Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical (...) results, given by the possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more (...) specifically, emphasises its refined composition and rigorous internal structure. (shrink)

Biological evolution is a complex blend of ever changing structural stability, variability and emergence of new phe- notypes, niches, ecosystems. We wish to argue that the evo- lution of life marks the end of a physics world view of law entailed dynamics. Our considerations depend upon dis- cussing the variability of the very ”contexts of life”: the in- teractions between organisms, biological niches and ecosys- tems. These are ever changing, intrinsically indeterminate and even unprestatable: we do not know ahead of (...) time the ”niches” which constitute the boundary conditions on selec- tion. More generally, by the mathematical unprestatability of the ”phase space”, no laws of mo- tion can be formulated for evolution. We call this radical emergence, from life to life. The purpose of this paper is the integration of variation and diversity in a sound concep- tual frame and situate unpredictability at a novel theoretical level, that of the very phase space. Our argument will be carried on in close comparisons with physics and the mathematical constructions of phase spaces in that discipline. The role of symmetries as invariant preserving transformations will allow us to under- stand the nature of physical phase spaces and to stress the differences required for a sound biological theoretizing. In this frame, we discuss the novel notion of ”enablement”. Life lives in a web of enablement and radical emergence. This will restrict causal analyses to differential cases. Mutations or other causal differ- ences will allow us to stress that ”non conservation princi- ples” are at the core of evolution, in contrast to physical dynamics, largely based on conservation principles as sym- metries. Critical transitions, the main locus of symmetry changes in physics, will be discussed, and lead to ”extended criticality” as a conceptual frame for a better understanding of the living state of matter. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the (...) interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or (...) theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

In this paper a class of languages which are formal enough for mathematical reasoning is introduced. Its languages are called mathematically agreeable. Languages containing a given MA language L, and being sublanguages of L augmented by a monadic predicate, are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of those languages. MTT makes them fully interpreted MA languages which posses their own truth predicates. MTT is shown to conform well with the eight norms formulated for theories (...) of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)', by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. Main tools used in proofs are Zermelo-Fraenkel (ZF) set theory and classical logic. (shrink)

In this paprer a class of so called mathematically acceptable (shortly MA) languages is introduced First-order formal languages containing natural numbers and numerals belong to that class. MA languages which are contained in a given fully interpreted MA language augmented by a monadic predicate are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of these languages. MTT makes them fully interpreted MA languages which posses their own truth predicates, yielding consequences to philosophy of mathematics. MTT (...) is shown to conform well with the eight norms presented for theories of truth in the paper 'What Theories of Truth Should be Like (but Cannot be)' by Hannes Leitgeb. MTT is also free from infinite regress, providing a proper framework to study the regress problem. (shrink)

This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by J.P. Burgess and (...) G. Rosen). Examples are discussed. The historical development of written mathematics from algorithms on clay tablets to theorems with proofs is said to show that application of algorithms to specific problems and theorems to scientific objects is like allegorical interpretation. Mathematics fits into the modern scientific context because the sciences, beginning with Galileo, have been constructed in imitation of mathematics. Viewing mathematics this way does not solve any ontological problems, but it does show how mathematics avoids them. Epistemological problems, insuperable for object realism, are simplified. For example, we have access to relations among any objects that we can consider objectively, first physical objects and then mathematical objects of greater and greater abstraction. (shrink)

Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and space–time (...) are touched upon. (shrink)

This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths (...) or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization. (shrink)

The paper discusses the possibility of constructing economic models using the methodology of model construction in classical mechanics. At the same time, unlike the "econophysical" approach, the properties of economic models are derived without involvement of any equivalent physical properties, but with account of the types of symmetry existing in the economic system. It has been shown that at this approach practically all known mechanical variables have their "economic twins". The variational principle is formulated on the basis of formal mathematical (...) construction without involving the subjective factor common to the majority of models in economics. The dynamics of interaction of two companies has been studies in details, on the basis of which we can proceed to modeling of more complex and realistic economic systems. Prediction of the possibility of constructing economic theory on the basis of primary principles analogously to physics has been made. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

