# Mathematical object

A **mathematical object** is an

## In philosophy of mathematics

### Nature of mathematical objects

In

^{[4]}

^{[5]}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.

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#### Quine-Putnam indispensability

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(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^{[8]} (or sometimes Predicativism)^{[9]} which states that the only authoritative standards on existence are those of science.

### Schools of thought

#### Platonism

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^{[11]}(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.^{[12]}

#### Nominalism

^{[13]}

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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.^{[15]}

#### Logicism

Logicism asserts that all mathematical truths can be reduced to logical truths, and a*ll* 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.^{[16]}

Some notable logicists include:

- modern logic and was highly influential, though it encountered difficulties, most notably Russell’s paradox, which revealed inconsistencies in Frege’s system.
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**
**#### 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 philosophers consider logicism to be a type of formalism.^{[19]}

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.^{[20]}**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.^{[21]}

#### Constructivism

#### 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.^{[25]}^{[26]}

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*.^{[27]}

### Objects versus mappings

Frege famously distinguished between *functions* and *objects*.^{[29]} 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.^{[30]} 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'.^{[31]}

## See also

- Abstract object
- Exceptional object
- Impossible object
- List of mathematical objects
- List of mathematical shapes
- List of shapes
- List of surfaces
- List of two-dimensional geometric shapes
- Mathematical structure

## References

**Cited sources**

**^***Oxford English Dictionary*, s.v. “Mathematical (*adj.*), sense 2,” September 2024. "*Designating or relating to objects apprehended not by sense perception but by thought or abstraction*."**^**Rettler, Bradley; Bailey, Andrew M. (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Object",*The Stanford Encyclopedia of Philosophy*(Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28- ISBN 978-0-521-82629-7.
**^**Falguera, José L.; Martínez-Vidal, Concha; Rosen, Gideon (2022), Zalta, Edward N. (ed.), "Abstract Objects",*The Stanford Encyclopedia of Philosophy*(Summer 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28**^**Horsten, Leon (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-29**^**Colyvan, Mark (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Indispensability Arguments in the Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28**^**Paseau, Alexander (2016), Zalta, Edward N. (ed.), "Naturalism in the Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Winter 2016 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28**^**Horsten, Leon (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28**^**Linnebo, Øystein (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Platonism in the Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27**^**"Platonism, Mathematical | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.**^**Roibu, Tib (2023-07-11). "Sir Roger Penrose".*Geometry Matters*. Retrieved 2024-08-27.**^**Bueno, Otávio (2020), Zalta, Edward N. (ed.), "Nominalism in the Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Fall 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27**^**"Mathematical Nominalism | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.- ISBN 978-0-19-877791-5.
**^**Tennant, Neil (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Logicism and Neologicism",*The Stanford Encyclopedia of Philosophy*(Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27**^**"Frege, Gottlob | Internet Encyclopedia of Philosophy". Retrieved 2024-08-29.- ISBN 978-0-521-87267-6. Retrieved 2023-08-28.
**^**Weir, Alan (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Formalism in the Philosophy of Mathematics",*The Stanford Encyclopedia of Philosophy*(Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28- ISBN 9780080930589.
**^**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**^**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- doi:10.1016/S0049-237X(08)71127-3
- ISBN 4-87187-714-0.
**^**"Structuralism, Mathematical | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.**^**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- ISBN 0-19-513930-5
**.****.****^**Hale, Bob, "Abstract objects",*Routledge Encyclopedia of Philosophy*, London: Routledge, retrieved 2024-08-28**^**Falguera, José L.; Martínez-Vidal, Concha; Rosen, Gideon (2022), Zalta, Edward N. (ed.), "Abstract Objects",*The Stanford Encyclopedia of Philosophy*(Summer 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28

**Further reading**

- Azzouni, J., 1994.
*Metaphysical Myths, Mathematical Practice*. Cambridge University Press. - Burgess, John, and Rosen, Gideon, 1997.
*A Subject with No Object*. Oxford Univ. Press. - Davis, Philip and Reuben Hersh, 1999 [1981].
*The Mathematical Experience*. Mariner Books: 156–62. - Gold, Bonnie, and Simons, Roger A., 2011.
*Proof and Other Dilemmas: Mathematics and Philosophy*. Mathematical Association of America. - Hersh, Reuben, 1997.
*What is Mathematics, Really?*Oxford University Press. - Sfard, A., 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P.,
*et al.*,*Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design*. Lawrence Erlbaum. - Stewart Shapiro, 2000.
*Thinking about mathematics: The philosophy of mathematics*. Oxford University Press.

## External links

*Stanford Encyclopedia of Philosophy*: "Abstract Objects"—by Gideon Rosen.- Wells, Charles. "Mathematical Objects".
- AMOF: The Amazing Mathematical Object Factory
- Mathematical Object Exhibit