Philosophy of Mathematics, Miscellaneous

Los Alephs de Borges se distinguen de los Alefs de Cantor. ¿Qué son aparte de infinitud? ¿Dónde viven? ¿Qué hacen? Este escrito es una invitación a la imaginación ante la dialéctica entre la Teoría de Conjuntos cantarina y la Poesía Múltiple borgesiana. Enfrentaremos entonces lo profundo, el corazón, la invención en esos dos autores, revisando en particular ideas de densidad, asintoticidad, fractalidad. Nuestro pensamiento se expande a partir de la Teoría de Conjuntos, pero solo Borges puede distorsionar algo tan sublime, (...) con sus deformaciones del espacio, del tiempo, de lo real, donde inventa más infinitos de los que podemos imaginar. Estudiaremos cómo, en la matemática y en la literatura, la expansión de la imaginación es fundamental para crear, descubrir, ligar pensamientos y llegar a lo más verdadero y profundo. (shrink)

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

The book is a philosophical refection on the possibility of mathematical history. Are poosible models of historical phenomena so exact as those of physical ones? Mathematical models borrowed from quantum mechanics by the meditation of its interpretations are accomodated to history. The conjecture of many-variant history, alternative history, or counterfactual history is necessary for mathematical history. Conclusions about philosophy of history are inferred.

This article, written in Bengali ('Gonit Dorshon' means `philosophy of mathematics' ), briefly reviews a few of the major points of view toward mathematics and the world of mathematical entities, and interprets the philosophy of mathematics as an interaction between these. The existence of these different points of view is indicative that mathematics, in spite of being of universal validity, can nevertheless accommodate alternatives. In particular, I review the alternative viewpoints of Platonism and Intuitionism and present the case that in (...) spite of their great differences, they are not mutually exclusive - that both can be accommodated within the infinite edifice of mathematics. This, in turn, is argued to be consistent with the viewpoint of Category Theory that holds the promise of an entirely new interpretation of the world of mathematics and the relation of that world to the world of our concepts and ideas: mathematics is a human enterprise and mathematical logic is a reflection of how our ideas and concepts are formed and combined with one another. I venture that this, perhaps, is the view of mathematics that Ludwig Wittgenstein would espouse. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

人们普遍认为，不可能性、不完整性、不一致性、不可度、随机性、可预见性、悖论、不确定性和理性极限是完全不同的科学物理或数学问题，在常见。我认为，它们主要是标准的哲学问题（即语言游戏），这些问题大多在80 多年前由维特根斯坦解决。 -/- "在这种情况下，我们'想说'当然不是哲学，而是它的原材料。因此，例如，数学家倾向于对数学事实的客观性和现实性说的，不是数学哲学，而是哲学处理的东西。维特根斯坦 PI 234 -/- "哲学家们经常看到科学的方法，他们不可抗拒地试图以科学的方式提问和回答问题。这种倾向是形而上学的真正源泉，将哲学家带入完全的黑暗之中。 维特根斯坦 -/- 我简要地总结了现代两位最杰出的学生路德维希·维特根斯坦和约翰·西尔关于故意的逻辑结构（思想、语言、行为）的一些主要发现，作为我的起点Wittgenstein 的基本发现——所有真正的"哲学"问题都是相同的——关于在特定上下文中如何使用语言的困惑，因此所有解决方案都是一样的——研究如何在相关上下文中使用语言，使其真实性条件（满意度或 COS 条件）是明确的。基本问题是，人们可以说什么，但一个人不能意味着（状态明确COS）任何任意的话语和意义只有在非常具体的上下文中才可能。 -/- 在两种思想体系的现代视角（被推广为"思维快，思维慢"）的框架内，我从维特根斯坦人的角度剖析了一些主要评论员关于这些问题的一些著作，并采用了一个新的表意向性和新的双系统命名法。 我表明，这是一个强大的启发式描述这些假定的科学，物理或数学问题的真实性质，这是真正最好的处理作为标准哲学问题，如何使用语言（语言游戏在维特根斯坦的术语）。 -/- 我的论点是，这里突出特征的意向表（理性、思想、思想、语言、个性等）或多或少地准确地描述了，或者至少作为启发式，我们思考和行为的方式，所以它包含不只是哲学和心理学，但其他一切（历史，文学，数学，政治等） 。特别要注意，我（以及西尔、维特根斯坦和其他人）认为，故意和理性包括有意识的审议语言系统2和无意识的自动预语言系统1行为或反射。 .

