Metamathematics

Alfred Tarski’s semantic conception of truth is arguably the most influential – certainly, most discussed - modern conception of truth. It has provoked many different interpretations and reactions, some thinkers celebrating it for successfully explicating the notion of truth, whereas others have argued that it is no good as a philosophical account of truth. The aim of this work is to offer a systematic and critical investigation of its nature and significance, based on the thorough explanation of its conceptual, technical as well as historical underpinnings.

The methodological strategy adopted in the thesis reflects the author’s belief that in order to evaluate the import of Tarski’s conception we need to understand what logical, mathematical and philosophical aspects it has, what role they play in his project of theoretical semantics, which of them hang in together, and which should be kept separate. Chapter 2 therefore starts with a detailed exposition of the conceptual and historical background of Tarski’s semantic conception of truth and his method of truth definition for formalized languages, situating it within his project of theoretical semantics, and Chapter 3 explains the formal machinery of Tarski’s truth definitions for increasingly more complex languages. Chapters 4 - 7 form the core of the thesis, all being concerned with the problem of significance of Tarski’s conception. Chapter 4 explains its logico-mathematical import, connecting it to the related works of Gödel and Carnap. Having explained the seminal ideas of the model-theoretic approach to semantics, Chapter 5 tackles the question to what extent Tarski’s ‘The Concept of Truth in Formalized Languages’ (and related articles from the 1930s) anticipates this approach, and what elements might be missing from it. Chapter 6 then deals with the vexed question of its philosophical import and value as a theory of truth, reviewing a number of objections and arguments that purport to show that the method fails as an explanation (explication) of the ordinary notion of truth, and, in particular, that it is a confusion to think that Tarski’s truth definitions have semantic import. Finally, Chapter 7 is devoted to the question whether Tarski’s theory of truth is a robust or rather a deflationary theory of truth.

On the basis of a careful analysis, the thesis aims to substantiate the following view. [A] Tarski’s theory with its associated method of truth definition was primarily designed to serve logico-mathematical purposes. [B] It can be regarded a deflationary theory of a sort, since it completely abstracts from meta- semantical issues concerning the metaphysical or epistemological basis or status of semantic properties. Indeed, [C] this can be interpreted as its laudable feature, since by separating formal (or logico-mathematical) from meta-semantical (or foundational) aspects it usefully divides the theoretical labour to be done in the area of meaning and semantic properties in general. [D] In spite of the fact that Tarski’s conception of truth has this deflationary flavour, the formal structure of its method of truth-definition is quite neutral in that it can be interpreted and employed in several different ways, some of them deflationary, others more robust.

Aξιωματική βάση διατεταγμένο σώμα
μεταμαθηματικά
Αξίωμα της πληρότητας (συνέχειας)
Δεκαδικά αναπτύγματα αριθμών
Ακολουθίες Κωσύ
Άνω φράγμα, ελάχιστο άνω φράγμα
Ιδιότητες της συνέχειας στο R
Αρχιμήδειος νόμος πραγματικών αριθμών
Ο θάνατος των απειροστών
Axiomatic base, ordered field
metamathematics
Axiom of Completeness (continuity)
Decimal expansions of numbers
sequences Cauchy
Upper bound, least upper bound
Properties of continuity in the R
Archimedean law of real numbers
the end of infinitesimals

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

The first theorem of Cantor's 1891 paper introduced the diagonal method by example; namely, as an argument for showing that any list of the reals is necessarily incomplete and concluding the reals are not denumerable. The present paper, one of three on the diagonal method, argues that the diagonal method is not a valid form of argument (i.e., not a valid proof schema). Based on any of three independent refutations presented here, the diagonal argument cannot logically lead to the conclusion(s) alleged for it. We generalize these arguments to apply to the diagonal method and its generalizations and variations, in a second, companion paper [13]. We review the profound and pervasive impact of the diagonal method on meta-mathematics, mathematics, logic and computer science, noting the central position arguments based on the diagonal method hold in the foundations of these subjects, in a third companion paper [14], thereby providing a context for the results presented in both this and the second papers. "And as of 'no-go theorems'… One always has to take the assumptions into consideration, just as the small print in a contract."-Gerard t'Hooft (1999 Nobelist in Physics) [18] I.

