Optimisation and mathematical explanation: doing the Lévy Walk

The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra-mathematical explanation (the explanation of physical facts by mathematical facts). In this paper, I identify a new case of extra-mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra-mathematical explanation in science.     

Evidence,explanation and enhanced indispensability

In this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument (EIA) proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects.     

LOT 2: The Language of Thought Revisited

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is.

I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.     

Book Review

The main purpose of the paper concerns the question of the existence of hard mathematical facts as truth-makers of mathematical sentences. The paper defends the standpoint according to which hard mathematical facts do not exist in semantic models of mathematical theories. The argumentative line in favour of the defended thesis proceeds as follows: (i) slingshot arguments supply us with some reasons to reject various ontological theories of mathematical facts; (ii) there are two ways of blocking these arguments: through the rejection of the principle of extensionality for individual terms (Ext) or through the rejection of the principle of Wittgenstein; (iii) the first way cannot be accepted because it leads to the practice of softening mathematical facts; (iv) the second way, called fine-graining facts, cannot be accepted because it also results in the practice of softening facts. (v) Hence, only soft mathematical facts (SMFs) can be introduced into semantic models of mathematical theories. Because they should be rather interpreted as mental representations of mathematical objects, they no longer satisfy the semantic role of truth-makers in relation to mathematical sentences. The argument formulated in the paper is based on the analysis of mathematical extensions of the basic non-Fregean logic. Sixty-two definitions of potential fact-identity connectives are also presented. The paper's conclusion may be interpreted as undermining the situational paradigm of building semantics for mathematical languages.     

Nominalistic content,grounding, and covering generalizations: Reply to ‘Grounding and the indispensability argument’

‘Grounding and the indispensability argument’ presents a number of ways in which nominalists can use the notion of grounding to rebut the indispensability argument for the existence of mathematical objects. I will begin by considering the strategy that puts grounding to the service of easy-road nominalists (“Nominalistic content meets grounding” section). I will give some support to this strategy by addressing a worry some may have about it (“A misguided worry about the grounding strategy” section). I will then consider a problem for the fast-lane strategy (“Grounding and generality: a problem for the fast lane” section) and a problem for easy-road nominalists willing to accept Liggins’ grounding strategy (“More on the grounding strategy and easy-road nominalism” section). Both are related to the problem of formulating nominalistic explanations at the right level of generality. I will then consider a problem that Liggins only hints at (“Mathematics and covering generalizations” section). This problem has to do with mathematics’ function of providing the sort of covering generalizations we need in scientific explanations.     

Normative explanation unchained

Normative theories aim to explain why things have the normative features they have. This paper argues that, contrary to some plausible existing views, one important kind of normative explanations which first-order normative theories aim to formulate and defend can fail to transmit downward along chains of metaphysical determination of normative facts by non-normative facts. Normative explanation is plausibly subject to a kind of a justification condition whose satisfaction may fail to transmit along the relevant kind of metaphysical hierarchy. A broader aim of the paper is to contribute to systematic theorizing about normative explanation: whereas there has been a great deal of work on scientific explanation, there has been little by way of systematic exploration of what sort of explanations normative theories aim to formulate and defend.     

Effects of self‐explanation as a metacognitive strategy for solving mathematical word problems1

Abstract: This study examined how a metacognitive strategy known as self‐explanation influences word problem solving in elementary school children. Participants were 79 sixth‐graders. They were assigned to one of three groups: the self‐explanation group, the self‐learning group, or the control group. Students in each group performed a ratio word problem test and a transfer test. The results showed that students in the self‐explanation group outperformed students in the other two groups on both the ratio word problem test and on the transfer test. In addition, high explainers who generated more self‐explanations relating to deep understanding of worked‐out examples outperformed low explainers on both ratio word problem and transfer tests. The self‐explanation effect is discussed.     

Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist.

     

Investigating the origins of political views: biases in explanation predict conservative attitudes in children and adults

We tested the hypothesis that political attitudes are influenced by an information‐processing factor – namely, a bias in the content of everyday explanations. Because many societal phenomena are enormously complex, people's understanding of them often relies on heuristic shortcuts. For instance, when generating explanations for such phenomena (e.g., why does this group have low status?), people often rely on facts that they can retrieve easily from memory – facts that are skewed toward inherent or intrinsic features (e.g., this group is unintelligent). We hypothesized that this bias in the content of heuristic explanations leads to a tendency to (1) view socioeconomic stratification as acceptable and (2) prefer current societal arrangements to alternative ones, two hallmarks of conservative ideology. Moreover, since the inherence bias in explanation is present across development, we expected it to shape children's proto‐political judgments as well. Three studies with adults and 4‐ to 8‐year‐old children (N = 784) provided support for these predictions: Not only did individual differences in reliance on inherent explanations uniquely predict endorsement of conservative views (particularly the stratification‐supporting component; Study 1), but manipulations of this explanatory bias also had downstream consequences for political attitudes in both children and adults (Studies 2 and 3). This work contributes to our understanding of the origins of political attitudes.     

