In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing the (...) unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques. (shrink)

The following definition of objective is proposed: A statement S is objective if and only if in S all parameters that are relevant to its truth value are made explicit. The objectivity of predicates and relations can be defined in a similar manner. This simple conception of objectivity-which could be called explicitness conception of objectivity-can be found in Hermann Weyl and plays a central part in the natural sciences. There are grades of objectivity depending on the quality and the number (...) of parameters our predicates are relativized to A relativistic Ockham principle has to be recognized: Relativization parameters are not to be multiplied beyond necessity. The explicitness conception of objectivity is accessible to mathematical specifications, is the core of the idea of invariance, has a lot of philosophical applications and leads to precise notions of subjectivity and a precise formulation of the problem of the limits of objectivity. (shrink)

Mit der vorliegenden Arbeit verfolgen wir drei Ziele. Erstens exemplifizieren wir einige wissenschaftstheoretische Fragen und die zugehörigen Antworten am sehr einfachen Beispiel der klassischen Stoßmechanik. Zweitens läßt sich an diesem Beispiel besonders klar der Begriff des Meßmodells darstellen; insbesondere erhalten wir eine vollständige Übersicht über alle Meßmodelle. Und drittens erhalten wir ein schönes Beispiel für den Begriff der Reduktion einer Theorie auf eine andere, denn die Stoßmechanik läßt sich auf einfache Weise auf eine Spezialisierung der Klassischen Partikelmechanik reduzieren.

The later Wittgenstein holds that the sole function of mathematical propositions is to determine the concepts they invoke. In the paper this view is discussed by means of a single example: Wittgenstein's investigation of the concept of a regular heptagon as used in Euclidean geometry (i.e., the Euclidean constructiongame with rulerand compass) andinCartesian analytic geometry. Going on from some well-known passages in Wittgenstein's Lectures on the Foundations of Mathematics, and completing these passages, it is shown that Wittgenstein'sview makes perfectly good (...) sense and can be very well defended. (shrink)

Peter Janichs konstruktiver Aufbau einer Chronometrie, wie er in der neuen Auflage seines Buches "Die Protophysik der Zeit" beschrieben ist, wird einer detaillierten Kritik unterzogen. Dabei kristallisieren sich drei ernste Probleme heraus: Das Problem der Eindeutigkeit der Leitlinie, das Problem der Transitivität der Begleitstellungen, und das Problem der eindeutigen Bestimmtheit der Uhren. Alle drei Probleme bleiben bei Janich unbewältigt, woraus folgt, daß seine Chronometrie in der vorliegenden Form zumindest unvollständig ist.

Severin Schroeder’s book Wittgenstein on Mathematics is reviewed and at the same time critically discussed by concentrating on its main aim: to show the coherence of Wittgenstein’s mature philosophy of mathematics. Although Schroeder is dealing with Wittgenstein’s philosophy of mathematics in its entirety, he is mainly interested in the mature philosophy which he sees as dominated by two central ideas: that mathematics is essentially algorithmic, called the calculus view, and that the results of mathematical proofs are grammatical propositions, called the (...) grammar view. According to Schroeder there is an initial tension between these two ideas which he then resolves in two steps: he first clarifies the specific sense of »grammatical« with respect to mathematical propositions and, secondly, he emphasizes the application of mathematical propositions. The reviewer’s criticism of Schroeder’s approach is mainly concerned with this second point. Discussing several specific cases, the reviewer shows that Wittgenstein is much less oriented towards applicability than is assumed by Schroeder. (shrink)

Sebastian Grève interviews Felix Mühlhölzer on his work on the philosophy of mathematics.

Name der Zeitschrift: Wittgenstein-Studien Jahrgang: 5 Heft: 1 Seiten: 145-162.

Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...) so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered. (shrink)

Not Even a Something. If one exclaims that a pain is not a Nothing, what one means is that one feels it and that, when felt, it is simply there. Why shouldn’t one then also say, contrary to PI 293 & 304, that it is ‘a Something’? One has to admit, however, that, as felt, it is only momentarily there, and when I ask whether the pain that I feel now is the same as the pain I felt earlier, I (...) realize that for me there are no objective criteria of sameness: for me, what counts as the same is what appears to be the same. In the sparse privacy-scenario of PI 258 this talking about ‘sameness’ is totally empty, and so also our talking about ‘pain’, ‘sensation’, and so on, quite generally. In our public language, however, the situation is different, because there our use of a word like “pain” and our talking about the sameness of our own pains is embedded in our intersubjectively assessable speaking about pains. Nevertheless, also in this familiar case Wittgenstein’s statement near the end of PI 258 persists, at least under normal circumstances, and is accepted with regard to my own pain: “whatever is going to seem correct to me is correct”. But now, as part of our normal way of speaking, to claim this correctness is not empty. At the same time, however, it is not justified in a substantial way. With assertions like “I am in pain” or like “This is the same pain as earlier” the language-game does not end, as Wittgenstein says; it begins with them. This view, expressed in PI 290, can appear outrageous and may lead to the reaction, scrutinized in the beetle-paragraph PI 293, that everyone can tell from her own case that she knows what pain is. To talk in this way involves a grammatical model, “the model of ‘object and name’”, that presumably underlies our reference to our own private objects and according to which our language-game with sensation words would ‘begin’ with the objects. However, by this severing of reference from our use of language, the objects referred to prove to be completely indeterminate. – In the two ways just described, regarding sameness and determinateness, our talk of a ‘Something’ in the case of sensations turns out to be problematic. (shrink)

Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...) so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered. (shrink)