Topological and Algebraic Patterns in Philosophical Analysis: Case Studies from Ockham’s Quodlibetal Quaestiones and Avenarius’ Kritik der Reinen Erfahrung

Arturo Tozzi

doi:10.20944/preprints202502.1518.v1

Submitted:

18 February 2025

Posted:

20 February 2025

You are already at the latest version

Abstract

The intersection of mathematics and philosophy has been extensively explored through logic and set theory, but the application of topological and algebraic tools to the analysis of philosophical arguments and conceptual structures has received less attention. By integrating key concepts from algebraic topology, homotopy theory and probability theory, we propose a framework for analysing epistemological and logical relationships across different philosophical traditions. Our approach classifies conceptual relations within mathematical spaces, allowing for systematic comparisons between frameworks of thought. The application of mathematical models contributes to a more comparative evaluation of epistemic dependencies, revealing local and global structures that might otherwise remain implicit. Within this framework, we consider as examples William of Ockham’s Quodlibetal Quaestiones and Richard Avenarius’ Kritik der Reinen Erfahrung, assessing their epistemological positions through the lens of formal mathematical tools. By utilizing theorems such as Seifert-van Kampen, Borel’s theorem and Kolmogorov’s zero-one law, we examine the logical foundations of Ockham’s rejection of metaphysical universals and Avenarius’ theory of pure experience. Our interdisciplinary analysis suggests that the two philosophical positions align with distinct but definable mathematical structures, reinforcing the applicability of topology and algebra to philosophical inquiry. This provides a refined model for historical and conceptual investigations in philosophy of science and epistemology.

Keywords:

epistemic invariants

;

metric structures

;

homotopy equivalence

;

categorical models

;

logical topology

Subject:

Arts and Humanities - Philosophy

Introduction

Philosophical concepts and arguments have been examined through mathematical tools by a relatively small number of scholars who have sought to formalize abstract reasoning using geometric, algebraic and set-theoretic approaches. Among them, Nicholas of Amiens (1956) applied ars geometrica in his Ars Fidei Catholicae to systematically structure theological doctrine. He sought to defend Catholic faith through axiomatic formulations, ensuring logical consistency while reinforcing orthodoxy. Spinoza (1996) used a geometric approach to reframe theology within a deterministic metaphysical system. Presenting definitions, axioms and propositions, he derived divine attributes through logical necessity rather than revelation. Descartes (1996) applied mathematical reasoning to metaphysics, developing analytic geometry and geometric clarity as a model for philosophical truth. Leibniz (1989) proposed a formalized calculus of reasoning and symbolic logic, aiming for a universal mathematical language, which he called the characteristica universalis. Bolzano (1972) applied mathematical rigor to logic and epistemology, prefiguring modern set theory and analytic philosophy. Comte (1896) advocated for the application of mathematical methods in social sciences, aiming to formalize philosophical and sociological analysis through positivism. Peirce (1931-58) integrated algebraic logic and topology into semiotics and pragmatism, emphasizing diagrammatic reasoning as a means to understand thought processes. Whitehead and Russell argued that mathematical structures underlie logical and metaphysical concepts, using set theory and symbolic logic to redefine philosophical problems (Whitehead and Russell, 1910-13). Carnap (1937) and other logical positivists applied formal semantics and mathematical logic to the philosophy of science. Gödel (1992) extended philosophical logic by proving incompleteness theorems, significantly influencing metaphysics, epistemology and mathematical philosophy. Badiou (2013) used set theory and mathematical ontology to redefine being, arguing that mathematics serves as the foundation of ontology and structured knowledge.

Although the intersection of mathematics and philosophy has been extensively explored through logic, set theory and the other mathematical concepts mentioned above, the application of structural, topological and algebraic tools to the analysis of philosophical arguments has received comparatively less attention. Recent research in algebraic topology, category theory and homotopy theory has provided novel methodologies for assessing conceptual structures beyond traditional formal logic (Hatcher 2005; Peters 2016). While studies in formal epistemology have incorporated mathematical frameworks to analyze knowledge representation and modal structures, few investigations have systematically applied these tools to historical philosophical texts. Specifically, mathematical approaches remain underutilized in examining the epistemological and metaphysical frameworks of figures like William of Ockham and Richard Avenarius. By applying advanced mathematical theorems to their philosophical works, we explore how concepts traditionally confined to qualitative analysis can be assessed through mathematical invariants and structural relations, establishing a novel method for investigating distinct epistemological paradigms. Our approach allows for a more precise examination of structural coherence and logical dependencies within philosophical texts, offering insights that may be difficult to obtain through purely textual analysis.