Phenomenological Naturalism

Naturalists seek to ground what exists in a set of fundamental metaphysical principles that they call ‘nature’. But metaphysical principles can’t function as fundamental explanatory grounds, since their ability to explain anything depends on the intelligibility granted by transcendental structures. What makes metaphysical principles intelligible, what unifies them, and allows them to characterize the being of worldly objects are the transcendental structures through which worldly objects are manifest. This means that the search for fundamental explanatory grounds must go deeper than the postulation of brute metaphysical facts. But this search cannot end with transcendental structures either, since the mode of being of transcendental subjects also calls out for explanation. Conceiving of transcendental subjects through the concept of being-in-the-world ties the mode of being of subjects to the world they inhabit. What grounds the existence of worldly objects, and what grounds our existence as being-in-the-world is nature: a principle that is neither an object, nor a subject – a principle that makes possible our encounters with intelligible worldly things.     

Expressivism and the Reliability Challenge

Suppose that there are objective normative facts and our beliefs about such facts are by-and-large true. How did this come to happen? This is the reliability challenge to normative realism. As has been recently noted, the challenge also applies to expressivist “quasi-realism”. I argue that expressivism is useful in the face of this challenge, in a way that has not been yet properly articulated. In dealing with epistemological issues, quasi-realists typically invoke the desire-like nature of normative judgments. However, this is not enough to prevent the reliability challenge from arising, given that quasi-realists also hold that normative judgments are truth-apt beliefs. To defuse this challenge, we need to isolate a deeper sense in which normative thought is not representational. I propose that we rely on the negative functional thesis of expressivism: normative thought does not have the function of tracking normative facts, or any other kind of facts. This thesis supports an argument to the effect that it is misguided to expect an explanation of our access to normative facts akin to the explanations available in regions of thought that have a tracking function. We should be content with explanations of our reliability that take for granted certain connections between our psychology and the normative truths.     

Moral Explanations of Moral Beliefs

Gilbert Harman and Judith Thomson have argued that moral facts cannot explain our moral beliefs, claiming that such facts could not play a causal role in the formation of those beliefs. This paper shows these arguments to be misguided, for they would require that we abandon any number of intuitively plausible explanations in non-moral contexts as well. But abandoning the causal strand in the argument over moral explanations does not spell immediate victory for the moral realist, since it must still be shown that moral facts do figure in our best global explanatory theory.     

The location problem in social ontology

Mental, mathematical, and moral facts are difficult to accommodate within an overall worldview due to the peculiar kinds of properties inherent to them. In this paper I argue that a significant class of social entities also presents us with an ontological puzzle that has thus far not been addressed satisfactorily. This puzzle relates to the location of certain social entities. Where, for instance, are organizations located? Where their members are, or where their designated offices are? Organizations depend on their members for their existence, but the members of an organization can be where the organization is not. The designated office of an organization, however, need be little more than a mailbox. I argue that the problem can be solved by conceptualizing the relation between social entities and non-social entities as one of constitution, a relation of unity without identity. Constituted objects have properties that cannot be reduced to properties of the constituting objects. Thus, my attempt to solve the Location Problem results in an argument in favor of a kind of non-reductive materialism about the social.     

Young children’s learning and generalization of teleological and mechanistic explanations

Young children often endorse explanations of the natural world that appeal to functions or purpose—for example, that rocks are pointy so animals can scratch on them. By contrast, most Western-educated adults reject such explanations. What accounts for this change? We investigated 4- to 5-year-old children’s ability to generalize the form of an explanation from examples by presenting them with novel teleological explanations, novel mechanistic explanations, or no explanations for 5 nonliving natural objects. We then asked children to explain novel instances of the same objects and novel kinds of objects. We found that children were able to learn and generalize explanations of both types, suggesting an ability to draw generalizations over the form of an explanation. We also found that teleological and mechanistic explanations were learned and generalized equally well, suggesting that if a domain-general teleological bias exists, it does not manifest as a bias in learning or generalization.     

