份或等价形式。最后，建议数学柏拉图主义是数学的一个真正分支，以便数学家避免陷入柏拉图式的“万物皆真实” /“万物皆不真实”的诱惑。