The Mathematical Universe [C1.I.1]

1. The Objects of Mathematical Science

1.1. Objects and Intuition

Traditionally, mathematicians have sought to conceptualize and theorize intuitive notions such as numbers, figures, magnitudes, shapes… In modern and contemporary times, the scope of mathematical objects has expanded alongside the development of mathematics itself, as well as the advent of experimental science and modern technology, to include spaces, functions, sets, formal languages, and more.

Moreover, mathematicians have always been interested in major topics such as infinity, continuity, and space.

Like any other science, mathematics is rooted in the intuition of its object(s). The aforementioned examples highlight the historical diversity of mathematical science’s objects and raise the question of the discipline’s unity.

It seems that what all these examples—and the new ones that emerge—have in common is that mathematics essentially deals, in one way or another, with multiplicities: numbers represent the form of finite multiplicities, figures and shapes are geometric multiplicities, magnitudes are ideal multiplicities, and so on.