Philosophy of Mathematics

We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical disciplines. This (...) diversity of approaches is founded on the manifold categories that we find in physical reality. We also propose, grounded on this idea, the use of Aristotelian categories as a first model for this division, generating from it a classification of mathematical branches. Finally we make a history review to show that there is consistency between our classification, and the historical appearance of the different branches of mathematics. (shrink)

मैं Wittgenstein और विकासवादी मनोविज्ञान के एक एकीकृत परिप्रेक्ष्य से Noson Yanofsky द्वारा 'कारण की बाहरी सीमा' की एक विस्तृत समीक्षा दे. मैं संकेत मिलता है कि भाषा और गणित में विरोधाभास के रूप में इस तरह के मुद्दों के साथ कठिनाई, अपूर्णता, अनिर्णयीयता, computability, मस्तिष्क और कंप्यूटर आदि के रूप में ब्रह्मांड, सभी विफलता से उठता है उचित में भाषा के हमारे उपयोग को ध्यान से देखने के लिए संदर्भ और इसलिए कैसे भाषा काम करता है के मुद्दों से (...) वैज्ञानिक तथ्य के मुद्दों को अलग करने में विफलता. मैं अधूरापन, paraconsistency और undecidability और गणना करने के लिए सीमा पर Wolpert के काम पर Wittgenstein के विचारों पर चर्चा की. यह योग अप करने के लिए: ब्रह्मांड ब्रुकलीन के अनुसार---अच्छा विज्ञान, नहीं तो अच्छा दर्शन. आधुनिक दो systems दृश्यसे मानव व्यवहार के लिए एक व्यापक अप करने के लिए तारीख रूपरेखा इच्छुक लोगों को मेरी पुस्तक 'दर्शन, मनोविज्ञान, मिनडी और लुडविगमें भाषा की तार्किक संरचना से परामर्श कर सकते हैं Wittgenstein और जॉन Searle '2 एड (2019). मेरे लेखन के अधिक में रुचि रखने वालों को देख सकते हैं 'बात कर रहेबंदर- दर्शन, मनोविज्ञान, विज्ञान, धर्म और राजनीति पर एक बर्बाद ग्रह --लेख और समीक्षा 2006-2019 3 एड (2019) और आत्मघाती यूटोपियान भ्रम 21st मेंसदी 4वें एड (2019) . (shrink)

'गोडेल के रास्ते' में तीन प्रख्यात वैज्ञानिकों ने अनिर्णय, अपूर्णता, यादृच्छिकता, गणनाऔरता और परासंगति जैसे मुद्दों पर चर्चा की। मैं Wittgensteinian दृष्टिकोण से इन मुद्दों दृष्टिकोण है कि वहाँ दो बुनियादी मुद्दों जो पूरी तरह से अलग समाधान है. वहाँ वैज्ञानिक या अनुभवजन्य मुद्दों, जो दुनिया के बारे में तथ्य है कि अवलोकन और दार्शनिक मुद्दों की जांच की जरूरत है के रूप में कैसे भाषा intelligibly इस्तेमाल किया जा सकता है (जो गणित और तर्क में कुछ सवाल शामिल हैं), (...) जो की जरूरत है एकटी कैसे हम वास्तव में विशेष संदर्भों में शब्दों का उपयोग देख कर फैसला किया. जब हम जो भाषा खेल हम खेल रहे हैं के बारे में स्पष्ट हो, इन विषयों को किसी भी अन्य की तरह साधारण वैज्ञानिक और गणितीय सवाल देखा जाता है. है Wittgenstein अंतर्दृष्टि शायद ही कभी बराबर किया गया है और कभी नहीं पार कर रहे हैं और के रूप में आज के रूप में प्रासंगिक हैं के रूप में वे 80 साल पहले थे जब वह ब्लू और ब्राउन पुस्तकें हुक्म दिया. अपनी असफलताओं के बावजूद-वास्तव में एक समाप्त पुस्तक के बजाय नोटों की एक श्रृंखला-यह इन तीन प्रसिद्ध विद्वानों के काम का एक अनूठा स्रोत है जो आधे से अधिक सदी से भौतिकी, गणित और दर्शन के खून बह रहा किनारों पर काम कर रहे हैं। दा कोस्टा और डोरिया Wolpert द्वारा उद्धृत कर रहे हैं (नीचे देखें या Wolpert पर मेरे लेख और Yanofsky 'कारण की बाहरी सीमा' की मेरी समीक्षा) के बाद से वे सार्वभौमिक गणना पर लिखा था, और उनके कई उपलब्धियों के बीच, दा कोस्टा में अग्रणी है paraconsistency. आधुनिक दो systems दृश्यसे मानव व्यवहार के लिए एक व्यापक अप करने के लिए तारीख रूपरेखा इच्छुक लोगों को मेरी पुस्तक 'दर्शन, मनोविज्ञान, मिनडी और लुडविगमें भाषा की तार्किक संरचना से परामर्श कर सकते हैं Wittgenstein और जॉन Searle '2 एड (2019). मेरे लेखन के अधिक में रुचि रखने वालों को देख सकते हैं 'बात कर रहेबंदर- दर्शन, मनोविज्ञान, विज्ञान, धर्म और राजनीति पर एक बर्बाद ग्रह --लेख और समीक्षा 2006-2019 3 एड (2019) और आत्मघाती यूटोपियान भ्रम 21st मेंसदी 4वें एड (2019) . (shrink)

