As bases do intuicionismo matemático de Brouwer a natureza do continuum intuicionista

• Autor(a) principal: Oliveira, Paulo Júnio de
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/00130000063m8
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/11384
• topic: Teoria intuicionista do continuum; Intuicionismo matemático; Idealismo neokantiano; Brouwer; Intuitionist continuum theory; Mathematical intuitionism; Neo-Kantian idealism; Brouwer; CIENCIAS HUMANAS::FILOSOFIA

This dissertation has as its aim the philosophical presentation and discussion of the nature of the intuitionist continuum of Luitzen Egbertus Jan Brouwer (1881-1966) and of its philosophical bases. This conception of the nature of the intuitionist continuum led to the development of the notion of “real numbers” distinct from classical analysis. Such a notion of “real numbers” does not accept the principle of excluded middle as universally valid and, as a result, it would not be possible to accept, for example, the law of trichotomy. The refusal of the principle of excluded middle does not arise from vacuum, and it is not the central focus of mathematical intuitionism. It is, as one would put it, a consequence of the conception of intuitionist “continuum”. The notions of “continuum” and “mathematical entity” are, as one would put it, the main focus of Brouwer’s analysis. From his point of view, the continuum is not a collection of absolutely individualized already given discrete points, which could in some way be extracted and used. The mathematical points in intuitionism are, so to speak, mentally constructed as sequences of infinitely converging intervals. Thus, no interval has an absolutely segmented, intrinsically formed, and/or absolutely individualized “limit”. For any “stage” of the interval there is always a space of infinite possibilities of other intervals. Hence there would be no point of infinitely distant cumulation of the intervals that could be found by the mathematician, as if the continuum were mappable such as a city is mappable by a geographer. From this point of view, the classical perspective would need to subscribe, even if not intentionally, to the existence of absolutely individualized discrete points in order to be able to make some kind of sense of its analysis of real numbers, of calculus, in short, of classical mathematics. This intuitionist notions of “continuum” and “real numbers” qua “infinitely converging intervals” became possible because of the type of philosophical bases that precede them. Such bases were developed from two fundamental acts that are understood in the context of an idealistic philosophy of Kantian inspiration. Thus, such acts are comprehended from a “Neo-Kantian” framework, in some sense or, more accurately, to use Brouwer’s expression, from an “up-to-date Kantism”. They are: (i) the act of mental recognition of the distinction between mathematical entities and linguistic entities and (ii) the act of mental recognition of the possibility of new mathematical entities. Both acts are interconnected and presuppose the same bases. In fact, the first act is connected with the radically intuitive aspect of the mental construction of mathematical entities, and the second act brings to light particularly important notes of the nature of the intuitionist continuum through the possibility of always emerging new mathematical entities, i.e., the continuum is a type of entity that is never determinable, it does not ever have an absolute form determined by a collection of discrete points that are absolutely individualizable. Therefore, the continuum is definitely indeterminate. In other words, the second act acknowledges that mathematical entities are intensionally “expansible” through the notion of “choice sequences”. Thus, these sequences are fundamental to the intuitionist continuum. In this dissertation, we present these philosophical bases in the first chapters and apply them to some specific mathematical contexts in the last chapters so as to try to elucidate the philosophical and mathematical nature of the intuitionist continuum and/or real numbers through the explicitness of properties of cohesion, viscosity, and infinitely converging intervals.