Applying mathematical formalization of conceptual structures, we introduce a method that situates Ockham’s and Avenarius’ theories within well-defined mathematical spaces, identifying structural features that align with homotopy, cohomology and categorical transformations. This approach allows for a comparative analysis of their epistemic structures, determining whether their conceptual frameworks exhibit formal analogies in terms of global and local dependencies. By employing probabilistic models alongside algebraic structures, we explore how the stability and variability of epistemic claims can be quantitatively assessed.

We aim to demonstrate that the rigor of topological and algebraic methods can be extended to philosophical inquiry, reinforcing their applicability across disciplines that traditionally rely on textual and conceptual analysis. We will proceed as follows. First, we outline the methodology by detailing the mathematical tools employed. Then, we present the structural analyses of Ockham’s and Avenarius’ texts, emphasizing the mathematical interdependencies revealed by our approach. Finally, we discuss the broader implications of applying topological and algebraic frameworks in philosophical research.

Mathematical Tools

In this section, we outline the methodology used to establish structural and topological relationships in the mathematical analysis of philosophical frameworks. By employing a range of mathematical tools, we aim to provide a systematic approach to examining conceptual structures with rigor and coherence. The mathematical concepts utilized in the subsequent philosophical analysis, including key principles from topology, algebra and probability theory, are briefly introduced in this section.

Topological Spaces

A topological space is a set equipped with a topology, which defines how subsets relate to each other in terms of openness and continuity. It serves as a fundamental structure in topology, providing the framework for continuity and convergence.

A connected space is a topological space that cannot be divided into two disjoint nonempty open subsets. This property ensures that the space is in a single piece and is crucial in understanding the structure of continuous functions and mappings.

The Borsuk-Ulam theorem, which states that any continuous function from an n-sphere to an Euclidean space maps at least one pair of antipodal points to the same value, has implications for symmetry detection in topological structures (Borsuk 1967).

Lusternik-Schnirelmann theorem, which deals with the number of critical points of functions, is applicable in optimizing segmentation algorithms (James 1992). This theorem determines the minimal number of significant features needed for effective classification, ensuring that a sufficient number of critical points are considered and preventing under-segmentation or oversimplification of complex structures.

A weak topology is the coarsest topology on a space that makes a given set of functions continuous (Vaidya et al., 2024). It is widely used in functional analysis and topology to study convergence properties.

A coarse topology is a topology that considers large-scale structure rather than pointwise details (Hanke et al., 2008). It is useful in geometric group theory.

Homology

Homology is an algebraic tool used to classify topological spaces based on their cycles, holes and higher-dimensional analogs (Hatcher 2005). It assigns a sequence of abelian groups or modules to a space, providing invariants for comparison.

The Eilenberg-Zilber theorem provides a way to compute the homology of a product space using the homologies of its individual components, ensuring compatibility between algebraic and topological structures (Golański and Lima Gonçalves, 1999).

The Künneth theorem allows the computation of the homology of a product space in terms of the homologies of its factors (Smith 1970). It plays a central role in algebraic topology by enabling the decomposition of complex spaces into simpler components.

Sheaf cohomology and cellular approximations are mathematical tools used to study local-to-global properties of topological spaces (Wedhorn 2016). They provide a means to track how local structures influence global topological and algebraic properties.

Poincaré duality, a fundamental result in algebraic topology, allows for the identification of relationships between topological features in different dimensions (Hilman et al., 2024). This approach can be applied to functional connectivity analysis, particularly in examining the interplay between local and global networks.

Homotopy

Homotopy is a continuous deformation between two continuous functions, serving as a foundational concept in algebraic topology. It allows classification of spaces based on their deformability without tearing or gluing.

The Freudenthal suspension theorem relates the homotopy groups of a space to those of its suspension (Whitehead 1953). It plays an important role in the study of stable homotopy theory and the behavior of homotopy groups under suspension.

The Whitehead theorem states that a weak homotopy equivalence between CW complexes is also a homotopy equivalence in general topology (Kan 1976). This result ensures that homotopy equivalences preserve topological structures.