मैं कंप्यूटर के रूप में गणना और ब्रह्मांड की सीमा के कई हाल ही में चर्चा पढ़ लिया है, polymath भौतिक विज्ञानी और निर्णय सिद्धांतकार डेविड Wolpert के अद्भुत काम पर कुछ टिप्पणी खोजने की उम्मीद है, लेकिन एक भी प्रशस्ति पत्र नहीं मिला है और इसलिए मैं यह बहुत संक्षिप्त मौजूद सारांश. Wolpert कुछ आश्चर्यजनक असंभव या अधूरापन प्रमेयों साबित कर दिया (1992 से 2008-देखें arxiv dot org) अनुमान के लिए सीमा पर (कम्प्यूटेशन) कि इतने सामान्य वे गणना कर (...) डिवाइस से स्वतंत्र हैं, और यहां तक कि भौतिकी के नियमों से स्वतंत्र, इसलिए वे कंप्यूटर, भौतिक विज्ञान और मानव व्यवहार में लागू होते हैं. वे कैंटर विकर्णीकरण का उपयोग करते हैं, झूठा विरोधाभास और worldlines प्रदान करने के लिए क्या ट्यूरिंग मशीन थ्योरी में अंतिम प्रमेय हो सकता है, और प्रतीत होता है असंभव, अधूरापन, गणना की सीमा में अंतर्दृष्टि प्रदान करते हैं, और ब्रह्मांड के रूप में कंप्यूटर, सभी संभव ब्रह्मांडों और सभी प्राणियों या तंत्र में, उत्पादन, अन्य बातों के अलावा, एक गैर क्वांटम यांत्रिक अनिश्चितता सिद्धांत और एकेश्वरवाद का सबूत. वहाँ Chaitin, Solomonoff, Komolgarov और Wittgenstein के क्लासिक काम करने के लिए स्पष्ट कनेक्शन कर रहे हैं और धारणा है कि कोई कार्यक्रम (और इस तरह कोई डिवाइस) एक दृश्य उत्पन्न कर सकते हैं (या डिवाइस) अधिक से अधिक जटिलता के साथ यह पास से. कोई कह सकता है कि काम के इस शरीर का अर्थ नास्तिकता है क्योंकि भौतिक ब्रह्मांड से और विटगेनस्टीनियन दृष्टिकोण से कोई भी इकाई अधिक जटिल नहीं हो सकती है, 'अधिक जटिल' अर्थहीन है (संतोष की कोई शर्त नहीं है, अर्थात, सत्य-निर्माता या परीक्षण)। यहां तक कि एक 'भगवान' (यानी, असीम समय/स्थान और ऊर्जा के साथ एक 'डिवाइस' निर्धारित नहीं कर सकता है कि क्या एक दिया 'संख्या' 'यादृच्छिक' है, और न ही एक निश्चित तरीका है दिखाने के लिए कि एक दिया 'सूत्र', 'प्रमेय' या 'वाक्य' या 'डिवाइस' (इन सभी जटिल भाषा जा रहा है) खेल) एक विशेष 'प्रणाली' का हिस्सा है. आधुनिक दो systems दृश्यसे मानव व्यवहार के लिए एक व्यापक अप करने के लिए तारीख रूपरेखा इच्छुक लोगों को मेरी पुस्तक 'दर्शन, मनोविज्ञान, मिनडी और लुडविगमें भाषा की तार्किक संरचना से परामर्श कर सकते हैं Wittgenstein और जॉन Searle '2 एड (2019). मेरे लेखन के अधिक में रुचि रखने वालों को देख सकते हैं 'बात कर रहेबंदर- दर्शन, मनोविज्ञान, विज्ञान, धर्म और राजनीति पर एक बर्बाद ग्रह --लेख और समीक्षा 2006-2019 2 ed (2019) और आत्मघाती यूटोपियान भ्रम 21st मेंसदी 4वें एड (2019) . (shrink)

