Mathematical Object

A Mathematical Object is an abstract concept that can be formally defined, studied, and manipulated within mathematical theories using mathematical reasoning.

• AKA: Mathematical Entity, Abstract Mathematical Object, Formal Mathematical Object, Mathematical Construct.
• Context:
  • It can typically be characterized through mathematical properties that define its behavior and relationships.
  • It can typically participate in mathematical relations with other mathematical objects through mathematical operations.
  • It can typically be studied through mathematical theories that formalize its structure and properties.
  • It can typically exist independently of physical instantiation, residing purely in abstract mathematical space.
  • It can typically be communicated through mathematical language using mathematical symbols and mathematical notations.
  • It can typically be subject to mathematical proofs that establish mathematical truths about its properties.
  • It can typically be constructed from simpler mathematical objects through mathematical constructions.
  • It can typically serve as a mathematical model for real-world phenomena despite its abstract nature.
  • It can often be classified into mathematical categories based on shared structures.
  • It can often be transformed through mathematical transformations while preserving mathematical invariants.
  • It can often exhibit mathematical patterns that reveal deeper mathematical principles.
  • It can often be generalized to create more abstract mathematical objects with broader applicability.
  • It can often have multiple equivalent mathematical representations or mathematical formulations.
  • It can often be studied through different branches of mathematics revealing different aspects.
  • It can often inspire new mathematical theories and discoveries.
  • It can range from being a Simple Mathematical Object to being a Complex Mathematical Object, depending on its structural complexity.
  • It can range from being a Finite Mathematical Object to being an Infinite Mathematical Object, depending on its cardinality.
  • It can range from being a Discrete Mathematical Object to being a Continuous Mathematical Object, depending on its topological structure.
  • It can range from being a Constructive Mathematical Object to being a Non-Constructive Mathematical Object, depending on its definability.
  • It can range from being an Elementary Mathematical Object to being an Advanced Mathematical Object, depending on its mathematical sophistication.
  • It can be analyzed by mathematicians using mathematical methods and mathematical tools.
  • It can be represented through mathematical formulae, mathematical diagrams, and formal notations.
  • It can be implemented in computer systems through mathematical software and symbolic computation.
  • It can be verified through formal verification systems and proof assistants.
  • ...
• Example(s):
  • Numbers, such as:
    • Natural Numbers: 0, 1, 2, 3, ... (counting and ordering).
    • Integers: ..., -2, -1, 0, 1, 2, ... (including negatives).
    • Rational Numbers: 1/2, 22/7, -3/4 (fractions).
    • Real Numbers: π, e, √2 (continuous quantities).
    • Complex Numbers: i, 3+4i, e^(iπ) (extending reals).
    • Quaternions: 1+2i+3j+4k (4D rotations).
    • Transfinite Numbers: ℵ₀, ℵ₁, ω (infinite cardinals and ordinals).
  • Sets, such as:
    • Empty Set: ∅ (containing no elements).
    • Power Sets: P(S) (all subsets of S).
    • Cantor Set: fractal with measure zero.
    • Mandelbrot Set: complex fractal boundary.
  • Functions, such as:
    • Polynomial Functions: f(x) = x² + 2x + 1.
    • Trigonometric Functions: sin, cos, tan.
    • Exponential Functions: e^x, 2^x.
    • Special Functions: Gamma, Zeta, Bessel.
  • Mathematical Structures, such as:
    • Groups: integers under addition.
    • Rings: polynomial rings.
    • Fields: real numbers, complex numbers.
    • Topological Spaces: metric spaces, manifolds.
  • Geometric Objects, such as:
    • Points, Lines, Planes (basic elements).
    • Circles, Ellipses, Hyperbolas (conic sections).
    • Polyhedra: tetrahedron, cube, dodecahedron.
    • Manifolds: sphere, torus, Klein bottle.
  • Mathematical Proofs as objects in proof theory.
  • Formal Systems: ZFC, Peano arithmetic, type theory.
  • ...
• Counter-Example(s):
  • Physical Objects, which exist in physical space rather than abstract space.
  • Informal Concepts, which lack precise formal definition.
  • Empirical Observations, which are concrete measurements rather than abstractions.
  • Natural Language Expressions, which lack mathematical precision.
  • Subjective Experiences, which cannot be formally mathematized.
• See: Mathematics, Mathematical Structure, Abstract Algebra, Number, Set Theory, Category Theory, Topology, Mathematical Foundation, Philosophy of Mathematics, Mathematical Logic, Formal System, Mathematical Proof, Mathematical Theory, Mathematical Space, Graph, Isomorphism, Element (Mathematics), Formal Function, Equality Relation, Mathematical Analysis Task, Symbolic Reasoning, Mathematical Algorithm.

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References

2022

• (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Mathematical_object Retrieved:2022-12-28.
  • 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. 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.

    The ontological status of mathematical objects has been the subject of much investigation and debate by philosophers of mathematics.

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