An account of distinctively mathematicalexplanation (DME) should satisfy three desiderata: it should account for the modal import of some DMEs; it should distinguish uses of mathematics in explanation that are distinctively mathematical from those that are not (Baron [2016]); and it should also account for the directionality of DMEs (Craver and Povich [2017]). Baron’s (forthcoming) deductive-mathematical account, because it is modelled on the deductive-nomological account, is unlikely to satisfy these desiderata. I provide a counterfactual (...) account of DME, the Narrow Ontic Counterfactual Account (NOCA), that can satisfy all three desiderata. NOCA appeals to ontic considerations to account for explanatory asymmetry and ground the relevant counterfactuals. NOCA provides a unification of the causal and the non-causal, the ontic and the modal, by identifying a common core that all explanations share and in virtue of which they are explanatory. (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)
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)
A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in (...) the context of their treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematicalexplanation is problematic. (shrink)
Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematicalexplanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematicalexplanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematicalexplanation is proposed and then redeveloped in the light (...) of two difficulties that the basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)
Mathematical models provide explanations of limited power of specific aspects of phenomena. One way of articulating their limits here, without denying their essential powers, is in terms of contrastive explanation.
PurposeIn this article, we aim to present and defend a contextual approach to mathematicalexplanation.MethodTo do this, we introduce an epistemic reading of mathematicalexplanation.ResultsThe epistemic reading not only clarifies the link between mathematicalexplanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematicalexplanation can be read in this approach.ConclusionThe contextual approach defended here clears up the (...) notion of explanation and pushes us towards a pluralist vision on mathematicalexplanation. (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 mathematicalexplanation 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.
In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematicalexplanation is flawed in that it fails to account for the implicit directionality in each of his examples. This (...) inadequacy is remediable in each case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (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 mathematicalexplanation. (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)
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)
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 philosophical conception of mechanistic explanation is grounded on a limited number of canonical examples. These examples provide an overly narrow view of contemporary scientific practice, because they do not reflect the extent to which the heuristic strategies and descriptive practices that contribute to mechanistic explanation have evolved beyond the well-known methods of decomposition, localization, and pictorial representation. Recent examples from evolutionary robotics and network approaches to biology and neuroscience demonstrate the increasingly important role played by computer simulations (...) and mathematical representations in the epistemic practices of mechanism discovery and mechanism description. These examples also indicate that the scope of mechanistic explanation must be re-examined: With new and increasingly powerful methods of discovery and description comes the possibility of describing mechanisms far more complex than traditionally assumed. (shrink)
A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with (...) the Lotka-Volterra equations. There are at least two distinct ways those equations might hold of a system, one of which yields straightforwardly causal explanations, but the other of which yields explanations that are distinctively mathematical in terms of nomological strength. In the first, one first picks out a system or class of systems, finds that the equations hold in a causal -explanatory way; in the second, one starts with the equations and explanations that must apply to any system of which the equations hold, and only then turns to the world to see of what, if any, systems it does in fact hold. Using this new way in which a model might hold of a system, I highlight four specific avenues by which causal and non- causal explanations can complement one another. (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)
Do multi-level selection explanations of the evolution of social traits deepen the understanding provided by single-level explanations? Central to the former is a mathematical theorem, the multi-level Price decomposition. I build a framework through which to understand the explanatory role of such non-empirical decompositions in scientific practice. Applying this general framework to the present case places two tasks on the agenda. The first task is to distinguish the various ways of suppressing within-collective variation in fitness, and moreover to evaluate (...) their biological interest. I distinguish four such ways: increasing retaliatory capacity, homogenising assortment, and collapsing either fitness structure or character distribution to a mean value. The second task is to discover whether the third term of the Price decomposition measures the effect of any of these hypothetical interventions. On this basis I argue that the multi-level Price decomposition has explanatory value primarily when the sharing-out of collective resources is `subtractable'. Thus its value is more circumscribed than its champions Sober and Wilson (1998) suppose. (shrink)
Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can we still (...) consider that the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory. (shrink)
The literature on the indispensability argument for mathematical realism often refers to the ‘indispensable explanatory role’ of mathematics. I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically committing, (...) others ‘thin’ and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments. (shrink)
Explaining the behaviour of ecosystems is one of the key challenges for the biological sciences. Since 2000, new-mechanicism has been the main model to account for the nature of scientific explanation in biology. The universality of the new-mechanist view in biology has been however put into question due to the existence of explanations that account for some biological phenomena in terms of their mathematical properties (mathematical explanations). Supporters of mathematicalexplanation have argued that the (...) class='Hi'>explanation of the behaviour of ecosystems is usually provided in terms of their mathematical properties, and not in mechanistic terms. They have intensively studied the explanation of the properties of ecosystems that behave following the rules of a non-random network. However, no attention has been devoted to the study of the nature of the explanation in those that form a random network. In this paper, we cover that gap by analysing the explanation of the stability behaviour of the microbiome recently elaborated by Coyte and colleagues, to determine whether it fits with the model of explanation suggested by the new-mechanist or by the defenders of mathematicalexplanation. Our analysis of this case study supports three theses: (1) that the explanation is not given solely in terms of mechanisms, as the new-mechanists understand the concept; (2) that the mathematical properties that describe the system play an essential explanatory role, but they do not exhaust the explanation; (3) that a non-previously identified appeal to the type of interactions that the entities in the network can exhibit, as well as their abundance, is also necessary for Coyte and colleagues’ account to be fully explanatory. From the combination of these three theses we argue for the necessity of an integrative pluralist view of the nature of behaviour explanation when this is given by appealing to the existence of a random network. (shrink)
A number of philosophers have recently suggested that some abstract, plausibly non-causal and/or mathematical, explanations explain in a way that is radically dif- ferent from the way causal explanation explain. Namely, while causal explanations explain by providing information about causal dependence, allegedly some abstract explanations explain in a way tied to the independence of the explanandum from the microdetails, or causal laws, for example. We oppose this recent trend to regard abstractions as explanatory in some sui generis way, (...) and argue that a prominent ac- count of causal explanation can be naturally extended to capture explanations that radically abstract away from microphysical and causal-nomological details. To this end, we distinguish di erent senses in which an explanation can be more or less abstract, and analyse the connection between explanations’ abstractness and their explanatory power. According to our analysis abstract explanations have much in common with counterfactual causal explanations. (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)
Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely (...)mathematical, according to Baker :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (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)
This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the (...) modal structure of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)
Philippe Huneman has recently questioned the widespread application of mechanistic models of scientific explanation based on the existence of structural explanations, i.e. explanations that account for the phenomenon to be explained in virtue of the mathematical properties of the system where the phenomenon obtains, rather than in terms of the mechanisms that causally produce the phenomenon. Structural explanations are very diverse, including cases like explanations in terms of bowtie structures, in terms of the topological properties of the system, (...) or in terms of equilibrium. The role of mathematics in bowtie structured systems and in topologically constrained systems has recently been examined in different papers. However, the specific role that mathematical properties play in equilibrium explanations requires further examination, as different authors defend different interpretations, some of them closer to the new-mechanistic approach than to the structural model advocated by Huneman. In this paper, we cover this gap by investigating the explanatory role that mathematics play in Blaser and Kirschner’s nested equilibrium model of the stability of persistent long-term human-microbe associations. We argue that their model is explanatory because: i) it provides a mathematical structure in the form of a set of differential equations that together satisfy an ESS; ii) that the nested nature of the ESSs makes the explanation of host-microbe persistent associations robust to any perturbation; iii) that this is so because the properties of the ESS directly mirror the properties of the biological system in a non-causal way. The combination of these three theses make equilibrium explanations look more similar to structural explanations than to causal-mechanistic explanation. (shrink)
We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to (...) key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as ex- planatory generality is concerned. (shrink)
In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I (...) outline my argument. In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections. (shrink)
Lange’s collection of expanded, mostly previously published essays, packed with numerous, beautiful examples of putatively non-causal explanations from biology, physics, and mathematics, challenges the increasingly ossified causal consensus about scientific explanation, and, in so doing, launches a new field of philosophic investigation. However, those who embraced causal monism about explanation have done so because appeal to causal factors sorts good from bad scientific explanations and because the explanatory force of good explanations seems to derive from revealing the relevant (...) causal (or ontic) structures. The taxonomic project of collecting examples and sorting their types is an essential starting place for a theory of non-causal explanation. But the title of Lange’s book requires something further: showing that the putative explanations are, in fact, explanatory and revealing the non-causal source of their explanatory power. This project is incomplete if there are examples of putative non-causal explanations that fit the form but that nobody would accept as explanatory (absent a radical revision of intuitions). Here we provide some reasons for thinking that there are such examples. (shrink)
Detailed examinations of scientific practice have revealed that the use of idealized models in the sciences is pervasive. These models play a central role in not only the investigation and prediction of phenomena, but in their received scientific explanations as well. This has led philosophers of science to begin revising the traditional philosophical accounts of scientific explanation in order to make sense of this practice. These new model-based accounts of scientific explanation, however, raise a number of key questions: (...) Can the fictions and falsehoods inherent in the modeling practice do real explanatory work? Do some highly abstract and mathematical models exhibit a noncausal form of scientific explanation? How can one distinguish an exploratory "how-possibly" model explanation from a genuine "how-actually" model explanation? Do modelers face tradeoffs such that a model that is optimized for yielding explanatory insight, for example, might fail to be the most predictively accurate, and vice versa? This chapter explores the various answers that have been given to these questions. (shrink)
Recently, many have argued that there are certain kinds of abstract mathematical explanations that are noncausal. In particular, the irrelevancy approach suggests that abstracting away irrelevant causal details can leave us with a noncausal explanation. In this paper, I argue that the common example of Renormalization Group explanations of universality used to motivate the irrelevancy approach deserves more critical attention. I argue that the reasons given by those who hold up RG as noncausal do not stand up to (...) critical scrutiny. As a result, the irrelevancy approach and the line between casual and noncausal explanation deserves more scrutiny. (shrink)
A way to argue that something plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part is an explanatorily relevant part. This can be very useful in many ontological debates. (...) My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it. (shrink)
For an Aristotelian observer, the halo is a puzzling phenomenon since it is apparently sublunary, and yet perfectly circular. This paper studies Aristotle's explanation of the halo in Meteorology III 2-3 as an optical illusion, as opposed to a substantial thing (like a cloud), as was thought by his predecessors and even many successors. Aristotle's explanation follows the method of explanation of the Posterior Analytics for "subordinate" or "mixed" mathematical-physical sciences. The accompanying diagram described by Aristotle (...) is one of the earliest lettered geometrical diagrams, in particular of a terrestrial phenomenon, and versions of it can still be found in modern textbooks on meteorological optics. (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)
A common kind of explanation in cognitive neuroscience might be called functiontheoretic: with some target cognitive capacity in view, the theorist hypothesizes that the system computes a well-defined function (in the mathematical sense) and explains how computing this function constitutes (in the system’s normal environment) the exercise of the cognitive capacity. Recently, proponents of the so-called ‘new mechanist’ approach in philosophy of science have argued that a model of a cognitive capacity is explanatory only to the extent that (...) it reveals the causal structure of the mechanism underlying the capacity. If they are right, then a cognitive model that resists a transparent mapping to known neural mechanisms fails to be explanatory. I argue that a functiontheoretic characterization of a cognitive capacity can be genuinely explanatory even absent an account of how the capacity is realized in neural hardware. (shrink)
Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to the (...) classical tradition shape prominent debates in philosophy of mathematics, and I initiate a project of reconstruction within this field. (shrink)
In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, (...) D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)
This essay uses a mental files theory of singular thought—a theory saying that singular thought about and reference to a particular object requires possession of a mental store of information taken to be about that object—to explain how we could have such thoughts about abstract mathematical objects. After showing why we should want an explanation of this I argue that none of three main contemporary mental files theories of singular thought—acquaintance theory, semantic instrumentalism, and semantic cognitivism—can give it. (...) I argue for two claims intended to advance our understanding of singular thought about mathematical abstracta. First, that the conditions for possession of a file for an abstract mathematical object are the same as the conditions for possessing a file for an object perceived in the past—namely, that the agent retains information about the object. Thus insofar as we are able to have memory-based files for objects perceived in the past, we ought to be able to have files for abstract mathematical objects too. Second, at least one recently articulated condition on a file’s being a device for singular thought—that it be capable of surviving a certain kind of change in the information it contains—can be satisfied by files for abstract mathematical objects. (shrink)
Using as case studies two early diagrams that represent mechanisms of the cell division cycle, we aim to extend prior philosophical analyses of the roles of diagrams in scientific reasoning, and specifically their role in biological reasoning. The diagrams we discuss are, in practice, integral and indispensible elements of reasoning from experimental data about the cell division cycle to mathematical models of the cycle’s molecular mechanisms. In accordance with prior analyses, the diagrams provide functional explanations of the cell cycle (...) and facilitate the construction of mathematical models of the cell cycle. But, extending beyond those analyses, we show how diagrams facilitate the construction of mathematical models, and we argue that the diagrams permit nomological explanations of the cell cycle. We further argue that what makes diagrams integral and indispensible for explanation and model construction is their nature as locality aids: they group together information that is to be used together in a way that sentential representations do not. (shrink)
What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the (...) transfinite deadlock of higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)
Are there arguments in mathematics? Are there explanations in mathematics? Are there any connections between argument, proof and explanation? Highly controversial answers and arguments are reviewed. The main point is that in the case of a mathematical proof, the pragmatic criterion used to make a distinction between argument and explanation is likely to be insufficient for you may grant the conclusion of a proof but keep on thinking that the proof is not explanatory.
Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...) basadas en modelos matemáticos que fundamentarían muchos aspectos de la realidad. Dos claros ejemplos provienen de Roger Penrose y Max Tegmark. Esto lleva a pensar en una posición no solo nomológica sino además nomogónica de la matemática. ¿Puede la Naturaleza estar fundada por las matemáticas como señalan algunos físico-matemáticos? Y en ese caso, ¿sería pertinente buscar una nomo-génesis de esta índole en la constitución de la conciencia? -/- At the end of his book “La conciencia inexplicada”, Juan Arana points out that nomology, explanation according to the laws of nature requires a nomogony, an account of the origin of the laws. This means that the order that we can observe in the natural World demands something prior to posit that specific order. Since the scientific revolution we know that the best way to explain that nomology has been through mathematics. Anyway, in recent decades a number of proposals based on mathematical models that found many aspects of reality has been offered. Two clear examples come from Roger Penrose and Max Tegmark. This drives us to think of a position of mathematics as not only nomological but also nomogonical. Can Nature be founded by mathematics as some physicists and mathematicians point out? And, in this case, would be relevant this kind of approach to search a nomo-genesis in the constitution of consciousness? (shrink)
Zeno’s Arrow and Nāgārjuna’s Fundamental Wisdom of the Middle Way Chapter 2 contain paradoxical, dialectic arguments thought to indicate that there is no valid explanation of motion, hence there is no physical or generic motion. There are, however, diverse interpretations of the latter text, and I argue they apply to Zeno’s Arrow as well. I also find that many of the interpretations are dependent on a mathematical analysis of material motion through space and time. However, with modern philosophy (...) and physics we find that the link from no explanation to no phenomena is invalid and that there is a valid explanation and understanding of physical motion. Hence, those arguments are both invalid and false, which banishes the MMK/2 and The Arrow under this and derivative interpretations to merely the history of philosophy. However, a view that maintains their relevance is that each is used as a koan or sequence of koans designed to assist students in spiritual meditation practice. This view is partly justified by the realization that both Nāgārjuna and Zeno were likely meditation masters in addition to being logicians. The works are, therefore, not works that should be assessed as having valid arguments and true conclusions by the standards of modern analytic philosophy—contrary to some of the literature—but rather are therapeutic and perhaps more appropriately considered as part of an experientially focused philosophy such as existentialism, phenomenology or religion. (shrink)
This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of multi-dimensional intensional semantics ought to be applied to the mathematical setting. The framework permits of a (...) formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the multi-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I demonstrate, finally, how the duality axioms of the epistemic logic for the semantics can be availed of, in order to defuse the paradox of knowability. (shrink)
Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a (...) number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim, John Bigelow, and John Bigelow and Robert Pargetter have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)
This paper focuses on a particular kind of non-naturalism: moral non-naturalism. Our primary aim is to argue that the moral non-naturalist places herself in an invidious position if she simply accepts that the non-natural moral facts that she posits are not explanatory. This has, hitherto, been the route that moral non-naturalists have taken. They have attempted to make their position more palatable by pointing out that there is reason to be suspicious of the explanatory criterion of ontological commitment. That is (...) because other perfectly respectable views fall foul of that criterion, most notably: mathematical realism. Since we don’t want to rule out mathematical realism, we should jettison the explanatory criterion of ontological commitment. Against this manoeuvre, we argue that many contemporary mathematical realists accept the explanatory criterion and provide an account of how mathematical objects are indeed indispensable to our best explanations. Thus, the moral non-naturalist will be left in an awkward dialectical position if she accepts that non-natural moral properties play no such explanatory role. (shrink)
This paper brings together Thompson's naive action explanation with interventionist modeling of causal structure to show how they work together to produce causal models that go beyond current modeling capabilities, when applied to specifically selected systems. By deploying well-justified assumptions about rationalization, we can strengthen existing causal modeling techniques' inferential power in cases where we take ourselves to be modeling causal systems that also involve actions. The internal connection between means and end exhibited in naive action explanation has (...) a modal strength like that of distinctively mathematicalexplanation, rather than that of causal explanation. Because it is stronger than causation, it can be treated as if it were merely causal in a causal model without thereby overextending the justification it can provide for inferences. This chapter introduces and demonstrate the usage of the Rationalization condition in causal modeling, where it is apt for the system(s) being modeled, and to provide the basics for incorporating R variables into systems of variables and R arrows into DAGs. Use of the Rationalization condition supplements causal analysis with action analysis where it is apt. (shrink)
The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what (...) I will call ‘optimal representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions. (shrink)
On a view implicitly endorsed by many, a concept is epistemically better than another if and because it does a better job at ‘carving at the joints', or if the property corresponding to it is ‘more natural' than the one corresponding to another. This chapter offers an argument against this seemingly plausible thought, starting from three key observations about the way we use and evaluate concepts from en epistemic perspective: that we look for concepts that play a role in explanations (...) of things that cry out for explanation; that we evaluate not only ‘empirical' concepts, but also mathematical and perhaps moral concepts from an epistemic perspective; and that there is much more complexity to the concept/property relation than the natural thought seems to presuppose. These observations, it is argued, rule out giving a theory of conceptual evaluation that is a corollary of a metaphysical ranking of the relevant properties. -/- conceptual ethics, explanation, naturalness, epistemic value, concept/property, semantic internalism. (shrink)
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