While there has been much discussion about what makes some mathematical proofs more explanatory than others, and what are mathematical coincidences, in this article I explore the distinct phenomenon of mathematical facts that call for explanation. The existence of mathematical facts that call for explanation stands in tension with virtually all existing accounts of “calling for explanation”, which imply that necessary facts cannot call for explanation. In this paper I explore what theoretical revisions are needed in order to accommodate this (...) phenomenon. One of the important upshots is that, contrary to the current consensus, low prior probability is not a necessary condition for calling for explanation. In the final section I explain how the results of this inquiry help us make progress in assessing Hartry Field's style of reliability argument against mathematical Platonism and against robust realism in other domains of necessary facts, such as ethics. (shrink)

पादरी Hofstadter द्वारा कट्टरपंथी प्रकृतिवाद के चर्च से नवीनतम उपदेश. अपने बहुत अधिक प्रसिद्ध (या अपने अथक दार्शनिक त्रुटियों के लिए कुख्यात) काम Godel, Escher, बाख की तरह, यह एक सतही प्रशंसनीयता है, लेकिन अगर एक समझता है कि यह बड़े पैमाने पर वैज्ञानिकता है जो दार्शनिक लोगों के साथ वास्तविक वैज्ञानिक मुद्दों घोला जा सकता है (यानी, केवल असली मुद्दों क्या भाषा का खेल हम खेलना चाहिए रहे हैं) तो लगभग सभी अपनी रुचि गायब हो जाता है. मैं विकासवादी (...) मनोविज्ञान और Wittgenstein के काम में आधारित विश्लेषण के लिए एक रूपरेखा प्रदान (के बाद से मेरे और अधिक हाल ही में लेखन में अद्यतन). आधुनिक दो systems दृश्यसे मानव व्यवहार के लिए एक व्यापक अप करने के लिए तारीख रूपरेखा इच्छुक लोगों को मेरी पुस्तक 'दर्शन, मनोविज्ञान, मिनडी और लुडविगमें भाषा की तार्किक संरचना से परामर्श कर सकते हैं Wittgenstein और जॉन Searle '2 एड (2019). मेरे लेखन के अधिक में रुचि रखने वालों को देख सकते हैं 'बात कर रहेबंदर- दर्शन, मनोविज्ञान, विज्ञान, धर्म और राजनीति पर एक बर्बाद ग्रह --लेख और समीक्षा 2006-2019 3 एड (2019) और आत्मघाती यूटोपियान भ्रम 21st मेंसदी 4वें एड (2019) . (shrink)

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I (...) argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. -/- (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, I have no general justification for, even restricted to Sigma-1 claims. This doesn't undermine the philosophical point being made there, and actually makes it stronger: omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences, for that.). (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. (...) Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

The present paper deals with the ontological status of numbers and considers Frege´s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path that, departing from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more Kantian (...) sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory (...) or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

