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**Abstract object theory** is a branch of metaphysics regarding abstract objects. Originally devised by metaphysicist Edward Zalta in 1999,^{ [1] } the theory was an expansion of mathematical Platonism.

**Metaphysics** is the branch of philosophy that examines the fundamental nature of reality, including the relationship between mind and matter, between substance and attribute, and between possibility and actuality. The word "metaphysics" comes from two Greek words that, together, literally mean "after or behind or among [the study of] the natural". It has been suggested that the term might have been coined by a first century CE editor who assembled various small selections of Aristotle’s works into the treatise we now know by the name *Metaphysics*.

**Abstract** and **concrete** are classifications that denote whether the object that a term describes has physical referents. Abstract objects have no physical referents, whereas concrete objects do. They are most commonly used in philosophy and semantics. Abstract objects are sometimes called * abstracta* and concrete objects are sometimes called

*Abstract Objects: An Introduction to Axiomatic Metaphysics* (1983) is the title of a publication by Edward Zalta that outlines abstract object theory.^{ [2] }

On Zalta's account, there are two modes of predication: some objects (the ordinary concrete ones around us, like tables and chairs) "exemplify" properties, while others (abstract objects like numbers, and what others would call "non-existent objects", like the round square, and the mountain made entirely of gold) merely "encode" them.^{ [3] } While the objects that exemplify properties are discovered through traditional empirical means, a simple set of axioms allows us to know about objects that encode properties.^{ [4] } For every set of properties, there is exactly one object that encodes exactly that set of properties and no others.^{ [5] } This allows for a formalized ontology.

In mathematical logic, a **predicate** is commonly understood to be a Boolean-valued function *P*: *X*→ {true, false}, called the predicate on *X*. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

**Ontology** is the philosophical study of being. More broadly, it studies concepts that directly relate to being, in particular becoming, existence, reality, as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy, and subdivided according to similarities and differences.

- Abstract particulars
- Abstractionism (philosophy of mathematics)
- Linnebo, Øystein
- Mally, Ernst
- Mathematical universe hypothesis
- Modal neo-logicism
- Observation
- Present
- Self

**Abstract particulars** are metaphysical entities which are both abstract objects and particulars.

**Øystein Linnebo** is a Norwegian philosopher. As of August 2012 he is currently employed in the Department of Philosophy at the University of Oslo, having earlier held a position as Professor of Philosophy at Birkbeck College, University of London. He is a fellow of the Norwegian Academy of Science and Letters.

**Ernst Mally** was an Austrian philosopher affiliated with the so-called Graz School of phenomenological psychology. A pupil of Alexius Meinong, he was one of the founders of deontic logic and is mainly known for his contributions in that field of research.

**Category theory** formalizes mathematical structure and its concepts in terms of a labeled directed graph called a *category*, whose nodes are called *objects*, and whose labelled directed edges are called *arrows*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

**Logical positivism** and **logical empiricism**, which together formed **neopositivism**, was a movement in Western philosophy whose central thesis was verificationism, a theory of knowledge which asserted that only statements verifiable through empirical observation are meaningful. The movement flourished in the 1920s and 1930s in several European centers.

In metaphysics, **nominalism** is a philosophical view which denies the existence of universals and abstract objects, but affirms the existence of general or abstract terms and predicates. There are at least two main versions of nominalism. One version denies the existence of universals – things that can be instantiated or exemplified by many particular things. The other version specifically denies the existence of abstract objects – objects that do not exist in space and time.

In metaphysics, the **problem of universals** refers to the question of whether properties exist, and if so, what they are. Properties are qualities or relations that two or more entities have in common. The various kinds of properties, such as qualities and relations, are referred to as universals. For instance, one can imagine three cup holders on a table that have in common the quality of *being circular* or *exemplifying circularity*, or two daughters that have in common *being the female offsprings of Frank.* There are many such properties, such as being human, red, male or female, liquid, big or small, taller than, father of, etc. While philosophers agree that human beings talk and think about properties, they disagree on whether these universals exist in reality or merely in thought and speech.

**Platonic realism** is a philosophical term usually used to refer to the idea of realism regarding the existence of universals or abstract objects after the Greek philosopher Plato. As universals were considered by Plato to be ideal forms, this stance is ambiguously also called Platonic idealism. This should not be confused with idealism as presented by philosophers such as George Berkeley: as Platonic abstractions are not spatial, temporal, or mental, they are not compatible with the later idealism's emphasis on mental existence. Plato's Forms include numbers and geometrical figures, making them a theory of mathematical realism; they also include the Form of the Good, making them in addition a theory of ethical realism.

**Set theory** is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

In metaphysics, a **universal** is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For example, suppose there are two chairs in a room, each of which is green. These two chairs both share the quality of "chairness", as well as greenness or the quality of being green; in other words, they share a "universal". There are three major kinds of qualities or characteristics: types or kinds, properties, and relations. These are all different types of universals.

* Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being* is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000,

The **philosophy of mathematics** is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.

**Foundations of mathematics** is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

Several ways have been proposed to construct the natural numbers using set theory. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Frege and by Russell.

In mathematical logic, an **axiom schema** generalizes the notion of axiom.

**Object theory** is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.

**Edward N. Zalta** is a senior research scholar at the Center for the Study of Language and Information. He received his PhD in philosophy from the University of Massachusetts Amherst in 1980. Zalta has taught courses at Stanford University, Rice University, the University of Salzburg, and the University of Auckland. Zalta is also the Principal Editor of the *Stanford Encyclopedia of Philosophy*.

The "**round square copula**" is a common example of the dual copula strategy used in reference to the problem of nonexistent objects as well as their relation to problems in modern philosophy of language. The issue arose, most notably, between the theories of Alexius Meinong, Bertrand Russell—Gilbert Ryle playing a minor part as well in the eventual dismissal of Meinong's object theory.

A **mathematical object** is an abstract object arising in mathematics. The concept is studied in philosophy of mathematics.

In algebra, which is a broad division of mathematics, **abstract algebra** is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term *abstract algebra* was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

**Structuralism** is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the integer 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any integer is defined by their respective place in this structure of the number line. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.

- ↑ "The Theory of Abstract Objects". February 10, 1999. Retrieved March 29, 2013.
- ↑ Zalta, Edward N.
*Abstract Objects: An Introduction to Axiomatic Metaphysics*. D. Reidel Publishing Company. 1983. - ↑ Edward N. Zalta,
*Abstract Objects,*33. - ↑ Edward N. Zalta,
*Abstract Objects,*36. - ↑ Edward N. Zalta,
*Abstract Objects*, 35.

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