The concept of sustainable development considers environmental, social and economic issues in general. And the goals of resource conservation and socio-economic development do not contradict each other, but contribute to mutual reinforcement. The purpose of this study is to build and test an economic and mathematical model for the formation of strategies for the behavior of an economic entity with an increase in the impact of negative environmental factors. The proposed strategies and their models are based on the income-expenditure balance (...) equation, which takes into account both quantitative and qualitative characteristics. The constructed models are considered in the state space. The research methodology is based on building models in the form of linear combinations of functions of a homogeneous external impact and various spatial combinations of economic sources (sinks). The study makes it possible to assess the dependence of the amount of resources used for life support on the chosen adaptive strategy. Within the framework of the proposed model, it was found that the criterion for the effectiveness of the applied strategy can be an indicator of satisfaction with the state, the preservation of which, simultaneously with the preservation of the size of resources used, corresponds to the direction of optimization. This approach is consistent with the concept of sustainable development. (shrink)

Overall, it is first rate with accurate, sensitive and penetrating accounts of his life and thought in roughly chronological order, but, inevitably (i.e., like everyone else) it fails, in my view, to place his work in proper context and gets some critical points wrong. It is not made clear that philosophy is armchair psychology and that W was a pioneer in what later became cognitive or evolutionary psychology. One would not surmise from this book that he laid out the foundations (...) of the modern concept of intentionality (roughly, personality or higher order thought) which has been further advanced by many (most notably in philosophy by John Searle in “The Construction of Social Reality” and “Rationality in Action”). -/- There is no clear explanation of how W defined the class of potential actions, which he called dispositions or inclinations, (now often called propositional attitudes), differentiating them from perceptions, memories and actions and showing how they lack truth value. He notes that W spent much of his time discussing the foundations of mathematics but fails to provide any explanation as to how this relates to his work on language and logic. In fact, as W came to realize, they are all names for groups of functions of our innate psychology with many differences and none are dependent on the others. It is not really made clear that all our behavior depends on the unquestionable axioms of our evolved psychology and thus differs totally from the testable empirical facts which they enable us to discover. It is not explained that W’s frequent references to “grammar” and to “language games” refer to our innate psychology. All these failings are the norm in behavioral studies. -/- He notes that W described thinking and other dispositions or inclinations (W’s terms) -- (i.e., judging, feeling, remembering, believing etc.) -- as behaviors and not as mental activities but I don’t see that he really makes it clear that another pioneering discovery of W’s was that dispositions describe public actions and cannot be mental phenomena for the same reason that he so famously rejected the possibility of a private language. -/- He repeatedly and correctly notes (e.g., p176) that the core of W’s work is the nature of language but (again the universal failing) does not make it clear that language is for humans (as opposed to animals) almost coextensive with thought (public behavior as W insisted) and thus with our evolved psychology. Like most people, philosophers or not, Kanterian has not followed W and taken the final step towards understanding and describing behavior from an evolutionary standpoint, the only viewpoint that makes sense of it, or indeed of anything. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the (...) interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. (...) The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.<br><br>According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete. <br><br>According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.<br><br>Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes. (shrink)

Aristotle first investigated different modes, or ways of being. Unfortunately, in the modern literature the discussion of this concept has been largely neglected. Only recently, the interest towards the concept of ways increased. Usually, it is explored in connection with the existence of universals and particulars. The approach we are going to follow in this chapter is different. It discusses Wittgenstein’s conception of higher ontological levels as ways of arranging elements of lower ontological levels. In the Tractatus, Wittgenstein developed his (...) ontology of ways (Art und Weise) in six steps: (i) Constructing states of affairs out of objects; (ii) Constructing propositions out of states of affairs; (iii) Constructing propositional signs; (iv) Constructing thoughts with the help of propositional signs; (v) Constructing truth / falsity; (vi) Constructing works of art. In Philosophical Investigations he added further five ways of producing new ontological levels: (vii) the meaning of a proposition is the way in which it is verified; (viii) the child gets command on language/calculus in the way it replicates demonstrations of the teacher; (ix) the products of mind are ways of doing something; (x) an action is a way of carrying out the instructions for acting. (shrink)