Ich gebe einen ausführlichen Überblick über 'The Outer Limits of Reason' von Noson Yanofsky aus einer einheitlichen Perspektive von Wittgenstein und Evolutionspsychologie. Ich weise darauf hin, dass die Schwierigkeit bei Themen wie Paradoxon in Sprache und Mathematik, Unvollständigkeit, Unbedenklichkeit, Berechenbarkeit, Gehirn und Universum als Computer usw. allesamt auf das Versäumnis zurückzuführen ist, unseren Sprachgebrauch im geeigneten Kontext sorgfältig zu prüfen, und daher das Versäumnis, Fragen der wissenschaftlichen Tatsache von Fragen der Funktionsweise von Sprache zu trennen. Ich bespreche Wittgensteins Ansichten über (...) Unvollständigkeit, Parakonsistenz und Unentschlossenheit und die Arbeit Wolperts an den Grenzen der Berechnung. Zusammengefasst: The Universe According to Brooklyn---Good Science, Not So Good Philosophy. Wer aus der modernen zweisystems-Sichteinen umfassenden, aktuellen Rahmen für menschliches Verhalten wünscht, kann mein Buch "The Logical Structure of Philosophy, Psychology, Mindand Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019) konsultieren. Diejenigen,die sich für mehr meiner Schriften interessieren, können 'Talking Monkeys--Philosophie, Psychologie, Wissenschaft, Religion und Politik auf einem verdammten Planeten --Artikel und Rezensionen 2006-2019 3rd ed (2019) und Suicidal Utopian Delusions in the 21st Century 4th ed (2019) und andere sehen. (shrink)

In "Godel es Way" diskutieren drei namhafte Wissenschaftler Themen wie Unentschlossenheit, Unvollständigkeit, Zufälligkeit, Berechenbarkeit und Parakonsistenz. Ich gehe diese Fragen aus Wittgensteiner Sicht an, dass es zwei grundlegende Fragen gibt, die völlig unterschiedliche Lösungen haben. Es gibt die wissenschaftlichen oder empirischen Fragen, die Fakten über die Welt sind, die beobachtungs- und philosophische Fragen untersuchen müssen, wie Sprache verständlich verwendet werden kann (die bestimmte Fragen in Mathematik und Logik beinhalten), die entschieden werden müssen, indem man sich anschaut,wie wir Wörter in bestimmten (...) Kontexten tatsächlich verwenden. Wenn wir klar werden, welches Sprachspiel wir spielen, werden diese Themen als gewöhnliche wissenschaftliche und mathematische Fragen angesehen, wie alle anderen auch. Wittgensteins Einsichten wurden selten übertroffen und sind heute so treffend wie vor 80 Jahren, als er die Blauen und Braunen Bücher diktierte. Trotz seiner Versäumnisse – wirklich eine Reihe von Notizen statt eines fertigen Buches – ist dies eine einzigartige Quelle für die Arbeit dieser drei berühmten Gelehrten, die seit über einem halben Jahrhundert an den blutenden Rändern von Physik, Mathematik und Philosophie arbeiten. Da Costa und Doria werden von Wolpert zitiert (siehe unten oder meine Artikel über Wolpert und meine Rezension von Yanofskys 'The Outer Limits of Reason'), da sie auf universelle Berechnung schrieben,, und unter seinen vielen Errungenschaften ist Da Costa ein Pionier in Parakonsistenz. Wer aus der modernen zweisystems-Sichteinen umfassenden, aktuellen Rahmen für menschliches Verhalten wünscht, kann mein Buch "The Logical Structure of Philosophy, Psychology, Mindand Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019) konsultieren. Diejenigen,die sich für mehr meiner Schriften interessieren, können 'Talking Monkeys--Philosophie, Psychologie, Wissenschaft, Religion und Politik auf einem verdammten Planeten --Artikel und Rezensionen 2006-2019 3rd ed (2019) und Suicidal Utopian Delusions in the 21st Century 4th ed (2019) und andere sehen. (shrink)

A short reconstruction of the Medieval and early modern idea of the analytic-syntetic method.

