Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein’s offered picture of mathematical statements as rules of description, it can be difficult to see the role of (...) mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein’s picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question. (shrink)

A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected.

Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the (...) same time, that Putnam's objections go too far. (shrink)

In the scholarship on Wittgenstein’s later philosophy of mathematics, the dominant interpretation is a theoretical one that ascribes to Wittgenstein some type of ‘ism’ such as radical verificationism or anti-realism. Essentially, he is supposed to provide a positive account of our mathematical practice based on some basic assertions. However, I claim that he should not be read in terms of any ‘ism’ but instead should be read as examining philosophical pictures in the sense of unclear conceptions. The contrast here is (...) that basic assertions that frame philosophical ‘isms’ are propositional such that they are subject to normal argumentative evaluation, while pictures in Wittgenstein’s sense are non-propositional—they lack a clear truth condition. They, therefore, need clarification rather than argumentation. In this paper, I provide a detailed analysis of Wittgenstein’s treatment of philosophical pictures with special focus on his argument on contradiction. I begin by explaining the problem with this trend of theoretical interpretation, taking Steve Gerrard’s otherwise excellent interpretation as a representative example and pointing out why it is problematic. Next, I will argue that those problems do not arise if we take Wittgenstein’s task as the clarification of philosophical pictures. I do this, first, by explaining Wittgenstein’s method using his argument concerning the Augustinian Picture in Philosophical Investigations and then pointing out that the same method can be identified in the crucial arguments in his philosophy of mathematics. Finally, in order to connect my interpretation with the current scholarship, I will explain the relation of my interpretation with those of New Wittgensteinian scholars. (shrink)

Computational complexity is a discipline of computer science and mathematics which classifies computational problems depending on their inherent difficulty, i.e. categorizes algorithms according to their performance, and relates these classes to each other. P problems are a class of computational problems that can be solved in polynomial time using a deterministic Turing machine while solutions to NP problems can be verified in polynomial time, but we still do not know whether they can be solved in polynomial time as well. A (...) solution for the so-called NP-complete problems will also be a solution for any other such problems. Its artificial-intelligence analogue is the class of AI-complete problems, for which a complete mathematical formalization still does not exist. In this chapter we will focus on analysing computational classes to better understand possible formalizations of AI-complete problems, and to see whether a universal algorithm, such as a Turing test, could exist for all AI-complete problems. In order to better observe how modern computer science tries to deal with computational complexity issues, we present several different deep-learning strategies involving optimization methods to see that the inability to exactly solve a problem from a higher order computational class does not mean there is not a satisfactory solution using state-of-the-art machine-learning techniques. Such methods are compared to philosophical issues and psychological research regarding human abilities of solving analogous NP-complete problems, to fortify the claim that we do not need to have an exact and correct way of solving AI-complete problems to nevertheless possibly achieve the notion of strong AI. (shrink)

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (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)

This paper argues that there are three reasons why we should regard Wittgenstein's Tractatus as a forerunner of formal semantics: Wittgenstein is convinced that we can apply formal notions to natur...

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)

My essay explains and examines Anscombe's disagreement with Wittgenstein about what the Tractatus supposedly excludes. I also discuss her apparent disagreement with Geach about propositions that lack an intelligible negation. My discussion of these disagreements leads to the topic of Anscombe on the relation between the “business of thinking” and truth. I suggest that she takes the business of thinking to include thinking that helps to keep thinking on track. Since there is a tie between thinking truly and the business (...) of thinking being done well and since helping to keep thinking on track belongs to the business of thinking, helpings-of-thinking can be described as true. (shrink)

Wittgenstein and the logical positivists attempted to explain logical necessity in terms of linguistic conventions. It is often thought that their respective accounts have been conclusively refuted by objections from Quine, Dummett and others. We argue that this verdict is premature. Several of the most popular anti‐conventionalist arguments fail, partly because they misconstrue the idea of truth by convention in Wittgenstein and/or logical positivism. Correctly understood, conventionalism claims that, given certain linguistic conventions, some sentences are unconditionally true, that is true (...) irrespective of how the world happens to be. This claim is difficult to deny, and the corresponding conventionalism about logical necessity remains a viable position. (shrink)

This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation.

