object (mathematics)

 

  • [20] Constructivism [edit] 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.

  • 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.

  • 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.

  • (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.

  • Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.

  • 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.

  • [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).

  • 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.

  • 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.

  • This latter use of ‘object’ is interchangeable with ‘entity.’ It is this more broad interpretation that mathematicians mean when they use the term ‘object’.

  • • 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.

  • 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.

  • [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.

  • Some authors make use of Frege’s notion of ‘object’ when discussing abstract objects.

  • Schools of thought [edit] Platonism [edit] 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.

  • Structuralism [edit] Structuralism suggests that mathematical objects are defined by their place within a structure or system.

  • 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.

  • [citation needed] Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas.

  • 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.

  • The focus is on the manipulation of these symbols according to specified rules, rather than on the objects themselves.

  • Instead, it suggests that they are merely convenient fictions or shorthand for describing relationships and structures within our language and theories.

  • In mathematics, objects are often seen as entities that exist independently of the physical world, raising questions about their ontological status.

 

Works Cited

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Photo credit: https://www.flickr.com/photos/alan-light/3511104954/’]