Univalent Foundations: The Intrinsic Geometry of Mathematical Knowledge Article Swipe
Revista, Zen
,
MATH, 10
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17792366
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17792366
This paper explores Univalent Foundations (UF) as a revolutionary paradigm for understanding the intrinsic geometry of mathematical knowledge. Rooted in Homotopy Type Theory (HoTT), UF provides a framework where mathematical objects are interpreted as types, and equalities are understood as paths in an underlying space. The univalence axiom, a cornerstone of UF, posits that isomorphic or equivalent objects are equal, thereby formalizing the pervasive practice in mathematics where structurally identical entities are treated as one. We delve into how UF reconfigures the foundational landscape, offering a constructive and computational approach to mathematics that inherently reflects its abstract and relational structure. By interpreting mathematical concepts as inhabitants of higher-dimensional spaces, UF reveals a deep geometric substratum to mathematical reasoning, where proof itself acquires a homotopical interpretation. The paper discusses the theoretical underpinnings, practical implications for formal verification and proof assistants, and the philosophical ramifications of this geometric perspective on mathematical truth and knowledge.
Related Topics
Concepts
Constructive
Perspective (graphical)
Mathematics
Mathematical structure
Homotopy
Algebra over a field
Mathematical practice
Mathematical theory
Mathematical proof
Calculus (dental)
Constructive proof
Non-Euclidean geometry
Analytic geometry
Geometric modeling
Type (biology)
Computer science
Proof assistant
Euclidean geometry
Pure mathematics
Mathematical problem
Formal proof
Type theory
Invariant (physics)
Geometric transformation
Geometry
Mathematical model
Mathematical logic
Philosophy of mathematics
Foundations of mathematics
Category theory
Differential geometry
Metadata
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- https://doi.org/10.5281/zenodo.17792366
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Univalent Foundations: The Intrinsic Geometry of Mathematical KnowledgeWork title
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articleOpenAlex work type
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2025Year of publication
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2025-12-02Full publication date if available
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Revista, Zen, MATH, 10List of authors in order
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https://doi.org/10.5281/zenodo.17792366Publisher landing page
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greenOpen access status per OpenAlex
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https://doi.org/10.5281/zenodo.17792366Direct OA link when available
- Concepts
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Constructive, Perspective (graphical), Mathematics, Mathematical structure, Homotopy, Algebra over a field, Mathematical practice, Mathematical theory, Mathematical proof, Calculus (dental), Constructive proof, Non-Euclidean geometry, Analytic geometry, Geometric modeling, Type (biology), Computer science, Proof assistant, Euclidean geometry, Pure mathematics, Mathematical problem, Formal proof, Type theory, Invariant (physics), Geometric transformation, Geometry, Mathematical model, Mathematical logic, Philosophy of mathematics, Foundations of mathematics, Category theory, Differential geometryTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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| abstract_inverted_index.equivalent | 56 |
| abstract_inverted_index.inherently | 93 |
| abstract_inverted_index.isomorphic | 54 |
| abstract_inverted_index.knowledge. | 17, 151 |
| abstract_inverted_index.landscape, | 83 |
| abstract_inverted_index.reasoning, | 117 |
| abstract_inverted_index.relational | 98 |
| abstract_inverted_index.structure. | 99 |
| abstract_inverted_index.substratum | 114 |
| abstract_inverted_index.underlying | 43 |
| abstract_inverted_index.understood | 38 |
| abstract_inverted_index.univalence | 46 |
| abstract_inverted_index.Foundations | 4 |
| abstract_inverted_index.assistants, | 138 |
| abstract_inverted_index.cornerstone | 49 |
| abstract_inverted_index.formalizing | 61 |
| abstract_inverted_index.homotopical | 123 |
| abstract_inverted_index.inhabitants | 105 |
| abstract_inverted_index.interpreted | 32 |
| abstract_inverted_index.mathematics | 66, 91 |
| abstract_inverted_index.perspective | 146 |
| abstract_inverted_index.theoretical | 129 |
| abstract_inverted_index.constructive | 86 |
| abstract_inverted_index.foundational | 82 |
| abstract_inverted_index.implications | 132 |
| abstract_inverted_index.interpreting | 101 |
| abstract_inverted_index.mathematical | 16, 29, 102, 116, 148 |
| abstract_inverted_index.reconfigures | 80 |
| abstract_inverted_index.structurally | 68 |
| abstract_inverted_index.verification | 135 |
| abstract_inverted_index.computational | 88 |
| abstract_inverted_index.philosophical | 141 |
| abstract_inverted_index.ramifications | 142 |
| abstract_inverted_index.revolutionary | 8 |
| abstract_inverted_index.understanding | 11 |
| abstract_inverted_index.underpinnings, | 130 |
| abstract_inverted_index.interpretation. | 124 |
| abstract_inverted_index.higher-dimensional | 107 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.95015713 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |