Explanation in Mathematics

(Longer version - work in progress) Various accounts of distinctively mathematical explanations (DMEs) of complex systems have been proposed recently which bypass the contingent causal laws and appeal primarily to mathematical necessities constraining the system. These necessities are considered to be modally exalted in that they obtain with a greater necessity than the ordinary laws of nature (Lange 2016). This paper focuses on DMEs of the number of equilibrium positions of n-tuple pendulum systems and considers several different DMEs of these (...) systems which bypass causal features. It then argues that there is a tension between the modal strength of these DMEs and their epistemic hooking, and we are forced to choose between (a) a purported DME with greater modal strength and wider applicability but poor epistemic hooking, or (b) a narrowly applicable DME with lesser modal strength but with the right kind of epistemic hooking. It also aims to show why some kind of DMEs may be unappealing for working scientists despite their strong modality, and why some DMEs fail to be modally robust because of making ill-informed assumptions about their target systems. The broader goal is to show why such tensions weaken the case for DMEs for pendulum systems in general. (shrink)

The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 sections; (...) 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (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)

At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...) nature, the construction of senses, and motile behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated intelligent behaviors. -/- Alongside this development is a further account that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure. -/- In support of this inquiry we work toward the development of a calculus for biophysical construction and its dynamics. If successful this mechanics mathematically characterizes sensory and motile behavior. -/- Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we develop a probabilistic theory that enables us to reason about inaccessible factors in group behavior. -/- The mechanics we propose suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. -/- We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience. (shrink)

Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' relation among (...) mathematical proofs. (shrink)

This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation (...) to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman–Hilton duality. (shrink)

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...) model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...) an older formulation, what do we gain by reformulating? Second, some reformulations are genuinely trivial or insignificant. Merely replacing one symbol with another does not lead to intellectual progress. What distinguishes significant reformulations from trivial ones? According to what I call "conceptualism", reformulations are intellectually significant when they provide a different plan for solving problems. Significant reformulations provide inferentially different routes to the same solution. In contrast, trivial reformulations provide exactly the same problem-solving plans, and hence they do not change our understanding. This answers the second question about what distinguishes trivial from significant reformulations. However, the first question remains: what makes a new way of solving an old problem valuable? Here, a bevy of practical considerations come to mind: one formulation might be faster, less complicated, or use more familiar concepts. According to "instrumentalism," these practical benefits are all there is to reformulating. Some reformulations are simply more instrumentally valuable for meeting the aims of science than others. At another extreme, "fundamentalism" contends that a reformulation is valuable when it provides a more fundamental description of reality. According to this view, some reformulations directly contribute to the metaphysical aim of carving reality at its joints. Conceptualism develops a middle ground between instrumentalism and fundamentalism, preserving their benefits without their costs. I argue that the epistemic value of significant reformulations does not reduce to either practical or metaphysical value. Reformulations are valuable because they are a constitutive part of problem-solving. Both science and mathematics aim at solving all possible problems within their respective domains. Meeting this aim requires being able to plan for any possible problem-solving context, and this requires reformulating. By reformulating, we clarify what we need to know to solve problems. Still, one might wonder whether the value of reformulations requires underlying differences in explanatory power. According to "explanationism," a reformulation is valuable only when it provides a better explanation. Explanationism stands as a rival middle ground position to my own. However, it faces numerous counterexamples. In many cases, two reformulations provide the same explanation while nonetheless providing different ways of understanding a phenomenon. Hence, reformulating can be valuable even when neither formulation is more explanatory. Methodologically, I draw on a variety of case studies to support my account of reformulation. These range from classical mechanics to quantum chemistry, along with examples from mathematics. Symmetry arguments provide a paradigmatic example: the mathematics of symmetry groups radically recasts quantum mechanics and quantum chemistry. Nevertheless, elementary approaches exist that eschew this additional mathematical apparatus, solving problems in a more tedious but less mathematically-demanding manner. Further examples include reformulations of quantum field theory, Arabic vs. Roman numerals, and Fermat's little theorem in number theory. In each case, my account identifies how reformulations change and improve our understanding of science and mathematics. (shrink)

In a paper titled, “The Unreasonable Effectiveness of Mathematics”, published 20 years after Wigner’s seminal paper, the mathematician Richard W. Hamming discussed what he took to be Wigner’s problem of Unreasonable Effectiveness and offered some partial explanations for this phenomenon. Whether Hamming succeeds in his explanations as answers to Wigner’s puzzle is addressed by other scholars in recent years I, on the other hand, raise a more fundamental question: does Hamming succeed in raising the same question as Wigner? The answer (...) is no. My goal is to show that Hamming’s reading misses Wigner’s highly original formulation of the problem.Through a close and contextual reading of Wigner’s work, as I will show, we are led in new directions in addressing and solving the applicability problem. (shrink)

A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna in the recent literature. (...) My answer to the question of this paper is negative. As I will argue, there are serious problems with the main reasons for believing in the first place that it is possible to have physical understanding of mathematical facts. (shrink)

Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.

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)

This paper is about the underlying logical principles of scientific theories. In particular, it concerns ex contradictione quodlibet (ECQ) the principle that anything follows from a contradiction. ECQ is valid according to classical logic, but invalid according to paraconsistent logics. Some advocates of paraconsistency claim that there are ‘real’ inconsistent theories that do not erupt with completely indiscriminate, absurd commitments. They take this as evidence in favor of paraconsistency. Michael (2016) calls this the non-triviality strategy (NTS). He argues that this (...) strategy fails in its purpose. I will show that Michael's criticism significantly over-reaches. The fundamental problem is that he places more of a burden on the advocate of paraconsistency than on the advocate of classical logic. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. He does, however, make important observations that allow us to clarify some of the complexities involved in giving a logical reconstruction of a theory. I will argue that there are abductive arguments deserving of further consideration for the claim that paraconsistent logic offers the best explanation of the practice of inconsistent science. In this sense, the debate is still very much open. (shrink)

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case (...) study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding. (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)

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)

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)

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)

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)

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...) rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (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)

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)

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)

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

Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that (...) the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of (...) what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

This paper explores whether there is any relation between mathematical proofs that specify the grounds of the theorem being proved and mathematical proofs that explain why the theorem obtains. The paper argues that a mathematical fact’s grounds do not, simply by virtue of grounding it, thereby explain why that fact obtains. It argues that oftentimes, a proof specifying a mathematical fact’s grounds fails to explain why that fact obtains whereas any explanation of the fact does not specify its ground. The (...) paper offers several examples from mathematical practice to illustrate these points. These examples suggest several reasons why explaining and grounding tend to come apart, including that explanatory proofs need not exhibit purity, tend not to be brute force, and often unify separate cases by identifying common reasons behind them even when those cases have distinct grounds. The paper sketches an account of what makes a proof explanatory and uses that account to defend the morals drawn from the examples already given. (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)

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In general, (...) I argue that the answer to this question is “no” and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange. (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)

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)

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)

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)

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)

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is no important asymmetry (...) between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle. (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)

The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)

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)

Meta-laws, including conservation laws, are laws about the form of more specific, phenomenological, laws. Lange distinguishes between meta-laws as coincidences, where the meta-law happens to hold because the more specific laws hold, and meta-laws as constraints to which subsumed laws must conform. He defends this distinction as a genuine metaphysical possibility, such that metaphysics alone ought not to rule one way or another, leaving it an open question for physics. Lange’s distinction marks a genuine difference in how a given meta-law (...) can be used in explanations. Yet, I argue, it is not simply an empirical matter as to whether a given conservation law is a constraint or a coincidence. There is no set matter of fact about the world that determines this, and physics alone will not be able to return a determinate verdict on a law-by-law basis, even while there is a genuine difference between any given law as constraint and as coincidence. Rather, the difference marks different ways of treating the same law in a theoretical setting: by shifting the explanatory context, treating the same law as part of a different mathematical structure, it can be a genuine constraint and a genuine coincidence. The difference between constraint and coincidence relates to the way in which we use a law in specific theoretical and explanatory settings. Because the same law can appear in multiple contexts, it can be used in these genuinely different ways, without itself ‘‘really’’ being either one or the other as some atomistic empirical fact. Conservation laws as constraints and conservation laws as coincidences are both genuine theoretical roles that the same law can play. I conclude by considering how this pragmatist construal of constraints versus coincidences reveals how two parts of Lange’s work in this section of the book are unexpectedly independent of one another. -/- . (shrink)

Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.

