The paper analyses the concept of ‘bad infinity’ in connection with Hegel’s critique of infinitesimal calculus and with the belittling of Hegel’s mathematical notions by the representatives of modern logic and the foundations of mathematics. The main line of argument draws on the observation that Hegel’s difference is only derivatively a mathematical one and is primarily of a broadly logico-epistemological nature. Because of this, the concept of bad infinity can be fruitfully utilized, by way of inversion, in an analysis of (...) the conceptual shortcomings of the most prominent foundational attempts at dealing with infinite quanta, such as Cantor’s set theory and Hilbert’s axiomatism. As such, the paper is an attempt at reconstructing Hegel’s philosophy of mathematics and its role in his philosophical system and, more importantly, as a contribution to logic in the more general and radical sense of the word. (shrink)

The aim of this paper is to sketch a theory of musical experience which takes the empirical research seriously without abandoning or neglecting music’s transcendental features. The tension between the recent empirical approach, as represented particularly by Huron’s ITPRA theory, and the transcendental fact that music as an instance of art is something one can understand and, moreover, can understand oneself through, should be overcome by elaborating on the concept of emotion and the role it can play in musical understanding. (...) This will be done against the narrower background of the pragmatists’ theories of meaning, as represented by the semantic work of Meyer and Brandom, and their broader link to the philosophy of Hegel and the great systems of German idealism. (shrink)

The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on (...) diagonal construction are caused by superficial understanding of what a name is. (shrink)

Článek se zabývá Wittgensteinovou tezí, že se logika musí postarat sama o sebe, na pozadí jeho filosofických metod. Důraz je kladen zejména na metodu “přehledného znázornění” s úmyslem podat takové znázornění pro celek Wittgensteinovy filosofie v jeho souvislém vývoji od studia logické formy znázornění k pluralitě forem životních. Jádro článku tvoří diskuse tří principů čistého rozumu, zákonů identity, indukce a sporu, z wittgensteinovského úhlu pohledu. Poslední oddíl je věnován Wittgensteinově úvodní tezi a jeho filosofii v širším kontextu filosofických trendů, které (...) je formovaly, se zvláštním zaměřením na Kantův idealismus a pragmatismus Williama Jamese. (shrink)

Both Brandom and Wittgenstein base their concepts of experience on the game metaphor and the associated concept of rule. In fact, what Brandom seems to do is further refine Wittgenstein’s vocabulary by specifying the game as the game of giving and asking for reasons and rules as the rules of inference. By replacing the plurality of “games” with the one and only “game”, though, Brandom also lays the ground for a possible discord. This relates particularly to the cognitive significance of (...) different forms of human experience, such as music or art in general, which are treated by Wittgenstein as language games despite their being rather independent of claims and commitments and despite their utterly lacking the representational dimension. In my paper, I will show that with respect to these objections, one can argue that Brandom is in fact true to Wittgenstein’s instruction to always read his Investigations against his Tractatus. The general idea is to look at the game and picture metaphor as parts of a single concept that both philosophers work on together by going back to the very idealist concept of reflexivity or self-consciousness. (shrink)

The article deals with the evolution of the foundations of arithmetic: from the auspicious beginnings with Frege and Dedekind through the cataclysm of Russell’s antinomy to the late writings of Hilbert and the work of Paul Lorenzen . The proposed historical and dialectic reconstruction does not consist in ad hoc attempts to overcome formal difficulties of the systems in question. It rather shows how Frege’s and Dedekind’s original intentions are to be understood and can partly be justified. For this, Gödel’s (...) famous results are of great importance. (shrink)

Článek se zabývá wittgensteinovskou koncepcí následování pravidla na pozadí starých i moderních epistemických distinkcí, od Platónova rozpomínání se přes Kantovu opozici názoru a pojmu po Brandomovo činění něčeho implicitního explicitním.

Reviewing the beginnings and the further development of logicistic idea I am mainly focusing on the tradition of Frege and Russell, i. e. the tradition which was in the closest relationship with the enterprise of modern logic. I am trying to assess its connections to other „logistic“ programs such as Dedekind’s structuralism or Peano‘s axiomatism, to underline its specific features and the share in the success of modern logical semantics, including both its substantial contribution and limitations.

