Philosophy Of Mathematics

This was a talk at "Theology in Mathematics?" (Kraków, Poland, June 8-10, 2014).
Abstract: A deep conviction of the majority of mathematicians on the brink of the 20th century was that mathematics is or at least must be infallible, consistent, rigorous, certain, necessary and universal, as well as free and applicable to the world without restriction. This common belief needs to be explained. This very conviction or belief was responsible for a heated argument provoked by set-theoretic paradoxes which were interpreted as an indication of the foundational crisis of mathematics (Grundlagenkrise der Mathematik). The crisis caused the emergence of diverse programs for the foundations of mathematics. Those programs gave birth, on the one hand, to the contemporary philosophy of mathematics and, on the other, to the formation of a new research field within mathematics: mathematical logic and the foundations of mathematics. This paper proposes the hypothesis that mathematics, seen from the foundationalist perspective, served at the time as a substitute for theology. According to this approach, philosophy of mathematics mediates an impact between theology and mathematics. To confirm this hypothesis I consider the prehistory of such an absolutist account of mathematics and pay a special attention to theological and quasi-theological ideas of the key figures of the three main foundationalist programs (logicism, intuitionism and formalism).

The thesis is an investigation into the logical pluralism debate, aiming to understand how the philosophical commitments sustaining each side to the debate connects to more general issues connected to the foundations of logic. My investigation centers on the following three notions: (1) Epistemic justification, (2) The metaphysical "ground" for logical truth, and (3) Normativity. Chapter 1 traces the monistic and pluralistic conception of logic back to its philosophical/mathematical roots, which we find in the writings of Rudolf Carnap and Gottlob Frege. I argue that logical pluralism - in its more plausible, epistemic (rather than ontic) form - was enabled by the semantic shift which Carnap seems to have anticipated and that, from a conventionalist perspective, his 'Principle of Tolerance' follows as a consequence of that shift. Chapter 2 concerns the issues ensuing from Willard V. O Quine’s critique of Carnap's conventionalism, which had a devastating effect for his foundationalist project. The aim is in particular to address the issue of meaning-variance, a crucial assumption for the conventionalist approach to pluralism. In chapter 3, I present another framework for pluralism, due to Stewart Shapiro’s [2014] ‘modelling’ conception of logic, according which logic is conceived as a mathematical model of natural language. Shapiro argues that our concept of logical consequence is vague and in need of a sharpening to attain a fixed meaning. Pluralism follows from there being two or more equally "correct" such sharpenings; i.e., relative to our theoretical aims. I argue that the modelling-conception is the best way to approach a justification of basic logical laws. However, since that conception also grounds Timothy Williamson’s [2017] argument for monism, I argue that the conception ultimately fails to establish logical pluralism. Since both Williamson and Shapiro take a pragmatic approach to justification, I conclude that the question of pluralism does not turn on epistemological commitments (i.e., on (1)), and suggest instead that it is a matter of (2), i.e., of one's conception of the "ground" for logical truth.

A selection of some of Novalis’s most important reflections on mathematics from the years 1798-1800. Introduced and translated into English by David W. Wood.

In: Symphilosophie: International Journal of Philosophical Romanticism 3 (2021): 273-290
https://symphilosophie.com/issue-4-2022/2/

"Michel Weber et Pierfrancesco Basile (sous la direction de), Chromatikon III. Annuaire de la philosophie en procès — Yearbook of Philosophy in Process, Louvain-la-Neuve, Presses universitaires de Louvain, 2007. (300 p. ; ISBN 978-2-8746-3083-5 ; 23 € ; http://www.i6doc.com/fr/livre/?GCOI=28001100028210 ; http://www.i6doc.com/en/livre/?GCOI=28001100028210; http://www.i6doc.com/doc/chromatikon3 & http://www.i6doc.com/doc/chromatikon3-eng)