The Seifert-van Kampen theorem describes the fundamental group of a space in terms of its decomposition into subspaces (Lee 2011). It is widely used in algebraic topology to compute the fundamental group of complex spaces by breaking them into simpler parts.

Geometric and Algebraic Structures

A Betti number is an integer invariant of a topological space that counts the number of independent cycles in different dimensions. It is used in algebraic topology to distinguish spaces up to homotopy equivalence.

A simplicial complex is a combinatorial object consisting of vertices, edges and higher-dimensional simplices (Wu et al, 2023). It is used in computational topology and algebraic topology to model topological spaces discretely.

A symplectic manifold is a smooth manifold equipped with a closed non-degenerate 2-form (Favretti 2020). It serves as the foundation of symplectic geometry.

An infinite genus surface is a two-dimensional surface with infinitely many handles (Arredondo and Ramírez Maluendas, 2017; Randecker 2018; Aougab et al., 2021). These surfaces arise in complex analysis and low-dimensional topology.

The Legendrian knot, a structure in contact geometry, provides a way to visualize phase space embeddings in three dimensions. This can be particularly useful for studying the nonlinear trajectories within topological structures (Etnyre, 2005).

The Nash embedding theorem states that any Riemannian manifold can be isometrically embedded into Euclidean space. By embedding structures into higher-dimensional spaces, the theorem preserves geometric structures in higher dimensions crucial for accurate classification (Nash 1956)s

Probability and Convergence Theorems

Coarse proximity theory quantifies large-scale structural relationships by analyzing global rather than local interactions (Shi and Yao, 2024). It provides a framework for studying large-scale geometric and topological properties without focusing on pointwise details.

Borel’s theorem states that every sequence of independent random variables converges in probability (Borel 1953). This theorem ensures that the probability of deviation from expected behavior diminishes over time, providing a foundation for probabilistic convergence.

Kolmogorov’s zero-one law states that certain tail events in probability spaces occur with probability either zero or one (Brzeźniak and Zastawniak, 2020). It is a fundamental result in measure theory that helps characterize events that are determined entirely by an infinite sequence of independent random variables.

In sum, this overview ensures a theoretical framework for understanding how these formal methods are applied to the study of epistemological and logical relationships within philosophical texts. The visualization in Figure highlights how algebraic and probabilistic tools may contribute to epistemic coherence, structural stability and formal conceptual analysis. In the following sections, we examine two practical examples from two relatively unexplored writings of Ockham and Avenarius, demonstrating how connections between different methodologies illustrate the integration of mathematical formalisms in the analysis of epistemological structures.

Figure. Structural and topological relationships in the mathematical analysis of philosophical frameworks. Nodes represent key philosophical (red) and mathematical (blue) concepts, while edges indicate conceptual dependencies between them.

William of OckhaM

The study of William of Ockham’s Quodlibetal Quaestiones through the lens of structural, topological and algebraic patterns offers a rigorous framework for analyzing his philosophical positions, particularly concerning relational ontology, causality and modal logic. In the following paragraphs, the two numbers in brackets correspond to the numbers of the Quodlibet and of the question in which the topic is addressed in the Quodlibetal Quaestiones (Ockham, 1991). The structural decomposition of Ockham’s propositions can be investigated through homotopy and simplicial complexes, where his hierarchical treatment of concepts parallels the algebraic operations in topological spaces.

Employing mathematical methods such as Borel’s theorem, coarse proximity and homological structures allows us to examine Ockham’s rejection of unnecessary ontological commitments and his nominalist approach to universals. This approach builds upon his assertion that concepts are mental qualities rather than real entities [6.8], thereby aligning with mathematical methods that emphasize local versus global properties, as seen in sheaf cohomology and cellular approximation. One crucial aspect of Ockham’s thought is his treatment of relational entities, which he denies as independent realities [6.23]. This perspective can be examined using Borel’s theorem, which asserts that measurable sets in topological spaces can be constructed from countable unions and intersections of open sets. Applying this to Ockham’s notion that relational terms are merely linguistic or mental constructs, a structural analogy can be found: the construction of complex relations from simpler elements does not necessitate treating them as independent entities. Instead, much like Borel’s approach to measure theory, Ockham’s treatment of relations as dependent on absolute things suggests a hierarchical approach, reducible to fundamental elements. This interpretation supports the claim that the ontological economy in Ockham’s work is structurally akin to modern mathematical reductionist techniques.

