This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.
Quine’s criticism of the notion of analyticity applies, at best, to Carnap’s notion, not to those of Frege or Husserl. The failure of logicism is also the failure of Frege’s definition of analyticity, but it does not even touch Husserl’s views, which are based on logical form. However, some relatively concrete number-theoretic statements do not admit such a formalization salva veritate. A new definition of analyticity based not on syntactical but on semantical logical form is proposed and argued for.
Frege's semantics of sense and reference and two husserlian alternatives are discussed. it is shown that husserl neither took his semantics of sense and reference from frege nor abandoned psychologism under his influence. frege's arguments on behalf of his choice of truth values as the reference of statements and of concepts as the reference of conceptual words are submitted to criticism. some algebraic considerations are sketched in the last part of the article.
Husserl’s refutation of psychologism one hundred years ago in his opus mag-num Logische Untersuchungen is a painfully detailed enterprise. After justi-fying the existence of logic as a separate practical discipline, Husserl first shows that normative and a fortiori practical disciplines are founded on theoretical ones. He then formulates the psychologistic theses, extracts empirical consequences from them and shows how psychologism distorts the content of logical laws. The nucleus of the refutation consists in six arguments showing that specific relativism and, in (...) particu-lar, anthropologism is a form of skepticism, and, finally, establishing that psycholo-gism is a specific relativism, an anthropologism. A more direct and brief refutation follows, in which Husserl brings to the fore the prejudices on which psychologism is based. (shrink)
Analytic philosophy has been the most influential philosophical movement in 20th century philosophy. It has surely contributed like no other movement to the elucidation and demarcation of philosophical problems. Nonetheless, the empiricist and sometimes even nominalist convictions of orthodox analytic philosophers have served them to inadequately render even philosophers they consider their own and to propound very questionable conceptions.
In this short paper I am concerned with basically two especially important issues in Oswaldo Chateaubriand’s Logical Forms II; namely, the dispute between first- and higher-order logic and his conception of logical truth and related notions, like logical property, logical state of affairs and logical falsehood. The first issue was also present in the first volume of the book, but the last is privative of the second volume. The extraordinary significance of both issues for philosophy is emphasized and, though there (...) is a basic agreement with Chateaubriand’s views, some critical remarks are interspersed.Neste pequeno artigo considero basicamente duas questões particularmente importantes em Logical Forms II de Oswaldo Chateaubriand; a saber, a disputa entre a lógica de primeira e de segunda ordem e sua concepção de verdade lógica e noções relacionadas, como as de propriedade lógica, estado de coisas lógico e falsidade lógica. A primeira questão também estava presente no primeiro volume do livro, mas a última apenas aparece no segundo volume. Enfatizo o significado extraordinário de ambas as questões para a filosofia e, embora haja uma concordância básica com as visões de Chateaubriand, algumas observações críticas são inseridas. (shrink)
In this paper six of the most important issues in the philosophy of logic are examined from a standpoint that rejects the First Commandment of empiricist analytic philosophy, namely, Ockham’s razor. Such a standpoint opens the door to the clarification of such fundamental issues and to possible new solutions to each of them.
In this critical study I try to highlight some of the most important issues discussed in Chateaubriand’s excellent book. In particular, I discuss in some detail Chateaubriand’s criticism of one of the icons of analytic philosophy, namely, Quine, as well as some of his own valuable contributions to philosophy in this book – for example, his refutation of the various forms of the slingshot argument and his characterization of logical truth.
Probably the best arguments for Platonism are those directed against its rival philosophies of mathematics. Frege's arguments against formalism, Gödel's arguments against constructivism and those against the so-called syntactic view of mathematics, and an argument of Hodges against Putnam are expounded, as well as some arguments of the author. A more general criticism of Quine's views follows. The paper ends with some thoughts on mathematics as a sort of Platonism of structures, as conceived by Husserl and essentially endorsed by the (...) author. (shrink)
In this paper on Oswaldo Chateaubriand’s book Logical Forms I, I am mostly concerned with the critical task of indicating some shortcomings and stressing my disagreements with the distinguished scholar. The most important shortcoming of the book is Chateaubriand’s unfamiliarity with Husserl’s views on logic and semantics, some of which anticipate views propounded by the former – e.g., the distinction between logical law and logical necessity-, whereas others are more subtle than Chateaubriand’s views – e.g., Husserl’s views on the referent (...) of statements. One of the most important contributions of Chateaubriand’s book is his analysis and rejection of all forms of the so-called “slingshot argument”. On the other hand, I disagree with Chateaubriand’s rendering of some of Frege’s views, though some of these are very common among Fregean scholars. Finally, I assess Chateaubriand’s criticism of Kripke’s views as well as those of Tarski. I tend to agree with his criticism of Kripke, but disagree with his assessment of Tarskian semantics. (shrink)
This paper discusses Husserl’s views on physical theories in the first volume of his Logical Investigations, and compares them with those of his contemporaries Pierre Duhem and Henri Poincaré. Poincaré’s views serve as a bridge to a discussion of Husserl’s almost unknown views on physical geometry from about 1890 on, which in comparison even with Poincaré’s—not to say Frege’s—or almost any other philosopher of his time, represented a rupture with the philosophical tradition and were much more in tune with the (...) physical geometry underlying the Einstein-Hilbert general theory of relativity developed more than two decades later. (shrink)
Husserl's contributions to the nature of mathematical knowledge are opposed to the naturalist, empiricist and pragmatist tendences that are nowadays dominant. It is claimed that mainstream tendences fail to distinguish the historical problem of the origin and evolution of mathematical knowledge from the epistemological problem of how is it that we have access to mathematical knowledge.
