Philosophy of Mathematics

本书收录了《科学文化人与审美意识》、《试论“展望数学的新时代”》、《略论数学真理及真理性程度》等文章。.

This book is the first presentation in Polish scientific literature of the views and concepts of social constructivists on the complex and varied problems of the philosophy of science with particular consideration of mathematics. It is based on numerous publications and comments of the founders and supporters of social constructivism.

All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. In the (...) paper its historical and mathematical version is reconstructed. We next reconstruct Eudoxos' theory of proportions in an axiomatic fashion. Finally, we show that Heath's thesis both in the historical and mathematical version is false. To this end a counterexample is given; it is based upon a specific interpretation of the uniform distribution theorem. (shrink)

In the contemporary philosophy of mathematics there is an ongoing discussion concerning the issue of the justification of mathematical axioms and independent sentences. The works of P. Maddy in which the author focuses on the set theory are extensive studies devoted to that problem. In one of the monographs she deals with the problem of the justification of set theory's sentences in a new way, assuming a different metaphilosophical standpoint. In the paper (which is the first part of the two-part (...) whole) Maddy's conception is faithfully presented. The author makes no comments of his own, except for the first - introductory - chapter and footnotes, in which some additional explanations and comments can be found. (shrink)

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and (...) also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process. (shrink)

The main issue André Porto raises in his paper concerns the use of dot notation to indicate an infinite set of hypotheses. Whereas I agree that one cannot extract a unique infinite expansion from a finite initial segment, in my response I argue that this holds for finite expansions as well. I further explain how my remarks on infinite proof structures are neither motivated by the impact of Gödel’s incompleteness theorems on Hilbert’s program, nor by a negative view of strict (...) finitism.O problema central que André Porto discute em seu artigo diz respeito ao uso da notação de pontos para indicar um conjunto infinito de hipóteses. Mesmo estando de acordo não ser possível extrair uma expansão infinita a partir de um segmento inicial finito, em minha réplica argumento que isto vale igualmente para expansões finitas. Explico também que minhas observações sobre estruturas de prova infinitas não são motivadas pelo impacto dos teoremas de incompletude de Gödel no programa de Hilbert, e tampouco por uma visão negativa do finitismo estrito. (shrink)

In his book Chateaubriand points out some differences between the mathematical and the formal notions of proof. I argue here that the contrast between both cannot be exaggerated, and that the latter fails to represent essential aspects of the former. I also sketch a view of the nature of mathematics that can accommodate one particular feature of mathematical proofs the formal notion, by its very nature, cannot: their freedom.Em seu livro, Chateaubriand aponta algumas diferenças entre a noção formal e a (...) noção matemática de demonstração. Eu argumento que o contraste entre ambas não pode ser maior, e que aquela é incapaz de capturar alguns aspectos essenciais desta. Eu apresento também um esboço de uma teoria sobre a natureza da matemática capaz de acomodar um aspecto particular das demonstrações matemáticas que a noção formal, pela sua própria natureza, não pode: a liberdade que por direto cabe àquelas. (shrink)

This article attempts to link the notion of absolute ego as the ultimate subjectivity of consciousness in continental tradition with a phenomenology of Mathematical Continuum as this term is generally established following Cantor’s pioneering ideas on the properties and cardinalities of sets. My motivation stems mainly from the inherent ambiguities underlying the nature and properties of this fundamental mathematical notion which, in my view, cannot be resolved in principle by the analytical means of any formal language not even by the (...) addition of any new axioms to a consistent first-order axiomatical system such as the Zermelo-Fraenkel Set Theory. In this phenomenologically motivated approach I deal, to some extent, with the undecidability of a fundamental statement about the cardinality of Continuum inside ZFC theory, namely the Continuum Hypothesis, and also with the underlying root of Gödel’s first incompleteness result.Este artigo procura conectar a noção de ego absoluto enquanto subjetividade última da consciência na tradição continental com a fenomenologia da matemática do contínuo tal como este termo foi estabelecido seguindo as idéias pioneiras de Cantor sobre as propriedades e cardinalidade de conjuntos. Minha motivação deriva basicamente das ambigüidades subjacentes à natureza e propriedades desta noção matemática fundamental as quais, no meu entender, não podem, em princípio, ser resolvidas pelos meios analíticos de qualquer linguagem formal e nem mesmo pela adição de novos axiomas a um sistema axiomático consistente de primeira ordem como a teoria de conjuntos de Zermelo Fraenkel . Neste tratamento fenomenológica-mente motivado eu lido em certa medida com a indecidibilidade de uma tese fundamental sobre a cardinalidade do contínuo em ZFC, a saber, a Hipótese do Contínuo, e também com aquilo que está na base do primeiro resultado de incompletude de Gödel. (shrink)

The aim of this study is to try to make use of real numbers for representing an infinite analysis of individual notions in an infinity of possible worlds.As an introduction to the subject, the author shows, firstly, the possibility of representing Boole's lattice of universal notions by an associate Boole's lattice of rational numbers.But, in opposition to the universal notions, definable by a finite number of predicates, an individual notion, cannot admits this sort of definition, because each state of an (...) individual subject is characterized by the values taken by an infinite number of predicates, each of whom may appear or disappear in the next state.The notion of “degree of identification of an individual notion” is then introduced and arithmetized by a rational number.As an individual notion can be defined by a convergent succession of degrees of identification, the real characteristic number of such an individual notion can be defined by the corresponding convergent succession of rational numbers, satisfying Cauchy's conditions for the convergence of successions. (shrink)

The author elucidates the ontological basis of elementary mathematical theories and thereby assesses their certainty as a foundation for the more complex theories of modern mathematics, such as mathematical analysis and set theory. He adduces arguments in favor of the position of Frege, who held that geometry can provide a sufficiently broad and certain foundation for mathematics.

Logical theory – and philosophical theory generally – is just that, theory. Generations of logic students felt a sort of unease about it without knowing what to do about it. Nowadays, students of mathematical logic feel a similar unease when faced with the fact that in standard predicate calculus, “All unicorns are sneaky” is true precisely because there are no unicorns. Blanché’s analysis reminds us that such feelings of unease may indicate a shortcoming in the theory rather than in the (...) student’s understanding. (shrink)

This is a useful new translation of Fr. Bochenski’s well–known Precis. It is based on the original French edition with some important additions from the later German version. It is very brief—only about 75 pages of print, in fact—yet it covers in outline a great part of modern formal logic. All sorts of special abbreviations are used; historical and bibliographical notes are terse but illuminating ; explanations and discussions are avoided altogether. Everything of this sort is left to the teacher: (...) the book is intended to be a student outline, recalling the main features of formal logic but in no way elaborating upon them. (shrink)

This anthology has had a fantastic success in the U.S., selling over 200,000 sets in a few months, and prompting its astounded editor to remark that the tasteful four-volume set must have been secretly pushed by interior decorators! We have had so much of anthologies in late years that one automatically inclines to discount their lasting value in a field where profound and hard thinking is so necessary. Let it be said at once that Mr. Newman has not attempted to (...) give us any “straight” mathematics. No, this is a book about mathematics, not of mathematics. But it is one which—if I may use an overworked phrase—no one who is interested in mathematics, be he a philosopher, economist, strategist, historian or even mathematician, can possibly afford to be without. Its generous excerpts from the works of great mathematicians and perceptive commentators on mathematics are stimulating reading. The excerpts number 133 and range in length from five to seventy pages. There is something here for anyone with any interest in one of the mightiest adventures of the human mind. (shrink)