Philosophy of Mathematics, Misc

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...) issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (shrink)

This paper introduces a novel object that has less structure than, and is ontologically prior to the natural numbers. As such it is a candidate model of the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of the non-Cantorian outlook.

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation. Outlined here is a method for programming relativistic mass, space and time at the Planck level as applicable for use in Planck Universe-as-a-Simulation Hypothesis. For the virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven (...) by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle motion may be dispensed with. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this 3-D space and the hyper-sphere co-ordinates, relativity resembling the mathematics of perspective. (shrink)

Defined are gravitational formulas in terms of Planck units and units of $\hbar c$. Mass is not assigned as a constant property but is instead treated as a discrete event defined by units of Planck mass with gravity as an interaction between these units, the gravitational orbit as the sum of these mass-mass interactions and the gravitational coupling constant as a measure of the frequency of these interactions and not the magnitude of the gravitational force itself. Each particle that is (...) in the mass-state (defined by a unit of Planck mass) per unit of Planck time is directly linked to every other particle also in the mass-state by a discrete unit of $m_P v^2 r = \hbar c$, the velocity of a gravitational orbit is summed from these individual $v^2$. As this approach presumes a digital time, it is suitable for use in programming Simulation Hypothesis models. As this link is responsible for the particle-particle interaction it is analogous to the graviton. Orbital angular momentum of the planetary orbits derives from the sum of the planet-sun particle-particle orbital angular momentum irrespective of the angular momentum of the sun itself and the rotational angular momentum of a planet includes particle-particle rotational angular momentum. (shrink)

Euclid's classic proof about the infinitude of prime numbers has been a standard model of reasoning in student textbooks and books of elementary number theory. It has withstood scrutiny for over 2000 years but we shall prove that despite the deceptive appearance of its analytical reasoning it is tautological in nature. We shall argue that the proof is more of an observation about the general property of a prime numbers than an expository style of natural deduction of the proof of (...) their infinitude. (shrink)

An updated survey of the research scope in Integral Biomathics.

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)

'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)

Último sermão da Igreja do naturalismo fundamentalista pelo pastor Hofstadter. Como o seu muito mais famoso (ou infame por seus erros filosóficos implacáveis) Godel, Escher, Bach, ele tem uma plausibilidade superficial, mas se se compreende que este é um cientificismo desenfreado que mistura questões científicas reais com os filosóficos (ou seja, o somente as edições reais são que jogos da língua nós devemos jogar) então quase todo seu interesse desaparece. Eu forneci um quadro para análise baseada na psicologia evolutiva e (...) no trabalho de Wittgenstein (desde que atualizado em meus escritos mais recentes). Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd (2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no século 21 6a Ed (2020), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. "Pode ser justamente perguntado qual a importância que a prova de Gödel tem para o nosso trabalho. Para um pedaço de matemática não pode resolver problemas do tipo que nos incomoda. --A resposta é que a situação, em que tal prova nos traz, é do nosso interesse. "O que devemos dizer agora?" -Esse é o nosso tema. No entanto, estranho soa, minha tarefa, tanto quanto diz respeito à prova de Gödel parece meramente consistir em deixar claro que tal proposição como: "suponha que isso poderia ser provado" significa em matemática. " Wittgenstein "observações sobre as fundações da matemática" p337 (1956) (escrito em 1937). "Meus teoremas só mostram que a mecanização da matemática, ou seja, a eliminação da mente e de entidades abstratas, é impossível, se alguém quiser ter uma base satisfatória e um sistema de matemática. Eu não tenho provado que existem questões matemáticas que são indecidíveis para a mente humana, mas só que não há nenhuma máquina (ou formalismo cego) que pode decidir todas as questões número-teórico, (mesmo de um tipo muito especial).... Não é a própria estrutura dos sistemas dedutivos que está a ser ameaçado com um demolir, mas apenas uma certa interpretação do mesmo, ou seja, a sua interpretação como um formalismo cego. " Gödel "obras coletadas" Vol. 5, p 176-177. (2003) "Toda inferência tem lugar a priori. Os acontecimentos do futuro não podem ser inferidos a partir do presente. A superstição é a crença no nexo causal. A liberdade da vontade consiste no fato de que as ações futuras não podem ser conhecidas agora. Só podíamos conhecê-los se a causalidade fosse uma necessidade interior, como a da dedução lógica. --A conexão do conhecimento e do que é sabido é aquela da necessidade lógica. ("A sabe que p é o caso" é sem sentido se p é um tautologia.) Se do fato que uma proposição é óbvia a nós, não segue que é verdadeiro, a seguir o obviedade não é nenhuma justificação para a crença em sua verdade. " TLP 5,133--5,1363 . (shrink)