Nous reconsidérons les arguments de Zénon d’Élée dits de l’« Achille » et de la « Dichotomie », en réunissant les perspectives de plusieurs disciplines, dont l’histoire de la philosophie ancienne, l’histoire et la philosophie des mathématiques, et la philosophie du temps.
Nous soutenons que les réponses ordinairement données à ces arguments au xx e siècle, d’après lesquelles la mathématique moderne nous donne les moyens de dissoudre l’aporie, sont erronées et s’accompagnent d’une vue faussée sur le problème originel, notamment sur le concept d’infini qu’il implique.
Dans la première partie, nous étudions les sources sur Zénon et sur son contexte de réception, pour établir que l’infini est chez lui second par rapport à l’idée d’inachevabilité, qui découle d’un mode de raisonnement nouveau qu’on peut nommer « itératif indéfini ». Nous examinons comment Zénon a utilisé ce raisonnement dans l’élaboration d’apories dialectiques, et comment l’ensemble des systèmes antiques étaient susceptibles de résoudre ces dernières.
Dans la seconde partie, nous défendons l’aporie zénonienne du mouvement. Nous montrons qu’elle repose sur un principe que nous nommons « principe d’achevabilité », lui-même ancré dans notre intuition temporelle du passage. À travers la considération de la littérature sur les “supertasks”, des problèmes concernant la réalité et la nature du temps, des différents concepts d’infini, et de la réflexion métamathématique, nous montrons à la fois pourquoi les théories de l’infini mathématique sont, de fait, la seule raison conduisant à rejeter le principe d’achevabilité, et pourquoi elles ne sont pas, de droit, en mesure de justifier ce rejet.

Historical analysis and new concepts enable understanding of Riemann's Hypothesis (RH), his zeta function, mathematics, the principles enabling them, and ontological proofs. They enable new, post-modern metamathematics. New terms and definitions support comprehensive proof of new number theory, set theory, and proof theory. They enable deep understanding of the full scope of the historic context and causes of RH. The "Results" section provides basics and the technical context of work on RH and related problems. The "History" section summarizes the background of useful work and attempted solutions. Section 2 enables understanding of the work, enabling metalogic, and the domain of discourse. Section 3 provides comprehensively definitive, deeply explanatory proof of RH, and an integral critique of prior metatheory. It includes proofs of closely related problems, explanations, and how primal numbers may be rapidly, economically located. The new metatheorems of ontology, mathematics, numbers, sets, and proof also support new conjectures and possibilities. Disproof of the P/NP problem is included. Section 4 provides summary comments and predictions, for inspiration and verification.

In Différence et répétition, Deleuze’s ontology is structured by his theory of dialectical Ideas or problems, which draws features from Plato, Kant, and classical calculus. Deleuze unifies these features through a theory of Ideas/problems developed by the mathematician and philosopher Albert Lautman. Lautman worked to explain the nature of the problems or dialectical Ideas mathematics engages and the solutions or mathematical theories endeavouring to understand them. Lautman drew upon Heidegger to do this. This article (1) clarifies Deleuze’s theory of dialectical Ideas/problems by analysing its debts to Lautman and Heidegger and (2) demonstrates a Heideggerian influence on Deleuze via Lautman.

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans.
Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique.
Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie).

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre's theory of pure reflection, the linchpin of the works of Sartre's early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection's relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou's Being and Event in rejecting both ontotheological foundationalisms and constructivist anti-foundationalisms.

ويعتقد عادة أن الاستحالة، وعدم اكتمال، وParaconsistency، وعدم تحديد، العشوائية، والحوسبة، والمفارقة، وعدم اليقين وحدود العقل هي قضايا علمية مادية أو رياضية متباينة وجود القليل أو لا شيء في المشتركه. أقترح أنها مشاكل فلسفية قياسية إلى حد كبير (أي ألعاب اللغة) التي تم حلها في الغالب من قبل فيتغنشتاين أكثر من 80years منذ.

"إن ما نميل إلى قوله في مثل هذه الحالة هو، بطبيعة الحال، ليس فلسفة، ولكنه مادة خام. وهكذا، على سبيل المثال، ما يميل عالم الرياضيات إلى قوله عن موضوعية وواقع الحقائق الرياضية، ليس فلسفة الرياضيات، بل شيء للعلاج الفلسفي". فيتغنشتاين بي 234

"يرى الفلاسفة باستمرار طريقة العلم أمام أعينهم ويميلون بشكل لا يقاوم إلى طرح الأسئلة والإجابة عليها بالطريقة التي يفعلبها العلم. هذا الاتجاه هو المصدر الحقيقي للميتافيزيقيا ويقود الفيلسوف إلى الظلام الكامل".  فيتغنشتاين

