I argue that the contrast between models and theories is important for public policy issues. I focus especially on the way a mathematical model explains just one aspect of the data.

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics (...) textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (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.

Mathematical models of biological patterns are central to contemporary biology. This paper aims to consider what these models contribute to biology through the detailed consideration of an important case: Hamilton’s selfish herd. While highly abstract and idealized, Hamilton’s models have generated an extensive amount of research and have arguably led to an accurate understanding of an important factor in the evolution of gregarious behaviors like herding and flocking. I propose an account of what these models are able to achieve (...) and how they can support a successful scientific research program. I argue that the way these models are interpreted is central to the success of such programs. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. (...) Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

We have developed a simple mathematical model with three physiologically significant states to describe the changes in intrauterine pressure associated with a contraction during human parturition. The myometrium is modelled as a set of smooth muscle cells, each of which is in one of three states (quiescent, contracted, refractory) at a given time. These states are occupied according to a cycle governed by three temporal parameters. The solutions of the equations describing the model show an oscillatory behavior for particular (...) values of these parameters, which is very similar to the time dependant development of intrauterine pressure during labor. Due to its non-linear terms, our model could lead to chaotic oscillations (in the mathematical sense), whose clinical counterpart may occur in cases of dystocia. Despite its simplicity, this model appears to be a useful guide to further investigations of the oscillatory behavior of the myometrium, or other smooth muscles, in normal and pathological situations. (shrink)

This article contributes to the revision of the procedure of robustness analysis of mathematical models in epistemic democracy using the systematic review method. It identifies the drawbacks of robustness analysis in epistemic democracy in terms of sample universality and inference from samples with the same results. To exemplify the effectiveness of systematic review, this article conducted a pilot review of diversity trumps ability theorem models, which are mathematical models of deliberation often cited by epistemic democrats. A review of (...) nine models extracted from 352 papers exemplifies the effectiveness of robustness analysis supplemented by systematic review in epistemic democracy. (shrink)

The in vitro spontaneous contractions of human myometrium samples can be described using a phenomenological model involving different cell states and adjustable parameters. In patients not receiving hormone treatment, the dynamic behavior could be described using a three-state model similar to the one we have already used to explain the oscillations of intra-uterine pressure during parturition. However, the shape of the spontaneous contractions of myometrium from patients on progestin treatment was different, due to a two-step relaxation regime including a latched (...) phase which cannot be simulated using the previous model without introducing an ad hoc mechanism to account for the extra energy involved in this sustained contraction. One way to do this is to introduce an anomalous rate of ATP consumption, the biochemical reasons for which have not yet been elucidated and which cannot be mathematically simulated using our experimental data. An alternative explanation is the reduced cycling rate of actin-myosin cross-bridges known to occur during the latch-phase. Our experimental findings suggest a third possibility, namely a sol-gel transition with a specific relaxation time constant, which would maintain a significant part of the cell population in the contracted-state until the intracellular-medium returns to its normal fluid behavior. (shrink)

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in (...) discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely (...) homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)

Tuberculosis has continued to retain its title as “the captain among these men of death”. This is evident as it is the leading cause of death globally from a single infectious agent. TB as it is fondly called has become a major threat to the achievement of the sustainable development goals (SDG) and hence require inputs from different research disciplines. This work presents a mathematical model of tuberculosis. A compartmental model of seven classes was used in the model formulation (...) comprising of the susceptible S, vaccinated V, exposed E, undiagnosed infectious I1, diagnosed infectious I2, treated T and recovered R. The stability analysis of the model was established as well as the condition for the model to undergo backward bifurcation. With the existence of backward bifurcation, keeping the basic reproduction number less than unity $$({R_{0}}<1)$$ is no more sufficient to keep TB out of the community. Hence, it is shown by the analysis that vaccination program, diagnosis and treatment helps to control the TB dynamics. In furtherance to that, it is shown that preference should be given to diagnosis over treatment as diagnosis precedes treatment. It is as well shown that at lower vaccination rate (0–20%), TB would still be endemic in the population. As such, high vaccination rate is required to send TB out of the community. (shrink)

