Mathematical object

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A tesseract or four-dimensional hypercube is an example of mathematical object Schlegel wireframe 8-cell.png
A tesseract or four-dimensional hypercube is an example of mathematical object

A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs.[ citation needed ] Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces. Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.

Contents

In philosophy of mathematics

Nature of mathematical objects

In Philosophy of mathematics, the concept of "objects" touches on topics of existence, identity, and the nature of reality. [1] In metaphysics, objects are often considered entities that possess properties and can stand in various relations to one another. [2] Philosophers debate whether 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 in the world, to the abstract, and it is in this latter which mathematical objects usually lie. 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. [3] [4] 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. [5]

Quine-Putnam indispensability

Quine-Putnam indispensability is an argument for the existence of mathematical objects based on their unreasonable effectiveness in the natural sciences. Every branch of science relies largely on large and often vastly different areas of mathematics. From physics' use of Hilbert spaces in quantum mechanics and differential geometry in general relativity to Biology's use of chaos thoery and combinatorics (see Mathematical biology), not only does mathematics help with predictions, it allows these areas to have an elegant language to express these ideas. Moreover, it is hard to imagine how areas like quantum mechanics and general relativity could have developed without their assistance from mathematics, and therefore, one could argue that mathematics is indispensable to these theories. It is because of this unreasonable effectiveness and indispensibility of mathematics that philosophers Willard Quine and Hilary Putnam argue that we should believe the mathematical objects for which these theories depend actually exist, that is, we ought to have an ontological commitment to them. The argument is described by the following syllogism: [6]

(Premise 1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(Premise 2) Mathematical entities are indispensable to our best scientific theories.

(Conclusion) We ought to have ontological commitment to mathematical entities

This argument resonates with a philosophy in applied mathematics called Naturalism [7] (or sometimes Predicativism) [8] which states that the only authoritative standards on existence are those of science.

Schools of thought

Platonism

Plato depicted in The School of Athens by Raphael Sanzio Plato-raphael.jpg
Plato depicted in The School of Athens by Raphael Sanzio

Platonism asserts that mathematical objects are seen as real, abstract entities that exist independently of human thought, often in some Platonic realm. Just as physical objects like electrons and planets exist, so do numbers and sets. And just as statements about electrons and planets are true or false as these objects contain perfectly objective properties, so are statements about numbers and sets. Mathematicians discover these objects rather than invent them. [9] [10] (See also: Mathematical Platonism)

Some some notable platonists include:

  • Plato : The ancient Greek philosopher who, though not a mathematician, laid the groundwork for Platonism by positing the existence of an abstract realm of perfect forms or ideas, which influenced later thinkers in mathematics.
  • Kurt Gödel : A 20th-century logician and mathematician, Gödel was a strong proponent of mathematical Platonism, and his work in model theory was a major influence on modern platonism
  • Roger Penrose : A contemporary mathematical physicist, Penrose has argued for a Platonic view of mathematics, suggesting that mathematical truths exist in a realm of abstract reality that we discover. [11]

Nominalism

Nominalism denies the independent existence of mathematical objects. Instead, it suggests that they are merely convenient fictions or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects don't have an existence beyond the symbols and concepts we use. [12] [13]

Some notable nominalists incluse:

  • Nelson Goodman : A philosopher known for his work in the philosophy of science and nominalism. He argued against the existence of abstract objects, proposing instead that mathematical objects are merely a product of our linguistic and symbolic conventions.
  • Hartry Field : A contemporary philosopher who has developed the form of nominalism called "fictionalism," which argues that mathematical statements are useful fictions that don't correspond to any actual abstract objects. [14]

Logicism

Logicism asserts that all mathematical truths can be reduced to logical truths, and all objects forming the subject matter of those branches of mathematics are logical objects. In other words, mathematics is fundamentally a branch of logic, and all mathematical concepts, theorems, and truths can be derived from purely logical principles and definitions. Logicism faced challenges, particularly with the Russillian axioms, the Multiplicative axiom (now called the Axiom of Choice) and his Axiom of Infinity, and later with the discovery of Gödel’s incompleteness theorems, which showed that any sufficiently powerful formal system (like those used to express arithmetic) cannot be both complete and consistent. This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program. [15]

Some notable logicists include:

