Issue #06. August 1996. Pp. 61-68.
Copyright (C) by SORITES and Jesús Padilla-Gálvez
Gödel, Kurt: Unpublished Philosophical Essays. (With a historico-philosophical introduction by Francisco A. Rodríguez-Consuegra (Ed.) and Foreword by W. V. Quine). Basel; Boston; Berlin: Birkhäuser, 1995, 235pp. + 8 Fotos.
The publication of essays by Gödel edited by Francisco A. Rodríguez-Consuegra comprises three very distinct sections. The first section containts a succint introduction by W. V. Quine (1994, 9-10 / 1995, 7-8) and the acknowledgements of the editor (1994, 11-14 / 1995, 9). In the second section (part I) one finds the introduction in which the editor writes on `Kurt Gödel and the philosophy of mathematics' (1994, 21-126 / 1995, 15-106). The third section (part II) is the most comprehensive one (1994, 127-240 / 1995, 107-235) consisting of the following five parts: 1. the character and origin of the manuscripts (1994, 129-144 (+ 4) / 1995, 109-123 (+ 4)); 2. the text `Some basic theorems on the foundations of mathematics and their philosophical implications' (1994, 129-147 / 1995, 149-169), `Gödel's footnotes' (1994, 171-178 / 1995, 149-157) and the `Appendix' (1994, 179-187 (+ 3) / 1995, 159-167 (+ 3)); 3. the text `Is mathematics syntax of language?, II' (1994, 191-207 / 1995, 171-189) and `Gödel's footnotes' (1994, 209-228 / 1995, 191-211); 4. the text `Is mathematics syntax of language?, VI' (1994, (+ 1) 231-236 / 1995, (+ 1) 213-218) and `Editorial footnotes of comparison with version V' (1994, 237-240 / 1995, 219-222); 5. and Index of themes and authors (1995, 223-235).
W. v. O. Quine presents the main objectives of Gödel's proposals: the theorem of incompleteness which shows that it is not possible to construct any formal method which it would be possible to prove all mathematical truths. He underlines the effort made in locating the manuscripts on philosophy of mathematics in state and their processing by Rodriguez-Consuegra. In the acknowledgements, the editor cites innumerable persons to whom he is indebted. In the introduction the reader finds a survey of the topics addressed in the texts.
F. Rodríguez-Consuegra analyzes the motives and intuitions that led Gödel to the results of the completeness theorem for basic logic and his incompleteness theorem for arithmetic. Gödel's mathematical realism is considered to be a philosophical consequence of these results. It is a heuristic principle that leads to them and a philosophical hypothesis which is verified by them. He them turns to an analysis of the ontological, semantic and epistemological components of mathematical realism. The autor believes that Gödel defended the analytical character apriori, but not in a tautological sense of the mathematical propositions founded on an obscure notion of meaning. He distinguishes four forms of understanding the term «analytical» with regard to mathematical statements, i. e., (i) logico-syntactical analytical; (ii) logico-semantical analytical; (iii) epistemological analytical; and (iv) theoretical analytical. G. Frege opts for (i) and (iii), that is, for what can be proven, what is logically true and known a priori, directly and intuitively. L. Wittgenstein is associated with (i) even if it is not stated whether this is the early or late Wittgenstein. R. Carnap is considered to accept (i) and (ii), that is what is true is considerd according to synonym and meaning. W. v. O. Quine accepts (iv) and thus what is not empirical, not observable without theory, that is, what receives the objective status of a theory or of language. Finally, K. Gödel accepts elements of (ii), (iii) and (iv). Gödel's and Quine's critique against Carnap is interpreted in a similar way. Whith regard to these critical remarks we will deal with some specific points in the published book.
In the article titled `Some basic theorems on the foundations of mathematics and their philosophical implications' the key intuitions and motivations of Gödel's metamathematical results are addressed. It is on them that his realist position is based. In a neutral way, we could understand realism to be the postulation of a model that applies, in a speculative way, the relation between a singular term and an individual object, which can be demonstrated empirically in generic terms. In this sense general terms are postulated to be universals, which in turn cannot be empirically demonstrated. In this connection Gödel formulates his convention that mathematical objects exist and that their reality is analogous to that of physical objects.
