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)

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)

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)

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 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)

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 philosophy of mathematics involves two highly controversial theses: the idea that mathematical propositions are not about (abstract) objects and the idea that no mathematical conjecture is ever answered as such, because the advent of the proof always determines a semantical shift of the meanings of the terms involved in the conjecture. The present article offers a reconstruction of Wittgenstein’s arguments supporting these theses within a very restricted setting: Archimedes’ discovery of an algorithm for calculating the number Pi.

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)

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the (...) class='Hi'>philosophy of mathematics by writing short essays and letters. (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)

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). -/- On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or (...) false. Mathematical truths are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. -/- The goal of the paper is not merely exegetical, however, and argues that radical conventionalism can withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics. (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.

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, (...) math is an artistic and moral activity that has an essential role to play in the joyful wisdom. (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)

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)

This essay offers a strategic reinterpretation of Kant’s philosophy of mathematics in Critique of Pure Reason via a broad, empirically based reconception of Kant’s conception of drawing. It begins with a general overview of Kant’s philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant’s constructivism takes up these issues, largely inspired by Hintikka’s unorthodox conception (...) of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics. (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)

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)

K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern (...) class='Hi'>mathematics and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking. (shrink)

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long (...) time, the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different (...) communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as (...) yield mathematical objects, she thinks we are sometimes licensed in drawing conclusions about physical things from mathematical premises. (shrink)

A superb effort but in my view Wittgenstein is not completely understood by anyone, so we can hardly expect Budd, writing in the mid 80’s, without the modern dual systems of thought view and no comprehensive logical structure of rationality to have grasped him completely. Like everyone, he does not get that W’s use of the word ‘grammar’ refers to our innate Evolutionary Psychology and the general framework of Wittgenstein’s and Searle’s work since laid out (e.g., in my recent articles) (...) was unavailable to him. Nevertheless he does a good job and nicely complements the work by Johnston (Wittgenstein: Rethinking the Inner) which I have also reviewed. Budd’s summary is a fitting end to the book(p165). “The repudiation of the model of ‘object and designation’ for everyday psychological words—the denial that the picture of the inner process provides a correct representation of the grammar of such words, is not the only reason for Wittgenstein’s hostility to the use of introspection in the philosophy of psychology. But it is its ultimate foundation.” An excellent study, but in my view, like them all, it falls short of a full appreciation of W as I explain here and in my other reviews. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle 59p(2016). For all my articles on Wittgenstein and Searle see my e-book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Wittgenstein and Searle 367p (2016). Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016’ 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

Indonesia has recently faced problems in various aspects of life. The results of a social media survey in Indonesia in early 2021 that the biggest threat to the Pancasila ideology is communism and other western ideologies. Communism has a dark history in the life of the Indonesian people. It shows the problem of thinking and philosophical views of the Indonesian people. This research is textbook research that aims to analyze philosophy books, namely mathematics education philosophy textbooks written (...) with a Western cultural background by Paul Ernest. This book was chosen as the object of analysis because this book is the essential reference textbook for the philosophy of mathematics education in universities providing prospective mathematics teachers in Indonesia. The research results show that this book contributes to developing Western thoughts that are contrary to Pancasila. It needs to eliminate the most critical thing in this book. It is the atheistic and individualistic philosophy underlying the ideology of mathematics education because this is not in line with the theistic collectivistic Pancasila. Therefore, this research recommends developing a textbook on the philosophy of mathematics education. It should be under the Pancasila ideology or written based on the Pancasila ideology. (shrink)

The author- following his own research on the subject- argues that Wittgenstein ignores argumentation theory and in general, the problems of rhetoric and argumentation. From this point of view, he frames Stephen Toulmin’s reading of Wittgenstein, arguing that the British philosopher- who was a student of the Austrian- advocates precisely the same thesis. He explains that this happens in a very peculiar (rhetorical) context on Toulmin’s part; a context in which, in essence, Wittgenstein’s philosophy is being rehabilitated.

