Ontology and the Foundations of Mathematics

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics.

SELF-REFERENCE, THEORY OF TYPES, AND CATEGORIZATION IN WITTGENSTEIN’S PICTURE THEORY OF STATEMENTS

Рассматривается источник логических парадоксов, выявленных Б. Расселом в системе обоснования математики, предложенной Г. Фреге. Самореферентность выражений, предложенная Б. Расселом как объяснение возникновения парадоксов, рассматривается с точки зрения разработанной им простой и разветвленной теории типов. Обосновывается, что теория типов, предложенная Б. Расселом, основана на онтологических предпосылках. Онтологические предпосылки зависят от предпочтения семантическому перед синтаксическим подходом, который принимается Б. Расселом. Рассмотрены синтаксические подходы к логическому символизму, которые позволяют устранить парадоксы с точки зрения языка современной символической логики. Анализируется подход к решению парадоксов Л. Витгенштейна, который основан на синтаксическом подходе. Показано, что этот подход отличается от способов построения языка, принятых в современной логике. The article analyzes the source of logical paradoxes Bertrand Russell identified in the foundations of mathematics proposed by Gottlob Frege. Russell proposed self-reference of expressions as the source of paradoxes. To solve paradoxes, he developed the simple and ramified theory of types. Ontological presuppositions are well substantiated for his theory; they depend on semantic, but not syntactic, preference. Contemporary approaches in symbolical logic prefer syntactic methods. But Wittgenstein’s approach in his Tractatus Logico-Philosophicus is more interesting, especially from the perspective of his picture theory of statements.

Taxonomy of Key Terms for Mathematics Education

We present the process of developing a taxonomy of key terms for Mathematics Education. We build on the existing taxonomy of key terms that has been used in an open access document repository. Additionally, we took into account terms that have been established in encyclopedias of the discipline and the frequency of use of keywords in specialized journals that were indexed in Scopus and Web of Science. We made a review of synonymy between these terms and the terms of the existing taxonomy. We included in our proposal the terms that are relevant given their frequency of use in the journals. We removed from the existing taxonomy the terms that are little used in practice. The new taxonomy is organized in six main categories: approach, educational level, foundations of Mathematics Education, research in Mathematics Education, pedagogical notions and mathematical content. This proposal was validated in three phases by researchers, innovators in Mathematics Education, and editors of specialized journals and experts who lead associations and events in the discipline.

Quantum information as the information of infinite collections or series

The quantum information introduced by quantum mechanics is equivalent to a certain generalization of classical information: from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The “qubit”, can be interpreted as that generalization of “bit”, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the transfinite ordinal number corresponding to the infinity series in question. The transfinite ordinal numbers can be defined as ambiguously corresponding “transfinite natural numbers” generalizing the natural numbers of Peano arithmetic to “Hilbert arithmetic” allowing for the unification of the foundations of mathematics and quantum mechanics.

Rules, understanding and language games in mathematics

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games.