In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...) a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor inconsistency, and thus contradictions do not have any special status as a thing to be avoided. (shrink)

We argue that all general concepts are family resemblance concepts. These include concepts introduced by ostension, such as colour. Concepts of colour and of each of the specific colours are family resemblance concepts because similarities concerning an open-ended range of colour or of appearance features crop up and disappear. After discussing the notion of “same colour” and Wittgenstein's use of the phrase “our colours”, we suggest family resemblance concepts in one tradition can often be extended to family resemblance concepts in (...) another tradition, illustrated by Wittgenstein's use of the words Spiel and “game”. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

The goal of the paper is to develop and propose a general model of the state space of AI. Given the breathtaking progress in AI research and technologies in recent years, such conceptual work is of substantial theoretical interest. The present AI hype is mainly driven by the triumph of deep learning neural networks. As the distinguishing feature of such networks is the ability to self-learn, self-learning is identified as one important dimension of the AI state space. Another dimension is (...) recognized as generalization, the possibility to go over from specific to more general types of problems. A third dimension is semantic grounding. Our overall analysis connects to a number of known foundational issues in the philosophy of mind and cognition: the blockhead objection, the Turing test, the symbol grounding problem, the Chinese room argument, and use theories of meaning. It shall finally be argued that the dimension of grounding decomposes into three sub-dimensions. And the dimension of self-learning turns out as only one of a whole range of “self-x-capacities” (based on ideas of organic computing) that span the self-x-subspace of the full AI state space. (shrink)

Name der Zeitschrift: Nordic Wittgenstein Review Jahrgang: 2 Heft: 1 Seiten: 33-56.

There is now an established literature on the link between later Wittgenstein and enactivist approaches in cognitive science. However, is this link not just a matter for card carrying Wittgensteinians? Can enactivists not manage perfectly well without Wittgenstein? In this paper, I show why some enactivists should care about Wittgenstein. Focusing on the enactivist view, “Sensorimotor Identity”. I argue that proponents of this view can use Wittgensteinian considerations to resolve an issue confronting their view and thereby shore up their proposed (...) dissolution of the explanatory gap. Some enactivists thus have in fact much to gain by engaging with Wittgenstein. (shrink)

Moral particularists have seen Wittgenstein as a close ally. One of the main reasons for this is that particularists such as Jonathan Dancy and John McDowell have argued that Wittgenstein's so-called "rule-following considerations" (RFCs) provide support for their skepticism about the existence and/or role of rules and principles in ethics. In this paper, I show that while Wittgenstein's RFCs challenge the notion that competence with language, i.e., the ability to apply concepts properly, is like mechanically following a rule, he does (...) not reject the idea that there are rules that govern proper use of language. I then argue that while the RFCs may, at best, support a weak form of particularism that denies that moral competence is dependent on an explicit grasp of rules, they do not support a stronger version of particularism that denies that there are any true rules or principles in ethics. (shrink)

This article critically examines Saul Kripke's (1972) argument for the separability of necessary truths from truths known a priori, focusing on his criticism of the standard meter case presented by Wittgenstein (1968). It attempts to show that Kripke's argument is unworkable on any of several readings. Wittgenstein's own broadly conventionalist account of necessary truth is then considered in the light of the standard meter example.

Appreciating the undeniable value of General Systems Theory, Alfred Locker considers the question whether or not GST is able to go beyond a mere scientific point of view. Locker’s own systems theoretical approach, Trans-Classical Systems Theory, proposes not only to include usual observations into a systems view, but likewise their theoretical presuppositions. Locker hereby creates two levels of observation; an ortho- and a meta-level, where otherwise incommensurable viewpoints are united into whole. In this way, Locker is able to articulate a (...) holistic systems theory of seeming opposites, like, for example, creation and evolution. (shrink)