أقدم ملخصاً موجزاً لبعض النتائج الرئيسية لاثنين من أبرز الطلاب في السلوك في العصر الحديث، لودفيغ فيتغنشتاين وجون سيرل، على البنية المنطقية للتعمد (العقل واللغة والسلوك)، مع الأخذ كنقطة انطلاق لي اكتشاف فيتغنشتاين الأساسي - أن جميع المشاكل "الفلسفية" حقاهي نفسها - الالتباسات حول كيفية استخدام اللغة في سياق معين، وبالتالي فإن جميع الحلول هي نفسها - النظر في كيفية استخدام اللغة في السياق في القضية بحيث حقيقتها الشروط (شروط الرضا أو COS) واضحة. المشكلة الأساسية هي أن المرء يمكن أن يقول أي شيء، ولكن لا يمكن للمرء أن يعني (الدولة واضحة كوس ل) أي كلام ومعنى تعسفي غير ممكن إلا في سياق محدد جدا.

أنا تشريح بعض كتابات عدد قليل من المعلقين الرئيسيين على هذه القضايا من وجهة نظر Wittgensteinian في إطار المنظور الحديث للنظاميين الفكر (شعبية باسم 'التفكير السريع، والتفكير بطيئة')، وتوظيف جدول جديد من التعمد وتسميات النظم المزدوجة الجديدة.  أنا تبين أن هذا هو سياحي قوي لوصف الطبيعة الحقيقية لهذه القضايا العلمية أو المادية أو الرياضية المفترضة التي يتم التعامل معها على أفضل وجه حقا كمشاكل فلسفية قياسية لكيفية استخدام اللغة (ألعاب اللغة في فيتغنشتاين المصطلحات).

هو زعمي أن جدول التعمد (العقلانية، العقل، الفكر، اللغة، الشخصية وما إلى ذلك) التي تبرز بشكل بارز هنا يصف أكثر أو أقل دقة، أو على الأقل بمثابة سياحية ل، وكيف نفكر وتصرف، وبالتالي فإنه لا يشمل مجرد فلسفة وعلم نفس، ولكن كل شيء آخر (التاريخ والأدب والرياضيات والسياسة الخ). لاحظ بشكل خاص أن التعمد والعقلانية كما أنا (جنبا إلى جنب مع سيرل، فيتغنشتاين وغيرها) عرض ذلك، ويشمل كل من النظام اللغوي التداولي الواعي 2 واللاوعي الآلي النظام قبل اللغوية 1 الإجراءات أو ردود الفعل.

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S results in true sentences of L. For two reasons, this theorem is relevant to issues relative to Quine's substitutional definition of logical truth. First, it makes it possible for Quine to reply to widespread objections raised against his account (the lexicon-dependence problem and the cardinality-dependence problem). These objections purport to show that Quine's account overgenerates: it would count as logically true sentences which intuitively or model-theoretically are not so. Second, since this theorem is a crucial premise in Quine's proof of the equivalence between his substitutional account and the model-theoretic one, it enables him to show that, from a metamathematical point of view, there is no need to favour the model-theoretic account over one in terms of substitutions. The purpose of that essay is thus to explore the philosophical bearings of the Löwenheim-Hilbert-Bernays theorem on Quine's definition of logical truth. This neglected aspect of Quine's argumentation in favour of a substitutional definition is shown to be part of a struggle against the model-theoretic prejudice in logic. Such an exploration leads to reassess Quine's peculiar position in the history of logic.

Nous reconsiderons les arguments de Zenon d’Elee dits de l’« Achille » et de la « Dichotomie », en reunissant les perspectives de plusieurs disciplines, dont l’histoire de la philosophie ancienne, l’histoire et la philosophie des mathematiques, et la philosophie du temps. Nous soutenons que les reponses ordinairement donnees a ces arguments au XXe siecle, d’apres lesquelles la mathematique moderne nous donne les moyens de dissoudre l’aporie, sont erronees et s’accompagnent d’une vue faussee sur le probleme originel, notamment sur le concept d’infini qu’il implique. Dans la premiere partie, nous etudions les sources sur Zenon et sur son contexte de reception, pour etablir que l’infini est chez lui second par rapport a l’idee d’inachevabilite, qui decoule d’un mode de raisonnement nouveau qu’on peut nommer « iteratif indefini ». Nous examinons comment Zenon a utilise ce raisonnement dans l’elaboration d’apories dialectiques, et comment l’ensemble des systemes antiques etaient suscepti...

