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Book chapter

Математика как «работа» с абстрактными объектами: онтолого-трансцендентальный статус математических абстракций

С. 417-449.
Катречко С. Л.
Moscow Philosophy of Mathematics Seminar’s sixth Collection of papers grew out of the 3rd Conference “Philosophy of Mathematics: Actual Problems” which was held in Lomonosov Moscow State University (Faculty of Philosophy) on September 27/28 2013. The 2013 meeting was focused on “Mathematics and Reality”.   This book may be of interest to philosophers and historians of mathematics and physics, as well as to logicians, mathematicians, psychologists, postgraduate and PhD students in all areas of mathematics and natural sciences.   Key words: philosophy of mathematics, philosophy of physics, ontology, realism,  Platonism,  antirealism,  the  applicability  of  mathematics,  mathematization, “unreasonable effectiveness of mathematics”.  
Language: Russian
Full text (PDF, 198 Kb)
Keywords: philosophy of mathematics

In book

Математика как «работа» с абстрактными объектами: онтолого-трансцендентальный статус математических абстракций
Математика и реальность. Труды Московского семинара по философии математики
М.: Издательство Московского университета, 2014.
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XXII World Congress of Philosophy
Vol. 41: Philosophy of mathematics. 2008.
Added: Jun 7, 2018
Transcendental Analysis of Mathematics: the Transcendental Constructivism (Pragmatism) as the Foundation of Mathematics
Katrechko S. L. Humanities. HUM. Basic Research Programme, 2015
Kant's transcendentalism is associated with the study and substantiation of objective value as a human way of cognition as whole and individual kinds of our knowledge. This article is devoted to Kant’s understanding (substantiation) of mathematics as cognition via constructing of concepts. Unlike the natural sciences the mathematics is an abstract – formal cognition, “[its] thoroughness is based on definitions and axioms” [B754]. The article consequently analyzes each of these components. Mathematical objects are determined by the principle of Hume. Transcendentalism considers the question of genesis and ontological status of concepts. To solve them Kant suggests the doctrine of schematism, which is compared with the modern concepts of mathematics. We develop the dating back to Kant original concept of the transcendental constructivism (pragmatism) as the programme of substantiation of mathematics. We also give a brief comparison of the axiomatic method of Kant and Hilbert.  “Constructive” understanding of mathematical constructions (calculations) is a significant innovation of Kant. Thus mathematical activity is considered as a two-level system, which supposes a “descent” from the level of rational understanding to the level of sensual contemplation and a return “rise”. The article also examines the difference between the mathematical structures and logical proofs. In his concept Kant highlights ostensive (geometric) and symbolic (algebraic) design. The article analyses each of them and shows that it is applicable to modern mathematics, in activity of which both types of Kant's constructing are intertwined. 
Added: Mar 19, 2015
Трансцендентальный анализ математики: абстрактная природа математического знания
Катречко С. Л. Кантовский сборник. 2015. Т. 52. № 2. С. 16-31.

Kant's transcendentalism is associated with the study and substantiation of objective value as a human way of cognition as whole and individual kinds of our knowledge. This article is devoted to Kant’s understanding (substantiation) of mathematics as cognition via constructing of concepts. Unlike the natural sciences the mathematics is an abstract – formal cognition, “[its] thoroughness is based on definitions and axioms” [B754].

The article consequently analyzes each of these components. Mathematical objects are determined by the principle of Hume. Transcendentalism considers the question of genesis and ontological status of concepts. To solve them Kant suggests the doctrine of schematism, which is compared with the modern concepts of mathematics. We develop the dating back to Kant original concept of the transcendental constructivism (pragmatism) as the programme of substantiation of mathematics. We also give a brief comparison of the axiomatic method of Kant and Hilbert.

 “Constructive” understanding of mathematical constructions (calculations) is a significant innovation of Kant. Thus mathematical activity is considered as a two-level system, which supposes a “descent” from the level of rational understanding to the level of sensual contemplation and a return “rise”. The article also examines the difference between the mathematical structures and logical proofs.

In his concept Kant highlights ostensive (geometric) and symbolic (algebraic) design. The article analyses each of them and shows that it is applicable to modern mathematics, in activity of which both types of Kant's constructing are intertwined. 

Added: Mar 19, 2015
Философия математики в контексте преподавания истории и теории философии и математики, науки и гуманитарных знаний
Барабашев А. Г., Перминов В. Я., Панфилов В. А. Вестник Днепропетровского университета. Серия: История и философия науки и техники. 2012. № 20. С. 27-39.

The significance of the education in the field of philosophy of mathematics as the part of both philosohpy and mathematics at the universities is the subject of the article.

Added: Nov 19, 2013