On the Philosophical Roots of the Naïve and Axiomatic Set Theories: Determinatio est Negatio

Öz

The principle determinatio est negatio—that determination is achieved through negation—has philosophical roots extending back to Plato and Aristotle, and it later influenced early modern thinkers such as Francisco Suárez and Spinoza. This paper has two aims. The first demonstrates how the principle of negation functions as a tool for conceptual determination across various philosophical frameworks, and the second demonstrates that the principle plays a key role in the analysis and resolution of the Burali-Forti paradox within the context of the naïve and axiomatic set theories. In the first section, the analysis focuses on the evolution of the principle from ancient philosophy to early modern metaphysics, examining Plato’s dialectics, Aristotle’s metaphysical distinctions, Suárez’s scholastic theories, and Spinoza’s monist metaphysics. The second section shifts to mathematics, where determinatio est negatio plays a key role in resolving set-theoretical paradoxes, particularly the Burali-Forti paradox. By exploring Cantor’s solution and its reliance on the distinction between consistent and inconsistent sets, this study demonstrates how this principle is essential for avoiding self-referential inconsistencies. The contributions of Zermelo and von Neumann, who developed axiomatic frameworks to address these paradoxes, further elaborate the philosophical foundations of set theory. Ultimately, the study reveals a deep connection between metaphysical principles and the mathematical treatment of paradoxes, emphasizing the ongoing relevance of determinatio est negatio in both domains.

Anahtar Kelimeler

• Determinatio est negatio
• The Burali-Forti paradox
• Cantor
• set theory
• self-reference

Naif ve Aksiyomatik Küme Teorilerinin Felsefi Kökenleri Üzerine: Determinatio est Negatio

Öz

Determinatio est negatio ilkesi—yani belirlemenin olumsuzlama yoluyla gerçekleştirildiği anlayışı—Platon ve Aristoteles’e kadar uzanan felsefi köklere sahiptir ve daha sonra Francisco Suárez ve Spinoza gibi erken modern düşünürleri etkilemiştir. Söz konusu ilkeyi analiz eden bu çalışmanın iki amacı vardır. İlki, olumsuzlama ilkesinin çeşitli felsefi sistemler boyunca kavramsal belirleme aracı olarak nasıl işlediğini göstermek; ikincisi ise ilkenin, naif ve aksiyomatik kümeler teorisi bağlamında Burali-Forti paradoksunun analizinde ve çözümünde önemli bir rol oynadığını ortaya koymaktır. İlk bölümde, analiz bu ilkenin antik felsefeden erken modern metafiziğe olan evrimine odaklanmakta ve Platon’un diyalektiği, Aristoteles’in metafiziksel ayrımları, Suárez’in skolastik teorileri ve Spinoza’nın monist metafiziği incelenmektedir. İkinci bölümde ise, determinatio est negatio ilkesinin özellikle Burali-Forti paradoksunu çözmedeki rolü vurgulanarak matematik alanına geçiş yapılmaktadır. Cantor’un çözümü ve tutarlı ve tutarsız kümeler arasındaki ayrımı determinatio est negatio bağlamında incelenerek, bu ilkenin özreferanssal paradokslardan kaçınmada oynadığı önemli rol gösterilmektedir. Zermelo ve von Neumann’ın Burali-Forti paradoksu için geliştirdikleri aksiyomatik çerçeveler, kümeler teorisinin felsefi temellerini daha da aydınlatmaktadır. Sonuç olarak, çalışma, metafizik ilkeler ile paradoksların matematiksel çözümü arasındaki derin bağlantıyı ortaya koyarak determinatio est negatio ilkesinin her iki alandaki devam eden önemini vurgulamaktadır.

Anahtar Kelimeler

• Determinatio est negatio
• Burali-Forti paradoksu
• Cantor
• kümeler teorisi
• öz-referans

Kaynakça

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