This paper presents SRVB, which consists of four approaches for knapsack-based public-key cryptosystems, the three first authored by prof. Daniel Santana Rocha and Yuri Villas Boas, and the other by the latter alone. It provides some evidences favorable to the nonexistence of trivial cryptanalysis (like some trivial adaptation of Shamir attack) against the proposed cryptosys-tems and also mentions a line of research of lattice-based signature schemes using some of these four approaches. We also make the case for the market of mathematical and scientific conjectures derivatives as a way to deal with the uncertainty inherent to the progress of Mathematics and Science and present our own contribution to this market: a cryptanalysis contest offering a prize for breaking SRVB.

INTRODUCING TARSKI'S 1983 LSM.docx
Editor's introduction revised edition.  Logic, Semantics, Metamathematics. Alfred Tarski, pages xv–xxvii.
I wish to express here my most genuine and cordial gratefulness to Professor John Corcoran for his work related to the publication of the present volume. The new edition of Logic, Semantics, Metamathematics is almost exclusively due to his initiative. In serving as the editor, he fulfilled his task with great fervor and enthusiasm. We collaborated in searching out and trying to remove various defects and weak spots of earlier texts. The new indices at the end of this volume, prepared by Corcoran, are much superior to those in the first edition and hopefully will facilitate the study and the use of the work. What is more valuable, he has provided the volume with his own introduction.--ALFRED TARSKI, Berkeley, California, January 1982
Corcoran’s 1983 edition of Tarski’s LOGIC, SEMANTICS, METAMATHEMATICS is available free. http://library1.org/_ads/DE3C6E8E30D380DC39D1057DB4BD8EE1
CORCORAN ON TARSKI
https://www.academia.edu/s/179246c7cd/corcoran-on-tarski-december-2016-pdf?source=link
Corcoran’s publications that interpret, criticize, explain, or build on Alfred Tarski’s theories and results.

Topological challenges to the geometry of globalization and polarization
Space-time crystals as fundamental to comprehension of global order
Via sphere, torus and hyperbola to space-time crystals of governance?
Comprehension of requisite complexity through game-ball design?
Contrasting forms of coherence framed by positive and negative curvature
Space-time crystals as space-time polyhedra fundamental to appropriate organization
Imagining complementary images of the shape of civilization
Logic of renormalization as enabling flying metaphorically understood?

This paper proposes two distinct types of imaginary (im) infinities ("imfinities") in mathematics and meta-mathematics (including meta-geometry), emphasizing the unlimited "diversity" of zero and infinity, with far-reaching implications in all these domains, but also in math-related domains like modern physics, including the help in redefining the basics of Einstein's General relativity theory (GRT), quantum field theory (QFT), superstring theories (SSTs) and M-theory (MT).

Keywords (including a list of main abbreviations): imaginary (im) infinities (“imfinities”), mathematics, metamathematics, metageometry, zero, infinity; Einstein’s General relativity theory (GRT); quantum field theory (QFT); superstring theories (SSTs); M-theory (MT);

#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

Hal ini sering berpikir bahwa kemustahilan, ketidaklengkapan, Paraconsistency, Undecidability, Randomness, komputasi, Paradox, ketidakpastian dan batas alasan yang berbeda ilmiah fisik atau matematika masalah memiliki sedikit atau tidak ada dalam Umum. Saya menyarankan bahwa mereka sebagian besar masalah filosofis standar (yaitu, Permainan bahasa) yang sebagian besar diselesaikan oleh Wittgenstein lebih dari 80years yang lalu.

"Apa yang kita ' tergoda untuk mengatakan ' dalam kasus seperti ini, tentu saja, bukan filsafat, tetapi bahan baku. Jadi, misalnya, apa yang seorang matematikawan cenderung mengatakan tentang objektivitas dan realitas fakta matematika, bukan filsafat matematika, tetapi sesuatu untuk pengobatan filosofis. " Wittgenstein PI 234

"Filsuf terus melihat metode ilmu di depan mata mereka dan tak tertahankan tergoda untuk bertanya dan menjawab pertanyaan dalam cara ilmu tidak. Kecenderungan ini adalah sumber nyata metafisika dan memimpin filsuf menjadi gelap gulita. "  Wittgenstein

Aku memberikan ringkasan singkat dari beberapa temuan utama dari dua siswa yang paling terkemuka perilaku zaman modern, Ludwig Wittgenstein dan John Searle, pada struktur Logis intensionality (pikiran, bahasa, perilaku), mengambil sebagai titik awal Penemuan fundamental Wittgenstein – bahwa semua masalah ' filosofis ' adalah sama — kebingungan tentang bagaimana menggunakan bahasa dalam konteks tertentu, sehingga semua solusi sama — melihat bagaimana bahasa dapat digunakan dalam konteks yang menjadi masalah sehingga kebenaranNya kondisi (kondisi kepuasan atau COS) jelas. Masalah dasar adalah bahwa seseorang dapat mengatakan apa-apa, tetapi orang tidak dapat berarti (negara yang jelas cos untuk) sembarang ucapan dan makna hanya mungkin dalam konteks yang sangat spesifik.

