Reconstruction in Philosophy of Mathematics

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 enquiry. 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

Supertasks and Numeral Systems

Physical supertasks are completed, infinite sequences of events or interactions that occur within a finite amount of time. Examples thereof have been constructed to show that infinite physical systems may violate conservation laws. It is shown in this paper that this conclusion may be critically sensitive to a selection of numeral system. Weaker numeral systems generate physical reports whose inaccuracy simulates the violation of a conservation law. Stronger numeral systems can confirm this effect by allowing a direct computation of the quantities conserved. The supertasks presented in [2], [4] are used to illustrate this phenomenon from the point of view of the new numeral system introduced in [6]

Numerical Methods for Infinite Decision-Making Processes

The new computational methodology due to Yaroslav Sergeyev (see [25–27]) makes it possible to evaluate numerically the terminal features of complete, sequential decision-making processes. By standard numerical methods, these processes have indeterminate features or seem to support paradoxical conclusions. We show that they are better regarded as a class of problems for which the numerical methods based on Sergeyev’s methodology provide a uniform technique of resolution

Investigating secondary school students' epistemologies through a class activity concerning infinity

In this paper, we report findings from a pilot study investigating school students' epistemologies of mathematics by using novel mathematics definitions. Students aged 17 and 18-year-old in Italy and the UK were asked to complete a worksheet that used a numerical approach to determine the sizes of infinite sets and were, then, invited to attend focus group interviews about their experience with the material. Thematic analysis of the interviews reveals that this approach is useful to distinguish between naïve and advanced epistemologies and using unseen mathematical definitions can help enrich our understanding of epistemologies held by students of school age

Applicability, idealization, and mathematization

In this thesis I provide a study of the applicability of mathematics. My starting point is the account of applicability offered in Hartry Field's book Science without numbers and arising from the nominalistic project carried out therein. By examining the limitations and shortcomings of Field's account, I develop a new one. My account retains the advantages and insights of Field's and avoids its difficulties, which are essentially due to its being incomplete and too restrictive. Field's account is incomplete because it does not deal with the nature and use of idealization in science. Field only describes how mathematics is applied to highly idealized physical theories (e.g. ones containing postulates which are untestable or contradicted by experiment) but he does not explain how idealization arises and why idealized theories are relevant to the actual experimental investigation of empirical phenomena. I offer such an explanation for an elementary scientific theory to which the more complex examples discussed by Field can be reduced. Even in presence of an analysis of idealization, Field's account of applicability remains problematic. The reason is that it characterizes the role of mathematics in applications in a very restrictive way, which neglects some of its most important uses. I show this by looking at several examples of applications. I then employ the resulting analysis of how mathematics enters them to give a characterization of applicability which does not suffer of the restrictiveness affecting Field's. This characterization encompasses Field's but also extends to applications he cannot adequately describe. I thus complete and extend Field's account of applicability, reaching a more comprehensive and realistic alternative.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Eulerian routing in practice

The K\"onigsberg bridge problem has played a central role in recent philosophical discussions of mathematical explanation. In this paper I look at it from a novel perspective, which is independent of explanatory concerns. Instead of restricting attention to the solved K\"onigsberg bridge problem, I consider Euler's construction of a solution method for the problem and discuss two later integrations of Euler's approach into a more structured methodology, arisen in operations research and genetics respectively. By examining Euler's work and its later developments, I achieve two main goals. First, I offer an analysis of the role played by mathematics as a problem-solving instrument within scientific enquiry. Second, I shed light on the broader significance of well known contributions to the debate on mathematical explanation. I suggest that these contributions, which are tied to a localised explanatory context, achieve a greater relevance and attain a sharper formulation when they are referred to scientific enquiry at large, as opposed to its possible explanatory outcomes alone

Mathematical problem-solving in scientific practice

In this paper I study the activity of mathematical problem-solving in scientific practice, focussing on enquiries in mathematical social science. I identify three salient phases of mathematical problem-solving and adopt them as a reference frame to investigate aspects of applications that have not yet received extensive attention in the philosophical literature