RésuméPaul Boghossian soutient contre Wittgenstein que le normativisme au sujet de la logique et des mathématiques est incompatible avec le fait de tenir les énoncés logiques et mathématiques pour vrais et que le normativisme entraîne une régression indue. Je soutiens, pour ma part, que le normativisme n'entraîne pas une telle régression, parce que les normes peuvent être implicites et que le normativisme peut bien être «factualiste» si l'on rejette ce que Rockney Jacobsen appelle le «cognitivisme sémantique». Je tiens en outre (...) que la position de Boghossian repose sur la façon trompeuse dont il découpe le terrain conceptuel. (shrink)

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real (...) number, as an 'extension', is a homeless fiction; 'homeless' in that it neither is supported by anything nor supports anything. The picture of a real number as an 'extension' is not supported by actual practice in calculus; calculus has nothing to do with 'extensions'. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being 'existential axiomatics'. And in real analysis, as existential axiomatics, a point on the real line need not be an 'extension'. (shrink)

The extensions of Goodman’s ‘grue’ predicate and Kripke’s ‘quus’ are constructed from the extensions of more familiar terms via a reinterpretation that permutes assignments of reference. Since this manoeuvre is at the heart of Putnam’s model-theoretic and permutation arguments against metaphysical realism (‘Putnam’s Paradox’), both Goodman’s New Riddle of Induction and the paradox about meaning that Kripke attributes to Wittgenstein are instances of Putnam’s. Evidence cannot selectively confirm the green-hypothesis and disconfirm the grue-hypothesis, because the theory of which the green-hypothesis (...) is a part has an unintended model in which the grue-hypothesis is equally confirmed; and there are no meaning-facts that determine reference, because the objects referred to by the referring terms of any language or set of intentional mental states are permutable in a way that is consistent with the truth-values of all other sentences in that language or beliefs in that set. The upshot is that the three paradoxes need to be solved in a unified way. (shrink)

This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conjectures involving effective procedures as well.

This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.

Central to Kuhn's notion of incommensurability are the ideas of meaning variance and lexicon, and the impossibility of translation of terms across different theories. Such a notion of incommensurability is based on a particular understanding of what a scientific language is. In this paper we first attempt to understand this notion of scientific language in the context of incommensurability. We consider the consequences of the essential multisemiotic character of scientific theories and show how this leads to even a single theory (...) being potentially 'internally incommensurable'. We then discuss Kuhn's lexicon-based approach to incommensurability and the problems associated with it. Finally we argue that this approach by Kuhn has interesting overlaps with the problem of meaning associated with multisemiosis, particularly the challenge of understanding the process of symbolization in scientific theories. (shrink)

The paper provides an interpretation of Wittgenstein’s claim that a mathematical proof must be surveyable. It will be argued that this claim specifies a precondition for the applicability of the word ‘proof’. Accordingly, the latter is applicable to a proof-pattern only if we can come to agree by mere observation whether or not the pattern possesses the relevant structural features. The claim is problematic. It does not imply any questionable finitist doctrine. But it cannot be said to articulate a feature (...) of our actual usage of the word ‘proof’. The claim can be dissociated, however, from two tenable doctrines of Wittgenstein, namely that proofs can be used as paradigms for corresponding proof concepts and that a proof is not an experiment. (shrink)

Although Wittgenstein’s most extensive discussion of aspect‐recognition appears in Part II of the Philosophical Investigations, aspect‐recognition was of interest to Wittgenstein almost from the beginning of his engagement with philosophy at Cambridge in 1912. However, the nature of that interest changes upon his return to Cambridge in 1929, and that change in turn is connected with the inter‐related ideas that philosophical clarity rests on recognising aspects of our grammar and that mathematical proof leads us to recognise new aspects of mathematical (...) expressions. (shrink)

This multifaceted essay emerges from a host of sources within diverse academic settings. Its central thesis is guided by physicist John A. Wheeler's thoughts on the quantum enigma. Wheeler concludes, following Niels Bohr, that we are co-participants within the universal self-organizing process. This notion merges with concepts from Peirce's process philosophy, Eastern thought, issues of topology, and border theory in cultural studies and social science, while surrounding itself with such key terms as complementarity, interdependence, interrelatedness, vagueness, generality, incompleteness, inconsistency, and (...) mestizaje. Ultimately, a sense of semiosic process pervades in light of combined homogenous and hetergenous tendencies. (shrink)

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...) of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)