In analytic philosophy, anti-realism is the position that the truth of a statement rests on its demonstrability through internal logic mechanisms, such as the context principle or intuitionistic logic, in direct opposition to the realist notion that the truth of a statement rests on its correspondence to an external, independent reality. In anti-realism, this external reality is hypothetical and is not assumed.
Existence is the state of having being or reality in contrast to nonexistence and nonbeing. Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does not know whether the entity exists.
Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the fundamental categories of human understanding. Some philosophers, including Aristotle, designate metaphysics as first philosophy to suggest that it is more fundamental than other forms of philosophical inquiry.
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals – that which can be instantiated or exemplified by many particular things. The other version specifically denies the existence of abstract objects as such – objects that do not exist in space and time.
Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every entity within it. To articulate the basic structure of being, ontology examines the commonalities among all things and investigates their classification into basic types, such as the categories of particulars and universals. Particulars are unique, non-repeatable entities, such as the person Socrates, whereas universals are general, repeatable entities, like the color green. Another distinction exists between concrete objects existing in space and time, such as a tree, and abstract objects existing outside space and time, like the number 7. Systems of categories aim to provide a comprehensive inventory of reality by employing categories such as substance, property, relation, state of affairs, and event.
The problem of universals is an ancient question from metaphysics that has inspired a range of philosophical topics and disputes: "Should the properties an object has in common with other objects, such as color and shape, be considered to exist beyond those objects? And if a property exists separately from objects, what is the nature of that existence?"
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 share the quality of "chairness", as well as "greenness" or the quality of being green; in other words, they share two "universals". There are three major kinds of qualities or characteristics: types or kinds, properties, and relations. These are all different types of universals.
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
In logic and philosophy, a property is a characteristic of an object; for example, a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property, however, differs from individual objects in that it may be instantiated, and often in more than one object. It differs from the logical and mathematical concept of class by not having any concept of extensionality, and from the philosophical concept of class in that a property is considered to be distinct from the objects which possess it. Understanding how different individual entities can in some sense have some of the same properties is the basis of the problem of universals.
In philosophy and the arts, a fundamental distinction is between things that are abstract and things that are concrete. While there is no general consensus as to how to precisely define the two, examples include that things like numbers, sets, and ideas are abstract objects, while plants, dogs, and planets are concrete objects. Popular suggestions for a definition include that the distinction between concreteness versus abstractness is, respectively: between (1) existence inside versus outside space-time; (2) having causes and effects versus not; 3) being related, in metaphysics, to particulars versus universals; and (4) belonging to either the physical versus the mental realm. Another view is that it is the distinction between contingent existence versus necessary existence; however, philosophers differ on which type of existence here defines abstractness, as opposed to concreteness. Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete, such that most interpretations agree, for example, that rocks are concrete objects while numbers are abstract objects.
Modal realism is the view propounded by philosopher David Lewis that all possible worlds are real in the same way as is the actual world: they are "of a kind with this world of ours." It is based on four tenets: possible worlds exist, possible worlds are not different in kind from the actual world, possible worlds are irreducible entities, and the term actual in actual world is indexical, i.e. any subject can declare their world to be the actual one, much as they label the place they are "here" and the time they are "now".
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundamental level, Platonism affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness, and is the opposite of nominalism. This can apply to properties, types, propositions, meanings, numbers, sets, truth values, and so on. Philosophers who affirm the existence of abstract objects are sometimes called Platonists; those who deny their existence are sometimes called nominalists. The terms "Platonism" and "nominalism" also have established senses in the history of philosophy. They denote positions that have little to do with the modern notion of an abstract object.
Edward Nouri Zalta is an American philosopher who is a senior research scholar at the Center for the Study of Language and Information at Stanford University. He received his BA from Rice University in 1975 and his PhD from the University of Massachusetts Amherst in 1981, both in philosophy. 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.
Penelope Maddy is an American philosopher. Maddy is Emerita UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism and mathematical naturalism.
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.
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterisation, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.
In philosophy and specifically metaphysics, the theory of Forms, theory of Ideas, Platonic idealism, or Platonic realism is a theory widely credited to the Classical Greek philosopher Plato. The theory suggests that the physical world is not as real or true as "Forms". According to this theory, Forms—conventionally capitalized and also commonly translated as "Ideas"—are the non-physical, timeless, absolute, and unchangeable essences of all things, which objects and matter in the physical world merely imitate, resemble, or participate in. Plato speaks of these entities only through the characters in his dialogues who sometimes suggest that these Forms are the only objects of study that can provide knowledge.
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 number 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 natural number is defined by its respective place in that theory. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.
The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Van Orman Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.