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

It is often said that ‘every logical truth is obvious’ (Quine 1970: 82), that the ‘axioms and rules of logic are true in an obvious way’ (Murawski 2014: 87), or that ‘logic is a theory of the obvious’ (Sher 1999: 207). In this chapter, I set out to test empirically how the idea that logic is obvious is reflected in the scholarly work of logicians and philosophers of logic. My approach is data-driven. That is to say, I propose that systematically (...) searching for patterns of usage in databases of scholarly works, such as JSTOR, can provide new insights into the ways in which the idea that logic is obvious is reflected in logical and philosophical practice, i.e., in the arguments that logicians and philosophers of logic actually make in their published work. (shrink)

Doy una revisión detallada de ' los límites externos de la razón ' por Noson Yanofsky desde una perspectiva unificada de Wittgenstein y la psicología evolutiva. Yo indiqué que la dificultad con cuestiones como la paradoja en el lenguaje y las matemáticas, la incompletitud, la indeterminación, la computabilidad, el cerebro y el universo como ordenadores, etc., surgen de la falta de mirada cuidadosa a nuestro uso del lenguaje en el adecuado contexto y, por tanto, el Error al separar los problemas (...) de hecho científico de las cuestiones de cómo funciona el lenguaje. Discuto las opiniones de Wittgenstein sobre la incompletitud, la paracoherencia y la indecisión y el trabajo de Wolpert en los límites de la computación. Resumiendo: el universo según Brooklyn---buena ciencia, no tan buena filosofía. Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych Berto, Floyd, Moyal-Sharrock y Yanofsky. (shrink)

Em "Godel's Way", três cientistas eminentes discutem questões como a undecidability, incompletude, aleatoriedade, computabilidade e paraconsistência. Eu abordar estas questões do ponto de vista Wittgensteinian que existem duas questões básicas que têm soluções completamente diferentes. Há as questões científicas ou empíricas, que são fatos sobre o mundo que precisam ser investigados observacionalmente e questões filosóficas sobre como a linguagem pode ser usada inteligìvelmente (que incluem certas questões em matemática e lógica), que precisam ser decidido por olhar uma como nós realmente (...) usar palavras em contextos específicos. Quando nós começ claros sobre que jogo da língua nós estamos jogando, estes tópicos são vistos para ser perguntas científicas e matemáticas ordinárias como qualquer outro. As idéias de Wittgenstein raramente foram igualadas e nunca ultrapassaram e são tão pertinentes hoje como eram 80 anos atrás, quando ele ditou os livros azul e marrom. Apesar de suas falhas-realmente uma série de notas em vez de um livro acabado-esta é uma fonte única do trabalho destes três estudiosos famosos que têm trabalhado nas bordas sangrantes da física, matemática e filosofia por mais de meio século. Da costa e Doria são citados por Wolpert (veja abaixo ou meus artigos sobre Wolpert e minha revisão de Yanofsky ' s "os limites exteriores da razão") desde que escreveu sobre a computação universal, e entre suas muitas realizações, da costa é um pioneiro em a paraconsistência. Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. (shrink)