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, (...) are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...) in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. (shrink)

Thomas Nagel provides a brief summary of Wittgenstein's thought, both early and late, for the general public. Summarizing the late Wittgenstein, Nagel writes: "The beginning, the point at which we run out of justifications for dividing up or organizing the world or experience as we do, is typically a form of life. Justification comes to an end within it, not by an appeal to it. This is as true of the language of experience as it is of the language (...) of physical objects or mathematic. Experience cannot be regarded as the absolute given on which the structure of the world is based, for the categories of experience itself—what constitutes sameness, difference, and structure in that domain—are as much dependent on the interpersonal responses of a communal form of life as any other conceptual or linguistic domain.". (shrink)

Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of people with on-field spatial and (...) ‘defensive-role’ relations”. “A structure is the abstract form of a system” ; it consists of “a collection of places [sometimes called positions or offices] and a finite collection of functions and relations on these places”. (shrink)

Walter Dubislav (1895–1937) was a leading member of the Berlin Group for scientific philosophy. This “sister group” of the more famous Vienna Circle emerged around Hans Reichenbach’s seminars at the University of Berlin in 1927 and 1928. Dubislav was to collaborate with Reichenbach, an association that eventuated in their conjointly conducting university colloquia. Dubislav produced original work in philosophy of mathematics, logic, and science, consequently following David Hilbert’s axiomatic method. This brought him to defend formalism in these (...) disciplines as well as to explore the problems of substantiating (Begründung) human knowledge. Dubislav also developed elements of general philosophy of science. Sadly, the political changes in Germany in 1933 proved ruinous to Dubislav. He published scarcely anything after Hitler came to power and in 1937 committed suicide under tragic circumstances. The intent here is to pass in review Dubislav’s philosophy of logic, mathematics, and science and so to shed light on some seminal yet hitherto largely neglected currents in the history of philosophy of science. (shrink)

Ludwig Wittgenstein, in the "Remarks on the Foundation of Mathematics", often refers to contradictions as deserving special study. He is said to have predicted that there will be mathematical investigations of calculi containing contradictions and that people will pride themselves on having emancipated themselves from consistency. This paper examines a way of taking this prediction seriously. It starts by demonstrating that the easy way of understanding the role of contradictions in a discourse, namely in terms of pure convention within (...) a specific linguistic community, is naive and therefore to be avoided. A number of steps will then be taken to formulate and justify what will be called a ‘containment-account’ of contradiction. This term refers to a way of understanding how a contradiction in a discourse can be contained without allowing it to infect the entire discourse with inconsistency. The account builds on work done by N. Rescher and involves complex ontologies that are either over-determined or under-determined at some points. These worlds are such that, for some P, they allow us to deny simultaneously both that P and that -P. They likewise allow us to hold both that P and that -P. The ontological status of P and that of -P are considered independent issues, so as to block the inference from P and -P to the conjunction (P & -P). In this way, the task of containment is carried out by the complexity of the ontology determining the possible worlds under consideration. The paper proceeds by countering the objection that such possible worlds are useless fictions. This is carried out by highlighting three significant areas of application, one in philosophy of science, one in philosophy of religion, and one in the area of epistemology. In general, one may say that, in some language-games, a contradiction can be a useful instrument to attest that rationality goes beyond ratiocination, in other words to indicate that there is more to discourse and practice than what can be expressed in precise algorithmic trees. The upshot of the entire argument is that the proposed ‘containment-account’ of contradiction can be an adequate illustration of how Wittgenstein’s prediction is not sterile, but a possible source of inspiration. (shrink)

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second-order or continuous phase transitions in which correlation length diverges. For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, (...) and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper. (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)

What happens when we are uncertain about what we want, feel or whish for? How should we understand uncertainty in introspection? This paper reconstructs and critically assess two answers to this question frequently found in the secondary literature on Wittgenstein: indecision and self-deception (Hacker 1990, 2012; Glock 1995, 1996). Such approaches seek to explain uncertainty in introspection in a way which is completely distinct from uncertainty about the ‘outer world’. I argue that in doing so these readings fail to account (...) for the substantial role the intellect seems to play in the process of resolving such uncertainties. I then attempt to show that Wittgenstein’s remarks connecting psychological vocabulary, behaviour and public criteria (e. g. PI 2009: 580) provide alternative ways for thinking about uncertainty in introspection which allow for a substantial role of the intellect. (shrink)

Wittgenstein’s comparison of philosophical methods to therapies has been interpreted in highly different ways. I identify the illness, the patient, the therapist and the ideal of health in Wittgenstein’s philosophical methods and answer four closely related questions concerning them that have often been wrongly answered by commentators. The results of this paper are, first, some answers to crucial questions: philosophers are not literally ill, patients of philosophical therapies are not always philosophers, not all philosophers qualify as therapists, the therapies are (...) not necessarily to be thought of as psychological therapies and the ideal of health does not consist in the end of philosophy. Second, the paper shows that the comparison has had a misleading effect, because properties of therapies have been illegitimately projected onto the philosophical methods advanced by Wittgenstein. (shrink)