Another relevant theorem is the Seifert–van Kampen theorem, which describes how the fundamental group of a topological space can be computed from its subspaces. Ockham’s assertion that no universal concepts exist independently [4.25] resonates with the decomposition of complex topological spaces into simpler, overlapping regions. In particular, if concepts are understood as structures derived from individual perceptual inputs, the theorem illustrates how their interconnection does not necessitate an overarching independent entity. The fundamental group’s ability to be reconstructed from parts aligns with Ockham’s nominalist view that universals do not exist outside of mental constructs but are instead derived from real individual entities. This provides a topological methodology for evaluating his critique of universals.

The Nash embedding theorem, which states that any Riemannian manifold can be isometrically embedded into a higher-dimensional Euclidean space, provides a further mathematical analogy for Ockham’s distinction between spoken, written and mental propositions [5.8]. If we consider propositions as lower-dimensional approximations of a broader epistemological structure, the embedding theorem suggests that even complex logical statements can be situated within a unifying conceptual framework without requiring additional ontological categories. Ockham’s claim that spoken propositions must correspond to mental propositions to be true [5.8] aligns with the idea that embedded structures preserve their intrinsic properties while existing within a larger representational space.

The Kolmogorov zero-one law, which states that certain probabilistic events occur with probability either 0 or 1, offers an insight into Ockham’s skepticism toward causal determinism [4.2]. Ockham’s argument that natural agents cause accidents of the same type in similar affected subjects [2.11] suggests a deterministic framework. However, he simultaneously argues that God, as an omnipotent agent, can act independently of secondary causes [6.12]. This apparent contradiction can be resolved by considering the zero-one law, where certain future contingents, when analyzed within a finite system, tend toward deterministic certainty while remaining formally indeterminate in an infinite setting. This aligns with Ockham’s notion that God knows the future contingently, not necessarily, maintaining logical coherence within his framework. Therefore, the application of probabilistic topology helps clarify Ockham’s nuanced stance on necessity and contingency.

Topological concepts such as coarse proximity and weak topology further illuminate Ockham’s treatment of relations. His claim that mutual relations are simultaneous but posterior to both cause and effect [6.12] suggests a structural hierarchy, where proximity between elements does not imply immediate causation. Coarse topology, which considers large-scale structural relations rather than fine-grained details, parallels Ockham’s view that entities exist independently of their relations. By applying coarse proximity to his theological arguments, particularly regarding the Trinity [2.3], we can model the relationship between divine persons as existing within a broad topological space where their connections emerge at a structural level without necessitating additional ontological commitments. This interpretation preserves Ockham’s insistence on minimal explanatory assumptions while allowing for Trinity’s relational complexity.

Homological tools such as the Künneth theorem and the Eilenberg–Zilber theorem allow us to examine how Ockham structures theological and logical distinctions. His argument that a relation is posterior to its foundation and terminus [6.12] can be mapped onto the homological decomposition of topological spaces, where cohomology groups represent interactions between separate components. The Eilenberg–Zilber theorem, which establishes an equivalence between singular homology and the homology of a product space, suggests that Ockham’s relational claims can be understood through algebraic topology, wherein independent entities generate relations rather than relations existing independently. This provides a rigorous mathematical basis for his nominalist framework.

Legendrian knot theory and symplectic topology also offer insights into Ockham’s treatment of causality and time. His assertion that causal relations are conceptually posterior to their terms [6.12] aligns with the structure of Legendrian knots, where the interaction of curves does not imply a fundamental connection but instead results from constraints imposed by the system. Similarly, the study of Betti numbers and infinite genus surfaces sheds light on Ockham’s rejection of necessary causal sequences [2.9]. If causal structures are not inherently ordered but instead arise from constraints, this reinforces his claim that causes and effects are not necessarily linked.

In sum, through the application of homotopy, cohomology and algebraic topology, Ockham’s claims regarding relations, causality and ontology can be systematically analyzed. His rejection of independently existing universals finds resonance in the decomposition theorems of topology, while his treatment of contingency aligns with probabilistic mathematical principles. This approach not only reinforces the coherence of Ockham’s philosophy, but also provides a novel structural methodology for examining medieval logical and theological frameworks. The mathematical techniques applied here demonstrate that Ockham’s nominalism and theological minimalism exhibit a robust structural and topological foundation, mirroring modern approaches to formal systems and category theory.