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” (...) in Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)

Wittgenstein comes up with his model simply through starting with the assumption that language can be an accurate picture of the world, and realizing the failings of that idea. This makes him a rather odd outsider in the sociology and politics of modern philosophy. He’s a trained engineer. A soldier. An architect. A logician (including being the guy who invented Truth tables for logic). In other words, a total geek. He’s still part of the analytic tradition, dismissed and rejected by (...) many in the continental tradition. But he ends up saying the kind of things that drive your average conservative culture warrior harrumphing about \"post-modernism\" and \"relativism\" up the wall. Simply because he’s thought about it a lot. -/- The other thing that’s striking is how much he is an \"anti-philosopher\". Always running away from philosophy to do other things in his life. And his work is often intended as \"therapeutic\" not trying to \"solve\" philosophical problems so much as \"cure\" us of worrying about them. He emphasizes that philosophical \"problems\" are often just misuses and misunderstandings of words rather than deeper issues. In his first philosophy, many problems come from us not understanding the meaning of the words well enough. If only we could pin them down better, the problems would disappear. In his second, the fact that we have a word for something doesn’t mean that the world really has that thing. And many philosophical problems, he asserts, are nothing but trying to take words that have a \"function\" in a particular context and abstract them out and using them in a different context where they have no useful function. (shrink)

This article analyses in detail Wittgenstein’s ‘Cambridge period’ from his return to Cambridge in 1929 until his decease in 1951. Within the ‘Cambridge period’, scholars usually distinguish the ‘middle’ (1929–1936) and the ‘late’ (1936–1951) periods. The trigger point of Wittgenstein’s return to Cambridge and philosophy was his visit to Brouwer’s lecture on ‘Mathematics, Science, and Language’ in Vienna in March 1928. Dutch mathematician Brouwer influenced not only Wittgenstein’s ability to do philosophy again but also the development of some of (...) his ideas. Namely, for Brouwer, language was a natural development of the social history of human beings. With the help of his friends, F. P. Ramsey, J. M. Keynes, G. E. Moore, and B. Russell, in 1929 Wittgenstein returned to Cambridge. The author argues that this was the crucial turning point in his philosophy that led to the revision of some of his ideas. Wittgenstein returned from self-isolation in remote villages of Lower Austria to the intellectual academic environment, where he could discuss his ideas with intellectual interlocutors and receive their valuable remarks and comments. This article is organised in chronological order: it describes the very ‘early’ middle period of 1929–1935 involving the development of Wittgenstein’s ‘phenomenology’ and the origin of the ‘language-games’ concept in the Blue and Brown Books; 1935–1936 period of ‘romantic’ enthusiasm for Soviet Russia and a trip there; Wittgenstein’s life and work during the WWII-time; resign from teaching in Cambridge in 1947 to finally finish his Philosophical Investigations. The author suggests that the ‘romantic areal’ about the USSR was formed by Wittgenstein’s predilection for Russian poetry and literature of the nineteenth century, i.e., Tolstoy, Dostoevsky, and Pushkin. Analysing the development of Wittgenstein’s ideas on language in his Cambridge period, the author mentions the influence of N. Bakhtin and P. Sraffa. Also, the author touches upon the topic of the possibility of Marxism’s influence on Wittgenstein’s life and thought. (shrink)

In his Tractatus and Notebooks 1914-1916, Wittgenstein develops some themes concerning the nature of the subject, transcendentalism, solipsism and mysticism. Though Wittgenstein rejects a naive, psychological understanding of the subject, he preserves the idea of the metaphysical subject, so-called “philosophical I”. The present investigations exhibit two ways of grasping the subject: (1) subject as a boundary (of the world); (2) subject (I) as the world. The author of the paper aims to analyze different methods of conceiving the subject, both logical (...) and transcendental. Then he discusses the naturalistic or reductionist consequences of solipsism which were derived by Wittgenstein. Moreover, he refers to the concept of ‘subject of will’ introduced in the Tractatus. Finally, the author puts a question whether the metaphysical subject is a boundary of the world identified with the subject of will. While trying to answer this question one can point to the essential difficulties of Wittgenstein’s standpoint. These difficulties become especially evident if we examine Wittgenstein’s statements concerning mysticism. The category of subject seems to gain a new dimension when reconsidered in this context. In the conclusion, the author offers an interpretation inspired by Schopenhauer’s conception of the double aspect of the subject that is to overstep these difficulties. (shrink)

