"Mathematicians do not study objects, but relations between objects"

Henri Poincaré’s assertion that mathematicians focus not on objects themselves but on the relations binding them together reveals a deep insight into the true nature of mathematical inquiry. Rather than treating numbers, shapes, or abstract structures as entities with intrinsic significance, mathematics concerns itself with the patterns, connections, and systematic links shared among these entities. For instance, the number five is not inherently interesting; its significance emerges in how it relates to other numbers, through operations such as addition, subtraction, or more abstractly, its behavior within the broader system of arithmetic.

This perspective clarifies why so much of mathematics abstracts away from the physical intuition of objects. A triangle, a group, or a function are not studied for their concrete reality but for how they fit into networks of definitions, theorems, and logical relationships. The actual symbols or objects might be subject to transformations, replacements, or redefinitions, but the structure of their interrelations remains consistent. By focusing on these relations, mathematicians are afforded incredible freedom to generalize from familiar situations, developing entire fields such as topology, group theory, or category theory, where the essence resides in the relationships and mappings, not the individual elements.

When studying geometry, for example, mathematicians no longer look solely at physical triangles and circles but at the properties preserved under transformations, such as congruence or similarity. Similarly, in algebra, the subject matter becomes the rules fields, rings, and groups obey, not the particular objects within them. This relational viewpoint is especially powerful when considering isomorphism: two apparently distinct structures are fundamentally the same if there exists a way to relate their components so that all operations and relations are preserved.

Through focusing on relations over objects, mathematics achieves a universality and scope far beyond the particulars. Mathematics becomes a study of possibility, how discrete elements can be woven into vast, interconnected structures, reclaiming its position as the language that describes not only quantities and forms but the very fabric of logical relationships.