Richard Avenarius

The study of Richard Avenarius’ Kritik der Reinen Erfahrung through structural, topological and algebraic methodologies provides a novel assessment of his philosophical positions on perception, dualism and cognition. In the following, the number in brackets corresponds to the paragraph number in which the topic is addressed in one of the two volumes of the Kritik der reinen Erfahrung (Avenarius 1880; 1890). To evaluate his work mathematically, we employ tools such as Borel’s theorem, homotopy, sheaf cohomology and the Nash embedding theorem. These techniques allow us to assess how his epistemology can be interpreted in terms of topological continuity, algebraic structures and differential geometry.

Avenarius rejects metaphysical distinctions between internal and external reality [XXV], proposing a unified experience-based world model. A key aspect of Avenarius’ critique is the notion that statements depend on environmental changes, which ultimately correspond to brain oscillations [40]. This dynamic dependency aligns with Borel’s theorem, which concerns measurable functions in probability spaces. Just as Borel sets ensure that probability measures remain stable under limits, Avenarius suggests that experiences, despite their variability, maintain coherence in a structured environment. The connection between probability and perception can be further examined through Kolmogorov’s zero-one law, which dictates that certain events in a probability space either almost surely occur or do not. Applying this to Avenarius’ rejection of metaphysical absolute distinctions, we can argue that cognitive experiences exhibit a probabilistic determinacy that aligns with his empirical world model. This connection helps ground Avenarius’ rejection of rigid categorical distinctions in a mathematical structure that highlights the stability of experiential perception.

The Seifert–van Kampen theorem, which allows for the computation of fundamental groups from topological subspaces, may serve as a model for Avenarius’ discussion of cognition and world-building. He argues that empirical knowledge is constructed from the integration of sensory inputs [35], much like how fundamental groups describe the global structure of a topological space through local information. This aligns with his claim that the human experience of reality is not mediated by metaphysical constructs but is instead built from fundamental, interconnected perceptual elements. Similarly, the Künneth theorem, which establishes how homology groups of product spaces decompose into simpler components, parallels Avenarius’ breakdown of world-concept formation. His distinction between elements (perceptions of things) and characters (emotions and subjective states) [40] mirrors the decomposition of homological structures into independent components, reinforcing his rejection of a monolithic internal-external distinction.

Avenarius describes nervous oscillations as fundamental to perception, with synchronized oscillations giving rise to identity and desynchronized patterns generating contrast [40]. This idea can be mapped onto Legendrian knot theory and symplectic geometry, where phase spaces describe the evolution of dynamical systems. The Legendrian approach captures the structural constraints of perception, as variations in phase-space structures can correspond to perceptual shifts, mirroring Avenarius’ assertion that experience is dynamically structured rather than passively received. Similarly, the Lusternik-Schnirelmann theorem, which deals with critical points of functionals on manifolds, offers a mathematical analogy for Avenarius’ notion that repeated cognitive patterns give rise to certainty [40]. The emergence of certainty through oscillatory repetition can be interpreted as the appearance of stable critical points in an evolving functional space, reinforcing the structured nature of cognition.

The Nash embedding theorem, which states that any Riemannian manifold can be isometrically embedded into Euclidean space, serves as an analogy for Avenarius’ claim that empirical experience retains structural coherence regardless of perspective. He posits that the world is not internally divided but remains a coherent entity regardless of subjective standpoint [22]. The theorem suggests that even if cognitive experience appears fragmented or perspectival, it can still be embedded within a unified structural framework, validating Avenarius’ rejection of dualism. This interpretation allows us to map his epistemological claims onto differential geometric structures, further reinforcing the topological stability of his world-model.

The Borsuk-Ulam theorem, which states that any continuous function from an n-sphere to a n-1 manifold must map some pair of antipodal points to the same value, supports Avenarius’ claim that subjective perspectives are necessarily interconnected [163]. He argues that differences in perceptual content, such as two individuals seeing different colors, do not imply a fundamental divide but are instead variations within a structured relational space [163]. The Borsuk-Ulam theorem reinforces this by demonstrating that variations in perception still preserve underlying structural coherence, thus supporting Avenarius’ assertion that distinctions between internal and external cognition are methodologically imposed rather than ontologically necessary (Tozzi and Peters, 2017). This insight strengthens the mathematical grounding of his rejection of classical dualism.

