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

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)

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)

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.

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)

Felix Mühlhölzer’s book Wissenschaft (“Science”) appeared in the series “Grundwissen Philosophie” (“Basic Knowledge in Philosophy”), whose aim is to give an introduction to different philosophical topics. Thus, Mühlhölzer’s Wissenschaft can be read with profit by beginners, while not providing a straightforward introduction to the subject. Rather, the author puts forward an argumentation of his own, and in so doing, chooses to ignore some essential questions and problems of the Philosophy of Science.On the one hand, some very influential positions are (...) not mentioned or only implied, e.g. Popper’s Falsificationism, Lakatos’ approach to research programs or van Fraassen’s Constructive Empiricism. On the other hand, a lot of more specific issues are also not considered, e.g. Nancy Cartwright’s dilemma concerning the truth of natural-law-assertions or the regress-argument concerning the relationship between experiments and theories. Therefore my review will not discuss the didactic qualitie. (shrink)

References to God. Some Remarks by Wittgenstein on Religion in the Years 1949 – 51. After a brief overview of Wittgenstein's stock of remarks on the subject of religion from 1949 – 1951, this article will focus on two particular points: supposedly nonsensical conceptions of God, for instance in the context of proofs of God, definitions of the term ”God” by hinting at something. Connections between and both systematically and exegetically within the framework of Wittgenstein's remarks are made.Ich danke Anja (...) Weiberg für sehr hilfreiche Kommentare. (shrink)

This paper discusses the pluralist theory of reparations for historical injustice offered by Daniel Butt (Ethical Theory and Moral Practice 24(5):1161–75, 2021). Butt attempts to vindicate purely past-regarding corrective duties in response to Alasia Nuti’s historical-structural model of reparations. I agree with Butt that reparative justice requires both past-regarding and future-looking structural duties. And I agree with him that Nuti’s model leaves out purely past-regarding duties. I argue, however, that Butt does not offer a genuinely pluralist account. I present minimal (...) necessary conditions for past-regarding (corrective) justice and demonstrate that the past-regarding duties Butt advocates do not meet these conditions. The past-regarding duties Butt offers collapse into the kinds of distributive (structural) duties from which he attempts to separate them. Yet, I suggest these shortcomings are instructive and sketch a path forward for a genuinely pluralist account of reparations. A genuinely pluralist account must follow this path in order to vindicate the intuitions that motivate both past-regarding duties and the structural injustice model. (shrink)

Der Band nimmt gewisse, mit dem Begriff Theurgie assoziierte Wandlungen innerhalb des spätantiken Neuplatonismus in den Blick. Theurgie wird vor allem als Suche nach immanenten Medien zur Verkörperung göttlicher Kräfte verstanden. Entgegen dem häufig kolportierten Bild vom weltflüchtigen Platonismus zielt die von Iamblichos und seinen Nachfolgern verfochtene Philosophie, so die Grundthese, auf Integration der körperlich-materiellen Wirklichkeitsebenen ab und das Endziel der Philosophie wird stärker unter dem Gesichtspunkt einer positiven Teilhabe am demiurgisch-kosmogonischen Werk begriffen. Diese Aufwertung von Körperlichkeit wird anhand verschiedener (...) Aspekte – z. B. dem Symbolbegriff, der „göttlichen Materie", aber auch des feinstofflichen Ochema-Pneuma-Konzepts – in ihrer Relevanz für das spirituelle Gesamtprofil der theurgischen Neuplatoniker erstmals umfassender untersucht. Abgerundet wird die Studie durch einen Vergleich mit ähnlichen Tendenzen im nicht-dualistischen Śivaismus von Kaschmir. (shrink)

Relying on aesthetic testimony seems problematic. For instance, it seems problematic for me to simply believe or assert that The Velvet Underground's debut album The Velvet Underground and Nico (1964) is amazing solely because you have told me so, even though I know you to be an honest and competent music critic. But why? After all, there do not seem to be similar reservations regarding testimony from many other domains. In this paper, I will argue that relying on aesthetic testimony (...) seems problematic because we are attached to an ideal of aesthetic authenticity and feel that living up to this ideal is anathema to simply relying on aesthetic testimony. (shrink)

This book proposes a holistic transdisciplinary approach to sustainability as a subject of social sciences. At the same time, this approach shows new ways, as perspectives of philosophy, political science, law, economics, sociology, cultural studies and others are here no longer regarded separately. Instead, integrated perspectives on the key issues are carved out: Perspectives on conditions of transformation to sustainability, on key instruments and the normative questions. This allows for a concise answer to urgent and controversial questions such as the (...) following: Is the EU an environmental pioneer? Is it possible to achieve sustainability by purely technical means? If not: will that mean to end of the growth society? How to deal with the follow-up problems? How will societal change be successful? Are political power and capitalism the main barriers to sustainability? What is the role of emotions and conceptions of normality in the transformation process? To which degree are rebound and shifting effects the reason why sustainability politics fail? How much climate protection can be claimed ethically and legally e.g. on grounds of human rights? And what is freedom? Despite all rhetoric, the weak transition in energy, climate, agriculture and conservation serves as key example in this book. It is shown how the Paris Agreement is weak with regard to details and at the same time overrules the growth society by means of a radical 1,5-1,8 degrees temperature limit. It is shown how emissions trading must – and can – be reformed radically. It is shown why CSR, education, cooperation and happiness research are overrated. And we will see what an integrated politics on climate, biodiversity, nitrogen and soil might look like. (shrink)