Saya membedah beberapa tulisan dari beberapa komentator utama pada isu ini dari sudut pandang Wittgensteinian dalam kerangka perspektif modern dari dua sistem pemikiran (Dipopulerkan sebagai ' berpikir cepat, berpikir lambat '), mempekerjakan meja baru intensionality dan baru sistem ganda nomenklatur.  Saya menunjukkan bahwa ini adalah heuristik yang kuat untuk menggambarkan sifat sebenarnya dari hal ini ilmiah, fisik atau matematika masalah yang benar-benar terbaik didekati sebagai masalah filosofis standar bagaimana bahasa yang akan digunakan (permainan bahasa di Wittgenstein's terminologi).

Ini adalah pendapat saya bahwa tabel intensionality (rasionalitas, pikiran, pikiran, bahasa, kepribadian dll) yang fitur mencolok di sini menggambarkan lebih atau kurang akurat, atau setidaknya berfungsi sebagai heuristic untuk, bagaimana kita berpikir dan berperilaku, dan sehingga mencakup tidak hanya filsafat dan psikologi, tetapi segala sesuatu yang lain (sejarah, sastra, matematika, politik dll). Perhatikan terutama bahwa intensionalitas dan rasionalitas sebagai I (bersama dengan Searle, Wittgenstein dan lain-lain) melihatnya, mencakup baik sistem linguistik pertimbangan sadar 2 dan tidak disadari otomatis sistem prelinguistik 1 tindakan atau refleks.

In this first part of a series of articles we discuss the torsion geometry of biology, physics, cognition and perception. We discuss the relations with the non-linear morphomechanics of organisms and the integration of the body's chiralities. We present the connections with the topology of neural networks as heterarchies and with multi-loci logics as a basis for the HyperKlein Bottle logophysics in genomics, systems theory and cognition. We discuss the link to the relational paradigm for biology, systems theory and physics. We introduce the relations of torsion with self-reference as a generating principle, and as a protosemiotic agency. We present the relations of heterarchies to cognition and systems theory. We discuss the relation between chaotic dynamics, blown-up systems, the non-linear logophysics of the KB and the paradoxical structure of the real numbers. In particular, we elaborate on the breakdown of the continuum hypothesis, the origin of singularities, particularly with respect to General Relativity, and the non-linear elasticity of biological development. We discuss the relation of computational error and the KB logophysics. We present the heterarchical logophysics and the constitution of the self as a model and of a sense or reality as a primary epistemic state further promoted to a dual ontology, which we discuss in terms of the foundations of mathematics in set theory and topoi, and the measurement problem in quantum mechanics. We introduce a fundamental Inside/Outside image-schema in cognitive semantics and discuss its pervasive role in cognition and the framing of the sciences, and give several examples of this. We introduce logical systems as formalizations of ontologies, their role in cognition, the neurosciences, cybernetics and biocomputation, and discuss their setting in terms of the positional HyperKlein Bottle logics and classical dual logic as a particular case. We develop a phenomenological approach to unified science. We discuss the time operator. We discuss the relations of the present logophysics to R.Rosen's proposal of an approach to systems theory based on the calculus of differential forms and with regards to the topological approach to theoretical physics and Chemical Topology. We discuss the relation between the present logophysics and Spencer-Brown's Laws of Form.

यह आमतौर पर सोचा जाता है कि असंभवता, अपूर्णता, Paraconsistency, अनिर्णितता, Randomness, Computability, विरोधाभास, अनिश्चितता और कारण की सीमा अलग वैज्ञानिक शारीरिक या गणितीय मुद्दों में कम या कुछ भी नहीं कर रहे हैं आम. मेरा सुझाव है कि वे काफी हद तक मानक दार्शनिक समस्याओं (यानी, भाषा का खेल) जो ज्यादातर 80years पहले Wittgenstein द्वारा हल किए गए थे.