Recently, a number of philosophers of biology have endorsed views about random drift that, we will argue, rest on an implicit assumption that the meaning of concepts such as drift can be understood through an examination of the mathematical models in which drift appears. They also seem to implicitly assume that ontological questions about the causality of terms appearing in the models can be gleaned from the models alone. We will question these general assumptions by showing how the same (...) equation — the simple 2 = p2 + 2pq + q2 — can be given radically different interpretations, one of which is a physical, causal process and one of which is not. This shows that mathematical models on their own yield neither interpretations nor ontological conclusions. Instead, we argue that these issues can only be resolved by considering the phenomena that the models were originally designed to represent and the phenomena to which the models are currently applied. When one does take those factors into account, starting with the motivation for Sewall Wright’s and R.A. Fisher’s early drift models and ending with contemporary applications, a very different picture of the concept of drift emerges. On this view, drift is a term for a set of physical processes, namely, indiscriminate sampling processes. (shrink)

John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays. Cambridge: Cambridge University Press, 2008. xiii + 301 pp. $90.00, £50.00. ISBN 978-0-521-88034-3. Adobe eBook, $...

Schistosomiasis, a vector-borne chronically debilitating infectious disease, is a serious public health concern for humans and animals in the affected tropical and sub-tropical regions. We formulate and theoretically analyze a deterministic mathematical model with snail and bovine hosts. The basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} is computed and used to investigate the local stability of the model’s steady states. Global stability of the endemic equilibrium is carried out by constructing a suitable Lyapunov (...) function. Sensitivity analysis shows that the basic reproduction number is most sensitive to the model parameters related to the contaminated environment, namely: shedding rate of cercariae by snails, cercariae to miracidia survival probability, snails-miracidia effective contact rate and natural death rate of miracidia and cercariae. Numerical results show that when no intervention measures are implemented, there is an increase of the infected classes, and a rapid decline of the number of susceptible and exposed bovines and snails. Effects of the variation of some of the key sensitive model parameters on the schistosomiasis dynamics as well as on the initial disease transmission threshold parameter R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document} are graphically depicted. (shrink)

The globalization in business and tourism becomes crucial more and more for the economical sustainability of local communities. In the presence of an epidemic outbreak, there must be such a decision on the policy by the host community as whether to accept visitors or not, the number of acceptable visitors, or the condition for acceptable visitors. Making use of an SIRI type of mathematical model, we consider the influence of visitors on the spread of a reinfectious disease in a (...) community, especially assuming that a certain proportion of accepted visitors are immune. The reinfectivity of disease here means that the immunity gained by either vaccination or recovery is imperfect. With the mathematical results obtained by our analysis on the model for such an epidemic dynamics of resident and visitor populations, we find that the acceptance of visitors could have a significant influence on the disease’s endemicity in the community, either suppressive or supportive. (shrink)

John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be of interest to a wide range of readers across (...) class='Hi'>philosophy of mathematics, logic, and philosophy of language. (shrink)

Mathematics from the electromagnetic field quantization procedure and the soliton models of photons are used to construct a new 3D model of photons. Besides the interaction potential between the charged particle and the photons, which contains the annihilation and creation operators of photons, the new function for a description of free propagating photons is derived. This function presents the vector potential of the field, the function is a product of the harmonic oscillator eigenfunction with the well-defined coordinate of the oscillator (...) and the Gaussian function of the polar radius in the transverse direction. In the article, the difference between the quantum mechanics of particles and photons is discussed. (shrink)