It is commonly thought that Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were mostly resolved by Wittgenstein over 80years ago. -/- “What we are ‘tempted to say’ in such a case is, of course, not philosophy, but it is its raw material. Thus, for example, what a mathematician is (...) inclined to say about the objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical treatment.” Wittgenstein PI 234 -/- "Philosophers constantly see the method of science before their eyes and are irresistibly tempted to ask and answer questions in the way science does. This tendency is the real source of metaphysics and leads the philosopher into complete darkness." Wittgenstein -/- I provide a brief summary of some of the major findings of two of the most eminent students of behavior of modern times, Ludwig Wittgenstein and John Searle, on the logical structure of intentionality (mind, language, behavior), taking as my starting point Wittgenstein’s fundamental discovery –that all truly ‘philosophical’ problems are the same—confusions about how to use language in a particular context, and so all solutions are the same—looking at how language can be used in the context at issue so that its truth conditions (Conditions of Satisfaction or COS) are clear. The basic problem is that one can say anything, but one cannot mean (state clear COS for) any arbitrary utterance and meaning is only possible in a very specific context. -/- I dissect some writings of a few of the major commentators on these issues from a Wittgensteinian viewpoint in the framework of the modern perspective of the two systems of thought (popularized as ‘thinking fast, thinking slow’), employing a new table of intentionality and new dual systems nomenclature. I show that this is a powerful heuristic for describing the true nature of these putative scientific, physical or mathematical issues which are really best approached as standard philosophical problems of how language is to be used (language games in Wittgenstein’s terminology). -/- It is my contention that the table of intentionality (rationality, mind, thought, language, personality etc.) that features prominently here describes more or less accurately, or at least serves as an heuristic for, how we think and behave, and so it encompasses not merely philosophy and psychology, but everything else (history, literature, mathematics, politics etc.). Note especially that intentionality and rationality as I (along with Searle, Wittgenstein and others) view it, includes both conscious deliberative linguistic System 2 and unconscious automated prelinguistic System 1 actions or reflexes. (shrink)

En ' Godel’s Way ', tres eminentes científicos discuten temas como la indecisión, la incompleta, la aleatoriedad, la computabilidad y la paraconsistencia. Me acerco a estas cuestiones desde el punto de vista de Wittgensteinian de que hay dos cuestiones básicas que tienen soluciones completamente diferentes. Existen las cuestiones científicas o empíricas, que son hechos sobre el mundo que necesitan ser investigados observacionalmente y cuestiones filosóficas en cuanto a cómo el lenguaje se puede utilizar inteligiblemente (que incluyen ciertas preguntas en matemáticas (...) y lógica), que necesitan decidirse por unt cómo realmente usar palabras en contextos concretos. Cuando tenemos claro sobre qué juego de idiomas estamos jugando, estos temas son vistos como preguntas científicas y matemáticas ordinarias como cualquier otra. Las percepciones de Wittgenstein rara vez se han igualado y nunca superado y son tan pertinentes hoy como lo fueron hace 80 años cuando dictó los libros azul y marrón. A pesar de sus fallas — realmente una serie de notas en lugar de un libro terminado —, esta es una fuente única de la obra de estos tres eruditos famosos que han estado trabajando en los bordes sangrantes de la física, las matemáticas y la filosofía durante más de medio siglo. Da Costa y Doria son citados por Wolpert (ver abajo o mis artículos sobre Wolpert y mi reseña de ' los límites de la razón ' de Yanofsky) desde que escribieron en el cómputo universal, y entre sus muchos logros, da Costa es pionera en paraconsistencia. Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein,. (shrink)

Eu dou uma revisão detalhada de "os limites exteriores da razão" por Noson Yanofsky de uma perspectiva unificada de Wittgenstein e psicologia evolutiva. Eu indico que a dificuldade com tais questões como paradoxo na linguagem e matemática, incompletude, undecidabilidade, computabilidade, o cérebro eo universo como computadores, etc., todos surgem a partir da falta de olhar atentamente para o nosso uso da linguagem no apropriado contexto e, consequentemente, a falta de separar questões de fato científico a partir de questões de como (...) a linguagem funciona. Discuto os pontos de vista de Wittgenstein sobre incompletude, paraconsistência e indecidabilidade e o trabalho de Wolpert sobre os limites para a computação. Para resumir: o universo de acordo com o Brooklyn---boa ciência, não tão boa filosofia. Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. (shrink)

Último sermón de la iglesia del naturalismo fundamentalista por el pastor Hofstadter. Al igual que su mucho más famoso (o infame por sus incesantemente errores filosóficos) trabajo Godel, Escher, Bach, tiene una plausibilidad superficial, pero si se entiende que se trata de un científico rampante que mezcla problemas científicos reales con los filosóficos (es decir, el sólo los problemas reales son los juegos de idiomas que debemos jugar) entonces casi todo su interés desaparece. Proporciono un marco para el análisis basado (...) en la psicología evolutiva y el trabajo de Wittgenstein (ya actualizado en mis escritos más recientes). Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych Berto, Floyd, Moyal-Sharrock y Yanofsky y otras. (shrink)