‘Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means.’ Along this line, in The Open World, Hermann Weyl contrasted the desire to make the infinite accessible through finite processes, which underlies any theoretical investigation of reality, with the intuitive feeling for the infinite ‘peculiar to the Orient,’ which remains ‘indifferent to the concrete manifold of reality.’ But a critical analysis may acknowledge a valuable dialectical opposition. Struggling to spell (...) out the infinity of real numbers mathematicians come to see the active role of emptiness. Pondering over the essence of self-awareness, the Japanese philosopher Nishida Kitarō comes to see the ‘place’ where it abides as absolute nothingness. Thus, the two ways of seeing coalesce into a perspective in which infinity and nothingness mirror each other. (shrink)

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics, and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics and in speech act theory. As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation (...) of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion. (shrink)

Records of online collaborative mathematical activity provide us with a novel, rich, searchable, accessible and sizeable source of data for empirical investigations into mathematical practice. In this paper we discuss how the resources of crowdsourced mathematics can be used to help formulate and answer questions about mathematical practice, and what their limitations might be. We describe quantitative approaches to studying crowdsourced mathematics, reviewing work from cognitive history (comparing individual and collaborative proofs); social psychology (on the prospects for a measure of (...) collective intelligence); human–computer interaction (on the factors that led to the success of one such project); network analysis (on the differences between collaborations on open research problems and known-but-hard problems); and argumentation theory (on modelling the argument structures of online collaborations). We also give an overview of qualitative approaches, reviewing work from empirical philosophy (on explanation in crowdsourced mathematics); sociology of scientific knowledge (on conventions and conversations in online mathematics); and ethnography (on contrasting conceptions of collaboration). We suggest how these diverse methods can be applied to crowdsourced mathematics and when each might be appropriate. (shrink)

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...) in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

The concept of truth has many aims but only one source. The article describes the primary concept of truth, here called the synthetic concept of truth, according to which truth is the objective result of the synthesis of us and nature in the process of rational cognition. It is shown how various aspects of the concept of truth -- logical, scientific, and mathematical aspect -- arise from the synthetic concept of truth. Also, it is shown how the paradoxes of truth (...) arise. (shrink)

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

「ゴーデルの道」では、3人の著名な科学者が、デシッド不能、不完全性、ランダム性、計算可能性、パラコンシステンションなどの問題について議論しています。私は、ウィトゲンシュタイニアンの視点から、全く異なる 解決策を持つ2つの基本的な問題があることをこれらの問題に取り組んでいます。科学的または経験的な問題は、言語がどのように理解的に使用できるか(数学と論理に特定の質問を含む)、特定の文脈で実際にどのように 単語を使用するかを調べて決定する必要がある、観察的および哲学的な問題を調査する必要がある世界に関する事実です。私たちがプレイしている言語ゲームについて明確になると、これらのトピックは他の人と同じように 普通の科学的、数学的な質問であると見なされます。ウィトゲンシュタインの洞察はめったに等しくなく、決して上回ることはなく、彼がブルーブックスとブラウンブックスを口述した80年前と同じくらい適切です。失敗 にもかかわらず、本当に完成した本ではなく一連のノートは、半世紀以上にわたって物理学、数学、哲学の出血エッジで働いてきたこれらの3人の有名な学者の作品のユニークな源です。ダ・コスタとドリアは、普遍的な計 算に書いて以来、ウォルパート(以下または私の記事を参照)によって引用されています(ウォルパートとヤナフスキーの「理由の外側の限界」の私のレビューを参照)、,そして彼の多くの成果の中で、ダ・コスタはパラ コンシタンションのパイオニアです。 現代の2つのシス・エムスの見解から人間の行動のための包括的な最新の枠組みを望む人は、私の著書「ルートヴィヒ・ヴィトゲンシュタインとジョン・サールの第2回(2019)における哲学、心理学、ミンと言語の論 理的構造」を参照することができます。私の著作の多くにご興味がある人は、運命の惑星における「話す猿--哲学、心理学、科学、宗教、政治―記事とレビュー2006-2019 第3回(2019)」と21世紀4日(2019年)の自殺ユートピア妄想st Century 4th ed (2019)などを見ることができます。 .

