Editing Mathematical Structuralism and the Nature of Mathematical Objects (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Mathematical Structuralism''' — "The View" that "Mathematics" "Studies" "Abstract Structures" — "Patterns of Relations" — rather than "Individual Objects."
* '''Benacerraf's Puzzle''' — "If Numbers Are Sets," **"Which Sets"** are "They"? (von Neumann: 0={}, 1={{}}, ... or Zermelo: 0={}, 1={0}, ...). "Both Work" — "So Neither" "Is Uniquely" "The Numbers."
* '''Eliminative Structuralism''' — "The View" that "There Are No" "Mathematical Objects" — "Only Structures" that "Can Be Instantiated" by "Concrete" "Systems."
* '''Non-Eliminative Structuralism (Ante Rem)''' — "Structures Exist" **"Independently"** of "Their Instantiations" — "Like Platonism" but "For Structures," not "Objects."
* '''In Re Structuralism''' — "Structures Exist" "Only" **"Within"** "Their Instantiations" — "No Instantiation," "No Structure."
* '''Category Theory''' — "A Branch" of "Mathematics" that "Studies" "Mathematical Structures" and "Their Relationships" at "The Highest Level" of "Abstraction" — "Often Proposed" as "A Foundation" for "Structuralism."
* '''The Identity Problem''' — "The Challenge" for "Structuralism": "In A Structure," "Can" "Two Distinct" "Objects" be "Structurally Identical"? (e.g. "In Euclidean Geometry," "Point A" and "Point B" "Have" "The Same" "Structural Properties").
* '''Isomorphism''' — "A Structure-Preserving" "Map" between "Two Mathematical Structures": "Structuralism" "Says" "Isomorphic" "Structures" are "The Same."
* '''The Skolem Paradox''' — "First-Order" "Set Theory" has "A Countable Model" — "Even Though" it "Proves" "Uncountable Sets" "Exist." "Points to" "The Relativity" of "Mathematical Structures" to "Their Foundations."
* '''Homotopy Type Theory''' (HoTT) — "A Modern" "Foundation" for "Mathematics" based on "The Idea" that "Mathematical" "Types" are "Like" **"Spaces"** — "Offering" a "New" "Structuralist" "Foundation."
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