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Objectivity in Mathematical Knowledge

José Ferreirós

in Mathematical Knowledge and the Interplay of Practices

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new ... More

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new hypotheses are embedded in the web of mathematical practices, they become systematically linked with previous strata of mathematical knowledge, and this forces upon us agents (for example, research mathematicians or students of math) certain results, be they principles or conclusions. The chapter first considers a simple case that illustrates objective features in the introduction of basic mathematical hypotheses. It then discusses Georg Cantor's “purely arithmetical” proofs of his set-theoretic results, along with the notion of arbitrary set in relation to the Axiom of Choice that has strong roots in the theory of real numbers. It also explores Cantor's ordinal numbers and the Continuum Hypothesis.Less

Neo-Fregeanism and the Burali-Forti Paradox

Ian Rumfitt

in Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the ... More

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle per se. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.Less

Numerals: Earrooyn

in Practical Manx

Manx cardinal and ordinal numerals can occur with or without a noun. This chapter discusses cardinal numbers; cardinal numbers with noun; duals; nouns of measure after numeral; numerals preceding ... More

Manx cardinal and ordinal numerals can occur with or without a noun. This chapter discusses cardinal numbers; cardinal numbers with noun; duals; nouns of measure after numeral; numerals preceding nouns; nouns embedded in compounds; numerals with dy; numerals used as nouns; ordinal numbers; fractions; telling the time; and mathematical terms.Less

Nationalism and the Process of Reception and Appropriation of the Periodic System in Europe and the Czech Lands

Soňa Štrbáňová

in Early Responses to the Periodic System

The 1870s marked the onset of an exceptionally fruitful and dynamic period in the development of chemistry in the Czech Lands. University education and ... More

The 1870s marked the onset of an exceptionally fruitful and dynamic period in the development of chemistry in the Czech Lands. University education and research in chemistry was taking place at several universities and technical universities, where the structure of the main chemical subjects developed gradually into organic, inorganic, analytical, physical, fermentation, and medical chemistry, just to mention the main specialties. At the same time, the process of the Czech National Revival led to the cultural, linguistic, social, and political emancipation of the modern Czech nation and stepwise almost entirely separated the linguistically Czech and German scientific communities in all their representations, including university education. In Prague, the divided German and Czech Polytechnics (and later Technical Universities) existed since 1869, whereas the Charles-Ferdinand University split into its Czech and German counterparts only in the years 1882 and 1883. The chemical community was organized in several professional associations that also reflected the ethnic division of the scientific scene. The Society of Czech Chemists, founded in 1866, had almost exclusively Czech membership, while a specialized German chemical association has never been created in the Czech Lands. This study deals with two closely intertwined themes: the reception of the periodic system in the Czech Lands and in Europe and the crucial role of the Czech chemist Bohuslav Brauner in this process. I am going to demonstrate a specific set of conditions that shaped the process of appropriation of this new scientific idea by not only scholarly argumentation, but also particular circumstances, in this case Slavic nationalism and Russophilia in the Czech society at the turn of the nine­teenth century. The course of dissemination and reception of the periodic system also showed linkage to the linguistic emancipation of the Czech nation as reflected in the controversy over the Czech chemical terminology, where the periodic system served as argument to one party of the dispute. Less