Logika racjonalności. W stronę modalnego platonizmu matematycznego

• Authors: Piotr Wilczek

Title variants

EN

The Logic of Rationality. Towards Modal Mathematical Platonism

• Languages of publication: PL

Abstracts

PL

In this article Whitehead’s philosophy of mathematics is characterized as a Structural Second-Order Platonism and it is demonstrated that the Whiteheadian ontology is consistent with modern formal approaches to the foundation of mathematics. We follow the pathway taken by model-theoretically and semantically oriented philosophers. Consequently, it is supposed that all mathematical theories (understood as deductively closed set of sentences) determine their own models. These models exist mind-independently in the realm of eternal objects. From the metatheoretical point of view the hypothesis (posed by Józef Życiński) of the Rationality Field is explored. It is indicated that relationships between different models can be described in the language of modal logics and can further be axiomatized in the framework of the Second Order Set Theory. In conclusion, it is asserted that if any model (of a mathematical theory) is understood, in agreement with Whitehead’s philosophy, as a collection of eternal objects, which can be simultaneously realized in a single actual occasion, then our external world is governed by the hidden pattern encoded in the field of pure potentialities which constitute the above mentioned Field of Rationality. Therefore, this work can be regarded as the first step towards building a Logic of Rationality.

Keywords

PL

Alfred N. Whitehead; Alfred Tarski; logical consequence; ZFC; second-order set theory; forcing; modal logics; field of rationality; structuralism; platonism

• Publisher: Copernicus Center Press
• Journal: Zagadnienia Filozoficzne w Nauce
• Year: 2011
• Issue: 49
• Pages: 98-122

Dates

published

2011-11-19

Contributors

author

Piotr Wilczek