Amathematical object is anabstract concept arising inmathematics.^[1] Typically, a mathematical object can be a value that can be assigned to asymbol, and therefore can be involved informulas. Commonly encountered mathematical objects includenumbers,expressions,shapes,functions, andsets. Mathematical objects can be very complex; for example,theorems,proofs, and evenformal theories are considered as mathematical objects inproof theory.

From left to right, top to bottom: Atesseract orfour-dimensional hypercube, thegraph of abinary function, atrefoil knot (a kind ofmathematical knot), and a common hierarchy of sets of numbers.^[a]

InPhilosophy of mathematics, the concept of "mathematical objects" touches on topics ofexistence,identity, and thenature ofreality.^[2] Inmetaphysics, objects are often consideredentities that possessproperties and can stand in variousrelations to one another.^[3] Philosophers debate whether mathematical objects have an independent existence outside ofhuman thought (realism), or if their existence is dependent on mental constructs or language (idealism andnominalism). Objects can range from theconcrete: such asphysical objects usually studied inapplied mathematics, to theabstract, studied inpure mathematics. What constitutes an "object" is foundational to many areas of philosophy, fromontology (the study of being) toepistemology (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of thephysical world, raising questions about their ontological status.^[4][5] There are varyingschools of thought which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.^[6]

Quine-Putnam indispensability is an argument for the existence of mathematical objects based on theirunreasonable effectiveness in thenatural sciences. Everybranch of science relies largely on large and often vastly different areas of mathematics. Fromphysics' use ofHilbert spaces inquantum mechanics anddifferential geometry ingeneral relativity tobiology's use ofchaos theory andcombinatorics (seemathematical biology), not only does mathematics help withpredictions, it allows these areas to have an elegantlanguage 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 isindispensable to these theories. It is because of this unreasonable effectiveness and indispensability of mathematics that philosophersWillard Quine andHilary Putnam argue that we should believe the mathematical objects for which these theories depend actually exist, that is, we ought to have anontological commitment to them. The argument is described by the followingsyllogism:^[7]

(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 inapplied mathematics calledNaturalism^[8] (or sometimes Predicativism)^[9] which states that the onlyauthoritative standards on existence are those ofscience.

Platonism asserts that mathematical objects are seen as real,abstract entities that exist independently of humanthought, often in somePlatonic realm. Just asphysical objects likeelectrons andplanets exist, so do numbers and sets. And just asstatements about electrons and planets are true or false as these objects contain perfectlyobjective properties, so are statements about numbers and sets. Mathematicians discover these objects rather than invent them.^[10][11] (See also:Mathematical Platonism)

Some some notable platonists include:

• Plato: The ancientGreek philosopher who, though not a mathematician, laid the groundwork for Platonism by positing the existence of an abstract realm of perfectforms 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 inmodel theory was a major influence onmodern platonism
• Roger Penrose: A contemporarymathematical 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 denies the independent existence of mathematical objects. Instead, it suggests that they are merelyconvenient fictions or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects do not have an existence beyond the symbols and concepts we use.^[13][14]

Some notable nominalists include:

• 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: Acontemporary philosopher who has developed the form of nominalism called "fictionalism," which argues that mathematical statements are useful fictions that do not correspond to any actual abstract objects.^[15]

Logicism asserts that all mathematical truths can be reduced tological truths, and all objects forming the subject matter of those branches of mathematics are logical objects. In other words, mathematics is fundamentally a branch oflogic, 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 theAxiom of Choice) and hisAxiom of Infinity, and later with the discovery ofGödel's incompleteness theorems, which showed that any sufficiently powerfulformal system (like those used to expressarithmetic) cannot be bothcomplete andconsistent. This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program.^[16]

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 ofmodern logic and was highly influential, though it encountered difficulties, most notablyRussell's paradox, which revealed inconsistencies in Frege's system.^[17]
• Bertrand Russell: Russell, along withAlfred North Whitehead, further developed logicism in their monumental workPrincipia Mathematica. They attempted to derive all of mathematics from a set oflogical axioms, using atype theory to avoid the paradoxes that Frege's system encountered. AlthoughPrincipia Mathematica was enormously influential, the effort to reduce all of mathematics to logic was ultimately seen as incomplete. However, it did advance the development ofmathematical logic andanalytic philosophy.^[18]

Mathematical formalism treats objects as symbols within aformal 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 playingludo orchess. 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 as a foundation of mathematics (seeHilbert's program). 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]Freeman Dyson wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century",Henri Poincaré andDavid Hilbert.^[22]

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 acontradiction from that assumption. Such aproof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of theexistential quantifier, which is at odds with its classical interpretation.^[23] There are many forms of constructivism.^[24] These includeBrouwer's program ofintutionism, thefinitism ofHilbert andBernays, the constructive recursive mathematics of mathematiciansShanin andMarkov, andBishop's program ofconstructive analysis.^[25] Constructivism also includes the study ofconstructive set theories such asConstructive Zermelo–Fraenkel and the study of philosophy.

Some notable constructivists include:

• L. E. J. Brouwer: Dutch mathematician and philosopher regarded as one of the greatest mathematicians of the 20th century, known for (among other things) pioneering theintuitionist movement to mathematical logic, and opposition of David Hilbert's formalism movement (see:Brouwer–Hilbert controversy).
• Errett Bishop: American mathematician known for his work on analysis. He is best known for developingconstructive analysis in his 1967Foundations of Constructive Analysis, where he proved most of the important theorems inreal analysis using constructivist methods.

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 ofarithmetic. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.^[26][27]

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 bookPhilosophy of Mathematics: Structure and Ontology.^[28]

Frege famously distinguished betweenfunctions andobjects.^[30] According to his view, a function is a kind of ‘incomplete’entity thatmaps 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 reducedproperties andrelations 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.^[31] 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 istype 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'.^[32]

• 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

Citations

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 andReuben 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.

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