"क्या हम 'इस तरह के एक मामले में कहने के लिए' कर रहे हैं, ज़ाहिर है, दर्शन नहीं है, लेकिन यह अपने कच्चे माल है. इस प्रकार, उदाहरण के लिए, क्या एक गणितज्ञ वस्तुपरकता और गणितीय तथ्यों की वास्तविकता के बारे में कहने के लिए इच्छुक है, गणित का दर्शन नहीं है, लेकिन दार्शनिक उपचार के लिए कुछ है। विटगेनस्टीन पीआई 234

"Philosophers लगातार उनकी आँखों के सामने विज्ञान की विधि को देखने और irresistibly पूछने के लिए और जिस तरह से विज्ञान करता है में सवालों के जवाब परीक्षा कर रहे हैं. यह प्रवृत्ति तत्वमीमांसा का वास्तविक स्रोत है और दार्शनिक को पूर्ण अंधकार में ले जाती है।  विटगेनस्टाइन

मैं आधुनिक समय के व्यवहार के सबसे प्रतिष्ठित छात्रों में से दो के प्रमुख निष्कर्षों में से कुछ का एक संक्षिप्त सारांश प्रदान करते हैं, लुडविग Wittgenstein और जॉन Searle, जानबूझकर की तार्किक संरचना पर (मन, भाषा, व्यवहार), मेरे प्रारंभिक बिंदु के रूप में ले रही Wittgenstein की मौलिक खोज - कि सभी सही मायने में 'लोकप्रिय' समस्याओं को एक ही हैं एक विशेष संदर्भ में भाषा का उपयोग करने के बारे में भ्रम, और इसलिए सभी समाधान एक ही हैं-कैसे भाषा के संदर्भ में इस मुद्दे पर इस्तेमाल किया जा सकता है पर देख इतना है कि इसकी सच्चाई शर्तें (संतोष या COS की शर्तें) स्पष्ट हैं. मूल समस्या यह है कि एक कुछ भी कह सकते हैं, लेकिन एक मतलब नहीं कर सकते (राज्य स्पष्ट COS के लिए) किसी भी मनमाने ढंग से कथन और अर्थ केवल एक बहुत ही विशिष्ट संदर्भ में संभव है.

मैं सोचा की दो प्रणालियों के आधुनिक परिप्रेक्ष्य के ढांचे में एक Wittgensteinian दृष्टिकोण से इन मुद्दों पर प्रमुख टिप्पणीकारों में से कुछ के कुछ लेखन विच्छेदन (के रूप में लोकप्रिय 'तेजी से सोच, धीमी गति से सोच'), की एक नई मेज रोजगार जानबूझकर और नई दोहरी प्रणाली नामकरण.  मैं बताता हूँ कि यह इन putative वैज्ञानिक, शारीरिक या गणितीय मुद्दों जो वास्तव में सबसे अच्छा कैसे भाषा का इस्तेमाल किया जा रहा है के मानक दार्शनिक समस्याओं के रूप में संपर्क कर रहे हैं की सही प्रकृति का वर्णन करने के लिए एक शक्तिशाली heuristic है (Witgenstein में भाषा का खेल शब्दावली)

यह मेरा तर्क है कि जानबूझकर की मेज (तर्कसंगतता, मन, सोचा, भाषा, व्यक्तित्व आदि) है कि यहाँ प्रमुखता से सुविधाएँ अधिक या कम सही वर्णन करता है, या कम से कम के लिए एक heuristic के रूप में कार्य करता है, हम कैसे सोचते हैं और व्यवहार करते हैं, और इसलिए यह शामिल नहीं केवल दर्शन और मनोविज्ञान, लेकिन सब कुछ (इतिहास, साहित्य, गणित, राजनीति आदि). ध्यान दें कि विशेष रूप से जानबूझकर और तर्कसंगतता के रूप में मैं (Searle, Wittgenstein और अन्य लोगों के साथ) यह देखने के लिए, दोनों सचेत विचार विमर्श भाषाई प्रणाली 2 और बेहोश स्वचालित prelinguistic प्रणाली 1 कार्रवाई या सजगता भी शामिल है.

This paper will focus on contradictions found between the Godel completeness and first incompleteness theorem in relation to Principia Mathematica (PM).

Analysis accepts the first result of the proof that shows that an undecideable proposition can exist in PM. However, the refutation finds that an undecideable proposition is neither true nor false and conflicts with the completeness theorem, and therefore, cannot be a theorem. As a non-theorem, an undecideable proposition does not meet the necessary requirement of incompleteness. However, a weak case of incompleteness is asserted and for an undecideable proposition, falsity is not provable. Hence, the proof can be refuted and the first incompleteness theorem not proven.

In the vacancy, an axiom is proposed  'The Completeness or Incompleteness Axiom' and a strict definition of incompleteness elaborated.