Freudenthal suspension and Whitehead’s theorem further illuminate Avenarius’ framework by demonstrating how topological structures remain homotopy equivalent even when modified. Just as suspension preserves homotopy groups, Avenarius argues that perception remains fundamentally structured despite variation in sensory input [40]. Whitehead’s theorem, which asserts that a weak homotopy equivalence suffices for a space to have the same homotopy type, mirrors his assertion that empirical world-experience does not require additional metaphysical categories for coherence. This reinforces the claim that experience-based cognition can be analyzed in terms of structural invariance rather than categorical distinction, offering a robust mathematical model for his epistemological framework.

In sum, through these topological, algebraic and differential geometric techniques, Avenarius’ critique of metaphysical assumptions can be systematically assessed. His insistence that experience is structured yet dynamic finds clear resonance in mathematical frameworks dealing with continuity, probability and topological invariance. By mapping his epistemology onto modern mathematical structures, we provide a foundation for his rejection of classical dualism and categorical divisions, reinforcing the coherence of his empirical model.

Conclusions

We show that structural, topological and algebraic methods may provide an analytical framework for examining philosophical arguments, with particular focus on the works of William of Ockham and Richard Avenarius. By applying homotopy theory, cohomology and probabilistic models, we identified formal mathematical structures underlying their epistemological frameworks. We argjue that Ockham’s nominalism, which traditionally resists formalization, exhibits homological invariants that align with localized dependency structures, while Avenarius’ empirical model can be mapped onto continuous transformations in topological spaces. Our results suggest that conceptual relationships, which are often examined through qualitative methods, can be systematically decomposed into algebraic components, revealing both points of convergence and divergence between their respective theories. These mathematical characterizations provide an additional layer of rigor to the comparative analysis of epistemological doctrines, illustrating the potential of this methodology for broader applications. In the introduction, we noted that numerous past scholars have examined the relationship between mathematical approaches and philosophical concepts, exploring how formal structures can be applied to epistemology, metaphysics and logic. While previous research has primarily focused on set theory, formal logic and computational models, our approach introduces a more comprehensive framework by systematically applying topological and algebraic techniques to analyse conceptual structures. Traditional logic-based methods, such as predicate logic or modal analysis, are highly effective for reconstructing arguments but often struggle to capture the relational and hierarchical nature of epistemological systems (Carnap 1937). Set-theoretic models provide a means of classifying philosophical concepts, yet they lack the continuity and dynamic structure afforded by homotopy and cohomology (Badiou 2013). Additionally, network-theoretic approaches, while useful in mapping conceptual relationships, do not offer the depth of topological invariants that our method provides (Kenna and MacCarron, 2017). Compared with other analytical techniques, our method offers a more formalized approach to evaluating philosophical structures.

Unlike traditional hermeneutic or purely logical methods, which focus on interpretive or axiomatic reconstructions, our approach translates abstract philosophical arguments into formal mathematical spaces, ensuring internal coherence and facilitating cross-framework comparisons. One key advantage is its ability to represent epistemological relations in terms of homotopy equivalence, allowing for a visualization of philosophical dependencies and transformations that remain obscure in conventional textual analysis. Additionally, our application of probability theory, particularly through Kolmogorov’s zero-one law, introduces a metric for assessing epistemic stability, providing quantitative measures of certainty and contingency in theoretical models. These approaches allow for a more precise classification of philosophical doctrines, making them accessible to computational analysis and automated reasoning systems. Furthermore, by embedding historical philosophical texts within a mathematical framework, we provide a method that extends beyond textual interpretation, enabling the application of algebraic topology to historical and conceptual research. The broader implications of our methodology extend into various fields, offering testable experimental hypotheses and practical applications. One potential application is the formalization of epistemological systems in artificial intelligence and knowledge representation, where homotopy equivalence could serve as a model for concept evolution in machine learning algorithms. Our findings also suggest that the application of algebraic topology to digital humanities could facilitate automated comparative analyses of philosophical texts, identifying structural patterns across different historical periods.

In sum, our algebraic and topological techniques ensure that both local and global properties of philosophical frameworks are preserved, offering a more nuanced approach to analyzing doctrinal structures. Our approach creates opportunities for interdisciplinary collaboration, bridging philosophy with mathematics, formal epistemology and computational linguistics.