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than (...) as a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)

Easy to understand philosophy papers in all areas. Table of contents: Three Short Philosophy Papers on Human Freedom The Paradox of Religions Institutions Different Perspectives on Religious Belief: O’Reilly v. Dawkins. v. James v. Clifford Schopenhauer on Suicide Schopenhauer’s Fractal Conception of Reality Theodore Roszak’s Views on Bicameral Consciousness Philosophy Exam Questions and Answers Locke, Aristotle and Kant on Virtue Logic Lecture for Erika Kant’s Ethics Van Cleve on Epistemic Circularity Plato’s Theory of Forms Can we trust (...) our senses? Yes we can Descartes on What He Believes Himself to Be The Role of Values in Science Modern Science Kant’s Moral Philosophy Plato’s Republic as Pol Potist Bureaucracy Schopenhauer on Human Suffering Bertrand Russell on the Value of Philosophy The Philosophical Value of Uncertainty Logic Homework: Theorems and Models Searle vs. Turing on the Imitation Game Hume, Frankfurt, and Holbach on Personal Freedom Manifesto of the University of Wisconsin, Madison Secular Society Michael’s Analysis of the Limits of Civil Protections Bentham and Mill on Different Types of Pleasure Set Theory Homework Aristotle on Virtue Nagel On the Hard Problem Wittgenstein on Language and Thought Camus and Schopenhauer on the Meaning of Life Camus’ Hero as Rebel without a Cause My Little Finger: Camus’ Absurdism Illustrated Are Late-term Abortions Ethical? Does Mathematics Assume the Truth of Platonism? The Self-defeating Nature of Utilitarianism and Consequentialism Generally What is The Good Life? Bentham and Mill regarding types of pleasures Kant’s Moral Philosophy Five Short Papers on Mind-body Dualism Tracy Latimer’s Father had the Right to Kill Her: Towards a doctrine of generalized self-defense Arguments Concerning God and Morality Goldman, Rousseau and von Hayek on the Ideal State J.S Mill on Liberty and Personal Freedom A Kantian Analysis of a Borderline Date-rape Situation Living Well as Flourishing: Aristotle’s Conception of the Good Life Three Essays on Medical Ethics: Answers to Exam Questions on Elective Amputation, Vaccination, and Informed Consent Hobbes, Marx, Rousseau, Nietzsche: Their Central Themes De Tocqueville on Egoism Mill vs. Hobbes on Liberty Exam-Essays on the Moral Systems of Mill, Bentham, and Kant Kant’s Moral System Aristotle on Virtue Plato’s Cave Allegory An Ethical Quandary Superorganisms The Tuskegee Experiment A Rawlsian Analysis Why Moore’s Proof of an External World Fails A Defense of Nagel’s Argument Against Materialism A Utilitarian Analysis of a Case of Theft The Paradox of the Self-aware Wretch: An Analysis of Pascal’s Moral Philosophy Jean-Paul Sartre: Decline and Fall of a Marxist Sell-out A One Page Proof of Plato’s Theory of Forms Plato’s Republic as Pol Potist Bureaucracy The problem of the one and the many Four Short Essays on Truth and Knowledge What is ‘the Good Life?’ The Ontological Argument Different Political Philosophies: Plato, Locke, Madison, Rousseau, Hayek, and Mill on the State What do I know with certainty? Skepticism about skepticism Neuroscience and Freewill Operant Conditioning What makes us special? Are Late-term Abortions Ethical? No Two Papers on Epistemology: Gettier and Bostrom Examination Nietzsche on Punishment God’s Foreknowledge and Moral Responsibility . (shrink)

Many philosophers have assumed, without argument, that Wittgenstein influenced Austin. More often, however, this is vehemently denied, especially by those who knew Austin personally. We compile and assess the currently available evidence for Wittgenstein’s influence on Austin’s philosophy of language. Surprisingly, this has not been done before in any detail. On the basis of both textual and circumstantial evidence we show that Austin’s work demonstrates substantial engagement with Wittgenstein’s later philosophy. In particular, Austin’s 1940 paper, ‘The Meaning of (...) a Word’, should be construed as a direct response to and development of ideas he encountered in Wittgenstein’s Blue Book. Moreover, we argue that Austin’s mature speech-act theory in How to Do Things with Words was also significantly influenced by Wittgenstein. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even (...) independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation,and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things,a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’ with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’ nor can find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle 59p(2016). For all my articles on Wittgenstein and Searle see my e-book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Wittgenstein and Searle 367p (2016). Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016’ 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio (...) of two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