In this essay I argue against I. Bernard Cohen's influential account of Newton's methodology in the Principia: the 'Newtonian Style'. The crux of Cohen's account is the successive adaptation of 'mental constructs' through comparisons with nature. In Cohen's view there is a direct dynamic between the mental constructs and physical systems. I argue that his account is essentially hypothetical-deductive, which is at odds with Newton's rejection of the hypothetical-deductive method. An adequate account of Newton's methodology needs to show how Newton's (...) method proceeds differently from the hypothetical-deductive method. In the constructive part I argue for my own account, which is model based: it focuses on how Newton constructed his models in Book I of the Principia. I will show that Newton understood Book I as an exercise in determining the mathematical consequences of certain force functions. The growing complexity of Newton's models is a result of exploring increasingly complex force functions (intra-theoretical dynamics) rather than a successive comparison with nature (extra-theoretical dynamics). Nature did not enter the scene here. This intra-theoretical dynamics is related to the 'autonomy of the models'. (shrink)

We propose simple nonlinear mathematical models for the legal concept of balancing of interests. Our aim is to bridge the gap between an abstract formalisation of a balancing decision while assuring consistency and ultimately legal certainty across cases. We focus on the conflict between the rights to privacy and to the protection of personal data in Art. 7 and Art. 8 of the EU Charter of Fundamental Rights (EUCh) against the right of access to information derived from Art. 11 (...) EUCh. These competing rights are denoted by ( \(i_1\) ) _right to privacy _ and ( \(i_2\) ) _access to information_; mathematically, their indices are respectively assigned by \(u_1\in [0,1]\) and \(u_2\in [0,1]\) subject to the constraint \(u_1+u_2=1\). This constraint allows us to use one single index _u_ to resolve the conflict through balancing. The outcome will be concluded by comparing the index _u_ with a prior given threshold \(u_0\). For simplicity, we assume that the balancing depends on only selected legal criteria such as the social status of affected person, and the sphere from which the information originated, which are represented as inputs of the models, called legal parameters. Additionally, we take “time” into consideration as a legal criterion, building on the European Court of Justice’s ruling on the right to be forgotten: by considering time as a legal parameter, we model how the outcome of the balancing changes over the passage of time. To catch the dependence of the outcome _u_ by these criteria as legal parameters, data were created by a fully-qualified lawyer. By comparison to other approaches based on machine learning, especially neural networks, this approach requires significantly less data. This might come at the price of higher abstraction and simplification, but also provides for higher transparency and explainability. Two mathematical models for _u_, a time-independent model and a time-dependent model, are proposed, that are fitted by using the data. (shrink)

This paper is concerned with the formulation and analysis of an epidemic model of COVID-19 governed by an eight-dimensional system of ordinary differential equations, by taking into account the first dose and the second dose of vaccinated individuals in the population. The developed model is analyzed and the threshold quantity known as the control reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{0}$$\end{document} is obtained. We investigate the equilibrium stability of the system, and the COVID-free equilibrium (...) is said to be locally asymptotically stable when the control reproduction number is less than unity, and unstable otherwise. Using the least-squares method, the model is calibrated based on the cumulative number of COVID-19 reported cases and available information about the mass vaccine administration in Malaysia between the 24th of February 2021 and February 2022. Following the model fitting and estimation of the parameter values, a global sensitivity analysis was performed by using the Partial Rank Correlation Coefficient (PRCC) to determine the most influential parameters on the threshold quantities. The result shows that the effective transmission rate (α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha )$$\end{document}, the rate of first vaccine dose (ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi )$$\end{document}, the second dose vaccination rate (σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma )$$\end{document} and the recovery rate due to the second dose of vaccination (η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta )$$\end{document} are the most influential of all the model parameters. We further investigate the impact of these parameters by performing a numerical simulation on the developed COVID-19 model. The result of the study shows that adhering to the preventive measures has a huge impact on reducing the spread of the disease in the population. Particularly, an increase in both the first and second dose vaccination rates reduces the number of infected individuals, thus reducing the disease burden in the population. (shrink)

