The structuralist approach represents the relation between a model and physical system as a relation between two mathematical structures. However, since a physical system is _prima facie_ _not_ a mathematical structure, the structuralist approach seemingly fails to represent the fact that science is about concrete, physical reality. In this paper, I take up this _problem of lost reality_ and suggest how it may be solved in a purely structuralist fashion. I start by briefly introducing both the structuralist approach and the (...) problem of lost reality and discussing the various (non-structuralist) solutions that have been proposed in the literature. Following this, I decompose the problem into the _ontological mismatch_ and _specification_ problems. In response to the former, I present a _metascientific dissolution argument_, according to which the difference in kind between mathematical structures and physical systems poses no deep obstacle to the structuralist approach, and consider some upshots of this argument for our views on representation. By way of conclusion, I argue that the metascientific dissolution argument paves the way for a solution to the specification problem as well. (shrink)

This paper argues for a representational semantic conception of scientific theories, which respects the bare claim of any semantic view, namely that theories can be characterised as sets of models. RSC must be sharply distinguished from structural versions that assume a further identity of ‘models’ and ‘structures’, which we reject. The practice-turn in the recent philosophical literature suggests instead that modelling must be understood in a deflationary spirit, in terms of the diverse representational practices in the sciences. These insights are (...) applied to some mathematical models, thus showing that the mathematical sciences are not in principle counterexamples to RSC. (shrink)

RESUMEN La sección 1 introduce lo que llamo la tesis del colapso de las estructuras matemáticas y las estructuras físicas. La sección 2 examina si acaso la indispensabilidad de las matemáticas para la física fundamental involucra la adopción del platonismo matemático, en este caso acerca de estructuras matemáticas, como argumenta el realismo estructural óntico. La sección 3 muestra que la adopción de la tesis del colapso arriesga introducir la hipótesis del universo matemático. Desde la perspectiva de la concepción inferencial en (...) filosofía de las matemáticas aplicadas, la sección 4 examina aspectos epistémicos del rol de las matemáticas en la teorización científica, derivando distinciones entre estructuras matemáticas y estructuras físicas. El argumento concluye en la sección 5 indicando algunos límites para la presente estrategia. ABSTRACT Section 1 introduces what I call the collapse thesis, which holds that the boundaries of mathematical and physical structures blur or even vanish in certain areas of fundamental physics. Section 2 examines the claim that the indispensability of mathematics to fundamental physics requires the adoption of mathematical Platonism, this time concerning the reality of mathematical structures, as ontic structural realism contends. Section 3 demonstrates that the collapse thesis introduces the threat of the mathematical universe hypothesis. Then, from the perspective of the inferential conception in the philosophy of applied mathematics, section 4 examines epistemic features of the role of mathematics in scientific theorizing, drawing distinctions between mathematical and physical structures. Lastly, the argument closes in section 5 pointing out some of the limits of the proposed strategy for a distinction. (shrink)

‘Mapping accounts’ of applied mathematics hold that the application of mathematics in physical science is best understood in terms of ‘mappings’ between mathematical structures and physical structures. In this paper, I suggest that mapping accounts rely on the assumption that the mathematics relevant to any application of mathematics in empirical science can be captured in an appropriate mathematical structure. If we are interested in assessing the plausibility of mapping accounts, we must ask ourselves: how plausible is this assumption as a (...) working hypothesis about applied mathematics? In order to do so, we examine the role played by mathematics in the multiscalar modelling of sea ice melting behaviour and examine whether we can indeed capture the mathematics involved in the kind of mathematical structure employed by the mapping account. Along the way, we note that the cases of applied mathematics that appear in discussions of mapping accounts almost exclusively involve the employment of a single clearly circumscribed mathematical field or domain. While the core assumption of mapping accounts may appear plausible in such situations, we ultimately suggest that the mapping account is not able to handle the important added complexities involved in our sea ice case study. In particular, the notion of mathematical structure around which such accounts are framed does not seem to be able to capture the way in which some applications of mathematics require that very different pieces of mathematics be related to one another on the basis of both mathematical and empirical information. (shrink)

Several philosophers have argued that to capture the generality of certain scientific explanations, we must count mathematical facts among their explanantia. I argue that we can better understand these explanations by adopting a more nuanced stance toward mathematical representations, recognizing the role of mathematical representation schemata in representing highly abstract features of physical systems. It is by picking out these abstract but nonmathematical features that explanations appealing to mathematics achieve a high degree of generality. The result is a rich conception (...) of the role of mathematics in scientific explanations that does not require us to treat mathematical facts as explanantia. (shrink)

The paper examines connections between ontology and finance. The ontological debates concerning the role of finance are examined between two opposing schools of thought that can be labelled, very broadly, ‘instrumentalist’ and ‘realist’. These two schools of thought have had momentous repercussions in understanding what is a good society. Each school defines Nature in particular ways which can be explored using ontology and philosophical insight. Our theoretical investigation aims to accommodate Nature in community financial deliberations. A positive role for government (...) is advocated to finance environmental infrastructure initiatives. For example, precautionary strategies to address climate change must be funded. New roles for finance and government are proposed to align human relationships with Nature. Environmental precautionary principles must be developed in conjunction with finance theory to maintain decent standards of living for all. Reliance on impersonal market forces will not be enough to save the planet given the power of some over the many in the neoliberal arena. (shrink)