Derrida's key concepts or pseudo-concepts of différance, the trace, and the undecidable suggest analogies to some of the most significant results of formal, symbolic logic and metalogic. As early as 1970, Derrida himself pointed out an analogy between his use of ‘undecidable’ and Gödel's incompleteness theorems, which demonstrate the existence, in any sufficiently complex and consistent system, of propositions which cannot be proven or disproven (i.e., decided) within that system itself. More recently, Graham Priest has interpreted différance as an instance (...) of the general metalogical procedure of diagonalisation. In this essay, I consider the extent to which Derrida's key terms and the essential operations of deconstruction can be formalised. I argue that, if formalisation is indeed the technique of writing par excellence, then the formalisation of deconstructive concepts tends to show how the auto-deconstruction of total systems arises from the problematic possibility of writing itself. For instance, since diagonalisation permits the ‘arithmetisation of syntax’ whereby a formal system is able to formulate claims about its own logico-grammatical properties, we can understand its potential to inscribe the undecidable within the systematicity of language as simply one instance of the potential of writing, in figuring itself, to render inscrutable the trace of its own origin. (shrink)

In his Tractatus Logico-Philosophicus Ludwig Wittgenstein (1889–1951) presents the concept of order in terms of a notational iteration that is completely logical but not part of logic. Logic for him is not the foundation of mathematical concepts but rather a purely formal way of reflecting the world that at the minimum adds absolutely no content. Order for him is not based on the concepts of logic but is instead revealed through an ideal notational series. He states that logic is “transcendental”. (...) As such it requires an ideal that his philosophical method eventually forces him to reject. I argue that Wittgenstein’s philosophy is more dialectical than transcendental. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This paper outlines a solution to what can be called “the problem of domination by reason”, “conceptual domination” or “clarificatorory injustice”, connected with how a philosopher may appear to be in a position to legitimately coerce, by means of arguments, an interlocutor who shares with her a concept or a conceptual system to accept a philosophical characterization of a concept or whatever the concept concerns. The proposed solution is based on a particular interpretation of what Wittgenstein means by agreement in (...) his later philosophy, when he says that philosophy only states what anyone grants to it. Wittgenstein’s view and the proposed solution are characterized by their continued recognition of the value of logic and reason, truth and knowledge, as opposed to attempting to solve the problem by embracing relativism and questioning the value of the logic, reason, truth and knowledge. Relevant kind of disagreements licence no relativistic conclusions, because problems relating to them can be solved without going this far. Keywords: domination, relativism, Wittgenstein, method, agreement. (shrink)

This paper seeks to defend, develop, and revise Edward Craig's “genealogy of knowledge”. The paper first develops the suggestion that Craig's project is naturally thought of as an important instance of “social cognitive ecology”. It then introduces the genealogy of knowledge and some of its main problems and weaknesses, suggesting that these are best taken as challenges for further work rather than as refutations. The central sections of the paper conduct a critical dialogue between Craig's theory and Wittgenstein's claim–familiar from (...) On Certainty–that common-sense certainties cannot be known. It turns out that Craig's distinction between different stages in the development of our concept of knowledge can illuminate and make plausible Wittgenstein's claim. But it can do so only if Craig's traditional commitment to a central “core” in our concept of knowledge is replaced with the idea of knowledge as a family-resemblance concept. (shrink)

This paper seeks to widen the dialogue between the “epistemology of peer disagreement” and the epistemology informed by Wittgenstein’s last notebooks, later edited as On Certainty. The paper defends the following theses: not all certainties are groundless; many of them are beliefs; and they do not have a common essence. An epistemic peer need not share all of my certainties. Which response to a disagreement over a certainty is called for, depends on the type of certainty in question. Sometimes a (...) form of relativism is the right response. Reasonable, mutually recognized peer disagreement over a certainty is possible.—The paper thus addresses both interpretative and systematic issues. It uses Wittgenstein as a resource for thinking about peer disagreement over certainties. (shrink)