In this article the author argues by reference to the impossibility of developing a reductionist program founded on mathematics and to the shortcomings of a mechanist and algorithmic vision of the human mind. This position is, above all, based on its platonicfoundation which is defined in the sense that «...mathematics describes a non-sensual reality, which exists independently both of the acts and the dispositions of the human mind and is only perceived, and probably perceivad very incompletely, by the human mind.» (1994, 169 / 1995, 147.)
Another important topic revolves around the thesis that the nature of philosophy of logic and mathematics is analytic even if tautological. Thus the propositions and mathematical axioms refer to the mathematical objects that are analyti but not tautological, that is, their truth depends on the meanings of the concepts that they include but not on their definitions. This thesis is buttressed by a specific epistemological conception. The intuition of the objects and mathematical axioms can lead to real knowledge. According to the author, mathematical intuition contributes to guaranteeing the truth of mathematical axioms. The axioms and the mathematical objects are necessary to systematize mathematics and to thus make it possible to explain propositions. Gödel declares the thesis to be refutable according to which mathematical intuition can be replaced by conventions. The argument goes as follows: if we accept, in a given case, that certain syntactical conventions can replace mathematical intuition, we are obliged to prove the consistency of such conventions, enabling us to deduce any proposition from them. A similar proof of consistency is based on the mathematical intuition by virtue of which it remains without validity. (see: 1994, 231-236 / 1995, 213-218).
In `Is mathematics syntax of language?, II' the autor presents H. Hahn's and R. Carnap's position as a combination of nominalism and conventionalism. Nominalism is supposed to deal with the universe of discourse in moderation. Regarding this positions, Gödel attacks the thesis according to which mathematics is reduced to a formal syntax of language, by virtue of which is nature would be tautological and void of meaning. Here the reader is confronted with a purely polemical text.
The third chapter presents the context and the arguments according to which mathematics is similar to physics, both in terms of goals and methods. The analogy with physics is based on sensual perception. We were able to show above that one of the realist conclusions was that objects and mathematical axioms alloved us to systematize mathematics and to thereby explain the propositions. Such propositions can also be understood as «sense data» whose generalizations require transcendent assumptions. In this way, one can understand that the effectivity of the propositions is relevant. Its relevance is equally based on the parallelism that is created with the physical objects.
In `Is mathematics syntax of language?, VI' Gödel declares the thesis that mathematical facts and objects do not exist to be refutable. To this end he develops the following five arguments: (a) if it is maintained that mathematics and logics do not have empirical consequences, then the same holds in a similar way for laws of nature, given the fact that to infer such consequences from them already requires mathematics and logic with an objective content not expressed in terms of laws. (b) Mathematical axioms are just as refutable as laws of nature: it suffices to derive an inconsistency from them. Consequently, they have content and their objects existence. (c) The vacuity of mathematics could not be maintained even if we were to admit that it could be based on symbolic conventions. Consequently, we would have to show that such conventions do not add anything to the theory in which they are formulated. (d) If mathematics were to be constructed on the base of syntactic rules this would not make it more conventional since if we admit that the definition of meaning of a concept consists in its rules of use, then different rules would introduce distinct concepts, and such liberty in the selection of concepts does not obtain only in mathematics. (e) Empirical perception and mathematical intuition are forms of experience whose difference only consists in the fact that the former relates concept and object and the latter only concepts. Both, however, have the same function of unifying the multiplicity of independent impressions. Gödel state that the thesis that conventionalism in mathematics would be compatible with empiricism is refutable. The counter-argument goes as follows: if the consistency of conventions were based on empirical induction, then mathematics could not be conceived a priori, however, to prove this consistency by means of mathematical intuitions is incompatible with empiricism.