This dissertation is a discussion of Wittgenstein's later philosophy. In it, Wittgenstein's answer to the "going on problem" will be presented: I will give his reply to the skeptic who denies that rule-following is possible. Chapter One will describe this problem. Chapter Two will give Wittgenstein's answer to it. Chapter Three will show how Wittgenstein used this answer to give the standards of mathematics. Chapter Four will compare Wittgenstein's answer to the going on problem to (...) Plato's. Chapter Five will describe Kant's influence on Wittgenstein's later philosophy. The leitmotif of these chapters will be that the way human beings typically apply a rule is the source of justification for an individual's usage thereof. (shrink)

Horwich gives a fine analysis of Wittgenstein (W) and is a leading W scholar, but in my view, they all fall short of a full appreciation, as I explain at length in this review and many others. If one does not understand W (and preferably Searle also) then I don't see how one could have more than a superficial understanding of philosophy and of higher order thought and thus of all complex behavior (psychology, sociology, anthropology, history, literature, society). In a (...) nutshell, W demonstrated that when you have shown how a sentence is used in the context of interest, there is nothing more to say. I will start with a few notable quotes and then give what I think are the minimum considerations necessary to understand Wittgenstein, philosophy and human behavior. -/- First one might note that putting “meta” in front of any word should be suspect. W remarked e.g., that metamathematics is mathematics like any other. The notion that we can step outside philosophy (i.e., the descriptive psychology of higher order thought) is itself a profound confusion. Another irritation here (and throughout academic writing for the last 4 decades) is the constant reverse linguistic sexism of “her” and “hers” and “she” or “he/she” etc., where “they” and “theirs” and “them” would do nicely. Likewise, the use of the French word 'repertoire' where the English 'repertory' will do quite well. The major deficiency is the complete failure (though very common) to employ what I see as the hugely powerful and intuitive two systems view of HOT and Searle’s framework which I have outlined above. This is especially poignant in the chapter on meaning p111 et seq. (especially in footnotes 2-7), where we swim in very muddy water without the framework of automated true only S1, propositional dispositional S2, COS etc. One can also get a better view of the inner and the outer by reading e.g., Johnston or Budd (see my reviews). Horwich however makes many incisive comments. I especially liked his summary of the import of W’s anti-theoretical stance on p65. He needs to give more emphasis to ‘On Certainty’, recently the subject of much effort by Daniele Moyal- Sharrock, Coliva and others and summarized in my recent articles. -/- Horwich is first rate and his work well worth the effort. One hopes that he (and everyone) will study Searle and some modern psychology as well as Hutto, Read, Hutchinson, Stern, Moyal-Sharrock, Stroll, Hacker and Baker etc. to attain a broad modern view of behavior. Most of their papers are on academia dot edu and philpapers dot org , but for PMS Hacker see his papers on his Oxford page. -/- He gives one of the most beautiful summaries of where an understanding of Wittgenstein leaves us that I have ever seen. -/- “There must be no attempt to explain our linguistic/conceptual activity (PI 126) as in Frege’s reduction of arithmetic to logic; no attempt to give it epistemological foundations (PI 124) as in meaning based accounts of a priori knowledge; no attempt to characterize idealized forms of it (PI 130) as in sense logics; no attempt to reform it (PI 124, 132) as in Mackie’s error theory or Dummett’s intuitionism; no attempt to streamline it (PI 133) as in Quine’s account of existence; no attempt to make it more consistent (PI 132) as in Tarski’s response to the liar paradoxes; and no attempt to make it more complete (PI 133) as in the settling of questions of personal identity for bizarre hypothetical ‘teleportation’ scenarios.” -/- Finally, let me suggest that with the perspective I have encouraged here, W is at the center of contemporary philosophy and psychology and is not obscure, difficult or irrelevant, but scintillating, profound and crystal clear and that to miss him is to miss one of the greatest intellectual adventures possible. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