Table
Michel Weber, Avant-Propos & Foreword
I. Séminaires de recherche — Research Seminars
Pierfrancesco Basile, The Reality of Forms
Jean-Marie Breuvart, Husserl et Whitehead, sur l’Intentionnalité
Didier Debaise, Life and Orders
Ronny Desmet, The Rebirth of the Ether
Guillaume Durand, Les objets perceptuels du Professeur Whitehead
Franck Robert, Penser la couleur avec Process and Reality
Xavier Verley, Sentir : Whitehead et Merleau-Ponty
II. Etudes critiques — Critical Studies
Jean-Pascal Alcantara, Le rôle des mathématiques dans la genèse whiteheadienne
Une Sœur de Saint-Jean, Un Principe que les religions appellent « Dieu »
Laura Ilinescu, Vérité et pouvoir dans le Léviathan de Thomas Hobbes
Marc Maesschalck, Raison et événement chez Jean Ladrière
Pierre Montebello, Nietzsche, une philosophie de la nature
Michel Weber, Contact Made Vision: The Apocryphal Whitehead
III. Comptes rendus — Critical Reviews
Stéphan Galetic : James, Essais d’empirisme radical
Ronny Desmet : Lacoste Lareymondie, Une philosophie pour la physique quantique
IV. Informations réticulaires — Reticular News
Discussion: David Ray Griffin, A Reply to Alexander Cockburn
Notes : Peter H. Hare, On Gertrude Stein and Mathematical Logic
Interview : Carlo Sini, Luca Gaeta and Luca Vanzago, The Milano School
Tribute: Leemon McHenry, T.L.S. Sprigge, the Last Idealist
Séminaires — Seminars
Publications
Table des matières — Table of Contents
"

The paper deals with the aesthetic and religious dimensions of mathematics. These dimensions are considered as closely connected, though reciprocally non-reducible. “Mathematical beauty” is already firmly established as a term in the philosophy of mathematics. Here, an attempt is made to bring forward two additional candidates: “mathematical sublime” and “numinous mathematics”. The last one is meant to designate the recognition of some mathematical practices as inspiring anticipation of the meeting with the divine reality or producing a feeling of its presence. The first one is used here to designate the related feelings in disguise, i.e., being reinterpreted or transferred from the straightforwardly religious to the aesthetic sphere. Taking Kant’s theory of the sublime as a starting point, the paper introduces a related account of it that treats mathematical beauty through mathematical sublimity as a more fundamental category. Within this account, religious experience, the aesthetics of the sublime and mathematical practice are closely interlinked through an appropriate interpretation of the idea of the infinite. Both mathematical and art symbolism are seen as an endeavour to represent the infinite within the finite, which correlates well with the definition of mathematics as “the science of the infinite” (Hermann Weyl).Keywords: philosophy of mathematics, philosophy of mathematical practice, mathematical aims, mathematical beauty, the sublime, numinous, the infinite, symbolism.

Mini-Curso dividido em três partes:

1. Introdução
Em que discutiremos o diagnóstico de Badiou sobre o século vinte e o mundo contemporâneo, bem como as razões pelas quais a filosofia parece ter se tornado impossível hoje, localizando o próprio Badiou no interior desse mapeamento histórico e conceitual.

2. Filosofia e Ontologia
Em que partiremos das críticas à ontologia clássica para então demonstrar como a decisão de equacionar ontologia e matemática responde a elas adequadamente. Em seguida, analisaremos alguns conceitos da teoria dos conjuntos a fim de entender as consequências filosóficas da decisão de Badiou.

3. Verdade e verdades:
Em que as críticas relativistas ao tema da verdade serão tomadas como condições para uma nova doutrina da verdade. Veremos que a verdade é entendida como uma capacidade imanente a diferentes campos do pensamento, mas não à própria filosofia, cuja tarefa é pensar a compossibilidade das verdades amorosas, científicas, artísticas e políticas de uma época.

An Otherwise Non-distinct Geneaology:"Mathematical Logic" as seen from the Perspective of Alain Badiou's Philosophical System: The aim of this essay is to map eleven transformations of mathematical logic in Badiou's philosophical system. These transformations otherwise organize the development of logic and mathematics within the context of French epistemology. We argue that to grasp Badiou's ontological argument, especially as set out in Being and Event, it is necessary to adopt a perspective critical of logicism and logical positivism. In other words, as naive set theory is created by Georg Cantor, it is independent of the project of logicism stemming from Russell and Wittgenstein's work on Frege. Indeed, set theory should not be reduced to the field of logic, but be understood as mathematical per se. This is the line of development philosophy of mathematics, epistemology and logic took within the structuralist background of French philosophy.