Understanding the mechanisms and the time and spatial evolution of penumbra following an ischemic stroke is crucially important for developing therapeutics aimed at preventing this area from evolving towards infarction. To help in integrating the available data, we decided to build a formal model. We first collected and categorised the major available evidence from animal models and human observations and summarized this knowledge in a flow-chart with the potential key components of an evolving stroke. Components were grouped in ten sub-models (...) that could be modelled and tested independently: the sub-models of tissue reactions, ionic movements, oedema development, glutamate excitotoxicity, spreading depression, NO synthesis, inflammation, necrosis, apoptosis, and reperfusion. Then, we figured out markers, identified mediators and chose the level of complexity to model these sub-models. We first applied this integrative approach to build a model based on cytotoxic oedema development following a stroke. Although this model includes only three sub-models and would need to integrate more mechanisms in each of these sub-models, the characteristics and the time and spatial evolution of penumbra obtained by simulation are qualitatively and, to some extent, quantitatively consistent with those observed using medical imaging after a permanent occlusion or after an occlusion followed by a reperfusion. (shrink)

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms (...) of scientific modelling. I argue against both the Conceptualist and Platonist interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics. (shrink)

This paper develops a simple, conceptual, mathematical model for the evolution of holobionts. Its purpose is to clarify how holobiont selection may cause evolutionary change in the traits of holobionts.

A holobiont is a composite organism consisting of a host together with its microbiome, such as a coral with its zooxanthellae. To explain the often intimate integration between hosts and their microbiomes, some investigators contend that selection operates on holobionts as a unit and view the microbiome’s genes as extending the host’s nuclear genome to jointly comprise a hologenome. Because vertical transmission of microbiomes is uncommon, other investigators contend that holobiont selection cannot be effective because a holobiont’s microbiome is an (...) acquired condition rather than an inherited trait. This disagreement invites a simple mathematical model to see how holobiont selection might operate and to assess its plausibility as an evolutionary force. This paper presents two variants of such a model. In one variant, juvenile hosts obtain microbiomes from their parents (vertical transmission). In the other variant, microbiomes of juvenile hosts are assembled from source pools containing the combined microbiomes of all parents (horizontal transmission). According to both variants, holobiont selection indeed causes evolutionary change in holobiont traits. Therefore, holobiont selection is plausibly an effective evolutionary force with either mode of microbiome transmission. The modeling employs two distinct concepts of inheritance, depending on the mode of microbiome transmission: collective inheritance whereby juveniles inherit a sample of the collected genomes from all parents, as contrasted with lineal inheritance whereby juveniles inherit the genomes from only their own parents. A distinction between collective and lineal inheritance also features in theories of multilevel selection. (shrink)

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a (...) long time, the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

Herbert A. Simon's reply (Inquiry, Vol. 25, No. 3) to my criticism of his 1954 paper is not to the point. He fails to respond to some of my arguments and misconceives others. One of his misconceptions is that any mathematical deduction from empirical premises which are formulated mathematically will necessarily lead to empirically valid conclusions. This claim is particularly unwarrantable in Simon's case since his mathematical premise, the continuity of the reaction function, is empirically meaningless.

We argue that dynamical and mathematical models in systems and cognitive neuro- science explain (rather than redescribe) a phenomenon only if there is a plausible mapping between elements in the model and elements in the mechanism for the phe- nomenon. We demonstrate how this model-to-mechanism-mapping constraint, when satisfied, endows a model with explanatory force with respect to the phenomenon to be explained. Several paradigmatic models including the Haken-Kelso-Bunz model of bimanual coordination and the difference-of-Gaussians model of visual receptive fields (...) are explored. (shrink)

A mathematical model of the natural origin of our universe is presented. The model is based only on well-established physics. No claim is made that this model uniquely represents exactly how the universe came about. But the viability of a single model serves to refute any assertions that the universe cannot have come about by natural means.