ノソン・ヤノフスキーの「理性の外側の限界」を、ウィトゲンシュタインと進化心理学の統一的な視点から詳しくレビューします。私は、言語や数学のパラドックス、不完全さ、デデシッド性、コンピュータとしての脳、宇 宙などの問題の難しさは、すべて適切な文脈での言語の使用を注意深く見なさなかったことから生じるため、科学的事実の問題を言語の仕組みの問題から切り離すことができなかったことを示しています。私は、不完全さ、 パラタンシ、不整合性に関するヴィトゲンシュタインの見解と、計算の限界に関するウォルパートの仕事について議論します。要約すると:ブルックリンによると宇宙---良い科学、それほど良い哲学ではありません。 現代の2つのシス・エムスの見解から人間の行動のための包括的な最新の枠組みを望む人は、私の著書「ルートヴィヒ・ヴィトゲンシュタインとジョン・サールの第2回(2019)における哲学、心理学、ミンと言語の論 理的構造」を参照することができます。私の著作の多くにご興味がある人は、運命の惑星における「話す猿--哲学、心理学、科学、宗教、政治―記事とレビュー2006-2019 第3回(2019)」と21世紀4日(2019年)の自殺ユートピア妄想st Century 4th ed (2019)などを見ることができます。 .

In a recent article in this journal [Phil. Math., II, v.4 (1989), n.2, pp.?- ?] J. Fang argues that we must not be fooled by A.J. Ayer (God rest his soul!) and his cohorts into believing that mathematical knowledge has an analytic a priori status. Even computers, he reminds us, take some amount of time to perform their calculations. The simplicity of Kant's infamous example of a mathematical proposition (7+5=12) is "partly to blame" for "mislead[ing] scholars in the direction of (...) neglecting the temporal element"; yet a brief instant of time is required to grasp even this simple truth. If Kant were alive today, "and if he had had a little more mathematical savvy", Fang explains, he could have used the latest example of the largest prime number (391,581 x 2 216,193 - 1) as a better example of the "synthetic a priori" character of mathematics. The reason Fang is so intent upon emphasizing the temporal character of mathematics is that he wishes to avoid "the uncritical mixing of ... a theology and a philosophy of mathematics." For "in the light of the Computer Age today: finitism is king!" Although Kant's aim was explicitly "to study the 'human' ... faculty", Fang claims that even he did not adequatley emphasize "the clearly and concretely distinguishable line of demarcation between the human and divine faculties.". (shrink)

This is a critical review of the book Variational Approach to Gravity Field Theories - From Newton to Einstein and Beyond (2017), written by the Italian astrophysicist Alberto Vecchiato. In his work, Vecchiato shows that physics, as we know it, can be built up from simple mathematical models that become more complex step by step by gradually introducing new principles. The reader is invited to follow the steps that lead from classical physics to relativity and to understand how this happens (...) and why. Moreover, while presenting the variational approach to gravity field theories in a clear and elegant manner, Vecchiato shows a constant worry in leading the readers in such a way that they understand the relevance of what the author considers to be the most powerful technique and unifying concept of theoretical physics - and how it works. Each chapter is enriched with exercises, of which a step-by-step solution is offered, so that the reader can gain a real insight into the topic presented and follow the reading as fruitfully as possible. The most innovative aspect of the book, however, is not the rigorousness and the clarity of the treatment of the variational approach, but rather the point of view that Vecchiato keeps and that the reader is led to endorse too: physics is a human enterprise, prone to error and open to improvement, and not a complete and definite system of truths about our world. (shrink)

This review of Alain Badiou’s The Pornographic Age—as well of the essays included in the book by William Watkin, A.J. Bartlett and Justin Clemens—illuminates that this is one of the few, if not only, texts where Badiou reverses the operational directionality of the event qua category theory, so as to “dis-image” power. In doing so, Badiou provides a theory of power based on intentionality and relation, rather than the more common Foucauldian genealogic-historical methodologies so often co-opted by contemporary thinkers of (...) biopolitics and power. Using Jean Genet’s play, The Balcony, as a fulcrum with which to strip apart the representative regime vis-à-vis the Platonic gesture, Badiou posits a politically exigent critique of contemporary neoliberal democracy. (shrink)