Jesús Padilla-Gálvez
Johannes Kepler Universität
Cirera Duocastella Ramón, Carnap i el Cercle de Viena. Empirisme i sintaxis lógica. Barcelona: Anthropos 1990, p. 406.
Cirera Ramón, Carnap and the Vienna Circle (Empiricism and Logical Syntax). Rodopi: Amsterdam, 1994, p. 398 + xvi.
Ramón Cirera Duocastella's book (1990) is the author's doctoral thesis and was first published in Catalan as «Carnap i el Cercle de Viena. Empirisme i sintaxis lógica». Later he overworked it, and it appeared in 1994 with a number of minor alterations as «Carnap and the Vienna Circle (Empiricism and Logical Syntax).» The autor believes that the standar history and philosophy of science textbooks used for teaching in Spain should be rewritten to avoid historical imprecisions and since the doctrines discussed in the Circle are more interesting that what is referred to as official history. The author proceeds to describe tha arguments which he tries to refute and which we will now summarized: 1) the effort made to present the Circle not as a group of individual thinkers (chapter 1). 2) The author insist that verificationism had little importance for Carnap's wirk and that it was not at all present in Neurath's work. 3) The integration of the Circle's work in its historical context. The author first focuses on the turbulent life of the protagonists in the socio-political world of Vienna in the thirties. The book criticizes the distorting image of present-day historiography, first pointing to the discrepant influences of Schlick through L. Wittgenstein (chapter 2), of Neurath from a socio-political perspective (chapter 3) and of Popper by way of his methodological approach.
The author clearly develops his arguments in chapters four through six, analyzing how Carnap breaks with the epistemological stance of Aufbau before reaching the central argument of Logische Syntax der Sprache in 1934. He cites the most relevant arguments on epistemological neutrality. The author also presents the ideas which buttressed the construction of models of physicalism, conventionalism, etc.
Jesus Padilla-Gálvez
Johannes Kepler Universität
Cirera, Ramón, / Ibarra, Andoni / Mormann, Thomas: El programa de Carnap. Ciencia, lenguaje, filosofía. Barcelona: Textos del bronce, 1996.
The authors under review here understand that Carnap's work can be interpreted from different angles. Either, they seek to reconstruct the place which the author himself ascribes to it or they view it taking into account its philosophical descendence. The former approach is taken, among others, by R.Cirera Duocastella, Reiner Hegselmann und Thomas Uebel who go to great pains to explain that Carnap's project is interesting, that one should adhere more to history and understand why Carnap was influstered by his critics. The second approach does not have anything to do with the proposal sketched by Carnap himself. It is one taken by most of the authors of the volume published this year. Generally speaking it is argued that Carnap is the most important representative of the so-called received view and that this approach was overcome by the approaches which preceded it. For somewhat sinister reasons this gives the impression that it is interesting to unearth the dead from time to time to make sure that they are still lifeless. Some were to engage themselves in the dialogue with the deaf from which we would like to draw some conclusions.
Another group of works mentioned at the beginning of this review are related to a survey of Carnap's thought as something at bit obsolete and worthy of being placed in the annals of the history of philosophy. In his artcle, Carlos Ulises Moulines suggests a computational interpretation of Aufbau. For him the epistemological notion of the ideal observer becomes formally elucidated, i.e., that the observer capable of proving any statement of empirical science. His second article, Las raices epistemologicas del Aufbau de Carnap, seeks to show the influences of Neokantianism on empiricism, logical constructionism and scientific naturalism in Carnap`s work.
In `Teoría de los signos en Carnap' Javier Echeverría argues that Aufbau reveals the only semiotic idea which Carnap himself did not develop. The later development of Carnapian philosophy shows a gradual disinterest in semiology (p. 99 ff.). In his early work, he makes certain corrections of Frege's theory with respect to extensionality and intersubjectivity (p. 107). The author believes that Carnap's theory of signs vanished from the philosophical scene and that it could be recovered through reinterpretation.