The objective of this paper is to analyze the broader significance of Frege’s logicist project against the background of Wittgenstein’s philosophy from both Tractatus and Philosophical Investigations. The article draws on two basic observations, namely that Frege’s project aims at saying something that was only implicit in everyday arithmetical practice, as the so-called recursion theorem demonstrates, and that the explicitness involved in logicism does not concern the arithmetical operations themselves, but rather the way they are defined. It thus represents the (...) attempt to make explicit not the rules alone, but rather the rules governing their following, i.e. rules of second-order type. I elaborate on these remarks with short references to Brandom’s refinement of Frege’s expressivist and Wittgenstein’s pragmatist project. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

The series publishes monographs and essay collections devoted to the history of philosophy as well as studies in the theory of writing the history of philosophy. A special emphasis is placed on the contextualization of philosophical historiography into the areas of the history of science, culture, and the wider scope of intellectual history.

Článek rozvíjí ideu Wittgensteinovy filosofické gramatiky a na příkladě její aplikace v hudbě ukazuje, jak Wittgenstein využívá pojmu "přehledného znázornění" ve vymezení filosofie jako něčeho, co nemůže být smysluplně vyřčeno. The paper elaborates on Wittgenstein’s idea of philosophical grammar by transposing it to the field of music and demonstrating how the concept of “perspicuous representation” can be used in delimiting philosophy as something that cannot be meaningfully said.

The article deals with Cantor’s diagonal argument and its alleged philosophical consequences such as that there are more reals than integers and, hence, that some of the reals must be independent of language because the totality of words and sentences is always count-able. My claim is that the main flaw of the argument for the existence of non-nameable objects or truths lies in a very superficial understanding of what a name or representation actually is.

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)

Starting with Wittgenstein’s remark about his allegedly frequent visits to the performance of Wagner’s Die Meistersinger von Nürnberg, the paper presents Wagner’s opera – being explicitly an opera about rules and rule-following – as a possible stimulus for the later Wittgenstein’s thinking about language. Besides Wittgenstein’s systematic interest in parallels between music and language, the paper draws on the choice of terminology and on Wittgenstein’s own examples of rule-following. More speculatively, the phrasing as well as the solution to what Kripke (...) called Wittgenstein’s sceptical paradox is used as a point of comparison that brings Wittgenstein’s aesthetic innuendos closer not only to mainstream philosophy of language, but due to the antithetical structure of Kripke’s argument, also to the broader philosophical and aesthetic tradition of German idealism. (shrink)

Ve svém příspěvku chci argumentovat pro takové chápání analytické filosofie, které ji nevymezuje novými tématy, ale specifickým přístupem, tzv. obratem k jazyku, jenž se ukáže být v souladu s principem pravé filosofie vůbec.

Warum schon bei Platon die Zahlen und ihr gutes Verständnis einen speziellen Platz in der philosophischen Bildung einnehmen, ist zunächst ein Rätsel. Eine Aufhebung der zukunftsweisenden Leistungen der Pythagoräer besonders in der Harmonielehre gegenüber mystifizierendem Verständnis eines Pythagoräismus ist daher nach wie vor interessant, auch noch im Blick auf Freges ‚drittes Reich‘ abstrakter Gegenstände oder Cantors Mengenlehre. Zahlen sind von philosophischem Interesse durch ihr enges Verhältnis zu den Formen von Rationalität und Sprache – und wegen der Möglichkeit, Aussagen nicht bloß (...) über Zahlen selbst, sondern auch über andere Verhältnisse durch Zahlen zu kodieren und dadurch zum Thema zu machen. Auf einfach nachvollziehbare Weise wird außerdem die Verschränkung von mathematischem Fortschritt, von Problemen und ihren Aufhebungen vorgeführt oder skizziert, etwa die Entdeckung inkommensurabler Größenverhältnisse und das Rechnen mit infinitesimalen Größen, Cantors Stufen des Unendlichen, Brouwers Intuitionismus, Gödels Unvollständigkeitsätze, u.a.m. Die Philosophie der Mathematik wird zum Lehrstück logischer Selbstreflexion überhaupt. (shrink)

V článku nejprve rekapituluji postoje, které k modalitám zaujali zakladatelé moderní logiky a analytické filosofie, preferující logickou či epistemickou redukci, poté přecházím k jejich soudobým metafyzickým oponentům, vycházejícím z novoleibnizovské ontologie možných světů, poukazuji na nevyjasněné předpoklady některých z jejich „argumentů“, k nimž pak – a k filosofické analýze možnosti vůbec – připojuji několik obecně-kritických tezí založených na Wittgenstein-Carnapově pojetí možného jako smysluplného.

[The continuum in the mirror of analytical philosophy].

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)