While our approach introduces a novel framework, it also has inherent limitations that must be considered. One challenge lies in the complexity of translating philosophical texts into formal mathematical structures, as this process requires a degree of abstraction that may overlook nuances specific to linguistic and historical contexts. Additionally, our reliance on algebraic and topological methods assumes that philosophical arguments maintain an underlying formal coherence, which may not always be the case, particularly in works that emphasize paradox or ambiguity. Another limitation concerns the computational feasibility of applying these methods to large-scale textual analyses, as algebraic topology and homotopy theory often require significant computational resources. Furthermore, our methodology does not replace traditional interpretive approaches but rather complements them, meaning that its full utility depends on integration with established philosophical methodologies. Addressing these limitations requires further refinement of our translation techniques and exploration of hybrid methods combining mathematical formalization with traditional philosophical analysis.

In conclusion, our study highlights the potential of structural, topological and algebraic approaches in philosophical analysis, demonstrating their ability to formalize and compare complex epistemological systems. We provide a framework that enhances interdisciplinary collaboration, offering a method for analyzing philosophical doctrines and refining conceptual methodologies across diverse domains.

Authors' contributions. The Author performed: study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical and material support, study supervision.

Funding. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Ethics approval and consent to participate. This research does not contain any studies with human participants or animals performed by the Author.

Consent for publication. The Author transfers all copyright ownership, in the event the work is published. The undersigned author warrants that the article is original, does not infringe on any copyright or other proprietary right of any third part, is not under consideration by another journal and has not been previously published.

Availability of data and materials. all data and materials generated or analyzed during this study are included in the manuscript. The Author had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.

Acknowledgements. none.

Competing interests. The Author does not have any known or potential conflict of interest including any financial, personal or other relationships with other people or organizations within three years of beginning the submitted work that could inappropriately influence or be perceived to influence, their work.

Declaration of generative AI and AI-assisted technologies in the writing process. During the preparation of this work, the author used ChatGPT to assist with data analysis and manuscript drafting. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the content of the publication.