It has been shown that completeness proofs were advanced for PM and also that incompleteness applies to PM, giving rise to a contradiction. To solve the contradiction, many strong arguments are presented to show that PM is complete, while the incompleteness of PM relies exclusively on the result of Godel " s first incompleteness theorem. It has been found conclusively that Principia Mathematica is complete.

Godel's first incompleteness theorem has been falsified on three points:
i) The proof by Godel can be refuted and the first incompleteness theorem not proven.
ii) The case for and against incompleteness and completeness in PM found conclusively that PM is complete.
iii) If Godel's proof methodology is valid then many basic mathematical and logical systems could be proven incomplete.

 In his paper "Finitism" (1981), W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA.

Historical analysis, new discoveries, and explanation enable understanding of Riemann's Hypothesis (RH), his zeta function, and the principles enabling them. The work provides comprehensively definitive, unconditional proofs of Riemann's hypothesis, Goldbach's conjecture, the 'twin primes' conjecture, the Collatz conjecture, the Newcomb-Benford theorem, and the Quine-Putnam Indispensability thesis. The proofs validate holonomic metamathematics & meta-ontology, new number theory, new proof theory, new philosophy of logic, and unconditional disproof of the P/NP problem. The proofs, metatheory, and definitions are also confirmed and verified with graphic proof of intrinsic enabling and sustaining principles of reality.

Differentiation arithmetic is a principal and accurate technique for the computational evaluation of derivatives of first and higher order. This article aims at recasting real differentiation arithmetic in a formalized theory of dyadic real differentiation numbers that provides a foundation for first and higher order automatic derivatives. After we set the stage by putting on a systematic basis certain fundamental notions of the algebra of differentiation numbers, we begin by setting up an axiomatic theory of real differentiation arithmetic, as a many-sorted extension of the theory of a continuously ordered field, and then establish the proofs for its consistency and categoricity. Next, we carefully construct the algebraic system of real differentiation arithmetic, deduce its fundamental properties, and prove that it constitutes a commutative unital ring. Furthermore, we describe briefly the extensionality of the system to an interval differentiation arithmetic and to an algebraically closed commutative ring of complex differentiation arithmetic. Finally, a word is said on machine realization of real differentiation arithmetic and its correctness, with an addendum on how to compute automatic derivatives of first and higher order.

1971. Notes on a Semantic Analysis of Variable Binding Term Operators (Co-author John Herring), Logique et Analyse 55, 646–57.  MR0307874 (46 #6989). RG

A variable binding term operator (vbto) is a non-logical
constant, say v, which combines with a variable y and a formula
F containing y free to form a term (vy:F) whose free
variables are exact ly those of F, excluding y.

Kalish-Montague proposed using vbtos to formalize definite descriptions, set abstracts {x: F}, minimalization in recursive function theory, etc. However, they gave no sematics for vbtos. Hatcher gave a semantics but one that has flaws. We give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture was later proved independently by the authors and by Newton da Costa.

The expression (vy:F) is called a variable bound term (vbt). In case F has only
y free, (vy:F) has the syntactic propreties of an individual
constant; and under a suitable interpretation of the language
vy:F) denotes an individual. By a semantic analysis of vbtos
we mean a proposal for amending the standard notions of (1)
"an interpretation o f a first -order language" and (2) " the denotation
of a term under an interpretation and an assignment",
such that (1') an interpretation o f a first -order language associates
a set-theoretic structure with each vbto and (2') under
any interpretation and assignment each vb t denotes an individual.

2008. Meanings of Form. Manuscrito 31, 223–266. P
Abstract
The expressions ‘form’, ‘structure’, ‘schema’, ‘shape’, ‘pattern’, ‘figure’, ‘mold’, and related locutions are used in logic both as technical terms and in metaphors. This paper juxtaposes, distinguishes, and analyses uses of such expressions by logicians. No similar project has been attempted previously. After establishing general terminology, we present a variant of traditional usage of the expression ‘logical form’ followed by a discussion of the usage found in the two-volume Chateaubriand book Logical Forms (2001 and 2005)—the most comprehensive work on the subject ever written and in many ways the focus of this paper.
Key Words: morphology, character, sense, sentence, proposition, form, structure, schema, alternative constituent, type, token