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from the formula for a virtual electron; $f_e = 4\pi^2r^3$ ($r = 2^6 3 \pi^2 \alpha \Omega^5$) where the fine structure constant $\alpha$ = 137.03599... and $\Omega$ = 2.00713494... (...) are mathematical constants and the MLTA geometries are; M = (1), T = ($2\pi$), L = ($2\pi^2\Omega^2$), A = ($4\pi \Omega)^3/\alpha$. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated according to $f_e$, we can replace designations such as ($kg, m, s, A$) with a rule set; mass = $u^{15}$, length = $u^{-13}$, time = $u^{-30}$, ampere = $u^{3}$. The formula $f_e$ is unit-less ($u^0$) and combines these geometries in the following ratio M$^9$T$^{11}$/L$^{15}$ and (AL)$^3$/T, as such these ratio are unit-less. To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via the 2 (fixed) mathematical constants ($\alpha, \Omega$), 2 dimensioned scalars and the rule set $u$. As all constants can be defined geometrically, the least precise constants ($G, h, e, m_e, k_B$...) can also be solved via the most precise ($c, \mu_0, R_\infty, \alpha$), numerical precision then limited by the precision of the fine structure constant $\alpha$. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

Lewis Carroll’s 1895 paper, 'What the Tortoise Said to Achilles' is widely regarded as a classic text in the philosophy of logic. This special issue of 'The Carrollian' publishes five newly commissioned articles by experts in the field. The original paper is reproduced, together with contemporary correspondence relating to the paper and an extensive bibliography.

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...) process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings. (shrink)

Why is there Philosophy of Mathematics at all? Ian Hacking. in Metascience (2015).

One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic (...) when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories. (shrink)

Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that term’s quotes (...) name ‘‘{x: x = 1}’’. The term has no quotes, the term’s name has one set of quotes, and the name of the term’s name has two sets of quotes. The trickiness is further compounded by failure to explicitly distinguish a variable’s values from it substituents. The variable ranges over its values but its occurrences are replaced by occurrences of its substituents. In arithmetic the values are numbers not numerals but the substituents are numerals not numbers. See https://www.academia.edu/s/1eddee0c62?source=link -/- Raymond Boute tries to criticize Daniel Velleman for mistakes in this area. However, Corcoran finds mistakes in Boute’s handling of the material. The reader is invited to find mistakes in Corcoran’s handling of this tricky material. -/- The paper and the review treat other issues as well. -/- Acknowledgements: Raymond Boute, Joaquin Miller, Daniel Velleman, George Weaver, and others. (shrink)

John P. Burgess kritisiert Kurt Gödels Begriff der mathematischen oder rationalen Anschauung und erläutert, warum heuristische Intuition dasselbe leistet wie rationale Anschauung, aber ganz ohne ontologisch überflüssige Vorannahmen auskommt. Laut Burgess müsste Gödel einen Unterschied zwischen rationaler Anschauung und so etwas wie mathematischer Ahnung, aufzeigen können, die auf unbewusster Induktion oder Analogie beruht und eine heuristische Funktion bei der Rechtfertigung mathematischer Aussagen einnimmt. Nur, wozu benötigen wir eine solche Annahme? Reicht es nicht, wenn die mathematische Intuition als Heuristik funktioniert? Für (...) Gödel sind Mengen Extensionen von Begriffen und er beharrt beispielsweise auf dem ontologischen Objektstatus von Mengen, weil Denken für ihn einen Input benötigt, den es selbst nicht zu liefern im Stande ist. Im Falle der mathematischen Anschauung darf dieser Input allerdings nicht subjektiv kontingent sein, wenn es sich um objektiv gültige Theorien handeln soll. (Zudem vertritt Gödel eine Korrespondenztheorie der Wahrheit.) Der Begriff der heuristischen Intuition, den Burgess expliziert, stammt hingegen nicht zuletzt aus der kognitiven Psychologie, der bei der Verarbeitung impliziter, also unbewusster Informationen ansetzt und so das menschliche Vermögen erklärt, Entscheidungen und Urteile zu fällen, ohne sich der Urteilsgrundlagen bewusst zu sein. Über den ontologischen oder erkenntnistheoretischen Status dieser Urteilsgrundlagen sagen diese Theorien nichts aus. Sie könnten auch subjektiv kontingent zustande gekommen sein. Ihr normativer Anspruch ergibt sich lediglich aus „dem Funktionieren“, nicht daraus, dass es sich um Gesetze des Wahrseins handelt. (shrink)

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

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not (...) only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. (...) By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

Review of Maddy, Penelope "Defending the Axioms".