According to the computational theory of mind , to think is to compute. But what is meant by the word 'compute'? The generally given answer is this: Every case of computing is a case of manipulating symbols, but not vice versa - a manipulation of symbols must be driven exclusively by the formal properties of those symbols if it is qualify as a computation. In this paper, I will present the following argument. Words like 'form' and 'formal' are ambiguous, as (...) they can refer to form in either the syntactic or the morphological sense. CTM fails on each disambiguation, and the arguments for CTM immediately cease to be compelling once we register that ambiguity. The terms 'mechanical' and 'automatic' are comparably ambiguous. Once these ambiguities are exposed, it turns out that there is no possibility of mechanizing thought, even if we confine ourselves to domains where all problems can be settled through decision-procedures. The impossibility of mechanizing thought thus has nothing to do with recherché mathematical theorems, such as those proven by Gödel and Rosser. A related point is that CTM involves, and is guilty of reinforcing, a misunderstanding of the concept of an algorithm. (shrink)

U tekstu izlažemo rješenje razlike između Wittgensteinove uporabe modela u TLP i modeliranja u PI, RFM i OC temeljem analize razlike u izboru samih riječi i opisa pojmova fenomena u njegovim napomenama. Metoda je pojmovna analiza Wittgensteinova tretiranja znanstvenih modela i modeliranja. Dodatno, tekstualna raščlamba i poredba korištena je pri istraživanju njegovih glavnih napomena na tu temu. Ako je otkrivena razlika između modela i modeliranja i kontinuitet ovisnosti oba pojma o pojmu slike, pri čemu je razlika slike i oslikavanja uzeta (...) kao zadana, zaključujemo kako je Wittgenstein svoj stari pojam modela nastojao inkorporirati u novi pojam modeliranja. Eksplicirana razlika i inkorporiranje pridonosi pojašnjenju teme Wittgensteinovog posrednog i neposrednog utjecaja na filozofiju znanosti. (shrink)

Wittgenstein’s “machines-as-symbols” are considered with respect to their historical sources and their symbolic and logical nature. Among these sources and precursors, along with Leonardo’s drawings of machines, there are illustrated “machine books”, a kind of book published in the period from the 16th to the 18th centuries which consist of pictures and descriptions of a variety of mechanical devices. Most probably, these books were one of Wittgenstein’s inspirations for his view of machines as components of language-games. The picture of homo (...) volans in Vrančić’s machine book possessed by Wittgenstein is taken as an example. In particular, homo volans is shown to contain patterns of logical laws and rules and to be abstractly interpretable as a logical symbol. A machine, taken as a symbol, is shown to be a precondition of a meaningful “overview” of a mechanical work that exceeds the limits of decidability, and to possess causal features if causality is understood teleologically and in a deeper sense of a “binding” life. (shrink)

This article focuses on the conventions that sustain social interaction and argues that they are central to Simon's decision-making theory. Simon clearly identifies two kinds of coordination by convention: behavioral mores that shape human actions, and shared mental models that govern human perceptions. This article argues that Poincaré–Carnap's conventionalism provides powerful support for Simon's theory; it contends that this theory offers a more convincing account of decision and coordination than Lewis' concept of convention. Simon's approach to applying conventionalist logic to (...) social interaction emphasizes the normative role played by mental models in solving coordination problems and considers rationality in terms of both cognitive and moral considerations. By connecting conventional phenomena to social identifications, Simon stresses the resulting complexity of coordination problems. (shrink)