This is a rough and ready mix of finished sections and notes for a master-class on Alain Badiou's set theory ontology focusing in particular on the initial Meditations of Being and Event and attempting to identify different options open to Badiou in his construction of an ontology of multiplicity. It was used as the basis for a workshop presentation along with a powerpoint - which, incidentally, Badiou himself attended!

We live in an age of movement. More than at any other time in history , people and things move longer distances, more frequently, and faster than ever before. All that was solid melted into air long ago and is now in full circulation around the world like dandelion seeds adrift on turbulent winds. We find ourselves, in the early twenty-first century, in a world where every major domain of human activity has become increasingly defined by motion. 1 We have entered a new historical era defined in large part by movement and mobility and are now in need of a new historical on-tology appropriate to our time. The observation that the end of the twentieth century and the beginning of the twenty-first was marked by an increasingly " liquid " and " mobile modernity " is now something widely recognized in the scholarly literature at the turn of the century. 2 Today, however, our orientation to this event is quite different. Almost twenty years into the twenty-first century we now find ourselves situated on the other side of this heralded transition. The question that confronts us today is thus a new one: how to fold all that has melted back up into new solids. 3

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting.

Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. I provide a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory.

Hilbert's program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862-1943), and was pursued by him and his collaborators at the University of Gottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert's proposal called for a new foundation of mathematics based on two pillars: the axiomatic method and finitary proof theory.

L'apport des connaissances dans la science est soumis à l’expérience. Dans la vision de Pascal les anciens les ont trouvées seulement ébauchées, et nous les laisserons à ceux qui viendront après nous en un état plus accompli que nous ne les avons reçues se voyant ainsi infinies en l’étendue de leurs recherches. Cet écrit propose d’explorer la tension philosophique autour de l’expérience de simulation et de l’expérience de pensée qui créent et explorent tous deux des mondes hypothétiques.

We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (. Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66].

Russell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero.

Although Russell gives none, proof is obtainable from page 270 of De Morgan 1831.
REFERENCE LINKS

Russell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381.
https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link
https://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf
Semiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3.
(Co-author: Kevin Tracy) https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC
HOW CAN THIS BE IMPROVED?

This paper will utilize advanced computational mathematics and Quantum statistics for an algorithmic view on our universe. We will look at the theoretical nature of wave propagation, different Quantum states and propose a mathematical analysis for how it ties all together. Sequential equations applicable to nature and a utilization of Quantum similarity as proposed in my previous papers will also be used. The purpose of this paper is to provide a new approach to algorithmic Quantization in terms of describing the biggest complexities in Quantum Mechanics.

Traducción del texto de Donald Gillies "“The Fregean Revolution in Logic”, en Revolutions in Mathematics , ed. Gillies, D.
Oxford: Oxford University Press, 1995).

Traducción de Marcela del Pilar Gómez y Ángel Rivera Novoa. En: "Saga", No. 20, 71-107, 2009.

The video is available here: https://www.youtube.com/watch?v=xU2uWCWNwsE&list=PL6pIaFRoXNGNEeVfSmknemamNlHzBAz2K

Abstract:
Degrowth is the literal translation of 'decroissance', a French word meaning reduction. Launched by activists in 2001 as a challenge to growth, it became a missile word that sparks a contentious debate on the diagnosis and prognosis of our society. 'Degrowth' became an interpretative frame for a new (and old) social movement where numerous streams of critical ideas and political actions converge. It is an attempt to re-politicise debates about desired socio-environmental futures and an example of an activist-led science now consolidating into a concept in academic literature. This article discusses the definition, origins, evolution, practices and construction of degrowth. The main objective is to explain degrowth's multiple sources and strategies in order to improve its basic definition and avoid reductionist criticisms and misconceptions. To this end, the article presents degrowth's main intellectual sources as well as its diverse strategies (oppositional activism, building of alternatives and political proposals) and actors (practitioners, activists and scientists). Finally, the article argues that the movement's diversity does not detract from the existence of a common path.