Overall, it is first rate with accurate, sensitive and penetrating accounts of his life and thought in roughly chronological order, but, inevitably (ie, like everyone else) it fails, in my view, to place his work in proper context and gets some critical points wrong. It is not made clear that philosophy is armchair psychology and that W was a pioneer in what later became cognitive or evolutionary psychology. One would not surmise from this book that he laid out the foundations (...) of the modern concept of intentionality (roughly, personality or higher order thought) which has been further advanced by many (most notably in philosophy by John Searle in “The Construction of Social Reality” and “Rationality in Action”). There is no clear explanation of how W defined the class of potential actions, which he called dispositions or inclinations, (now often called propositional attitudes), differentiating them from perceptions, memories and actions and showing how they lack truth value. He notes that W spent much of his time discussing the foundations of mathematics but fails to provide any explanation as to how this relates to his work on language and logic. In fact, as W came to realize, they are all names for groups of functions of our innate psychology with many differences and none are dependent on the others. It is not really made clear that all our behavior depends on the unquestionable axioms of our evolved psychology and thus differs totally from the testable empirical facts which they enable us to discover. It is not explained that W’s frequent references to “grammar” and to “language games” refer to our innate psychology. All these failings are the norm in behavioral studies. -/- He notes that W described thinking and other dispositions or inclinations (W’s terms)-- (ie, judging, feeling, remembering, believing etc)-- as behaviors and not as mental activities but I don’t see that he really makes it clear that another pioneering discovery of W’s was that dispositions describe public actions and cannot be mental phenomena for the same reason that he so famously rejected the possibility of a private language. -/- He repeatedly and correctly notes (eg, p176) that the core of W’s work is the nature of language but (again the universal failing) does not make it clear that language is for humans (as opposed to animals) almost coextensive with thought (public behavior as W insisted) and thus with our evolved psychology. Like most people, philosophers or not, Kanterian has not followed W and taken the final step towards understanding and describing behavior from an evolutionary standpoint, the only viewpoint that makes sense of it, or indeed of anything. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

In his highly perceptive, if underappreciated introduction to Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Wittgenstein’s theory of number, relating specifically to the topic of transfinite number. The goal of this paper is two-fold. The first is to show that Russell’s concerns cannot be dismissed on the grounds that they are external to the Tractarian project, deriving, perhaps, from logicist ambitions harbored by Russell but not shared by Wittgenstein. The extensibility of Wittgenstein’s theory of number to the case of transfinite (...) cardinalities is, I shall argue, a desideratum generated by concerns intrinsic, and internal to Wittgenstein’s logical and semantic framework. Second, I aim to show that Wittgenstein’s theory of number as espoused in the Tractatus is consistent with Russell’s assessment, in that Wittgenstein meant to leave open the possibility that transfinite numbers could be generated within his system, but did not show explicitly how to construct them. To that end, I show how one could construct a transfinite number line using ingredients inherent in Wittgenstein’s system, and in accordance with his more general theories of number and of operations. (shrink)

Sören Halldén’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to it; on (...) our way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger. (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.

The book explores the impact of manuscript remarks during the year 1929 on the development of Wittgenstein’s thought. Although its intention is to put the focus specifically on the manuscripts, the book is not purely exegetical. The contributors generate important new insights for understanding Wittgenstein’s philosophy and his place in the history of analytic philosophy. -/- Wittgenstein’s writings from the years 1929-1930 are valuable, not simply because they marked Wittgenstein’s return to academic philosophy after a seven-year absence, but because these (...) works indicate several changes in his philosophical thinking. The chapters in this volume clarify the significance of Wittgenstein’s return to philosophy in 1929. In Part 1, the contributors address different issues in the philosophy of mathematics, e.g. Wittgenstein's understanding of certain aspects of intuitionism and his commitment to verificationism, as well as his idea of "a new system". Part 2 examines Wittgenstein's philosophical development and his understanding of philosophical method. Here the contributors examine particular problems Wittgenstein dealt with in 1929, e.g. the colour-exclusion problem, and the use of thought experiments as well as his relationship to Frank Ramsey and philosophical pragmatism. Part 3 features essays on phenomenological language. These chapters address the role of spatial analogies and the structure of visual space. Finally, Part 4 includes one chapter on Wittgenstein’s few manuscript remarks about ethics and religion and relates it to his Lecture on Ethics. -/- Wittgenstein’s Philosophy in 1929 will be of great interest to scholars and advanced students working on Wittgenstein and the history of analytic philosophy. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- (...) We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

