Educational Projection Systems, Levels And Prerequisites Of Mathematical Ontology OntoMathEdu

The developed educational projections, levels and prerequisites of the mathematical educational multilingual ontology OntoMathEdu are presented. Educational projection is viewed as the formalization of a certain system of subject training in mathematics. It is a subset of OntoMathEdu ontology concepts, which are structured at this stage of ontology development using two didactic relationships – educational level and prerequisites. Educational levels are allocated on the basis of the teaching standards of the corresponding education system, the relation of prerequisites is determined by the sequence of the studied concepts in a particular education system. The OntoMathEdu ontology defines two projections representing the educational systems of Russia and Great Britain. The algorithm for constructing an ontology through linking various projections allows it to be further replenished with new educational projections, which can later be used in the system of multilingual teaching of mathematics.

Sets, Set Sizes, and Infinity in Badiou's Being and Event

This paper argues that Cantorian transfinite cardinality is not a necessary assumption for the ontological claims in Badiou’s L’Être et l’Événement (Vol. 1). The necessary structure for Badiou’s mathematical ontology in this work was only the ordinality of sets. The method for reckoning the sizes of sets was only assumed to follow the standard Cantorian measure. In the face of different and compelling forms of measuring non-finite sets (following Benci and Di Nasso, and Mancosu), it is argued that Badiou’s project can indeed accommodate this pluralism of measurement. In turn, this plurality of measurement implies that Badiou’s insistence on the “subtraction of the one”, the move to affirm the unconditioned being of the “inconsistent multiple”, results in the virtuality of the one, a pluralism of counting that further complicates the relationship between the one and the multiple in the post-Cantorian era.

A New Hope for the Symbolic, for the Subject

This paper is perhaps an impressionistic response to accounts of the extraordinary set-theoretical activity being undertaken by W. Hugh Woodin (mathematician) and colleagues in the present moment, in the context of the mathematical ontology proposed and elaborated by Alain Badiou (philosopher). The argument presented is that the prevailing and sustained incoherence of the mathematical ontology (i.e. set theory) underscores a contemporary deficit of humanity’s symbolic organization which, in turn, yields confusion and conflict in terms of subjective orientation. But a new axiom (conjectured as yet) promises to realize a coherent set theory, i.e. stable, consistent and complete. This remarkable (and completely unexpected) development offers hope for the pursuit of a modern (i.e. non-hierarchical) symbolic, and a consequent resolution of the general subjective disorientation.

The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning.

La crítica de Berkeley al Cálculo de Newton [Berkeley’s critique to the Newton’s calculus]

En este artículo expongo las críticas que presenta George Berkeley, filósofo y Obispo de Cloyne, a la noción de fluxón que Newton introduce en su desarrollo del cálculo fluxional. En su propuesta Isaac Newton considera que se puede hablar de dos tipos de puntos: los puntos sin dimensiones y aquellos que surgen del movimiento y que pueden tener alguna clase de medida/magnitud/métrica, es gracias a la existencia de estos últimos que Newton puede realizar el cálculo de la fluxión aun cuando al final del procedimiento, vuelva a considerarlo como un punto sin dimensiones. Una vez expuesta la propuesta de Newton sobre las fluxiones, presentaré la crítica de Berkeley a diversos elementos tanto conceptuales como metodológicos del trabajo de su predecesor, Finalmente ofreceré algunas conclusiones acerca de la validez de las críticas del obispo de Cloyne a Newton. Palabras clave Newton, Berkeley, punto, medida, magnitud, movimiento, monadas. Preferencias Berkeley, George. “El Analista”. En Los escritos matemáticos de George Berkeley. editado por José A. Robles, 55-129. Traducido por José Antonio Robles. Instituto de Investigaciones Filosóficas, 2006. Euclides. The Thirteen Books of Euclid’s Elements. Dover Books on Mathematics, v. I. Traducido por Heath, T.L. Dover Publications, 1956. Guicciardini, N. The Development of Newtonian Calculus in Britain 1700-1800. Cambridge University Press, 2003. Jamblico también conocido como Iamblichus. The theology of arithmetic: on the mystical, mathematical and cosmological symbolism of the first ten numbers. Traducido por Robin Waterfield. A Kairos book. Phanes Press, 1988. Muntersbjorn, Madeline M. “Representational Innovation and Mathematical Ontology”. Synthese 134, no 1-2 (2003): 159-180. Newton, Isaac, “De gravitatione et æquipondio fuidorum”. En De Newton y los Newtonianos. Editado por: Lauria Benítez Grobet y José A. Robles García, 29-60. Traducción de Robles García J. Universidad Nacional de Quilmes, 2006. _____________________ . Itroduction to the Quadrature of Curves. Traducción al inglés por John Harris. Editado por Wilkins, D.R. en 2002 y 2014. School of Mathematics Trinity College, Dublin, 1710. Robles, José Antonio. “Comentarios a ‘De gravitatione et æquipondio fuidorum’”. En De Newton y los Newtonianos, 61-122. Universidad Nacional de Quilmes, 2006.

Continuity and Mathematical Ontology in Aristotle

In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle's understanding of continuity can only be saved if he holds a certain kind of mathematical ontology.

The Noble Family as “Singular Multiplicity”? Redefining the Smoczynski–Zarycki’s Totemic Definition of Nobility through the Lenses of Alain Badiou’s Mathematical Ontology

In our paper, we redefine the category of “family” denoting the relationship of selected members of a post-noble/post-aristocratic milieu in Poland using Alain Badiou’s terminology. Badiou’s ontology based on a mathematical set theory and a generic theory is the most developed, complex, and revolutionary ontology of the 20th and 21st centuries. However, it is rarely adapted to new empirical studies probably because of its novelty and complexity. We do not intend to use the empirical case study made by Smoczynski–Zarycki to inform our argument but instead perform a translation of the Durkheim–Lacanian theoretical standpoint from “Totem…” into the category of “singularity” [singularité] in its relation to “the state of situation” [état de la situation] from “Being and Event” (Badiou 2005). This approach seeks to find a universalizing potential of nobility that will allow it to become a relevant subject for truth procedure analysis.