To make sense of what Gilles Deleuze understands by a mathematical concept requires unpacking what he considers to be the conceptualizable character of a mathematical theory. For Deleuze, the mathematical problems to which theories are solutions retain their relevance to the theories not only as the conditions that govern their development, but also insofar as they can contribute to determining the conceptualizable character of those theories. Deleuze presents two examples of mathematical problems that operate in this way, which he considers (...) to be characteristic of a more general theory of mathematical problems. By providing an account of the historical development of this more general theory, which he traces drawing upon the work of Weierstrass, Poincaré, Riemann, and Weyl, and of its significance to the work of Deleuze, an account of what a mathematical concept is for Deleuze will be developed. (shrink)

Strong Composition as Identity is the thesis that necessarily, for any xs and any y, those xs compose y iff those xs are non-distributively identical to y. Some have argued against this view as follows: if some many things are non-distributively identical to one thing, then what’s true of the many must be true of the one. But since the many are many in number whereas the one is not, the many cannot be identical to the one. Hence is mistaken. (...) Although I am sympathetic to this objection, in this paper, I present two responses on behalf of the theorist. I also show that once the defender of accepts one of these two responses, that defender will be able to answer The Special Composition Question. (shrink)

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in the category of sets is identifiable with the elementary embeddings of large cardinal axioms. A modal coalgebraic automata's mappings are further argued to account for both reinterpretations of quantifier domains as well as the ontological expansion effected by the elementary embeddings in the category of sets. The interaction between the interpretational and (...) objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebraic automata are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

Abstract -/- The concept of infinity is of ancient origins and has puzzled deep thinkers ever since up to the present day. Infinity remains somewhat of a mystery in a physical world in which our comprehension is largely framed around the concept of boundaries. This is partly because we live in a physical world that is governed by certain dimensions or limits – width, breadth, depth, mass, space, age and time. To our ordinary understanding, it is a seemingly finite world (...) under those dimensions and we may find it difficult to comprehend something that by definition can have no beginning and no end, no limit or boundary. The article argues that this concept can have a meaning different from that normally envisaged in science, philosophy or mathematics, a meaning that transcends all boundaries and which proceeds from a non-material or metaphysical perspective. It examines the features and implications of that concept. -/- . (shrink)

It is argued that Gabriel Uzquiano's approach to set-theoretic indefinite extensibility is a version of in rebus structuralism, and therefore suffers from a vacuity problem.

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early development of (...) algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument shows (...) that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience. (shrink)

The is the Editorial of the 2015 JPBMB Special Issue on Integral Biomathics: Life Sciences, Mathematics and Phenomenological Philosophy.

An updated survey of the research scope in Integral Biomathics.

Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...) logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics. (shrink)

In philosophical logic, a certain family of model constructions has received particular attention. Prominent examples are the cumulative hierarchy of well-founded sets, and Kripke's least fixed point models of grounded truth. I develop a general formal theory of groundedness and explain how the well-founded sets, Cantor's extended number-sequence and Kripke's concepts of semantic groundedness are all instances of the general concept, and how the general framework illuminates these cases. Then, I develop a new approach to a grounded theory of proper (...) classes. -/- However, the general concept of groundedness does not account for the philosophical significance of its paradigm instances. Instead, I argue, the philosophical content of the cumulative hierarchy of sets is best understood in terms of a primitive notion of ontological priority. -/- Then, I develop an analogous account of Kripke's models. I show that they exemplify the in-virtue-of relation much discussed in contemporary metaphysics, and thus are philosophically significant. I defend my proposal against a challenge from Kripke's “ghost of the hierarchy”. (shrink)

This paper focuses on the distinction between methods which are mathematically "clever", and those which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out answers to problems. Examples of the latter include simulated probability distributions and resampling methods in statistics, and iterative methods for solving equations or optimisation problems. Most of these methods require software support, but this is easily provided by a PC. The paper argues that the crunchier methods often have substantial advantages from the perspectives (...) of user-friendliness, reliability (in the sense that misuse is less likely), educational efficiency and realism. This means that they offer very considerable potential for simplifying the mathematical syllabus underlying many areas of applied mathematics such as management science and statistics: crunchier methods can provide the same, or greater, technical power, flexibility and insight, while requiring only a fraction of the mathematical conceptual background needed by their cleverer brethren. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim of (...) this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively. (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)

A review of Carl Huffman's new edition of the fragments of Archytas of Tarentum. Praises the extensive commentary on four fragments, but argues that at least two dubious works not included in the edition ("On Law and Justice" and "On Wisdom") deserve further consideration and contain important information for the interpretation of Archytas. Provides a complete translation for the fragments of those works.

