Definiteness, and Semantic Completeness. An Analysis of E. Husserl’s Göttinger Doppelvortrag
Abstract
The significance of E. Husserl’s Göttingen Lectures of 1901 lies in his attempt to address the problem of the imaginary in mathematics, which he approached through his findings on the concepts of variety and completeness (Vollständigkeit) of an axiomatic system. According to Husserl, solving the problem of the imaginary involved finding answers to three fundamental questions: 1) when does an element become imaginary from the perspective of a formal axiomatic system? 2) how can the use of imaginary elements in mathematics be justified? and 3) is a syntactically complete axiomatic system compatible with the extension of the concept of number? In this article, I will present Husserl’s answers to these questions and provide a partial justification for their correctness. I will divide the content of both lectures into two parts. The first part will present the problem of the imaginary, and the second will examine Husserl’s proposal for addressing the aforementioned problems.References
Aranda, V. (2020). Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl. Bulletin of the Section of Logic, 49(2): 109-125 http://dx.doi.org/10.18778/0138-0680.2020.07
(2021). Completeness: From Husserl to Carnap. Logica Universalis, 16: 57-83. https://doi.org/10.1007/s11787-021-00283-4.
Canela Morales, L. A. (2019) From Grassmann, Riemann to Husserl. A brief history of concept of Manifold, META: Research in Hermeneutics, Phenomenology and Practical Philosophy, XI(2): 473-500.
(2023) Ser y calcular. El problema de las entidades matemáticas en la fenomenología temprana de Edmund Husserl. Editorial Aula de Humanidades.
Centrone, S. (2010). Logic and Philosophy of Mathematics in the Early Husserl. Springer.
Da Silva, J. J. (2000a). Husserl’s two Notions of Completeness, Husserl and Hilbert on Completeness and Imaginary Elements in Mathematics. Synthese, 125(3): 417-438.
(2000b). The Many Senses of Completeness. Manuscrito, 23(2): 41-60.
(2010). Beyond Leibniz: Husserl’s Vindication of Symbolic Knowledge. M. Hartimo (ed.), Phenomenology and Mathematics. Springer: 123-145.
(2017). Mathematics and Its Applications. A Transcendental-Idealist Perspective. Springer.
Gustavo Isaac, M. (2015). L’idée de la logique formelle dans les appendices VI à X du volume 12 des Husserliana (1970). History and Philosophy of Logic, 36(4): 321-345.
Hartimo, M. (2007). Towards Completeness: Husserl on Theories of Manifolds 1890-1901. Synthese, 156(2): 281-310.
(2021). Husserl and Mathematics. Cambridge University Press.
Husserl, E. (1970). Philosophie der Arithmetik. Mit ergänzenden Texten (1890-1901). Husserliana, Band XII. L. Eley (ed.). Martinus Nijhoff.
(1974). Formale und Transzendentale Logik: Versuch einer Kritik der logischen Vernunft. Husserliana, Band XVII. P. Janssen (ed.). Martinus Nijhoff.
(1976). Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Ergänzende Texte (1912-1929). Husserliana, Band III/1. K. Schuhmann (ed.). Martinus Nijhoff.
(1984). Logische Untersuchungen. Zweiter Band – I. Teil: Untersuchungen zur Phänomenologie und Theorie der Erkenntnis. Husserliana, Band XIX/1. U. Panzer (ed.) Martinus Nijhoff.
(1985) Einleitung in die Logik und Erkenntnistheorie. Vorlesungen 1906/07. Husserliana, Band XXIV. U. Melle (ed.) Martinus Nijhoff.
(2001). Husserls Manuskripte zu seinem Göttinger Doppelvortrag von 1901. K Schuhmann y E. Schuhmann (eds.), Husserl Studies, Band 17. Kluwer Academic Publishers: 87-123.
Majer, U. (1997). Husserl and Hilbert on Completeness. A Neglected Chapter in Early Twenty Century Foundation of Mathematics. Synthese, 110(1): 37-56.
Okada, M. (2013). Husserl and Hilbert on Completeness and Husserl’s Term Rewrite-based Theory of Multiplicity (Invited paper). F. van Raamsdonk (ed.), 24th International Conference on Rewriting Techniques and Applications, RTA’13. Eindhoven.: 4-19.
Ortiz Hill, C. (1995). Husserl and Hilbert on Completeness. J. Hintikka (ed.), From Dedekind to Gödel. Kluwer Academic Publishers: 143-163.
