Concept

Mathematical object

Overview

A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical objects in proof theory.

In philosophy of mathematics, the concept of "mathematical objects" touches on topics of existence, identity, and the nature of reality. In metaphysics, objects are often considered entities that possess properties and can stand in various relations to one another. Philosophers debate whether mathematical objects have an independent existence outside of human thought (realism), or if their existence is dependent on mental constructs or language (idealism and nominalism). Objects can range from the concrete: such as physical objects usually studied in applied mathematics, to the abstract, studied in pure mathematics. What constitutes an "object" is foundational to many areas of philosophy, from ontology (the study of being) to epistemology (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the physical world, raising questions about their ontological status. There are varying schools of thought which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

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The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
States what Mathematical object is; the author is tied to Mathematical object in the knowledge graph.
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The notion of continuity depends upon that of _order_, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry…
Bertrand Russell, Mysticism and Logic and Other Essays
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In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned. Poincare, the great French mathematician, held that our actual choice among these geometries is guided purely by convention, and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature. By 'convention' I understand…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Mathematical object is; the author is tied to Mathematical object in the knowledge graph.
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Related ideas

Opposing perspectives

Physical object

Place in the tradition

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