The dominant approach to analyzing the meaning of natural language sentences that express mathematical knowl- edge relies on a referential, formal semantics. Below, I discuss an argument against this approach and in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits, including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma. The new perspective also promises to facilitate discussion between philosophers of mathematics and cognitive scientists (...) working on topics of common interest. (shrink)

Philosophy of science in the 20th century is to be considered as mostly characterized by a fundamentally systematic heuristic attitude, which looks to mathematics, and more generally to the philosophy of mathematics, for a genuinely and epistemologically legitimate form of knowledge. Rooted in this assumption, the book provides a formal reconsidering of the dynamics of scientific theories, especially in the field of the physical sciences, and offers a significant contribution to current epistemological investigations regarding the validity of using formal (especially: (...) model-theoretic) methods of analysis, as developed principally by Stegmüller, Sneed, Suppes, Moulines, “to bring the airy flights of analytical philosophy back down to earth”, to borrow Stephan Hartmann’s provocative statement. At the same time, the volume represents a comprehensive account of the epistemic content of physical theories, the logic of theory change in science, and specific (inter-)theoretical core aspects of scientific progress, particularly in the form suggested informally by Thomas Kuhn. As C. Ulises Moulines writes in the preface, “there is no other example in present-day literature (in any language) on this topic, i.e. the formal analysis of the ideographic characterization of the dynamics of theories between Kuhn’s theory of science and structural epistemology, that is as systematic and complete as Perrone’s work”. (shrink)

This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the duality (...) of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. (shrink)

The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...) terminology, and of the radicalization of Bolzano's anti‐Kantianism. Bolzano's evolution is understood as a coherent move, once the criticism expressed in the Beyträge on the notion of quantity is compared with a different and larger notion of quantity that Bolzano developed already in 1816. This discussion is enriched by the discovery that two unknown texts mentioned by Bolzano in the Beyträge can be identified with works by von Spaun and Vieth respectively. Bolzano's evolution is interpreted as a radicalization of the criticism of the Kantian definition of mathematics and as an effect of Bolzano's unaltered interest in the Leibnizian notion of mathesis universalis. As a conclusion, the author claims that Bolzano never abandoned his original idea of considering mathematics as a scientia universalis, i.e. as the science of quantities in general, and suggests that the question of ideal elements in mathematics, apart from being a main reason for the development of a new logical theory, can also be considered as a main reason for developing a different definition of quantity. (shrink)

I distinguish between a metaphysical and a logical reading of Generality Relativism. While the former denies the existence of an absolutely general domain, the latter denies the availability of such a domain. In this paper I argue for the logical thesis but remain neutral in what concerns metaphysics. To motivate Generality Relativism I defend a principle according to which a collection can always be understood as a set-like collection. I then consider a modal version of Generality Relativism and sketch how (...) this version avoids certain revenge problems. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

In the chapter of Difference and Repetition entitled ‘Ideas and the synthesis of difference,’ Deleuze mobilizes mathematics to develop a ‘calculus of problems’ that is based on the mathematical philosophy of Albert Lautman. Deleuze explicates this process by referring to the operation of certain conceptual couples in the field of contemporary mathematics: most notably the continuous and the discontinuous, the infinite and the finite, and the global and the local. The two mathematical theories that Deleuze draws upon for this purpose (...) are the differential calculus and the theory of dynamical systems, and Galois’ theory of polynomial equations. For the purposes of this paper I will only treat the first of these, which is based on the idea that the singularities of vector fields determine the local trajectories of solution curves, or their ‘topological behaviour’. These singularities can be described in terms of the given mathematical problematic, that is for example, how to solve two divergent series in the same field, and in terms of the solutions, as the trajectories of the solution curves to the problem. What actually counts as a solution to a problem is determined by the specific characteristics of the problem itself, typically by the singularities of this problem and the way in which they are distributed in a system. Deleuze understands the differential calculus essentially as a ‘calculus of problems’, and the theory of dynamical systems as the qualitative and topological theory of problems, which, when connected together, are determinative of the complex logic of different/ciation. (DR 209). Deleuze develops the concept of a problematic idea from the differential calculus, and following Lautman considers the concept of genesis in mathematics to ‘play the role of model ... with respect to all other domains of incarnation’. While Lautman explicated the philosophical logic of the actualization of ideas within the framework of mathematics, Deleuze (along with Guattari) follows Lautman’s suggestion and explicates the operation of this logic within the framework of a multiplicity of domains, including for example philosophy, science and art in What is Philosophy?, and the variety of domains which characterise the plateaus in A Thousand Plateaus. While for Lautman, a mathematical problem is resolved by the development of a new mathematical theory, for Deleuze, it is the construction of a concept that offers a solution to a philosophical problem; even if this newly constructed concept is characteristic of, or modelled on the new mathematical theory. (shrink)

This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.

