Visual Thinking and the Socio-Historical Aspects of Richard Dedekind’s Contributions to the Foundations of Mathematics

Karolina Tytko

doi:10.14394/filnau.2023.0012

Autor

Karolina Tytko Uniwersytet Papieski Jana Pawła II w Krakowie https://orcid.org/0000-0003-2477-9090

DOI:

https://doi.org/10.14394/filnau.2023.0012

Słowa kluczowe:

Richard Dedekind, real numbers, natural numbers, PMP, philosophy of mathematical practice, visual thinking, socio-historical practices

Abstrakt

This paper presents a new interpretation of Dedekind’s philosophy of mathematics, based on an analysis of a selected part of his mathematical practice. The article consists of three parts. In the first part, I describe selected interpretations of Dedekind’s philosophy of mathematics, such as fictionalism, creationism, or realism on the one hand, and the ontology of the intentional object or structuralism on the other. In the second part, I introduce the tools and methods that I use in the third part of the article, such as Giaquinto’s proposed use of visual thinking in mathematical practice, as well as the socio-historical perspective of the development of mathematical knowledge. I also explain why a perspective of the philosophy of mathematical practice is useful here. In the last part of the article, I analyze how Dedekind introduced two number systems and present a new and original suggestion for the interpretation of his philosophy.
The main object of the research is Dedekind’s mathematical texts and the mathematical concepts and objects they describe, but I also try to reinterpret the non-mathematical statements contained in these texts. Finally, I argue that in addition to certain “technical” arguments that can be made against fictionalism, realism, and creationism in Dedekind’s case, the interpretation presented here
can also be used as an argument from the perspective of cognitive psychology and the socio-historical perspective of mathematical practices.

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