Mathematical proofs aim to establish the truth of mathematical propositions by means of logical rules. Some recent literature in philosophy of mathematics alleges that some mathematical proofs also reveal why the proved mathematical propositions are true. These mathematical proofs are called explanatory mathematical proofs. In this paper, I present and discuss some salient problems around mathematical explanation: the existence problem, the normative problem, the explanandum problems of truth value and psychological value, the logical structure problem, the regress problem and the (...) modelling problem. At the end, I sum up two contemporary models for mathematical explanation – the deductive-nomological model and the model of Steiner. I analyse these models against the previous problems. (shrink)

In the recent philosophy of explanation, a growing attention to and discussion of non-causal explanations has emerged, as there seem to be compelling examples of non-causal explanations in the sciences, in pure mathematics, and in metaphysics. I defend the claim that the counterfactual theory of explanation captures the explanatory character of both non-causal scientific and metaphysical explanations. According to the CTE, scientific and metaphysical explanations are explanatory by virtue of revealing counterfactual dependencies between the explanandum and the explanans. I support (...) this claim by illustrating that CTE is applicable to Euler’s explanation and Loewer’s explanation. (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)

In this paper we discuss three interrelated questions. First: is explanation in mathematics a topic that philosophers of mathematics can legitimately investigate? Second: are the specific aims that philosophers of mathematical explanation set themselves legitimate? Finally: are the models of explanation developed by philosophers of science useful tools for philosophers of mathematical explanation? We argue that the answer to all these questions is positive. Our views are completely opposite to the views that Mark Zelcer has put forward recently. Throughout this (...) paper, we show why Zelcer’s arguments fail. (shrink)

Enhanced indispensability arguments claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation. Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism to achieve platonism? We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things we should (...) be Scientific Realists about are arrived at by any inferential route which eschews causes, and nor is there any direct pressure for Scientific Realists to change their inferential methods. We suggest that in order to maintain inferential parity with Scientific Realism, proponents of EIA need to give details about how and in what way the presence of mathematical entities directly contribute to explanations. (shrink)

How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. The first—debunking arguments—aims to put pressure on realism by emphasising the (...) seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the 'Benacerraf-Field' problem. The second strategy—indispensability arguments—aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodology which we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination are also broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area. (shrink)

The author of “Parsimony and inference to the best mathematical explanation” argues for platonism by way of an enhanced indispensability argument based on an inference to yet better mathematical optimization explanations in the natural sciences. Since such explanations yield beneficial trade-offs between stronger mathematical existential claims and fewer concrete ontological commitments than those involved in merely good mathematical explanations, one must countenance the mathematical objects that play a theoretical role in them via an application of the relevant mathematical results. The (...) nominalist’s challenge is thus to undermine the platonistic force of such explanations by way of alternative nominalistic ones. The author’s contention is that such nominalistic explanations should provide a paraphrase of the proofs of the mathematical results being applied. There are reasons to doubt that proofs, construed here as formal derivations, actually contribute to the platonistc force to be undermined and, by parity, that nominalized proofs should bear responsability for the corresponding undermining. A discussion of two examples and of associated arguments by Lange, Pincock, Steiner and Tallant, point to a a wealth of worries concerning the construal of this explanatory role. Among those figure the distinction between the weak and strong role of proofs, the distinction between causal or “ordinary” explanations and genuine mathematical ones, and the unifying role of optimization explanations. More generally, the very idea that the explanatory advantages yielded by applied mathematical claims may be construed as gradual or progressive and the associated notion that the feasibility of their nominalistic paraphrases decreases as the generality and force of these claim increases, deserves a closer attention. (shrink)