Working Memory Capacity and Self‐Explanation Strategy Use Provide Additive Problem‐Solving Benefits

The present study examined the impact of working memory capacity (WMC) on college students' ability to solve probability problems while using a self‐explanation strategy. Participants learned to solve probability problems in one of three conditions: a backward‐faded self‐explanation condition, an example problem pairs self‐explanation condition, or a control (no self‐explanation) condition. Even when accounting for the impact of WMC, learning to problem‐solve using self‐explanation led to superior problem‐solving performance. Conditions that prompted self‐explanation during problem‐solving resulted in significantly better problem‐solving performance than the control condition. These findings provide insight into the influence of individual differences on problem‐solving when strategies are provided, as well as information about the effectiveness of the self‐explanation strategy during mathematical problem‐solving. Copyright © 2016 John Wiley & Sons, Ltd.     

“Looking Up” and “Looking Down”: On the Dual Character of Mechanistic Explanations

Mechanistic explanation is at present the received view of scientific explanation. One of its central features is the idea that mechanistic explanations are both “downward looking” and “upward looking”: they explain by offering information about the internal constitution of the mechanism as well as the larger environment in which the mechanism is situated. That is, they offer both constitutive and contextual explanatory information. Adequate mechanistic explanations, on this view, accommodate the full range of explanatory factors both “above” and “below” the target phenomenon. The aim of this paper is to demonstrate that mechanistic explanation cannot furnish both constitutive and contextual information simultaneously, because these are different types of explanation with distinctly different aims. Claims that they can, I argue, depend on several intertwined confusions concerning the nature of explanation. Particularly, such claims tend to conflate mechanistic and functional explanation, which I argue ought to be understood as distinct. Conflating them threatens to oversell the explanatory power of mechanisms and obscures the means by which they explain. I offer two broad reasons in favor of distinguishing mechanistic and functional explanation: the first concerns the direction of explanation of each, and the second concerns the type of questions to which these explanations offer answers. I suggest an alternative picture on which mechanistic explanation is understood as fundamentally constitutive, and according to which an adequate understanding of a phenomenon typically requires supplementing the mechanistic explanation with a functional explanation.     

Reasons as explanations

Can a normative reason be understood as a kind of explanation? I here consider and argue against two important analyses of reasons as explanations. John Broome argues that we can analyze reasons in terms of the concepts of explanation and ought. On his view, reasons to ? are either facts that explain why one ought to ? (what he calls “perfect reasons”) or facts that play a for-? role in weighing explanations (what he calls “pro tanto reasons”). I argue against Broome’s account of both perfect and pro tanto reasons. Other philosophers, including Joseph Raz, analyze reasons in terms of the concepts of explanation and good. On this view, some fact is a reason to ? if and only if that fact explains why ?-ing would be good in some respect, to some degree. This view avoids the objections to Broome’s view, but should be rejected since not all explanations of why ?-ing would be good constitute reasons to ?.     

In Search of the Underlying Mechanism of the Disposition Effect

Using comprehensive data from the Finnish stock market, we assess the explanatory value of the three most commonly cited explanations for the disposition effect: prospect theory, belief in mean reversion, and escalation of commitment. In general, the results provide evidence for the presence of the disposition effect. More importantly, the effect appears to be significantly more pronounced when investors are personally responsible for the initial investment decision. This finding suggests that investor behavior is influenced above all by self‐justificatory concerns, an interpretation that is consistent with the escalation of commitment‐based explanation of the disposition effect. Copyright © 2011 John Wiley & Sons, Ltd.     

“I Want to Know More!”: Children Are Sensitive to Explanation Quality When Exploring New Information

When someone encounters an explanation perceived as weak, this may lead to a feeling of deprivation or tension that can be resolved by engaging in additional learning. This study examined to what extent children respond to weak explanations by seeking additional learning opportunities. Seven‐ to ten‐year‐olds (N = 81) explored questions and explanations (circular or mechanistic) about 12 animals using a novel Android tablet application. After rating the quality of an initial explanation, children could request and receive additional information or return to the main menu to choose a new animal to explore. Consistent with past research, there were both developmental and IQ‐related differences in how children evaluated explanation quality. But across development, children were more likely to request additional information in response to circular explanations than mechanistic explanations. Importantly, children were also more likely to request additional information in direct response to explanations that they themselves had assigned low ratings, regardless of explanation type. In addition, there was significant variability in both children's explanation evaluation and their exploration, suggesting important directions for future research. The findings support the deprivation theory of curiosity and offer implications for education.     

The Affective Experience of Aesthetic Properties

It is widely agreed upon that aesthetic properties, such as grace, balance, and elegance, are perceived. I argue that aesthetic properties are experientially attributed to some non‐perceptible objects. For example, a mathematical proof can be experienced as elegant. In order to give a unified explanation of the experiential attribution of aesthetic properties to both perceptible and non‐perceptible objects, one has to reject the idea that aesthetic properties are perceived. I propose an alternative view: the affective account. I argue that the standard case of experiential aesthetic property attribution is affective experience.