Recent debates about mathematical ontology are guided by the view that Platonism's prospects depend on mathematics' explanatory role in science. If mathematics plays an explanatory role, and in the right kind of way, this carries ontological commitment to mathematical objects. Conversely, the assumption goes, if mathematics merely plays a representational role then our world-oriented uses of mathematics fail to commit us to mathematical objects. I argue that it is a mistake to think that mathematical representation is necessarily ontologically innocent and (...) that there is an argument from mathematics' representational capacity to Platonism. Given that it is common ground between the Platonist and nominalist that mathematics plays a representational role in science, this representationalist argument is to be preferred over the explanatory, or enhanced, indispensability argument. (shrink)

Gibbsian statistical mechanics (GSM) is the most widely used version of statistical mechanics among working physicists. Yet a closer look at GSM reveals that it is unclear what the theory actually says and how it bears on experimental practice. The root cause of the difficulties is the status of the Averaging Principle, the proposition that what we observe in an experiment is the ensemble average of a phase function. We review different stances toward this principle, and eventually present a coherent (...) interpretation of GSM that provides an account of the status and scope of the principle. (shrink)

Defenders of the enhanced indispensability argument argue that the most effective route to platonism is via the explanatory role of mathematical posits in science. Various compelling cases of mathematical explanation in science have been proposed, but a satisfactory general philosophical account of such explanations is lacking. In this paper, I lay out the framework for such an account based on the notion of “the mathematical stance.” This is developed by analogy with Dennett’s well-known concept of “the intentional stance.” Roughly, adopting (...) the mathematical stance towards a particular physical phenomenon involves treating it as an abstract mathematical structure for the purposes of prediction and explanation. Interestingly, Dennett himself frequently draws analogies between his intentional stance towards beliefs and desires and scientists’ stance towards centers of gravity. I explore the theoretical role played by centers of gravity within science and discuss how an indispensabilist platonist ought to categorize the ontological status of this type of posit. I conclude with some thoughts on how an approach based on the mathematical stance might be developed into a more general philosophical account of the application of mathematics in science. (shrink)

Much has been written about the free energy principle (FEP), and much misunderstood. The principle has traditionally been put forth as a theory of brain function or biological self-organisation. Critiques of the framework have focused on its lack of empirical support and a failure to generate concrete, falsifiable predictions. I take both positive and negative evaluations of the FEP thus far to have been largely in error, and appeal to a robust literature on scientific modelling to rectify the situation. A (...) prominent account of scientific modelling distinguishes between model structure and model construal. I propose that the FEP be reserved to designate a model structure, to which philosophers and scientists add various construals, leading to a plethora of models based on the formal structure of the FEP. An entailment of this position is that demands placed on the FEP that it be falsifiable or that it conform to some degree of biological realism rest on a category error. To this end, I deliver first an account of the phenomenon of model transfer and the breakdown between model structure and model construal. In the second section, I offer an overview of the formal elements of the framework, tracing their history of model transfer and illustrating how the formalism comes apart from any interpretation thereof. Next, I evaluate existing comprehensive critical assessments of the FEP, and hypothesise as to potential sources of existing confusions in the literature. In the final section, I distinguish between what I hold to be the FEP—taken to be a modelling language or modelling framework—and what I term “FEP models.”. (shrink)

The principal target of this article is the reification Bruineberg et al. perceive of formalism within the literature on the variational free energy minimization (VFEM) framework. The authors do not provide a definition of reification, as none yet exists. Here I offer one. On this definition, the objects of the authors' critiques fall short of full-blown reification – as do the authors themselves.

This thesis starts with three challenges to the structuralist accounts of applied mathematics. Structuralism views applied mathematics as a matter of building mapping functions between mathematical and target-ended structures. The first challenge concerns how it is possible for a non-mathematical target to be represented mathematically when the mapping functions per se are mathematical objects. The second challenge arises out of inconsistent early calculus, which suggests that mathematical representation does not require rigorous mathematical structures. The third challenge comes from renormalisation group (...) (RG) explanations of universality. It is argued that the structural mapping between the world and a highly abstract minimal model adds little value to our understanding of how RG obtains its explanatory force. I will address the first and second challenges from the similarity perspective. The similarity account captures representations as similarity relations, providing a more flexible and broader conception of representation than structuralism. It is the specification of the respect and degree of similarity that forges mathematics into a context of representation and directs it to represent a specific system in reality. Structuralism is treatable as a tool for explicating similarity rela-tions set-theoretically. The similarity account, combined with other approaches (e.g., Nguyen and Frigg’s extensional abstraction account and van Fraassen’s pragmatic equivalence), can dissolve the first challenge. Additionally, I will make a structuralist response to the second challenge, and suggestions regarding the role of infinitesimals from the similarity perspective. In light of the similarity account, I will propose the “hotchpotch picture” as a method-ological reflection of our study of representation and explanation. Its central insight is to dissect a representation or an explanation into several aspects and use different theories (that are usually thought of competing) to appropriate each of them. Based on the hotchpotch picture, RG explanations can be dissected to the “indexing” and “inferential” conceptions of explanation, which are captured or characterised by structural mappings. Therefore, structuralism accommodates RG explanations, and the third challenge is resolved. (shrink)