mathematical object
In
mathematics and its
philosophy, a
mathematical object is an
abstract object arising in
mathematics. Commonly encountered mathematical objects include
numbers,
permutations,
partitions,
matrices,
sets,
functions, and
relations.
Geometry as a branch of mathematics has such objects as
points,
lines,
triangles,
circles,
spheres,
polyhedra,
topological spaces and
manifolds.
Algebra, another branch, has
groups,
rings,
fields,
group-theoretic lattices and
order-theoretic lattices.
Categories are simultaneously homes to mathematical objects and mathematical objects in their own right.The
ontological status of mathematical objects is difficult to pin down, and has been the subject of much investigation and debate by
philosophers of mathematics. On this debate, see the monograph by Burgess and Rosen (1997). One view that emerged around the turn of the 20th century with the work of
Cantor is that all mathematical objects can be defined as
sets. The set {0,1} is a relatively clear-cut example. On the face of it the
group Z2 of integers mod 2 is also a set with two elements. However it cannot simply be the set {0,1} because this does not mention the additional structure imputed to
Z2 by its
operations of
addition and
negation mod 2: how are we to tell which of 0 or 1 is the
additive identity, for example? To organize this group as a set it can first be coded as the quadruple ({0,1},+,−,0), which in turn can be coded using one of several conventions as a set representing that quadruple, which in turn entails encoding the operations + and − and the constant 0 as sets.This approach to the ontology of mathematics raises a fundamental philosophical question of whether the ontology of mathematics needs to be beholden to either its practice or its pedagogy. Mathematicians do not work with such codings, which are neither canonical nor practical. They do not appear in any algebra texts, and neither students nor instructors in algebra courses have any familiarity with such codings. Hence if ontology is to reflect practice, mathematical objects cannot be reduced to sets in this way.If however the goal of mathematical ontology is taken to be the internal consistency of mathematics, it is more important that mathematical objects be definable in some uniform way, for example as sets, regardless of actual practice in order to lay bare the
essence of its
paradoxes. This has been the viewpoint taken by
foundations of mathematics, which has traditionally accorded the management of paradox higher priority than the faithful reflection of the details of mathematical practice as a justification for defining mathematical objects to be sets.Much of the tension created by this foundational identification of mathematical objects with sets can be relieved without unduly compromising the goals of foundations by allowing two kinds of objects into the mathematical universe, sets and
relations, without requiring that either be considered merely an instance of the other. These form the basis of
model theory as the
domain of discourse of
predicate logic. In this viewpoint mathematical objects are entities satisfying the
axioms of a formal theory expressed in the language of predicate logic.A variant of this approach replaces relations with
operations, the basis of
universal algebra. In this variant the axioms often take the form of
equations, or implications between equations.A more abstract variant is
category theory, which abstracts sets as objects and the operations thereon as
morphisms between those objects. At this level of abstraction mathematical objects reduce to mere
vertices of a
graph whose
edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the
composition law for morphisms.
Categories may arise as the models of some axiomatic theory and the
homomorphisms between them (in which case they are usually
concrete, meaning equipped with a faithful
forgetful functor to the category
Set or more generally to a suitable
topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their
provenance.
References
- Azzouni, J., 1994. Metaphysical Myths, Mathematical Practice. Cambridge University Press.
- Burgess, John, and Rosen, Gideon, 1997. A Subject with No Object. Oxford Univ. Press.
- Philip Davis and Reuben Hersh, 1999 [1981]. The Mathematical Experience. Mariner Books: 156-62.
- Hersh, R., 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
Mathematisches ObjektWiskundig object
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