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)

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)

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)

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)

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)

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)

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. (shrink)

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)

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 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 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)

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)

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)

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)

A superb effort, but in my view Wittgenstein (i.e., philosophy or the descriptive psychology of higher order thought) 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 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)

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.

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)

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)

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.

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)

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 Wit...

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)

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)

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)

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)

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)

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)

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)

This article examines the possibility of philosophizing about mathematics with children. It aims at outlining the nature of the practice of philosophy of mathematics with children in a mainly theoretical and exploratory way. First, an attempt at a definition is proposed. Second, I suggest some reasons that might motivate such a practice. My thesis is that one can identify an intrinsic as well as two extrinsic goals of philosophizing about mathematics with children. The intrinsic goal is (...) related to a presumed inherent importance of presenting children with some philosophical questions about mathematics. The extrinsic goals consist of first the positive effects such a practice can have on mathematical learning and abilities and second the fostering of children's understanding of philosophical method of inquiry and thinking and therefore of their philosophical thinking competences. Third, some examples found in the literature of previously developed ways of practising philosophy of mathematics with children are presented. This article aims at giving a general outlining picture of the issues surrounding the practice of philosophy of mathematics with children and should therefore be read as an encouragement to further development and studies. (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)

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)

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)

Pears is an eminent philosopher, notable among W scholars for his “The False Prison: a study of the development of Wittgenstein’s philosophy” in 2 volumes published 20 years ago. Based on these facts I expected some deep insights into W in the current volume. There were certainly some good points but overall it was profoundly disappointing. All of behavioral science is about our innate human nature and since W was the first to elucidate the axioms of our universal psychology, (...) I expected this to be front and center in a work written during the golden age of evolutionary and cognitive psychology and with much good recent work on W appearing. However one would never guess from this book that W or philosophy had any connection with psychology or indeed that there is such a thing as evolutionary psychology. Hence, I cannot recommend Pears works and recommend a framework for rationality totally lacking in Pears (and most writing on human behavior). -/- 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)

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)

Remodel[l]ing Reality is an inquiry into Wittgenstein's notion of übersichtliche Darstellung and the phenomenon of installation in visual art. In a sense, both provide a perspicuous overview of a particular part of our complex world, but the nature of the overview differs. Although both generate knowledge, philosophy via the übersichtliche Darstellung gives us a view of how things stand for us, while the installation shows an unexpected, exiting point of view. The obvious we tend to forget and the (...) ambiguity of reality are related to each other in a dynamic way. It is in this reflexive dynamics that we constantly remodel our reality. Tools we use are our creative abilities and our powers of imagination. In our choices and solutions we show which aspects of reality we find important and how we communicate these values. The outcome of this investigation is a new perspective on the art of installation and a new insight in Wittgenstein's notion of übersichtliche Darstellung. Because of this combination, the book is itself an artwork: an InstallationPackage. The book is distributed by Ideabooks Amsterdam, Isbn 978 90 804240 3 6. Remodel[l]ing Reality is included in Special Collections, Allard Pierson Museum Amsterdam. (shrink)

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)

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)

Pears is an eminent philosopher, notable among W scholars for his “The False Prison: a study of the development of Wittgenstein’s philosophy” in 2 volumes published 20 years ago. Based on these facts I expected some deep insights into W in the current volume. There were certainly some good points but overall it was profoundly disappointing. All of behavioral science is about our innate human nature and since W was the first to elucidate the axioms of our universal psychology, (...) I expected this to be front and center in a work written during the golden age of evolutionary and cognitive psychology and with much good recent work on W appearing. However, one would never guess from this book that W or philosophy had any connection with psychology or indeed that there is such a thing as evolutionary psychology. Hence, I cannot recommend Pears works and instead provide a framework for rationality totally lacking in Pears (and most writing on human behavior). -/- 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)

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)

Wittgenstein’s interpreters are undivided that the method plays a central role in his philosophy. This would be no surprise if we have in mind the Tractarian dictum: “philosophy is not a body of doctrine but an activity” (4.112). After 1929, Wittgenstein’s method evolved further. In its final form, articulated in Philosophical Investigations, it was formulated as different kinds of therapies of specific philosophical problems that torment our life (§§ 133, 255, 593). In this paper we follow the changes (...) in Wittgenstein’s thinking in four subsequent phases and in three dimensions: (i) in logic and ontology; (ii) in method proper; (iii) in style. (shrink)

A significant discrepancy in Wittgenstein's studies is whether Philosophical Investigations contains any trace of positive philosophy, notwithstanding the author's apparent anti-theoretic position. This study argues that the so-called ‘Chapter on philosophy’ in the Investigations §§89–133 contains negative and positive vocabulary and the use of various voices through which Wittgenstein employs his primary method of language-games, thus providing a surveyable understanding of several philosophical concepts, such as knowledge and time. His positive philosophy aims to reorient our attention (...) from understanding the theories on these concepts to understanding the concepts themselves, regardless of any theorisation. (shrink)

In this book, Rupert Read offers the first outline of a resolute reading, following the highly influential New Wittgenstein 'school', of the Philosophical Investigations. He argues that the key to understanding Wittgenstein's later philosophy is to understand its liberatory purport. Read contends that a resolute reading coincides in its fundaments with what, building on ideas in the later Gordon Baker, he calls a liberatory reading. Liberatory philosophy is philosophy that can liberate the user from compulsive patterns (...) of thought, freeing one for possibilities that were previously obscured. Such liberation is our prime goal in philosophy. This book consists in a sequential reading, along these lines, of what Read considers the most important and controversial passages in the Philosophical Investigations 1, 16, 43, 95 & 116 & 122, 130-3, 149-151, 186, 198-201, 217, and 284-6. Read claims that this liberatory conception is simultaneously an ethical conception. The PI should be considered a work of ethics in that its central concern becomes our relation with others. Wittgensteinian liberations challenge widespread assumptions about how we allegedly are independent of and separate from others. Wittgenstein's Liberatory Philosophy will be of interest to scholars and advanced students working on Wittgenstein, and to scholars of the political philosophy of liberation and the ethics of relation. (shrink)

The supposition that Wittgenstein's Tractatus advances a certain metaphysics has given rise to a controversy over the ontological status of his Tractarian objects. It has been debated, for instance, whether these objects consist only of particulars or of both particulars and universals; whether they are physical, phenomenal, or phenomenological entities; and whether they correspond to Russell's objects of acquaintance or Kant's phenomena and substance. In this essay, I endorse Ishiguro's view that these objects, being formal concepts, are ontologically neutral (...) and thus are not identifiable with any ontological kind of entities. I elaborate on the coherence of this view with the propositional dependence of the meaning of Tractarian names. After showing why some arguments for ascribing a Russellian theory of meaning to these names do not work, I demonstrate why Ishiguro's account of Tractarian objects and names provides a better explanation of the unalterability of these objects. (shrink)

In this review, as well as discussing the pedagogical of this text book, I also discuss Linnebo's approach to the Caesar problem and the use of metaphysical notions to explicate mathematics.

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking (...) to Aristotle for inspiration. (shrink)