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)

_ Source: _Volume 6, Issue 2-3, pp 120 - 142 This paper aims to contribute to the debate over epistemic versus non-epistemic readings of the ‘hinges’ in Wittgenstein’s _On Certainty_. I follow Marie McGinn’s and Daniele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are (...) true and are propositions; that some of them can be epistemically justified; that in some senses they are not prior to empirical knowledge; that they are not ineffable; and that their primary function is epistemic as much as it is semantic. (shrink)

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same (...) critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse. (shrink)

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of (...) constructive mathematics Bishop style. The aim of this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer. I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics. This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion to look anew at Bishop's own remarks for inspiration. (shrink)

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was (...) not moved by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)

Wittgenstein responds in his Notes on Logic to a discussion of Russell's Principles of Mathematics concerning assertion. Russell writes: "It is plain that, if I may be allowed to use the word assertion in a non-psychological sense, the proposition "p implies q" asserts an implication, though it does not assert p or q. The p and the q which enter into this proposition are not strictly the same as the p or the q which are separate propositions." (PoM p35) (...) Wittgenstein replies: "Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental." (NB p95) Wittgenstein's response is intriguing, not least because of the centrality to his Tractatus of the idea that a proposition says something. This paper will examine that idea, distinguishing it from 'merely psychological' assertion, and explore in this context how we should understand the occurrence of a Tractarian proposition within another. (shrink)

The objective of this article is to try to elucidate Wittgenstein’s ex-travagant thesis that each and every mathematical advancement involves some “semantical mutation”, i.e., some alteration of the very meanings of the terms involved. To do that we will argue in favor of the idea of a “modal incompati-bility” between the concepts involved, as they were prior to the advancement, and what they become after the new result was obtained. We will also argue that the adoption of this thesis profoundly (...) alters the traditional way of con-struing the idea of “progress” in mathematics. (shrink)

Gottlob Frege and Ludwig Wittgenstein (the later Wittgenstein) are often seen as polar opposites with respect to their fundamental philosophical outlooks: Frege as a paradigmatic "realist", Wittgenstein as a paradigmatic "anti-realist". This opposition is supposed to find its clearest expression with respect to mathematics: Frege is seen as the "arch-platonist", Wittgenstein as some sort of "radical anti-platonist". Furthermore, seeing them as such fits nicely with a widely shared view about their relation: the later Wittgenstein is supposed to have developed (...) his ideas in direct opposition to Frege. The purpose of this paper is to challenge these standard assumptions. I will argue that Frege's and Wittgenstein's basic outlooks have something crucial in common; and I will argue that this is the result of the positive influence Frege had on Wittgenstein. (shrink)

Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that this endeavor (...) can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