  • Gottlob Frege : Frege is often regarded as the founder of logicism. In his work, Grundgesetze der Arithmetik (Basic Laws of Arithmetic), Frege attempted to show that arithmetic could be derived from logical axioms. He developed a formal system that aimed to express all of arithmetic in terms of logic. Frege’s work laid the groundwork for much of modern logic and was highly influential, though it encountered difficulties, most notably Russell’s paradox, which revealed inconsistencies in Frege’s system. [16]
  • Bertrand Russell : Russell, along with Alfred North Whitehead, further developed logicism in their monumental work Principia Mathematica . They attempted to derive all of mathematics from a set of logical axioms, using a type theory to avoid the paradoxes that Frege’s system encountered. Although Principia Mathematica was enormously influential, the effort to reduce all of mathematics to logic was ultimately seen as incomplete. However, it did advance the development of mathematical logic and analytic philosophy. [17]

Formalism

Mathematical formalism treats objects as symbols within a formal system. The focus is on the manipulation of these symbols according to specified rules, rather than on the objects themselves. One common understanding of formalism takes mathematics as not a body of propositions representing an abstract piece of reality but much more akin to a game, bringing with it no more ontological commitment of objects or properties than playing ludo or chess. In this view, mathematics is about the consistency of formal systems rather than the discovery of pre-existing objects. Some philosphers consider logicism to be a type of formalism. [18]

Some notable formalists include:

  • David Hilbert : A leading mathematician of the early 20th century, Hilbert is one of the most prominent advocates of formalism. He believed that mathematics is a system of formal rules and that its truth lies in the consistency of these rules rather than any connection to an abstract reality. [19]
  • Hermann Weyl : German mathematician and philosopher who, while not strictly a formalist, contributed to formalist ideas, particularly in his work on the foundations of mathematics. [20]

Constructivism

Mathematical constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. [21] There are many forms of constructivism. [22] These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of mathematicians Shanin and Markov, and Bishop's program of constructive analysis. [23] Constructivism also includes the study of constructive set theories such as Constructive Zermelo–Fraenkel and the study of philosophy.

Structuralism

Structuralism suggests that mathematical objects are defined by their place within a structure or system. The nature of a number, for example, is not tied to any particular thing, but to its role within the system of arithmetic. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature. [24] [25]

Some notable structuralists include:

  • Paul Benacerraf : A philosopher known for his work in the philosophy of mathematics, particularly his paper "What Numbers Could Not Be," which argues for a structuralist view of mathematical objects.
  • Stewart Shapiro : Another prominent philosopher who has developed and defended structuralism, especially in his book Philosophy of Mathematics: Structure and Ontology. [26]

Objects versus mappings

In mathematics, a map or mapping, is a function in the general sense; here as in the association of any of the four colored shapes in X to its color in Y. Function color example 3.svg
In mathematics, a map or mapping, is a function in the general sense; here as in the association of any of the four colored shapes in X to its color in Y.

Frege famously distinguished between functions and objects. [28] According to his view, a function is a kind of ‘incomplete’ entity that maps arguments to values, and is denoted by an incomplete expression, whereas an object is a ‘complete’ entity and can be denoted by a singular term. Frege reduced properties and relations to functions and so these entities are not included among the objects. Some authors make use of Frege’s notion of ‘object’ when discussing abstract objects [29] . But though Frege’s sense of ‘object’ is important, it is not the only way to use the term. Other philosophers include properties and relations among the abstract objects. And when the background context for discussing objects is type theory, properties and relations of higher type (e.g., properties of properties, and properties of relations) may be all be considered ‘objects’. This latter use of ‘object’ is interchangeable with ‘entity.’ It is this more broad interpretation that mathematicians mean when they use the term 'object'. [30]

List of mathematical objects by branch

Algebra

Elementary algebra

Linear algebra

Abstract algebra

Calculus and Mathematical Analysis

Nonstandard analysis

Differential equations

Category theory

Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In proof theory, proofs and theorems are also mathematical objects

Combinatorics

Computer science and Theoretical computer science

Geometry

Euclidian geometry

Fractal geometry

Graph theory

Mathematical logic

Number theory

Set theory

Topology

Algebraic topology

Differential topology

See also

Related Research Articles

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.

<span class="mw-page-title-main">Existence</span> State of being real

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.