Speech mainly based upon our article:
Demaria, F., Schneider, F., Sekulova, F., Martinez-Alier, J. (2013). What is degrowth? From an activist slogan to a social movement. Environmental Values 22 (2): 191-215.

More info at: budapest.degrowth.org

RESUMO

O presente trabalho argumenta que o conhecimento matemático não é redu-tível a uma coleção de teoremas demonstrados e verificados (e, portanto, verdadeiros para todos os fins). Defende-se que as ideias de entendimento e explicação matemáticas são essenciais não apenas para a prática e heurística matemáticas, mas também para as fundações da disciplina. Para tanto, exa-mina-se o caso da clássica demonstração computacional do Teorema das Quatro Cores, de Appel-Haken, que é discutida sob uma perspective filo-sófica, considerando-se a rica bibliografia sobre o tópico. Este foi um dos primeiros teoremas originais demonstrados com ajuda considerável do ma-quinário computacional, em um esforço que envolveu checagem exaustiva de caso, e foi responsável por fomentar uma discussão relevante sobre di-versos tópicos da epistemologia matemática. Palavras-chave: Conhecimento Matemático. Demonstração. Validade. En-tendimento.

ABSTRACT

In this paper, I argue that mathematical knowledge is not reduced to a collection of proved and verified (hence, true to all practical matters) theorems. I defend that the ideas of mathematical understanding and mathematical explanation are essential not only to mathematical practice and heuristics but also to the very foundations of the discipline. To accomplish that, I examine the classic Appel-Haken proof of the Four-Color Theorem and discuss it from a philosophical perspective, taking on account the rich literature on the topic. The Four-Color Theorem was one of the first original theorems to be proved with considerable help from computational machinery, employed for performing exhaustive case checking, and that was responsible for encouraging a relevant discussion on several topics of mathematical epistemology.

사회 세계 (MSW)를 aking에 자세히 언급하기 전에 m나는 먼저 철학 (설명 심리학)과 Searle (S)와 비트 겐슈타인 (W)의 작품에서 예시로 현대 심리학 연구와의 관계에 대한 몇 가지 의견을 제공 할 것입니다, 나는 이것이 행동에 Searle 또는 어떤 해설자를 배치하는 가장 좋은 방법이라고 생각하기 때문에, 적절한 관점에서. 그것은 크게 설명 심리학의이 두 천재에 의해 PNC, TLP, PI, OC, TARW 및 기타 책의 내 리뷰를 볼 도움이 될 것입니다.

S는 TLP의 메커니즘으로 W의 선견지명 진술과 그의 후기 작업에서 그의 파괴에 대한 언급을하지 않습니다. W, S는 행동의 이러한 기계적 견해의 주요 해체자가되었다, 그리고 가장 중요한 설명 심리학자 (철학자), 하지만 W가 그를 완전히 예상하고, 크게, 다른 사람을 수행 (하지만 W, 튜링과 AI에 프라우드 풋과 코플랜드의 많은 논문과 책을 참조하십시오). S의 작품은 W보다 훨씬 쉽게 따라할 수 있으며, 일부 전문 용어가 있지만 올바른 방향으로 접근하면 대부분 극적으로 분명합니다. 자세한 내용은 W S 및 기타 책에 대한 내 리뷰를 참조하십시오.

전반적으로, MSW는 작업의 S의 반세기에서 발생하는 Wittgenstein을 통해 많은 실질적인 발전의 좋은 요약이지만, 내 보기에, W는 여전히 그가 말하는 것을 파악하면 기본 심리학에 대한 타의 추종을 불허합니다 (내 리뷰 참조). 이상적으로,  그들은 함께 읽어야한다 : S1 / S2의 운영W의 기발한 예와 함께 설명 S2 / S3의 운영에 명확한 일관된 산문과 일반화에 대한 Searle, 그의 화려한 격언. 제가 훨씬 어렸을 때, 저는 그 일을 하는 책을 쓰곤 했습니다.