Prior to statistical mechanics and especially before the advent of the new quantum mechanics, it was traditionally held, following in the main the Kantian philosophy, that the task of science is to attain a unique quantitative representation of reality. It was thought—and with justifiable zeal—that a scientific discipline is exact to the extent to which a mathematical pattern yielding quantitative relations can be applied to its subject matter. This view was based on the implicit assumptions that functional relatedness (...) conducive to quantitative evaluation is bound to reveal the most generic, and hence, the most essential traits of events. (shrink)

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact (...) of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

Rather than defend each instance of problem suggested by Williams, what I propose to do is to respond by making two clarificatory points: first, I rule out two ways of understanding mathematical problems that might be clouding the water; and then, second, I further characterise how Deleuze thinks some mathematical problems, two in particular, are not just examples of mathematical problems, but provide mathematical models for what a mathematical problem in general can be understood to (...) be. This is important because it is how mathematical problems can be modelled in mathematics that provides Deleuze with a model for how problems can be modelled in other discourses. This will then get to the bottom of what is at stake in my approach to Deleuze and to Deleuze studies in general. (shrink)

This article argues that many, if not most, behavior descriptions and sequencing are in essence an interpretation of signs, and are evaluated as sequences of signs by researchers. Thus, narrative analysis, as developed by Barthes and others, seems best suited to be used in behavioral/biosemiotic studies rather than mathematical modeling, and is very similar to some classic ethology methods. As our brain interprets behaviors as signs and attributes meaning to them, narrative analysis seems more suitable than mathematical modeling (...) to describe and study behavior. Mathematical models are, on many occasions, extremely reductionist and simplifying because of computational and/or numerical representation limitations that lead to errors and straitjacketing interpretations of reality. Since actual animals are not as optimal in real life as our mathematical models, here it is proposed that we should consider logical/verbal models and semantic interpretations as equally or even better suited for behavioral analysis, and refrain from enforcing mathematical modeling as the only way to study and understand biological problems, especially those of a behavioral and biosemiotic nature. (shrink)

In this paper I consider some data of Covid-19 infection in Italy from the 20th of February to the 29th of June 2020. Data are analyzed using some fits based on mathematical models. This analysis may be proposed to students of the last class of the Liceo Scientifico in order to debate a real problem with mathematical tools.

The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...) with successful representation, scientists often rely on the existence of a ‘mature mathematical formalism’, where the latter refers to a—mathematically formulated and physically interpreted—notational system of locally applicable rules that derive from (but need not be reducible to) fundamental theory. As mathematical formalisms undergo a process of elaboration, enrichment, and entrenchment, they come to embody theoretical, ontological, and methodological commitments and assumptions. Since these are enshrined in the formalism itself, they are no longer readily obvious to either the novice or the proficient user. At the same time as formalisms constrain what may be represented, they also function as inferential and interpretative resources. (shrink)

Although every emerging infectious disease occurs in a unique context, the behaviour of previous pandemics offers an insight into the medium- and long-term outcomes of the current threat. Where an informative historical analogue exists, epidemiologists and policymakers should consider how the insights of the past can inform current forecasts and responses.

An introduction to the model-theoretic approach in the philosophy of science is given and it is argued that this program is further enhanced by the introduction of partial structures. It is then shown that this leads to a natural and intuitive account of both "iconic" and mathematical models and of the role of the former in science itself.

Philosophy can shed light on mathematical modeling and the juxtaposition of modeling and empirical data. This paper explores three philosophical traditions of the structure of scientific theory—Syntactic, Semantic, and Pragmatic—to show that each illuminates mathematical modeling. The Pragmatic View identifies four critical functions of mathematical modeling: (1) unification of both models and data, (2) model fitting to data, (3) mechanism identification accounting for observation, and (4) prediction of future observations. Such facets are explored using a recent (...) exchange between two groups of mathematical modelers in plant biology. Scientific debate can arise from different modeling philosophies. (shrink)