The German physicist Heinrich Hertz played a decisive role for Wittgenstein's use of a unique philosophical method. Wittgenstein applied this method successfully to critical problems in logic and mathematics throughout his life. Logical paradoxes and foundational problems including those of mathematics were seen as pseudo-problems requiring clarity instead of solution. In effect, Wittgenstein's controversial response to David Hilbert and Kurt Gödel was deeply influenced by Hertz and can only be fully understood when seen in this context. To comprehend the arguments (...) against the metamathematical programme, and to appreciate how profoundly the philosophical method employed actually shaped the content of Wittgenstein's philosophy, it is necessary to make an intellectual biographical reconstruction of their philosophical framework, tracing the Hertzian elements in the early as well as in the later writings. In order to write Wittgenstein's biography, we have to take seriously the coherence of his thought throughout his life, and not let convenient philosophical ideologies be our guidance in drawing up a “Wittgensteinian philosophy”. To do so, we have to take a second look upon what he actually wrote, not only in the already published material, but in the entire Nachlass. Clearly, this is not easily done, but it is a necessary task in the historical reconstruction of Wittgenstein's life and work. (shrink)

This essay argues that there is room for a third position between moral particularism and moral generalism in their orthodox forms. The view proposed in this essay is inspired by the later Wittgenstein's conception of grammar and holds that formulations of ethical principles can be interpreted as grammatical statements, while ethical problems can be interpreted as instances of grammatical tension. On this reading, situations in which ethical principles turn out to conflict come out as moments in the evolution of language. (...) Three consequences are discussed, and welcomed as general insights into the workings of language. (shrink)

According to a standard criticism, Robert Brandom's “normative pragmatics”, i.e. his attempt to explain normative statuses in terms of practical attitudes, faces a dilemma. If practical attitudes and their interactions are specified in purely non-normative terms, then they underdetermine normative statuses; but if normative terms are allowed into the account, then the account becomes viciously circular. This paper argues that there is no dilemma, because the feared circularity is not vicious. While normative claims do exhibit their respective authors' practical attitudes (...) and thereby contribute towards establishing the normative statuses they are about, this circularity is not a mark of Brandom's explanatory strategy but a feature of social practice of which we theorists partake. (shrink)

Wright’s judgement-dependent account of intention is an attempt to show that truths about a subject’s intentions can be viewed as constituted by the subject’s own best judgements about those intentions. The judgements are considered to be best if they are formed under certain cognitively optimal conditions, which mainly include the subject’s conceptual competence, attentiveness to the questions about what the intentions are, and lack of any material self-deception. Offering a substantive, non-trivial specification of the no-self-deception condition is one of the (...) main problems for Wright. His solution is to view it as a positive presumption, which is violated only if there is strong evidence to the effect that the subject is self-deceived. In this paper, I will argue that the concern about self-deception in Wright’s account is misplaced and generally unmotivated. (shrink)

The paper aims at analyzing the production of creditworthiness within the context of commercial banking in international banks. Taking the interim financing in the Polish automobile sector as an example, the paper reconstructs the process between legal framing of the financial instrument, marketing, and risk management. Firstly, it shows that changes in the state vehicle registry function as a prerequisite upon which the bank uses the newly introduced vehicle registration document as a security. Secondly, it analyzes the change of perspectives (...) which takes place between the marketing of the product and the analysis of documents . Thirdly, it argues that creditworthiness cannot be deduced from the economic representation of the enterprise but is the very result of negotiations between the actors involved. (shrink)

This paper introduces the Generalized Argumentation Theory which takes argumentation as a locally rational socio-cultural interaction governed by social norms and carried out through discourse between the members of a socio-cultural community in order to reason things out. Then we bring in the basic structure of generalized argumentation and the localized procedure of Generalized Argumentation Theory for studying the argumentative rules. On the basis of above introduction, we use the localized procedure to analyze a case of political argumentation by reciting (...) poems in ancient China. By doing so, it’s indicated that political argumentation by reciting poems, in essence, takes cultural principle of Li of the Spring and Autumn Period as argumentative rules and uses poems to express ideas, in virtue of which arguments are accepted as locally rational by the politicians at that time. At last, the Little Red Book quotation fight, a similar political activity, is discussed briefly to show the significance that ancient Chinese political argumentation has on contemporary argumentation. (shrink)