Husserl developed – independently of Frege – a semantics of sense and reference. There are, however, some important differences, specially with respect to the references of statements. According to Husserl, an assertive sentence refers to a state of affairs, which was its basis what he called a situation of affairs. Situations of affairs could also be considered as an alternative referent for statements on their own right, although for Husserl they were simply a sort of referential basis. Both Husserlian states (...) of affairs and situations of affairs are extensional. Tarskian semantics can be rendered as a sort of state of affairs semantics. However, to assess adequately the existence of dual theorems in mathematics and, more generally, seemingly unrelated interderivable statements like the Axiom of Choice and its many equivalents, states of affairs are not enough. We need a sort of refinement of the notion of a situation of affairs, namely what we have called elsewhere an abstract situation of affairs. We are going to introduce abstract situations of affairs as equivalence classes of states of affairs denoted by closed sentences of a given language which are true in the same models. We first sketch the procedure for a first-order many-sorted language and then for a second-order many-sorted language.Husserl desenvolveu – independentemente de Frege – uma semântica do sentido e da referência. Contudo, há algumas diferenças importantes, especialmente com respeito às referências de enunciados. De acordo com Husserl, uma frase assertiva refere-se a um estado de coisas, que era sua base, o que ele chamou uma situação de coisas. As situações de coisas também poderiam ser consideradas como um referente alternativo para enunciados embora, para Husserl, elas fossem simplesmente um tipo de base referencial. Tanto os estados de coisas como as situações de coisas husserlianas são extensionais. A semântica tarskiana pode ser traduzida como um tipo de semântica de estado de coisas. Entretanto, para avaliar adequadamente a existência de teoremas duais em matemática e, de maneira mais geral, enunciados interderiváveis e aparentemente sem conexão, como o Axioma da Escolha e seus muitos equivalentes, os estados de coisas não são suficientes. Nós precisamos de um tipo de refinamento da noção de uma situação de coisas, a saber, o que nós chamamos de uma situação abstrata de coisas. Nós vamos introduzir as situações abstratas de coisas como classes de equivalência de estados de coisas denotadas por sentenças fechadas de uma determinada linguagem, as quais são verdadeiras nos mesmos modelos. Nós esboçamos o procedimento primeiro para uma linguagem multi-sortida de primeira ordem e então para uma linguagem multi-sortida de segunda ordem. (shrink)
Este estudio crítico se ocupa de la tesis doctoral de Rudolf Carnap, Der Raum. El mismo ofrece una breve exposición de esta obra juvenil, frecuentemente ignorada, de Carnap, e intenta corregir algunas interpretaciones incorrectas de dicha obra. Se muestra convincentemente que la principal influencia filosófica en Der Raum no es ni Kant ni los ne-okantianos, sino Edmund Husserl, y que la defensa que hace Carnap en esa obra de lo sintético a priori es claramente no kantiana, sino mucho más cercana (...) a lo que Carnap interpretaba que eran las concepciones de Husserl sobre lo sintético a priori.This critical study is concerned with Rudolf Carnap’s disserta-tion, Der Raum. It offers a brief exposition of Carnap’s often neglected youth work, and tries to correct some misunderstandings about that work. It is convincingly shown that the main philosophical influence on Der Raum is neither Kant, nor the neoKantians but Edmund Husserl, and that Carnap’s defense of the synthetic a priori in that work is clearly neither Kantian nor neoKantian, but much closer to what Carnap inter-preted as Husserl’s views on the synthetic a priori. (shrink)
In this critical study of the valuable collection of essays edited by Cirera, Ibarra and Mormann, the present author not only critically assesses the different renderings of Carnap’s writings propounded by the different authors therein represented, but also sketches his own interpretation and subjects to criticism some of the presumed consequences of the demise of logical empiricism.
This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.
Most areas of philosopher Edmund Husserl’s thought have been explored, but his views on logic, mathematics, and semantics have been largely ignored. These essays offer an alternative to discussions of the philosophy of contemporary mathematics. The book covers areas of disagreement between Husserl and Gottlob Frege, the father of analytical philosophy, and explores new perspectives seen in their work.