ホフスタッター牧師による原理主義自然主義教会からの最新の説教。彼のはるかに有名な(または容赦ない哲学的誤りで悪名高い)作品ゴーデル、エッシャー、バッハのように、それは表面的な妥当性を持っていますが、こ れが哲学的なものと実際の科学的問題を混ぜ合わせた横行するサイエンティズムであることを理解すれば(つまり、唯一の本当の問題は、私たちがプレイすべき言語ゲームです)、その後、ほとんどすべての関心が消えます 。進化心理学とヴィトゲンシュタインの仕事に基づく分析のフレームワークを提供しています(最近の著作で更新されて以来)。 現代の2つのシス・エムスの見解から人間の行動のための包括的な最新の枠組みを望む人は、私の著書「ルートヴィヒ・ヴィトゲンシュタインとジョン・サールの第2回(2019)における哲学、心理学、ミンと言語の論 理的構造」を参照することができます。私の著作の多くにご興味がある人は、運命の惑星における「話す猿--哲学、心理学、科学、宗教、政治―記事とレビュー2006-2019 第3回(2019)」と21世紀4日(2019年)の自殺ユートピア妄想st Century 4th ed (2019)などを見ることができます .

This article is mainly a critique of Philip Kitcher's book, The Nature of Mathematical Knowledge. Four weaknesses in Kitcher's objection to Kant arise out of Kitcher's failure to recognize the perspectival nature of Kant's position. A proper understanding of Kant's theory of mathematics requires awareness of the perspectival nuances implicit in Kant's theory of pure intuition.

Quentin Meillassoux est l’un des principaux philosophes français d’aujourd’hui. Son premier livre, Après la finitude. Essai sur la nécessité de la contingence (2006, traduit en anglais en 2008), est déjà un classique. Il comporte une préface de son ancien mentor, Alain Badiou. L’un des princi- paux objectifs de Meillassoux est de réhabiliter la distinction entre qualités premières et qualités secondes, typique des philosophies prékantiennes. Plus précisément, il affirme que les mathématiques sont capables de révéler les qualités premières de tout objet (...) : « Tout ce qui de l’objet peut être formulé en termes mathématiques, il y a sens à le penser comme propriété de l’objet en soi. » Nous allons utiliser la philosophie mathématique de Bunge pour remettre en question l’hypothèse précédente. (shrink)

This paper is an introduction to the volume Language, Logic and Mathematics in Schopenhauer. It shows the basic interpretations discussed in Schopenhauer’s research, explains the aims and tasks of Schopenhauer’s philosophy and shows the importance of language, logic and mathematics in Schopenhauer’s system.

Quentin Meillassoux is one of the leading French philosophers of today. His first book, Après la finitude : Essai sur la nécessité de la contingence, (2006, translated into English in 2008), has already become a cult classic. It features a préface by his former mentor, Alain Badiou. One of Meillassoux’s main goals is to rehabilitate the distinction between primary and secondary qualities, typical of pre-Kantian philosophies. Specifically, he claims that mathematics is capable of disclosing the primary qualities of any object (...) : “all those aspects of the object that can be formulated in mathematical terms can be meaningfully conceived as properties of the object in itself”. Here we will use Bunge’s philosophy of mathematics in order to challenge the preceding assumption. (shrink)

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...) abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those definitions. (shrink)

J’ai lu de nombreuses discussions récentes sur les limites du calcul et de l’univers en tant qu’ordinateur, dans l’espoir de trouver quelques commentaires sur le travail étonnant du physicien polymathe et théoricien de la décision David Wolpert, mais n’ont pas trouvé une seule citation et je présente donc ce résumé très bref. Wolpert s’est avéré quelques théoricaux d’impossibilité ou d’incomplétude renversants (1992 à 2008-voir arxiv dot org) sur les limites de l’inférence (computation) qui sont si généraux qu’ils sont indépendants de (...) l’appareil faisant le calcul, et même indépendamment des lois de la physique, ainsi ils s’appliquent à travers les ordinateurs, la physique, et le comportement humain. Ils utilisent la diagonalisation de Cantor, le paradoxe menteur et les worldlines (lignes du monde) pour fournir ce qui peut être le théorème ultime dans Turing Machine Theory, et apparemment fournir des aperçus de l’impossibilité, l’incomplétude, les limites du calcul, et l’univers comme ordinateur, dans tous les univers possibles et tous les êtres ou mécanismes possibles, générant, entre autres, un principe d’incertitude mécanique non quantique et une preuve de monothéisme. Il existe des connexions évidentes à l’œuvre classique de Chaitin, Solomonoff, Komolgarov et Wittgenstein et à l’idée qu’aucun programme (et donc aucun dispositif) ne peut générer une séquence (ou un dispositif) avec une plus grande complexité qu’il ne possède. On pourrait dire que cet ensemble de travaux implique l’athéisme puisqu’il ne peut y avoir d’entité plus complexe que l’univers physique et du point de vue wittgensteinien, « plus complexe » n’a aucun sens (n’a pas de conditions de satisfaction, c’est-à-dire véridique ou test). Même un « Dieu » (c’est-à-dire un « dispositif » avec un temps/ espace et une énergie illimité) ne peut pas déterminer si un « nombre » donné est « aléatoire», ni trouver un certain moyen de montrer qu’une « formule » donnée, un « théorème » ou une « phrase » ou un « dispositif » (tous ces jeux de langage complexes) fait partie d’un « système » particulier. Ceux qui souhaitent un cadre complet à jour pour le comportement humain de la vue moderne de deux systemes peuvent consulter mon livre 'The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019). Ceux qui s’intéressent à plus de mes écrits peuvent voir 'Talking Monkeys --Philosophie, Psychologie, Science, Religion et Politique sur une planète condamnée --Articles et revues 2006-2019 2ème ed (2019) et Suicidal Utopian Delusions in the 21st Century 4th ed (2019) et autres. (shrink)