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian “infinite book copies” (...) objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue (...) that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)

Beginning with an examination of the deep history of making things and thinking about making things made-up in our minds, I argue that the resultant declarative understanding of the procedural knowledge of abstracting theories and building models—the essence(s) of the practice of science—embodied in Conceptual Mathematics is worth learning beginning with high school, along with grammar and calculus. One of the many profound scientific insights introduced—in a manner accessible to total beginners—in Lawvere and Schanuel's Conceptual Mathematics textbook is: the way (...) we ought to think about things varies with the nature of things. In light of the universal yearning for understanding, it may be pedagogically unethical not to teach the eminently learnable Conceptual Mathematics. Summing it all, we need to champion the cause of taking Conceptual Mathematics to classrooms. (shrink)

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

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

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

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

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, (...) but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (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)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim (...) and discusses its significance. The second part of the paper considers whether viewing-as explanations occur in the empirical sciences, focusing on the case of so-called fictional models. It’s sometimes suggested that fictional models can be explanatory even though they fail to represent actual worldly dependence relations. Whether or not such models explain, I suggest, depends on whether we think scientific explanations necessarily give information relevant to intervention and control. Finally, I argue that counterfactual approaches to explanation also have trouble accommodating viewing-as cases. (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)

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)

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)

This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...) they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

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. (shrink)

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. (shrink)

Принято считать, что невозможность, неполнота, Парапоследовательность, Несоответствие, Случайность, вычислительность, парадокс, неопределенность и пределы разума являются разрозненными научными физическими или математическими вопросами, имеющими мало или ничего общего. Я полагаю, что они в значительной степени стандартные философские проблемы (т.е. языковые игры), которые были в основном решены Витгенштейном более 80 лет назад. -/- Я предоставляю краткое резюме некоторых из основных выводов двух из самых выдающихся студентов поведения о Fсовременности, Людвиг Витгенштейн и Джон Сирл, на логическую структуру преднамеренности (ум, язык, поведение), принимая в качестве (...) отправной точки фундаментальное открытие Витгенштейна, что все действительно "философские" проблемы одинаковы-путаницы AB,как использовать язык в частности контекст, и поэтому все решения одинаковы, глядя на то, как язык может быть использован в рассматриваемом контексте, так что его условия истины (Условия удовлетворенности или COS) ясны. Основная проблема заключается в том, что можно сказать что угодно, но нельзя означать (государство ясно COS для) любое произвольное высказывание и смысл возможен только в очень специфическом контексте. -/- Я вскрыть некоторые писания некоторых из основных комментаторов по этим вопросам с точки зрения Витгенштейна в framework современной точки зрения двух систем мысли (популяризировал как "мышление быстро, думая медленно"), используя новую таблицу преднамеренности и новых двойных систем номенклатуры. Я показываю, что это мощная эвристическая для описания истинной природы этих предположенных научных, физических или математических вопросов, которые действительно лучше всего подходить как стандартные философские проблемы использования языка (языковые игры в терминологии Витгенштейна). (shrink)

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)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

This is an introduction to the volume "Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations", edited by A. Reutlinger and J. Saatsi (OUP, forthcoming in 2017). -/- Explanations are very important to us in many contexts: in science, mathematics, philosophy, and also in everyday and juridical contexts. But what is an explanation? In the philosophical study of explanation, there is long-standing, influential tradition that links explanation intimately to causation: we often explain by providing accurate information about the causes of the (...) phenomenon to be explained. Such causal accounts have been the received view of the nature of explanation, particularly in philosophy of science, since the 1980s. However, philosophers have recently begun to break with this causal tradition by shifting their focus to kinds of explanation that do not turn on causal information. The increasing recognition of the importance of such non-causal explanations in the sciences and elsewhere raises pressing questions for philosophers of explanation. What is the nature of non-causal explanations - and which theory best captures it? How do non-causal explanations relate to causal ones? How are non-causal explanations in the sciences related to those in mathematics and metaphysics? This volume of new essays explores answers to these and other questions at the heart of contemporary philosophy of explanation. The essays address these questions from a variety of perspectives, including general accounts of non-causal and causal explanations, as well as a wide range of detailed case studies of non-causal explanations from the sciences, mathematics and metaphysics. (shrink)

The central topic of this article is de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that numerals are eligible for existential quantification in epistemic contexts, whereas other names for natural numbers are not. In other words, numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am looking for an explanation of this phenomenon. It (...) is argued that the standard induction scheme plays a key role. (shrink)

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the (...) attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.