ALTERNATIVE CONSTITUENT EXERCISES IN USE OF ‘FORM’.
https://www.academia.edu/11784901/New_formats_for_presenting_and_generating_language_data
Q1. Some Christians believe that (Satan * God) appeared to (Eve * Mary) (as * in the form of) a (snake * dove).
Q2. Some Christians believe that (Satan * God) came to (Eve * Mary) in the (form * appearance* manifestation) of a (snake * dove).
Q3. Some Christians believe that (Christ * God) came to (Christopher * Jacob) in the (form * appearance* manifestation) of a (child * man * traveler * wrestler).
Q4. Some Christians believe that (Christ * God) came to (Christopher * Jacob) (as *in the form of) a (child * man * traveler * wrestler).
Q5. Some Christians believe that Christ’s (body * blood) is consumed by communicants (as * in the form of) (bread * wine * wafers * grape juice).
Q6. (The theophanies * A theophany) (are appearances of gods* is an appearance of a god) in visible form.
Q7. (Christophanies * A Christophany) (are appearances * is an appearance) of Christ in visible forms.
Q8. Every (even * odd) number (is of* has) the form ( 2n * 2n + 1).

Q9. Every number (of* having) the form n(n + 1) (is of* has) the form ( 2n * 2n + 1).

Three contrasting approaches to the epistemology of argument are presented. Each one is naturalistic, drawing upon successful practices as the basis for epistemological virtue. But each looks at very different sorts of practices and they differ greatly as to the manner with which relevant practices may be described. My own contribution relies on a metamathematical reconstruction of mature science, and as such, is a radical break with the usual approaches within the theory of argument.

Este voluminoso libro es una recopilación de algunos trabajos del matemático y filósofo William Boos (1943-2014), situados en la zona fronteriza entre matemáticas y filosofía. El estilo que comparten todos los trabajos es un ir y venir alegórico, en el que, por un lado, la metamatemática se usa como metáfora para entender los grandes temas de la historia de la filosofía y, por otro lado, los problemas filosóficos sirven para iluminar el sentido de los resultados de la metamatemática del siglo XX.

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive.  But if so, then given various fundamental results in mathematical logic and the theory of computation, the set of humanly known mathematical truths is axiomatizable.  Furthermore, given Godel's First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics must be either inconsistent or incomplete.  Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine.  We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle.  So the Lucas-Penrose argument cannot be sound.

This research aims at studying engineering students’ perceptions of their mathematics courses. We present the methodology of data collection, the main themes that the questionnaire investigates and the results. The population on which we base this study are partners in two Tempus projects, MetaMath in Russia and MathGeAr in Georgia and Armenia. Pedro Lealdino Filho, Christian Mercat, Mohamed El-Demerdash. MetaMath and MathGeAr Projects: Students’ perceptions of mathematics in engineering courses.

In Différence et répétition, Deleuze's ontology is structured by his theory of dialectical Ideas or problems, which draws features from Plato, Kant, and classical calculus. Deleuze unifies these features through a theory of Ideas/problems developed by the mathematician and philosopher Albert Lautman. Lautman worked to explain the nature of the problems or dialectical Ideas mathematics engages and the solutions or mathematical theories endeavouring to understand them. Lautman drew upon Heidegger to do this. This article (1) clarifies Deleuze's theory of dialectical Ideas/problems by analysing its debts to Lautman and Heidegger and (2) demonstrates a Heideggerian influence on Deleuze via Lautman.

This piece isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments on physics. It's about deflating other people's applications of Gödel's theorems to physics.

Indeed Gödel himself wasn't too keen on applying his findings to physics – especially to quantum physics. According to John D. Barrow:

“Godel was not minded to draw any strong conclusions for physics from his incompleteness theorems. He made no connections with the Uncertainty Principle of quantum mechanics.”

More broadly, Gödel's theorems may not have the massive and important applications to physics which some philosophers and scientists believe they do have.

'explain' is a game of exploring space(s); #Logica is what it is programmed in.  It is a language game of sorts, with a metaphysical twist and a lot of computation. It's a virtual machine of 2 bits fitting into 117 lines of shell script.  It's an infinite space game in a tiny little space using only natural language consisting of just 3 words: tri swap and not.  With that, you can say anything...if you define 'meta'.  But that's up to you...

Good luck.

Este artigo é uma versão traduzida, revisada e expandida do artigo de G. J. Chaitin publicado em língua inglesa no American Philosophical Association Newsletter on Philosophy and Computers em 2009, uma palestra proferida em junho de 2008 na Universidade de Roma "Tor Vergata", cujos trechos preservados aparecem aqui em primeira pessoa. Discutimos as ideias de Leibniz sobre a complexidade, e os desenvolvimentos três séculos mais tarde.

Palavras-chave: Complexidade. Incompletude. Leibniz. Teoria da informação algorítmica (TIA). H. Weyl.