Divergent Mathematical Treatments in Utility Theory

Abstract In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper. Model Theory and Scientific Models In a 1960 paper, Patrick Suppes claimed that: [...] in the exact statement of the theory or in the exact analysis of data the notion of model in the sense of logicians provides the appropriate intellectual tool for making the analysis both precise and clear. (Suppes 1960: 295) This claim was defended against the background thesis that the meaning of the concept of model is the same in mathematics and the empirical sciences (Suppes 1960: 289). Suppes' view of models is too restrictive in two distinct ways: one of these has become clear through the recent literature on modelling, whereas the other has been neglected and provides the main motivation for the discussion presented in Arguments about the model as a theory, or a method, or a distortion of reality, all focus on the model as a scientific object and how it functions in science. Without denying this dimension of the model at all, this paper wants to broaden the perspective by claiming that the model is also a social and political device. The model will be understood as a practice connecting data, index numbers, national accounts, equations, institutes, trained personnel, laws, and policy-making. (van den Bogaard 1999: 283) It is clear that, if one is ready to accept the qualification of models for methods, distortions of reality and social devices, Suppes' more stringent semantic qualification must appear to impose very narrow, perhaps unrealistic, constraints on the study of modelling practices. It does not follow that Suppes' appeal to notions and techniques from mathematical logic should be deemed irrelevant to the study of modelling practices in general. In this paper, I seek to defend the opposite point of view by applying a model-theoretic approach to the study of mathematical modelling within utility theory. While doing so, I depart from Suppes' original aims, which presuppose, in my opinion, too strict a delimitation of the ways in which model-theoretic considerations may support philosophical investigations of scientific models. The quotation opening this section spells out the delimitation in question by restricting the mobilisation of set-theoretical semantics to the purposes of formulating scientific theories (typically as classes of models defined by a settheoretical predicate, an approach whose abstract development has been presented in Da Costa and Chuaqui 1988) or carrying out an exact analysis of data (e.g. by the embedding of a data structure into a representing structure, a strategy adopted in Da 123 (for continua) to infinity. 1 Niederée's results have shown, among other things, how one can identify measuring numbers with sets of experimental data, as well as the equivalence between certain mathematical assumptions (e.g. the Archimedean property) and properties of experimental procedures. These results shed light on important features of scientific models (especially measurement models) by modeltheoretic means, without pursuing any of the tasks recommended by Suppes, i.e., theory formulation or data analysis. The main objective of this paper is to extend along a further direction the same model-theoretic style of investigation, which recognises the value of Suppes' original proposal but transcends its limited scope. The model-theoretic machinery I shall rely upon comes from Nonstandard Analysis: it is briefly surveyed in Sect. 2 (more details are found in the ''Appendix''). I shall apply Nonstandard Analysis to two models from utility theory in order to construct an alternative mathematical treatment for the economic setups they are supposed to describe. This will allow me to show that the fragments of economic theory based on these models are crucially sensitive to a choice of mathematical treatment, more precisely, a selection of set-theoretic parameters. What this suggests is that economic theory is, at an abstract level, significantly sensitive to the choice of mathematical resources employed in its articulation. The existence of distinct choices leads to bifurcations in the kind of result one may hope to obtain. In particular, if one wishes to uphold certain normative constraints or introduce certain formal approaches, it is sometimes mandatory to drop traditional mathematical environments based on the real numbers. These remarks will be illustrated in full detail in Sects. 4-7, after a brief semi-technical preliminary. Classical and Nonstandard Analysis A vast amount of work in mathematical social science (especially economics) relies on the availability of the objects of classical analysis in the semantic metatheory. For example, in consumer theory utilities are real numbers, bundles of goods are real-valued vectors and their totality is canonically a subset of some Euclidean space. In many interesting cases there is no particular empirical motivation to select certain specific analytical objects in model-building, either because they (e.g. the metric structure on a set of alternatives ranked by a preference relation) support abstract models without having any empirical interpretation or because, even when they represent some non-mathematical content, they enter a model also as carriers of properties that have no particular connection with this content (e.g. the topological separability of the real numbers representing utilities) and yet influence what can be established about the given model. Because of this, it is of interest to consider what happens if one replaces certain canonically employed analytical objects with alternative objects. In this paper, I consider the objects of Nonstandard Analysis, 123 which share a number of properties with classical objects but are, at the same time, significantly different. I shall focus on their application to two mathematical models from utility theory. In each case, I study the consequences of using certain extensions of classical numerical sets within a Nonstandard universe as codomains of functions that are canonically selected to be real-valued. While a standard mathematisation based on real-valued functions gives rise to negative results, a Nonstandard mathematisation replaces them by positive results (which may hold under stronger conditions than were sufficient to deduce the negative results by standard means). This divergence highlights the essential relativity (i.e., with respect to a selection of mathematical resources) of negative results in economic theory, since the remarks that hold for the utility models discussed in detail admit of a general reformulation. Such a reformulation will be presented in Sect. 7, after a full discussion of the main examples has taken place, in Sects. 