현대 의 두 시스템 보기에서인간의 행동에 대한 포괄적 인 최신 프레임 워크를 원하는 사람들은 내 책을 참조 할 수 있습니다'철학의 논리적 구조, 심리학, 민d와 루드비히  비트겐슈타인과 존 Searle의언어'  2nd ed (2019). 내 글의 더 많은 관심있는 사람들은 '이야기 원숭이를 볼 수 있습니다-철학, 심리학, 과학, 종교와 운명 행성에 정치 - 기사 및 리뷰 2006-2019 3 rd 에드 (2019) 및 21st  세기 4번째  에드 (2019)  및 기타에서 자살 유토피아 망상.

In the article, the philosophical significance of quantum computation theory for philosophy of mathematics is discussed. In particular, I examine the notion of "quantum-assisted proof" (QAP); the discussion sheds light on the problem of the nature of mathematical proof; the potential empirical aspects of mathematics and the realism-antirealism debate (in the context of the indispensability argument). I present a quasi-empiricist account of QAP's, and discuss the possible impact on the discussions centered around the Enhanced Indispensabity Argument (EIA).

Atomic numbers of Hydrogen, Helium, Boron, Neon, Calcium, Technetium, and Protactinium are members of a cryptic numerical progression series. Contextually, this research paper involves postulation and hypothesis formulation around some elements from the periodic table. The atomic numbers of these selected elements form a numerical series. Let’s call it an arithmetico-exponential series. The main assumption to decipher the existence of such a series is instigated by the occurrence of certain elements that behave very exceptionally.

Copyright registration number: cg5QcXsO0hQScErG

Im Kontext mit allen benannten Phänomenen in diesem Post, ergibt Alles ein schlüssiges Bild zu
„United States INC“ Konkurs?
Wieviel davon stimmt kann ich nicht sagen. Da ich die Wahrscheinlichkeit als eher hoch erachte, dass da im Hintergrund in diese Richtung schon was passiert ist und laufend "gewerkelt" wird, möchte ich diese interessante Meldung bringen. Interessant auch vor dem Hintergrund des buchstäblichen
Impfwahnsinns
und
was in Afghanistan passiert
und
der neuen Goldenen Bulle der Erlösung Proklamation
und
im Kontext mit der Papstrede im Deutschen Bundestag 2011,
sowie
neues von Prof. Franz Hörmann zu Zukunft der Wissenschaften und dem Gelsystem
und
Botschaften vom Botschafter der physisch lebendigen Erde zu Recht und Kräfte
über
der Wiederanbindung zum ein 1 einzig All-Bewusstsein
mit
der Wiederanbindung verloren gegangener Wahrnehmung der 12 Sinnesorgane.
und der physischen Lebenordnung kosmischer Gesetze mit den Naturgesetzen.
zuerst Video ansehen
https://www.academia.edu/video/kAwrO1

In the framework of the philosophy of contemporary mathematics, Hellman and Awodey both hold an interesting discussion on the role of Zermelo-Fraenkel Set Theory and Category Theory in the perspective of a good foundation for mathematics. For Hellman, neither Set Theory nor Category Theory constitutes a good foundational framework for mathematics and, in addition, Categories does not achieve a strong autonomy regarding Sets. Awodey’s claim is that Category Theory is a best option in the frame of a new way of understanding what a foundation of mathematics means. In this sense, the aim of this paper is to highlight the philosophical main features of this discussion, to establish some related positions and to show some interesting consequences for the philosophy of mathematics.

God is a mathematician and He loves doing it by the numbers. Your future is tomorrow and your past we can not find and today you learn; so your future will be secure in the hands of an amazing God, but all this would not be without numbers!

No creation, no day or night, no sun, moon, or stars; not even you would be reading this today because you would not exist!

Numbers are something; a creation of God and not man. Numbers are more than something created to count to see how much of something or how little of something we have.

Numbers are not something that just exist in our minds; they are a reality of life, of spiritual life and even the scriptures we read and without them even God's written word would not exist.