Neutral monism is an umbrella term for a class of metaphysical theories in the philosophy of mind, concerning the relation of mind to matter. These theories take the fundamental nature of reality to be neither mental nor physical; in other words it is "neutral".

Ontology is the philosophical study of being. 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 what all entities have in common and how they are divided into fundamental classes, known as categories. An influential distinction is between particular and universal entities. Particulars are unique, non-repeatable entities, like the person Socrates. Universals are general, repeatable entities, like the color green. Another contrast is between concrete objects existing in space and time, like 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, employing categories such as substance, property, relation, state of affairs, and event.

<span class="mw-page-title-main">Problem of universals</span> Philosophical question of whether properties exist and, if so, what they are

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?"

Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.

<span class="mw-page-title-main">Gottlob Frege</span> German philosopher, logician, and mathematician (1848–1925)

Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.

In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, human beings and planets while things like numbers, sets and propositions are abstract objects. There is no general consensus as to what the characteristic marks of concreteness and abstractness are. Popular suggestions include defining the distinction in terms of the difference between (1) existence inside or outside space-time, (2) having causes and effects or not, (3) having contingent or necessary existence, (4) being particular or universal and (5) belonging to either the physical or the mental realm or to neither. Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete. So under most interpretations, all these views would agree that, for example, plants are concrete objects while numbers are abstract objects.

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<span class="mw-page-title-main">Platonism</span> Philosophical system

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.

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In metaphysics and ontology, nonexistent objects are a concept advanced by Austrian philosopher Alexius Meinong in the 19th and 20th centuries within a "theory of objects". He was interested in intentional states which are directed at nonexistent objects. Starting with the "principle of intentionality", mental phenomena are intentionally directed towards an object. People may imagine, desire or fear something that does not exist. Other philosophers concluded that intentionality is not a real relation and therefore does not require the existence of an object, while Meinong concluded there is an object for every mental state whatsoever—if not an existent then at least a nonexistent one.

<span class="mw-page-title-main">Quietism (philosophy)</span> View on the purpose of philosophy

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In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation. In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.

Meinong's jungle is the name given by Richard Routley (1980) to the repository of non-existent objects in the ontology of Alexius Meinong.

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<span class="mw-page-title-main">Quine–Putnam indispensability argument</span> Argument in the philosophy of mathematics

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 Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.

This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation.

References

Cited sources

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  3. ↑ Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics. Oxford University Press. ISBN   0198236158
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  11. ↑ Roibu, Tib (2023-07-11). "Sir Roger Penrose". Geometry Matters. Retrieved 2024-08-27.
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  14. ↑ Field, Hartry (2016-10-27). Science without Numbers. Oxford University Press. ISBN   978-0-19-877791-5.
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  19. ↑ Simons, Peter (2009). "Formalism". Philosophy of Mathematics. Elsevier. p. 292. ISBN   9780080930589.
  20. ↑ Bell, John L.; Korté, Herbert (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Hermann Weyl", The Stanford Encyclopedia of Philosophy (Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
  21. ↑ Bridges, Douglas; Palmgren, Erik; Ishihara, Hajime (2022), Zalta, Edward N.; Nodelman, Uri (eds.), "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
  22. ↑ Troelstra, Anne Sjerp (1977a). "Aspects of Constructive Mathematics". Handbook of Mathematical Logic. 90: 973–1052. doi:10.1016/S0049-237X(08)71127-3
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  25. ↑ Reck, Erich; Schiemer, Georg (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Structuralism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
  26. ↑ Philosophy of Mathematics: Structure and Ontology. Oxford University Press, 1997. ISBN 0-19-513930-5
  27. ↑ Halmos, Paul R. (1974). Naive set theory. Undergraduate texts in mathematics. New York: Springer-Verlag. p. 30. ISBN   978-0-387-90092-6.
  28. ↑ Marshall, William (1953). "Frege's Theory of Functions and Objects". The Philosophical Review. 62 (3): 374–390. doi:10.2307/2182877. ISSN   0031-8108.
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  31. ↑ Hu, Sze-Tsen (1971) [1959]. "Chapter VII. Cohomotopy Groups" (PDF). Homotopy theory (PDF). Academic Press. pp. 205–228. LCCN   59-11526 . Retrieved August 28, 2024.

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