Reiner Hegselmann gives an account of the ideological position of the Vienna Circle in his article titled `The Scientific Conception of the World'. The exploration of similarities and differences between R. Carnap and other authors allows analogies and distinctions to be drawn which is why this is a very common method for delimiting an argument. All of the following approaches are based on this position. Through the distinction between R. Carnap and L. Wittgenstein suggested by J. M. Terricabras in his `The Logic of Tractatus and the logical Construction of Carnap' different approaches are thus revealed. To this end he analyzes the thesis of extensionality (p. 152 ff) and the conceptions of method and the objective of philosophy (p. 160 ff).
The article by Thomas Uebel is titled `Physicalism in Wittgestein and Carnap'. The author tries to show that the controversy between Wittgenstein and Carnap over the priority of physicalism could in large measure be resolved by pursuing Hector Neri Castaneda's idea that, with regard to the problem of private language, Wittgenstein's analysis was not the only one that it evolved within the context of analytical philosophy in the thirties and fourties. He believes that the ideas of Carnap and Wittgenstein were quite different from those of physicalism. Whereas Wittgenstein rejects a phenomenon-based language while maintaining an interest phenomenology, Carnap retains a primitive protocol language for epistemological reasons. According to the author, J.Hintikka's idea that Wittgenstein had reasons right in becoming angry over Carnap's physicalism because it incorporated his idea of 1929 is wrong. Carnap was considerably less radical than Wittgenstein in his physicalism. Moreover, what Carnap elaborates was not what spurred on Wittgenstein (in pace McGuinnes). His doctrines on physicalism were not identical because in the early thirties, their respective physicalism were based on different versions of private language.
Thomas Mormann takes it upon himself to analyze language in Neurath and Carnap. He uses an inconsistent and vapid account of analytical philosophy as a framework for examining both philosophers. Accordingly, the former views language as a universal means which suggests that he followed the same line of argumentation as Heidegger and Hintikka (p. 216). The second considered language as a calculus which is whyhis thought bore resemblance with the project of Husserl and van Heijenoort. With his point (i) uniformity of logical empiricism and (ii) that which impliesan «antiphilosophy» as opposed to the western tradition is called into question.
Dirk Koppelber deals with empiricism and pragmatism in Carnap and Quine. The author focuses on a study that elaborates on the distinction between analytcal and synthetic statements and the distinction between internal and external questions. By means of such distinctions, the author delimits Quine's position vis-a-vis Carnap's ideas. According to Carnap both resolve central difficulties which are encountered in all of the conceptions of classical empiricism. They permit an epistemologically satisfactory explanation of the xistence of logical and mathematical truths, while also furnishing a theoretical clarification of the interrelation between philosophy and science. By contrast, Quine believes that it is not possible to obtain anysatisfactory form of empiricism by means of the two distinctions.
Andres Rivadulla analyzes Bayesian probability, frequential probability and the Carnapian theory of statistical inference. The author gives a number of reasons why the viability of Carnap's attempt to convert theoretical statistics into part of inductive logics can be called into question. The first is the obscurity of the concept of induction, the second the equivocity in the use of the concept of estimation. Third, the limited applicability of the explicatum c* of probability, restricted to a very simple artifial language, and finally the opacity of the interpretation of logical concept of probability as a degree of confirmation. According to the author, identifying Carnap with logical probability with a degree of confirmation means transforming the theory of probability into an inductive logic. This is hardly able to offer a logical foundation of theoretical statistics, but it does constitute a serious attempt to logically reconstruct existing statistical methods.
The book closes with a bibliography of works on Carnap, which is aleatorical and lacks all criteria of selection. There are too many works published in Castilian which are not listed for obscure reasons. The works in German are not listed according to their importance and reference is only made to books of certain authors which were published or complete in the years these volumes appeared, while essential works in English and Italian are completely missing. There also remain some overly superficial works. Bibliographical references could have also been made to Austrian philosophy published in Graz and other related works.
Jesús Padilla-Gálvez
Johannes Kepler Universität