This study which utilized the critical analysis, analyzed Wittgenstein’s concept of language-game and its implications in philosophical discipline and contemporary society. To delineate the origin of the problem of language in terms of epistemological dimension, the researcher analyzed the related concepts on classical philosophy. To determine the origin of the concept of language-game, the researcher used the historical method. To constructively criticize the end-goal of language-game, the hermeneutical approach of Hans Georg Gadamer was employed. From the findings and with the (...) aid of the above-mentioned methods, the researcher had come up with these conclusions. Language is inalienable in the sociological nature of man. Man is always related or connected to the other. In this sense, man shares his very subjectivity with the other. Unless he realizes this, he cannot actualize the essential part of his nature. In every aspect of man’s life, it is imperative for him to negotiate, communicate, and collaborate with the members of his household and society. Wittgenstein is the one of the philosophers who gave deeper commitment and emphasis on the concerns of language in man’s life. Mastering our use of language is not just for the sake of conventional understanding, but, more importantly, we use it in its proper context in order to express our thoughts clearly and in the process to understand ourselves. It is because, as we speak or write using language as medium of communication, we are projecting ourselves to the other. If we come up with the c common understanding, then we will be able to create a harmonious society. This study provided the readers with a better understanding of the importance of proper use of words or terminologies in their proper context. It is because this research delved on the conceptual and analytical mechanisms of using language in order to give a systematic pattern of corrective thinking and application for word’s use. Aside from affirming the importance of language in connection with our contemporary activities, this research looked into a deeper realization on the workings of language. This realization originated from Wittgensteinian context of understanding. As affirmation for Wittgenstein, it is appropriate to say that to understand a word in its proper context is to understand the sentence. To understand a sentence is to understand a language. To understand a language means to be master of technique. This signifies that understanding is to know how to do something; in the case of language, understanding language means knowing how to use it. The researcher realized that understanding the meaning of a word is not something private to the mental life of an individual, but something which exist out in the open, in the public domain. This means that understanding the meaning is objectively understandable by people. However, the researcher did not agree with Wittgenstein’s view that philosophy should stop once we clear the way of understanding how our language works. Instead, the researcher adheres to the fact that philosophy as enterprise is an undying process in search for truth. For the researcher, we do not stop philosophizing because our imaginations continue to desire progress in terms of creativity. Thus, he affirmed Gadamer in claiming that we should not hinder our philosophical endeavors by breaking it. Even if we can clear the confusions and misunderstanding of language, still our active minds cling to what is new in the field of learning things. This research led to the formulation of other theories which can be useful tool in understanding the complexities with regards to ourselves and our society as a whole. Henceforth, all the laborious efforts of this research and the regimented time invested in the writing of this work are meaningfully compensated. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic (...) logic and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (shrink)

Wittgenstein's Nachlass: The Bergen Electronic Edition is the only CD-ROM to give you instant facsimile and text access to the 20,000 pages of the philosopher's Nachlass as catalogued by Professor von Wright in his 1982 publication The Wittgenstein Papers. -/- The result of 10 years of academic research and editorial work by the Wittgenstein Archives at the University of Bergen this electronic edition is the first scholarly resource to apply a uniform, well-documented, consistent set of editorial principles to the (...) writings. The CD-ROM is a joint publication between Oxford University Press and the Wittgenstein Archives and is published in Text & Facsimile and Text Only formats. (shrink)

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of (...) class='Hi'>Wittgenstein's thought and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929. (shrink)

Although the later Wittgenstein appears as one of the most influential figures in Davidson’s later works on meaning, it is not, for the most part, clear how Davidson interprets and employs Wittgenstein’s ideas. In this paper, I will argue that Davidson’s later works on meaning can be seen as mainly a manifestation of his attempt to accommodate the later Wittgenstein’s basic ideas about meaning and understanding, especially the requirement of drawing the seems right/is right distinction and the way this requirement (...) must be met. These ideas, however, are interpreted by Davidson in his own way. I will then argue that Davidson even attempts to respect Wittgenstein’s quietism, provided that we understand this view in the way Davidson does. Having argued for that, I will finally investigate whether, for Davidson at least, his more theoretical and supposedly explanatory projects, such as that of constructing a formal theory of meaning and his use of the notion of triangulation, are in conflict with this Wittgensteinian quietist view. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of (...) class='Hi'>mathematics are considered and rejected. Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

In 1956, a few writings of Wittgenstein that he didn't publish in his lifetime were revealed to the public. These writings were gathered in the book Remarks on the Foundations of Mathematics (1956). There, we can see that Wittgenstein had some discontentment with the way philosophers, logicians, and mathematicians were thinking about paradoxes, and he even registered a few polemic reasons to not accept Gödel’s incompleteness theorems.