Mathematical Object

A Mathematical Object is an mathematical concept that can be formally defined and manipulated through mathematical operations within a mathematical theory.

• AKA: Mathematical Entity, Mathematical Structure, Abstract Mathematical Object, Formal Mathematical Object.
• Context:
  • It can typically be characterized through Mathematical Properties that define its mathematical behavior and mathematical 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 mathematical structure and properties.
  • It can typically exist independently of physical instantiation, residing purely in the realm of mathematical abstraction.
  • 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 and relationships.
  • ...
  • It can often be constructed from simpler mathematical objects through mathematical construction processes.
  • It can often serve as a Mathematical Model for real-world phenomena despite its abstract nature.
  • It can often be classified into Mathematical Categories based on shared mathematical structures.
  • It can often be transformed through Mathematical Transformations while preserving certain 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 range from being a Simple Mathematical Object to being a Complex Mathematical Object, depending on its mathematical structure complexity.
  • It can range from being a Finite Mathematical Object to being an Infinite Mathematical Object, depending on its mathematical cardinality.
  • It can range from being a Discrete Mathematical Object to being a Continuous Mathematical Object, depending on its mathematical topology.
  • It can range from being a Concrete Mathematical Object to being a Highly Abstract Mathematical Object, depending on its mathematical abstraction level.
  • It can range from being a Well-Understood Mathematical Object to being a Mysterious Mathematical Object, depending on current mathematical knowledge.
  • ...
  • It can be analyzed by Mathematicians using mathematical methods and mathematical tools.
  • It can be represented through Mathematical Representations including mathematical formulae and mathematical diagrams.
  • It can be implemented in Computer Systems through mathematical software and symbolic computation.
  • ...
• Example(s):
  • Basic Numbers, such as:
    • Natural Numbers like 1, 2, 3, representing counting quantities.
    • Real Numbers like π and e, encoding continuous quantities.
    • Complex Numbers like i = √-1, extending the number system.
    • Transfinite Numbers like ℵ₀, measuring infinite quantities.
  • Fundamental Sets, such as:
    • The Empty Set ∅, containing no elements.
    • Power Sets, containing all subsets of a given set.
    • Cantor Set, demonstrating fractal properties.
    • Russell's Set, illustrating paradoxes in naive set theory.
  • Essential Functions, such as:
    • Sine Function, mapping angles to ratios.
    • Exponential Function, modeling growth processes.
    • Dirac Delta Function, representing point masses.
    • Weierstrass Function, continuous everywhere but differentiable nowhere.
  • Geometric Objects, such as:
    • Euclidean Spaces of various dimensions.
    • Klein Bottle, a non-orientable surface.
    • Mandelbrot Set, exhibiting infinite complexity.
    • Hypercubes in n-dimensional space.
  • Advanced Structures, such as:
    • Groups like the Monster Group.
    • Topological Spaces encoding continuity.
    • Categories abstracting mathematical relationships.
    • Grothendieck Universes containing other mathematical objects.
  • Mathematical Proofs themselves, when viewed as objects in Proof Theory.
  • Theories like ZFC Set Theory, treated as formal objects.
  • ...
• Counter-Example(s):
  • Physical Objects, which exist in physical space rather than purely abstract mathematical space.
  • Informal Concepts, which lack the precise formal definition required of mathematical objects.
  • Empirical Observations, which are concrete measurements rather than abstract entities.
  • Natural Language Expressions, which lack the formal precision of mathematical objects.
  • Subjective Experiences, which cannot be formally defined in mathematical terms.
  • Undefined Terms in mathematics, which serve as primitives but are not themselves mathematical objects.
• See: Mathematics, Mathematical Structure, Abstract Algebra, Mathematical Foundation, Philosophy of Mathematics, Mathematical Ontology, Formal System, Mathematical Universe, Platonism (Mathematics), Mathematical Constructivism, Set Theory, Category Theory, Type Theory, Mathematical Logic.

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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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