Numbers are everyone's reality. True you don't see them but they are not arbitrary products of one person's imagination. They are as real as you and

The article outlines a thesis that, from the cognitive point of view, natural numbers share common structure with the physical experiences encountered by students during their early childhood years.
This common structure displays conceptual duality which is described in the context of numbers as magnitude and multitude. The issue is examined by using tangible examples of walking, cutting bread and pouring water into beakers. The article presents a cognitive view of learning.
Key words: mathematical thinking, mathematical practices, cognition of mathematics, conceptual development

Mathematics education includes beliefs about the nature of mathematics and the ways about mathematics teaching and
learning. Many thoughts brings as a result of looking at the philosophy of mathematics, which is necessary for understanding
the nature of mathematics. These thoughts today might make mathematics fallible rather than absolute correct information.
Beliefs about the nature of mathematics also are leading learning and teaching activities in mathematics classrooms. Reuben
Hersh, a mathematician, while explaining mathematics as a social-cultural-historical reality with mental, physical aspects and
as a part of the humanities that achieving a compromise similar to science and constructing reproducible results; reveals the
philosophy of his own mathematical experience in his studies of philosophy of mathematics. Reuben Hersh's mathematician
with his academic roles reflected in his studies of philosophy of mathematics that he emphasized experience. At this point,
the experiences of Reuben Hersh as a person who does mathematics and come from within mathematics set an example for
mathematics teachers. This study aims to provide an approach about the nature of mathematics based on Reuben Hersh and to
give suggestions for mathematics teachers.

``People say again and again that philosophy doesn´t really progress, that we are still occupied with the same philosophical problems as were the Greeks. But the people who say this don´t understand why it has to be so. It is because our language has remained the same and keeps seducing us into asking the same questions. As long as there continues to be a verb ´to be´ that looks as if it functions in the same way as ´to eat and to drink´, as long as we still have the adjectives ´identical´, ´true´, ´false´, ´possible´, as long as we continue to talk of a river of time, of an expanse of space, etc., etc., people will keep stumbling over the same puzzling difficulties and find themselves staring at something which no explanation seems capable of clearing up. And what´s more, this satisfies a longing for the transcendent, because, insofar as people think they can see the ‘limits of human understanding´, they believe of course that they can see beyond these.`` 

This quote is from Ludwig Wittgenstein who redefined philosophy some 70 years ago (but most people have yet to find this out). Dennett, though he has been a philosopher for some 40 years, is one of them. It is also curious that both he and his prime antagonist, John Searle, studied under famous Wittgensteinians (Searle with John Austin, Dennett with Gilbert Ryle) but Searle more or less got the point and Dennett did not, (though it is stretching things to call Searle or Ryle Wittgensteinians). Dennett is a hard determinist (though he tries to sneak reality in the back door), and perhaps this is due to Ryle, whose famous book ´The Concept of Mind´(1949) continues to be reprinted. That book did a great job of exorcising the ghost, but it left the machine.

Dennett enjoys making the mistakes Wittgenstein, Ryle (and many others since) have exposed in detail. Our use of the words consciousness, choice, freedom, intention, particle, thinking, determines, wave, cause, happened, event (and so on endlessly) are rarely a source of confusion, but as soon as we leave normal life and enter philosophy (and any discussion detached from the environment in which language evolved—i.e., the exact context in which the words had meaning) chaos reigns. Like most, Dennett lacks a coherent framework - which Searle has called the logical structure of rationality. I have expanded on this considerably since I wrote this review and my recent articles show in detail what is wrong with Dennett's approach to philosophy, which one might call Scientism on steroids. Let me end with another quote from Wittgenstein--´Ambition is the death of thought´.

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).

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. The crucial idea of a continuum is used to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.

An investigative attempt at explaining Euclid's first definition in the "Elements," and answering the question "what is a point"? I assert that a point is an indefinitely small quantity of space, and thus can "have no part" because it is incapable of being quantified in order to be divided into parts.

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended their 'speculations' when he declared that time was the fourth dimension. Yet our commonly perceived space is truly four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five-dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.

We give a pictorial, and absurdly simple, proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show, moreover, why the proof cannot be expressed within classical graph theory.

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.