Mathematical object

A mathematical object is an abstract concept used in mathematics. These objects can be values we give to a symbol and use in formulas. Common examples include numbers, expressions, shapes, functions, and sets.

Mathematical objects can also be very complex. Things like theorems, proofs, and formal theories are studied as mathematical objects in areas such as proof theory.

Thinking about what mathematical objects really are leads to big questions in the philosophy of mathematics. Some believe these objects exist on their own, apart from our thinking (realism), while others think they depend on our ideas and language (idealism and nominalism). Objects can be simple, like things we see in the real world and study in applied mathematics, or very abstract, as in pure mathematics. Understanding what an “object” means is important in many areas of philosophy, such as ontology (the study of what exists) and epistemology (the study of knowledge). In mathematics, these objects are often thought to exist beyond the physical world.

In philosophy of mathematics

The Quine-Putnam indispensability argument says that mathematical objects must exist because they are important in science. Many areas of science, from physics to biology, use math to make predictions and share ideas. Without math, it would be hard to develop theories like quantum mechanics.

There are different ideas about what mathematical objects are. Platonism says they are real things that exist on their own, like numbers and shapes. Nominalism says math objects are just helpful ideas we make to describe things. Logicism says that math is really part of logic. Formalism sees math as a game of symbols and rules. Constructivism says we need to find a specific example to prove something in math exists. Structuralism says math objects are defined by their role in a system, not by anything special about them.