I demonstrate that analogies, both explicit and implicit, between Wittgenstein’s discussion of rituals, aesthetics, and psychoanalysis (and, indeed, his own philosophical methodology) suggest that he entertained the idea that Freud’s psychoanalytic project, when understood correctly—that is, as a descriptive project rather than an explanatory-hypothetical one—provides a “surveyable representation” (übersichtliche Darstellung) of certain psychological facts (as opposed to psychological concepts). The consequences of this account are that it offers an explanation of Wittgenstein’s admiration for and self-perceived affinity to Freud, as well (...) as of his apparent “know-nothing approach” to empirical psychology. (shrink)

Wittgenstein composed five original musical fragments during his transitional middle period, in which he employs musical notation as a means by which to convey his philosophical thoughts. This is an overlooked aspect of the importance of aesthetics, and musical thinking in particular, in the development of Wittgenstein’s philosophy. We explain and evaluate the way the music interlinks with Wittgenstein’s philosophical thoughts. We show the direct relation of these musical examples as precursors to some of Wittgenstein’s most celebrated ideas (the (...) push beyond the inner/outer picture, the problem of rule-following, and aspect perception). We also show how important broad aesthetic notions such as style and composition concerning the very possibility of philosophizing were to Wittgenstein. (shrink)

This collection of articles was written over the last 10 years and the most important and longest within the last year. Also I have edited them to bring them up to date (2016). The copyright page has the date of this first edition and new editions will be noted there as I edit old articles or add new ones. All the articles are about human behavior (as are all articles by anyone about anything), and so about the limitations of having (...) a recent monkey ancestry (8 million years or much less depending on viewpoint) and manifest words and deeds within the framework of our innate psychology as presented in the table of intentionality. As famous evolutionist Richard Leakey says, it is critical to keep in mind not that we evolved from apes, but that in every important way, we are apes. If everyone was given a real understanding of this (i.e., human ecology and psychology) in school, maybe civilization would have a chance. -/- In my view these articles and reviews have many novel and highly useful elements, in that they use my own version of the recently (ca. 1980’s) developed dual systems view of our brain and behavior to lay out a logical system of rationality (personality, psychology, mind, language, behavior, thought, reasoning, reality etc.) that is sorely lacking in the behavioral sciences (psychology, philosophy, literature, politics, anthropology, history, economics, sociology etc.). The philosophy centers around the two writers I have found the most important, Ludwig Wittgenstein and John Searle, whose ideas I combine and extend within the dual system (two systems of thought) framework that has proven so useful in recent thinking and reasoning research. As I note, there is in my view essentially complete overlap between philosophy, in the strict sense of the enduring questions that concern the academic discipline, and the descriptive psychology of higher order thought (behavior). Once one has grasped Wittgenstein’s insight that there is only the issue of how the language game is to be played, one determines the Conditions of Satisfaction (what makes a statement true or satisfied etc.) and that is the end of the discussion. -/- Now that I think I understand how the games work I have mostly lost interest in philosophy, which of course is how Wittgenstein said it should be. But since they are the result of our innate psychology, or as Wittgenstein put, it due to the lack of perspicuity of language, the problems run throughout all human discourse, so there is endless need for philosophical analysis, not only in the ‘human sciences’ of philosophy, sociology, anthropology, political science, psychology, history, literature, religion, etc., but in the ‘hard sciences’ of physics, mathematics, and biology. It is universal to mix the language game questions with the real scientific ones as to what the empirical facts are. Scientism is ever present and the master has laid it before us long ago, i.e., Wittgenstein (hereafter W) beginning principally with the Blue and Brown Books in the early 1930’s. -/- "Philosophers constantly see the method of science before their eyes and are irresistibly tempted to ask and answer questions in the way science does. This tendency is the real source of metaphysics and leads the philosopher into complete darkness." (BBB p18) -/- Nevertheless, a real understanding of Wittgenstein’s work, and hence of how our psychology functions, is only beginning to spread in the second decade of the 21st century, due especially to P.M.S. Hacker (hereafter H) and Daniele Moyal-Sharrock (hereafter DMS),but also to many others, some of the more prominent of whom I mention in the articles. -/- When I read ‘On Certainty’ a few years ago I characterized it in an Amazon review as the Foundation Stone of Philosophy and Psychology and the most basic document for understanding behavior, and about the same time DMS was writing articles noting that it had solved the millennia old epistemological problem of how we can know anything for certain. I realized that W was the first one to grasp what is now characterized as the two systems or dual systems of thought, and I generated a dual systems (S1 and S2) terminology which I found to be very powerful in describing behavior. I took the small table that John Searle (hereafter S) had been using, expanded it greatly, and found later that it integrated perfectly with the framework being used by various current workers in thinking and reasoning research. -/- Since they were published individually, I have tried to make the book reviews and articles stand by themselves, insofar as possible, and this accounts for the repetition of various sections, notably the table and its explanation. I start with a short article that presents the table of intentionality and briefly describes its terminology and background. Next, is by far the longest article, which attempts a survey of the work of W and S as it relates to the table and so to an understanding or description (not explanation as W insisted) of behavior. -/- The key to everything about us is biology, and it is obliviousness to it that leads millions of smart educated people like Obama, Chomsky, Clinton and the Pope to espouse suicidal utopian ideals that inexorably lead straight to Hell On Earth. As W noted, it is what is always before our eyes that is the hardest to see. We live in the world of conscious deliberative linguistic System 2, but it is unconscious, automatic reflexive System 1 that rules. This is Searle’s The Phenomenological Illusion (TPI), Pinker’s Blank Slate and Tooby and Cosmides Standard Social Science Model. Democracy and equality are wonderful ideals, but without strict controls, selfishness and stupidity gain the upper hand and soon destroy any nation and any world that adopts them. The monkey mind steeply discounts the future, and so we sell our children’s heritage for temporary comforts. -/- The astute may wonder why we cannot see System 1 at work, but it is clearly counterproductive for an animal to be thinking about or second guessing every action, and in any case there is no time for the slow, massively integrated System 2 to be involved in the constant stream of split second ‘decisions’ we must make. As W noted, our ‘thoughts’ (T1 or the thoughts of System 1) must lead directly to actions. It is my contention that the table of intentionality (rationality, mind, thought, language, personality etc.) that features prominently here describes more or less accurately, or at least serves an heuristic for how we think and behave, and so it encompasses not merely philosophy and psychology but everything else (history, literature, mathematics, politics etc.). Note especially that intentionality and rationality as I (along with Searle, Wittgenstein and others) view it,includes both conscious deliberative System 2 and unconscious automated System 1 actions or reflexes. -/- Thus all the articles, like all behavior, are intimately connected if one knows how to look at them. As I note, The Phenomenological Illusion (oblivion to our automated System 1) is universal and extends not merely throughout philosophy but throughout life. I am sure that Chomsky, Obama, Zuckerberg and the Pope would be incredulous if told that they suffer from the same problem as Hegel, Husserl and Heidegger, but it’s clearly true. While the phenomenologists only wasted a lot of people’s time, they are wasting the earth. -/- My writings are available in updated versions as paperbacks and Kindles on Amazon. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) ASIN B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) ASIN B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) 2nd printing with corrections (Feb 2018) ASIN B0711R5LGX -/- Suicide by Democracy: an Obituary for America and the World (2018) ASIN B07CQVWV9C . (shrink)