Two seemingly contradictory tendencies have accompanied the development of the natural sciences in the past 150 years. On the one hand, the natural sciences have been instrumental in effecting a thoroughgoing transformation of social structures and have made a permanent impact on the conceptual world of human beings. This historical period has, on the other hand, also brought to light the merely hypothetical validity of scientific knowledge. As late as the middle of the 19th century the truth-pathos in the natural (...) sciences was still unbroken. Yet in the succeeding years these claims to certain knowledge underwent a fundamental crisis. For scientists today, of course, the fact that their knowledge can possess only relative validity is a matter of self-evidence. The present analysis investigates the early phase of this fundamental change in the concept of science through an examination of Hermann von Helmholtz's conception of science and his mechanistic interpretation of nature. Helmholtz (1821-1894) was one of the most important natural scientists in Germany. The development of this thoughts offers an impressive but, until now, relatively little considered report from the field of the experimental sciences chronicling the erosion of certainty. (shrink)

This book is important for philosophy of mathematics and for the study of French philosophy. French philosophers are more concerned than most Anglo-American with mathematical practice outside of foundations. This contradicts the fashionable claim that French intellectuals get science all wrong and we return below to a germane example from Sokal and Bricmont [1999]. The emphasis on practice goes back to mid-20th century French historians of science including those Kuhn cites as sources for his orientation in philosophy of science [Kuhn (...) 1996: p. viii]. And the French often look to the experience, rather than the ideology, of Bourbaki. (shrink)

If the allusive stratagems can claim to define a new type of systematicity, it is because they give access to a space where the singularity, the diagram and the metaphor may interlace, to penetrate further into the physico-mathematic intuition and the discipline of the gestures which precede and accompany ‘formalisation’. This interlacing is an operation where each component backs up the others: without the diagram, the metaphor would only be a short-lived fulguration because it would be unable to operate: without (...) the metaphor, the diagram would only be a frozen icon, unable to jump over its bold features which represent the images of an already acquired knowledge; without the subversion of the functional by the singular, nothing would come to oppose the force of habit. (shrink)

If Tahiti suggested to theorists comfortably at home in Europe thoughts of noble savages without clothes, those who paid for and went on voyages there were in pursuit of a quite opposite human ideal. Cook's voyage to observe the transit of Venus in 1769 symbolises the eighteenth century's commitment to numbers and accuracy, and its willingness to spend a lot of public money on acquiring them. The state supported the organisation of quantitative researches, employing surveyors and collecting statistics to..

While the appeal of scientific materialism has been weakened by developments in theoretical physics, chemistry and biology, Pythagoreanism still attracts the allegiance of leading scientists and mathematicians. It is this doctrine that process philosophers must confront if they are to successfully defend their metaphysics. Peirce, Bergson and Whitehead were acutely aware of the challenge of Pythagoreanism, and attempted to circumvent it. The problem addressed by each of these thinkers was how to account for the success of mathematical physics if the (...) world consists of creative processes. In this paper I critically examine the nature of the challenge posed by Pythagoreanism to process philosophy and examine the efforts by process philosophers, particularly Whitehead, to overcome it, and offer some suggestions for advancing these efforts. (shrink)

Geometry was a main source of inspiration for Carnap’s conventionalism. Taking Poincaré as his witness Carnap asserted in his dissertation Der Raum (Carnap 1922) that the metrical structure of space is conventional while the underlying topological structure describes "objective" facts. With only minor modifications he stuck to this account throughout his life. The aim of this paper is to disprove Carnap's contention by invoking some classical theorems of differential topology. By this means his metrical conventionalism turns out to be indefensible (...) for mathematical reasons. This implies that the relation between to-pology and geometry cannot be conceptualized as analogous to the relation between the meaning of a proposition and its expression in some language as logical empiricists used to say. (shrink)

After sketching the main lines of Hilbert's program, certain well-known and influential interpretations of the program are critically evaluated, and an alternative interpretation is presented. Finally, some recent developments in logic related to Hilbert's program are reviewed.