References

Aougab T, Patel P, Vlamis NG. 2021. Isometry groups of infinite-genus hyperbolic surfaces. Math. Ann. 381, 459–498. [CrossRef]
Arredondo JA, Ramírez Maluendas C. 2017. On the Infinite Loch Ness monster. Commentationes Mathematicae Universitatis Carolinae, 58 (4): 465–479. [CrossRef]
Avenarius, Richard. Kritik der reinen Erfahrung. Erster Band. Leipzig: Fues’s Verlag (R. Reisland), 1888.
Avenarius, Richard. Kritik der reinen Erfahrung. Zweiter Band. Leipzig: Fues’s Verlag (R. Reisland), 1890.
Badiou, Alain. Being and Event. Translated by Oliver Feltham. London: Bloomsbury, 2013.
Bolzano, Bernard. Theory of Science: Attempt at a Detailed and in the Main Novel Exposition of Logic with Constant Attention to Earlier Authors. Translated by Rolf George. Berkeley: University of California Press, 1972.
Borel, Armand. “Sur la Cohomologie des Espaces Fibrés Principaux et des Espaces Homogènes de Groupes de Lie Compacts.” Annals of Mathematics 57, no. 1 (1953): 115–207. [CrossRef]
Borsuk, Karol. Theory of Retracts. Monografie Matematyczne 44. Warsaw: Państwowe Wydawnictwo Naukowe, 1967.
Brzeźniak, Zdzisław and Tomasz Zastawniak. Basic Stochastic Processes. Springer, 2000. ISBN 3-540-76175-6.
Carnap, Rudolf. The Logical Syntax of Language. Translated by Amethe Smeaton. Chicago: University of Chicago Press, 1937.
Comte, Auguste. The Positive Philosophy of Auguste Comte. Translated by Harriet Martineau. London: George Bell & Sons, 1896.
Descartes, René. Meditations on First Philosophy. Translated by John Cottingham. Cambridge: Cambridge University Press, 1996.
Etnyre, John B. “Legendrian and Transversal Knots.” In Handbook of Knot Theory, 105–185. Elsevier, 2005. [CrossRef]
Favretti, M. “Lagrangian Submanifolds of Symplectic Structures Induced by Divergence Functions.” Entropy 22, no. 9 (September 3, 2020): 983. [CrossRef]
Golański, Marek and Daciberg Lima Gonçalves. “Generalized Eilenberg–Zilber Type Theorem and Its Equivariant Applications.” Bulletin des Sciences Mathématiques 123 (1999): 285–298.
Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Translated by B. Meltzer. New York: Dover, 1992.
Hanke, B. Kotschick, J. Roe, and T. Schick. “Coarse Topology, Enlargeability, and Essentialness.” Annales Scientifiques de l’École Normale Supérieure 41 (2008): 471–93. [CrossRef]
Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2005. ISBN 978-0-521-79540-1.
Hilman, Kaif, Dominik Kirstein, and Christian Kremer. “Parametrised Poincaré Duality and Equivariant Fixed Points Methods.” Preprint, submitted May 27, 2024. arXiv:2405.17641 [math.AT]. https://doi.org/10.48550/arXiv.2405.17641.
James, I. M. “The Lusternik-Schnirelmann Theorem Reconsidered.” Topology and Its Applications 44, no. 1–3 (May 22, 1992): 197–202. [CrossRef]
Kan, D. M. “A Whitehead Theorem.” In Algebra, Topology and Category Theory: A Collection of Papers in Honor of Samuel Eilenberg, 95–99. Academic Press, 1976.
Kenna, Ralph, and Pádraig MacCarron. “A Networks Approach to Mythological Epics.” In Maths Meets Myths: Quantitative Approaches to Ancient Narratives, edited by Ralph Kenna, Máirín MacCarron, and Pádraig MacCarron, 45–60. Understanding Complex Systems. Cham: Springer, 2017.
Lee, John M. “The Seifert–Van Kampen Theorem.” In Introduction to Topological Manifolds, 202: 277–292. Graduate Texts in Mathematics. Springer, New York, NY, 2011. [CrossRef]
Leibniz, Gottfried Wilhelm. Philosophical Essays. Translated by Roger Ariew and Daniel Garber. Indianapolis: Hackett, 1989.
Nash, John. "The Imbedding Problem for Riemannian Manifolds." Annals of Mathematics 63, no. 1 (1956): 20–63. [CrossRef]
Nicholas of Amiens. Ars Fidei Catholicae. Edited by Friedrich Stegmüller. Rome: Bibliotheca Apostolica Vaticana, 1956.
Ockham, William of. Quodlibetal Questions. Translated by Alfred J. Freddoso and Francis E. Kelley. New Haven: Yale University Press, 1991.
Peirce, Charles Sanders. Collected Papers of Charles Sanders Peirce. Edited by Charles Hartshorne and Paul Weiss. Cambridge: Harvard University Press, 1931-1958.
Peters, James F. Computational Proximity: Excursions in the Topology of Digital Images. 1st ed. Intelligent Systems Reference Library. Cham: Springer International Publishing, 2016. [CrossRef]
Randecker, A. 2018. Wild translation surfaces and infinite genus. Algebraic & Geometric Topology 18, 2661–2699. [CrossRef]
Russell, Bertrand. The Philosophy of Logical Atomism. La Salle, IL: Open Court, 1985.
Smith, L. “On the Künneth Theorem. I.” Mathematische Zeitschrift 116 (1970): 94–140. [CrossRef]
Spinoza, Baruch. Ethics. Translated by Edwin Curley. London: Penguin Classics, 1996.
Tozzi A, Peters JF. 2017. Towards Topological Mechanisms Underlying Experience Acquisition and Transmission in the Human Brain. Integr Psychol Behav Sci. 51(2), 303–323. [CrossRef]
Vaidya, S., M. C. Rechtsman, and W. A. Benalcazar. “Polarization and Weak Topology in Chern Insulators.” Physical Review Letters 132, no. 11 (March 15, 2024): 116602. [CrossRef]
Wedhorn, Torsten. Manifolds, Sheaves and Cohomology. Springer Studium Mathematik – Master. Springer, 2016. [CrossRef]
Whitehead, Alfred North and Bertrand Russell. Principia Mathematica. 3 vols. Cambridge: Cambridge University Press, 1910-1913.
Wedhorn, Torsten. Manifolds, Sheaves, and Cohomology. Springer Studium Mathematik – Master. Springer, 2016. [CrossRef]
Whitehead, G. W. “On the Freudenthal Theorems.” Annals of Mathematics 57, no. 2 (1953): 209–228. [CrossRef]
Wu, H., A. Yip, J. Long, J. Zhang, and M. K. Ng. “Simplicial Complex Neural Networks.” IEEE Transactions on Pattern Analysis and Machine Intelligence 46, no. 1 (January 2024): 561–75. [CrossRef]

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