This is an important work of scholarship with regard to Machiavelli and the development of political thought in England. It charts the reactions of successive English thinkers to Machiavelli’s challenge, and the different aspects of Machiavelli’s thought which were perceived in the changing context of English history. There is the Machiavelli of Catholic and Protestant reformers, the Machiavelli of Raleigh and Bacon, of the royalist Clarendon and the republican Harrington. Through their eyes the reader can see the gradual process whereby (...) the atheistical monster repudiated by the subjects of Henry VIII was quietly absorbed by the politically sophisticated subjects of William III. (shrink)

Recently, Jennifer Saul (“Racial Figleaves, the Shifting Boundaries of the Permissible, and the Rise of Donald Trump”, 2017; “Racist and Sexist Figleaves”, 2021) has explored the use of what she calls “figleaves” in the discourse on race and gender. Following Saul, a figleaf is an utterance that, for some portion of the audience, blocks the conclusion that some other utterance, R, or the person who uttered R is racist or sexist. Such racial and gender figleaves are pernicious, says Saul, because, (...) among other things, they can shift the boundaries of what is deemed acceptable to think or say. This paper expands on Saul’s picture in a threefold way: It is first argued that appeals to statistics can function as figleaves. It is then argued that—as far as figleaves go—there is reason to believe that statistics-as-figleaves are especially pernicious. Finally, the paper explores some strategies for counterspeech. (shrink)

Should refugees govern refugee camps? This paper argues that they should. It draws on normative political thought in consulting the all-subjected principle and an instrumental defense of democratic rule. The former holds that all those subjected to rule in a political unit should have a say in such rule. Through analyzing the conditions that pertain in refugee camps, the paper demonstrates that the all-subjected principle applies there, too. Refugee camps have developed as near distinct entities from their host states. They (...) have formed their own economic, legal and even political systems within which refugees are subjected to political rule. The paper then demonstrates that democratic rule should be preferred over any other decision-making procedure. No amount of experts can replace the institutions that would lead to the accountability of decision-makers and to the incorporation of refugees as situated and epistemically diverse knowers of the problems they face and the solutions that would work best. (shrink)

For quite some time, cognitive science has offered philosophy an opportunity to address central problems with an arsenal of relevant theories and empirical data. However, even among those naturalistically inclined, it has been hard to find a universally accepted way to do so. In this article, we offer a case study of how cognitive-science input can elucidate an epistemological issue that has caused extensive debate. We explore Jason Stanley’s idea of the practical grasp of a propositional truth and present naturalistic (...) arguments against his reductive approach to knowledge. We argue that a plausible interpretation of cognitive-science input concerning knowledge—even if one accepts that knowledge how is partly propositional—must involve an element of knowing how to act correctly upon the proposition; and this element of knowing how to act correctly cannot itself be propositional. (shrink)

Many philosophers, following Williamson (The Philosophical Review 105(4): 489–523, 1996), Williamson (Knowledge and its Limits, Oxford, Oxford Univer- sity Press, 2000), subscribe to the constitutive rule account of assertion (CRAA). They hold that the activity of asserting is constituted by a single constitutive rule of assertion. However, in recent work, Maitra (in: Brown & Cappelen (ed). Assertion: new philosophical essays, Oxford, Oxford University Press, 2011), Johnson (Acta Analytica 33(1): 51–67, 2018), and Kelp and Simion (Synthese 197(1): 125–137, 2020a), Kelp and (...) Simion (in: Goldberg (ed) The Oxford Handbook of Assertion, Oxford, Oxford University Press, 2020b) aim to show that, for all the most popular versions of the constitutive rule of assertion proposed in the literature, asserting is not an activity constituted by a single constitutive rule and that therefore CRAA is very likely false. To reach this conclusion, they all present a version of what can be dubbed the engagement condition objection. That is, they each propose a necessary condition on engaging in rule-constituted activities. Then they argue that, for all the most popular versions of the constitutive rule of assertion proposed in the literature, one can make assertions without satisfying this condition. In response, I present a counterexample that shows that the proposed engagement conditions lead to coun- terintuitive results, and I propose an alternative that better captures our intuitions. Then I argue that this alternative engagement condition is compatible with all the most popular versions of the constitutive rule of assertion. (shrink)

What exactly do we need to do in order to make a promise, or to exercise some other normative power? On a view relied on by many philosophers writing on promising, consent, and related phenomena, the answer is that we must communicate a suitable kind of intention. On this view, power-conferring principles assert that specific normative consequences, determined in part by the content of the communicated intention, attach to such communicative acts, and these principles need not be socially practised or (...) accepted to be true. The paper offers a defense of this convention-independent view against the forceful challenge developed by Jed Lewinsohn in ‘The “Natural Unintelligibility” of Normative Powers’. Lewinsohn appeals to action-theoretic considerations to show that the relevant type of communicative act could not be performed under conditions of rationality and full information, and that therefore promissory power and other normative powers require the existence of social rules conferring such powers. The defense of the view targeted by Lewinsohn turns partly on the exact content of plausible constraints on the intelligibility of actions done with a particular aim, and partly on the question just how the social acceptance of power-conferring rules should be thought to matter. (shrink)

This book traces the emergence of a modern culture of exploration, as reflected in the role of institutions such as the Royal Geographical Society and the ...