How does mathematical Platonism describe numbers and geometric objects?

Mathematical Platonism describes numbers, geometric objects, and other mathematical entities as abstract objects. According to this view, these objects possess the following characteristics:12

• Nonspatiotemporal and Nonphysical: They are not located anywhere in the physical universe and are not composed of matter.12
• Nonmental: They exist independently of human minds and thoughts, much like the Moon exists independently of our ideas about it.12
• Unchanging and Noncausal: Because they lack physical extension and location, they are eternal and cannot enter into cause-and-effect relationships.12

Platonists maintain that mathematics is the study of these abstract structures. While numbers are often the primary example, this classification extends to functions, sets, vectors, and geometric objects like circles.12

Some contemporary philosophers, such as those advocating for structuralism, refine this by suggesting that individual mathematical objects are not "objects" in the ordinary sense but are instead positions within larger patterns or structures. In this view, a number like 4 is simply a position within the sequence of positive integers, defined by its relations to other numbers rather than by internal properties.2