This article analyses the essential role of language in rational cognition. The approach is functional -- I only look at the effects of the connection between language, reality and thinking. I begin by analysing rational cognition in everyday situations. Then I show that the whole scientific language is an extension and improvement of everyday language. The result is a uniform view of language and rational cognition which solves many epistemological and ontological problems. I use some of them -- the nature (...) of ontology, truth, logic, thinking, scientific theories and mathematics, to demonstrate that the view of language and rational cognition developed in this article is fruitful and effective. (shrink)

HellmanGeoffrey ** and ShapiroStewart. **** Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, RushPenelope and ShapiroStewart, eds. Cambridge University Press, 2019. Pp. iv + 94. ISBN 978-1-108-45643-2, 978-1-108-69728-6. doi: 10.1017/9781108582933.

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of (...) this empirical study suggest that mathematical explanations do occur in research articles published in mathematics journals, as indicated by the occurrence of explanation indicators. When compared with the use of justification indicators, however, the data suggest that justifications occur much more frequently than explanations in scholarly mathematical practice. The results also suggest that justificatory proofs occur much more frequently than explanatory proofs, thus suggesting that proof may be playing a larger justificatory role than an explanatory role in scholarly mathematical practice. (shrink)

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the (...) matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into 10 different values. (shrink)

Dernier Sermon de l’Église du naturalisme fondamentaliste par le pasteur Hofstadter. Comme son travail beaucoup plus célèbre (ou infâme pour ses erreurs philosophiques implacables) Godel, Escher, Bach, il a une plausibilité superficielle, mais si l’on comprend que c’est le scientisme rampant qui mélange les vrais problèmes scientifiques avec les questions philosophiques (c’est-à-dire, les seules vraies questions sont ce que les jeux linguistiques que nous devrions jouer), alors presque tout son intérêt disparaît. Je fournis un cadre d’analyse basé sur la psychologie (...) évolutive et le travail de Wittgenstein (depuis mis à jour dans mes écrits plus récents). Ceux qui souhaitent un cadre complet à jour pour le comportement humain de la vue moderne de deux systemes peuvent consulter mon livre 'The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019). Ceux qui s’intéressent à plus de mes écrits peuvent voir «Talking Monkeys --Philosophie, Psychologie, Science, Religion et Politique sur une planète condamnée --Articles et revues 2006-2019 3e ed (2019) et Suicidal Utopian Delusions in the 21st Century 4th ed (2019) et autres. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

Although Peirce frequently insisted that continuity was a core component of his philosophical thought, his conception of it evolved considerably during his lifetime, culminating in a theory grounded primarily in topical geometry. Two manuscripts, one of which has never before been published, reveal that his formulation of this approach was both earlier and more thorough than most scholars seem to have realized. Combining these and other relevant texts with the better-known passages highlights a key ontological distinction: a collection is bottom-up, (...) such that the parts are real and the whole is an ens rationis, while a continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for Peirce’s topical continuum: rationality, divisibility, homogeneity, contiguity, and inexhaustibility. (shrink)