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.

On the basis of historical and textual evidence, this paper claims that after his Tractatus, Wittgenstein was actually influenced by Einstein's theory of relativity and, the similarity of Einstein's relativity theory helps to illuminate some aspects of Wittgenstein's work. These claims find support in remarkable quotations where Wittgenstein speaks approvingly of Einstein's relativity theory and in the way these quotations are embedded in Wittgenstein's texts. The profound connection between Wittgenstein and relativity theory concerns not only Wittgenstein's “verificationist” (...) phase , but also Wittgenstein's later philosophy centred on the theme of rule‐following. (shrink)

I demonstrate that analogies, both explicit and implicit, between Wittgenstein’s discussions of rituals, aesthetics, and aspect-perception, have important payoffs in terms of understanding his notion of a “surveyable representation” (übersichtliche Darstellung) as it applies to phenomena that are not exclusively grammatical in nature. In particular, I argue that a surveyable representation of certain anthropological and aesthetic facts allows us to see, qua form of aspect-perception, internal relations and formal connections, so that the inner nature of a ritual or the solution (...) of an aesthetic puzzle is exhibited. This particular form of seeing both explains why Wittgenstein thought that hypothetical explanations about the origins of a ritual are irrelevant to appreciating its meaning and elucidates his understanding of rituals, which involves seeing “internal relations” and “formal connections.” The upshots of the account for work in anthropology are discussed. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article (...) lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)