To Be or Not To Be: Vector ontologies as a truly formal ontological framework (2505.14940v1)

Abstract: Since Edmund Husserl coined the term "Formal Ontologies" in the early 20th century, a field that identifies itself with this particular branch of sciences has gained increasing attention. Many authors, and even Husserl himself have developed what they claim to be formal ontologies. I argue that under close inspection, none of these so claimed formal ontologies are truly formal in the Husserlian sense. More concretely, I demonstrate that they violate the two most important notions of formal ontology as developed in Husserl's Logical Investigations, namely a priori validity independent of perception and formalism as the total absence of content. I hence propose repositioning the work previously understood as formal ontology as the foundational ontology it really is. This is to recognize the potential of a truly formal ontology in the Husserlian sense. Specifically, I argue that formal ontology following his conditions, allows us to formulate ontological structures, which could capture what is more objectively without presupposing a particular framework arising from perception. I further argue that the ability to design the formal structure deliberately allows us to create highly scalable and interoperable information artifacts. As concrete evidence, I showcase that a class of formal ontology, which uses the axioms of vector spaces, is able to express most of the conceptualizations found in foundational ontologies. Most importantly, I argue that many information systems, specifically artificial intelligence, are likely already using some type of vector ontologies to represent reality in their internal worldviews and elaborate on the evidence that humans do as well. I hence propose a thorough investigation of the ability of vector ontologies to act as a human-machine interoperable ontological framework that allows us to understand highly sophisticated machines and machines to understand us.

Vector Ontologies: A Formalist Approach to Ontological Modeling

The paper presents an in-depth exploration of vector ontologies as a proposition to establish truly formal ontological frameworks, diverging from traditionally accepted foundational ontologies. The author, Kaspar Rothenfusser, through meticulous argumentation, posits that commonly recognized ontologies, despite being formal in logical expression, fall short of Husserl’s stringent criteria for formalism: a priori validity and an absolute absence of empirical content. Rothenfusser seeks to correct this by developing vector ontologies—rooted in the mathematical structure of vector spaces—as a pure embodiment of formal ontology that aligns philosophically with Husserl's foundational ideas.

Critical Analysis of Traditional Formal Ontologies

Rothenfusser critiques existing works positing them as "formal ontologies," highlighting their reliance on empirical categories and human perception, which intrinsically compromises their formality as per Husserl’s standards. By emphasizing that foundational ontologies abstract categories such as properties or relations from observed reality, he demonstrates that they are inductively derived, thus lacking the true independence from empirical content necessary for a Husserlian approach. In contrast, formal logic's basis lies in axiomatic systems that maintain validity through internal logical consistency alone, without reference to empirical perception—an attribute Rothenfusser aims to recapture through vector ontologies.

Proposed Framework: Vector Ontologies

Vector ontologies are introduced by leveraging the conceptual framework of vector spaces, characterized by axioms of commutativity, associativity, and existence of identity elements, among others. This framework facilitates an axiomatic structure independent of empirical particulars, enabling ontological modeling that is both scalable and interoperable. Notably, Rothenfusser advances the notion that AI, particularly artificial neural networks (ANNs), inherently operates within vectorial frameworks, thus implicitly employing vector ontologies. This inherent adoption underscores the potential for vector ontologies to serve as a robust intermediary for human-machine ontological interaction, given their shared foundation.

Implications and Future Directions

The paper has significant ramifications for how ontological structures could be re-envisioned, proposing a model that naturally synchronizes with AI's prevalent computational methods. Rothenfusser further implicates that by adopting such formal ontologies, there lies an opportunity to achieve deeper interpretability of AI systems—a notable challenge in current computation paradigms.

The theoretical premise set forth invites subsequent empirical investigations to elucidate the extent to which vector ontologies can encapsulate complex domains and offer unifying structures across diverse ontological systems. Additionally, Rothenfusser calls for exploration into the alignment of human cognitive processes with vector space conceptualizations, noting earlier parallel efforts like Gardenförs' work on conceptual spaces, which also suggest cognitive alignment with multidimensional scaling paradigms.

Conclusion

The proposition and argumentative pathway outlined by Rothenfusser present vector ontologies as a potentially pivotal shift toward realizing formal ontological frameworks consistent with Husserlian philosophy. By aligning the principles of mathematical formalism with ontological modeling, there emerges an intriguing avenue for future research to holistically integrate human, theoretical, and artificial ontological constructions. This paper sets the stage for discussions that challenge entrenched ontological constructs, inviting the scientific discourse to consider formalisms that are mathematically rigorous yet pragmatically substantial for ontological and AI advancements.