4-6. It is appropriate to note at this point, by way of a concluding remark, that applications of Nonstandard Analysis are not new to the field mathematical economics (see for instance, Skala 1975; Fishburn and Lavalle 1991; Lehmann 2001). However, all those of which I am aware adopt a local point of view, i.e., they construct an ultraproduct of some real structure suitable to specific modelling purposes. Moreover, they are not concerned with showing how canonical and nonstandard resources affect in divergent ways the results of modelling. My approach, on the contrary, is global in the sense that it relies on a Nonstandard universe in which several results involving nonstandard models are simultaneously obtained (this point will be clarified in Sect. 3). Moreover, it is primarily concerned with showing how canonical and nonstandard resources affect the results of modelling. A Formal Preliminary The Nonstandard universe I shall make use of in the next sections can be constructed from a collection S 0 of Urelemente (informally speaking, non-sets) that contains a copy of the set of real numbers R. One can use S 0 to generate a hierarchy of set-theoretical objects by means of the following inductive condition: where PðS n Þ is the powerset of S n . The union U of all the S i (with i a natural number) is a very rich object. It clearly contains the real numbers and all of their subsets, but it also contains the Cartesian product R n , for any natural number n, and, as a consequence, every finitary relation on the real numbers, all functions of one or several real variables and so on. The object U can then be described by means of a first-order language with identity L which contains, apart from connectives and quantifiers, the symbol 2 for set-theoretical membership and a name for every entity Footnote 2 continued refinements, D. Rizza 123 in U (including all relations, functions, sets of relations or functions etc.). 3 Thus, in particular, there are L-names for N, the set of natural numbers, or S, the set of all sequences of real numbers, both of which will be considered later. One may then use the compactness theorem of first-order logic to obtain an enlargement U 0 of the structure U ¼ hU; 2i (how this can be done is outlined in the ''Appendix''). What matters for present purposes is that an arbitrary enlargement will contain extended numerical sets à N and à R that are richer than their respective counterparts N; R, in the sense that the latter numerical sets can be embedded in their starred extensions and these include additional elements. The only additional numerical elements that will be extensively relied upon in what follows are the infinitely large numbers in à N, i.e., the elements in this set that are greater than any n 2 N (where 'greater than' is a binary relation on à N that agrees with the usual ordering on the standard numbers). The language L introduced above will also play an important role, since it allows one to state sentences that may contain names for N or R as parameters: if these sentences are true in the standard universe U, then they are true in its enlargement, with respect to à N and à R. In Sect. 6, this will make it possible to 'extend' properties of arbitrary, standard numbers to infinitely large numbers in particular. This preliminary is sufficient to introduce the models alluded to in the previous section. Ranking Information States My first example is taken from choice theory. A typical aim in this setting is to show that the choice-behaviour of an agent can be represented by a utility function. Formally, one introduces a space A of alternatives over which the agent is supposed to express or reveal her preferences by way of binary comparisons. Thus, a preference is understood to be a binary relation P on A, in particular a complete preorder, i.e., a relation that is transitive and complete. The aim is then to show the existence of a function u: A ! R, from A into the set of real numbers R such that, for any x; y 2 A: xPy iff uðxÞ uðyÞ: It is not unusual to encounter in the literature models where P is defined on an uncountably large set. A fairly recent collection of models based on this setting, the simplest of which I am going to discuss, is found in Divergent Mathematical Treatments in Utility Theory 123 main aim is to retain them in order to show that, once they have been deployed, the highly idealised character of the resulting models makes them amenable to distinct mathematical treatments that, in turn, give rise to vastly different results. In brief, substantive appeals to mathematical idealisation give rise to divergent modelling trajectories. Although I shall focus on the most basic model studied by Dubra end Echenique, it is worth noting that my remarks about it also apply, essentially unchanged, to more sophisticated variants presented in the same paper. The model in question involves an uncountably large set X of possible 'states of nature' endowed with the family of its partitions. Each partition divides the whole X into disjoint subsets and a partition P is finer than a partition Q if every element of P is included in some element of Q (if strictly included, then P is strictly finer than Q). This formal model is meant to describe in mathematical terms the informational states of a decision-maker: one interprets an arbitrary partition of X as resulting from an equivalence relation of informational indifference. For an agent in possession of Q, any two possible states of nature within an element of Q carry the same amount of information and are therefore indistinguishable from this point of view. It is also assumed that any decision-maker would find the transition from Q to a strict refinement P desirable, since it leads from a less informative to a more informative state. With this description of the model in place, taking PðXÞ to be the set of all partitions of X, Dubra and Echenique consider the complete preorders on PðXÞ that rank any partition strictly below any strict refinement (a condition they call monotonicity). According to them, any rational decision-maker must rank PðXÞ on one such preorder. It can however be shown that none of them is representable by a real-valued utility function. Dubra and Echenique draw the following conclusion: Our result is important because it shows that utility theory is not likely to be a useful tool in the analysis of the value of information. This finding should be contrasted with the existing literature on the value of information, where utility representations are used. The use of a utility implies that preferences are not monotone (Dubra and Echenique 2001: 1). The point of the above quotation is that the existence of a utility representation is incompatible with the assumption that an agent should prefer more information to less. This remark is certainly correct, but it hides the following dilemma: is the problem inherent in utility theory as a formal approach to the study of idealised rankings or is it an effect of restricting attention to real-valued utility functions and, thus, of certain properties of R? This dilemma refines the formulation of the problem highlighted by Dubra and Echenique because it does not presuppose that the codomain of a utility function should be R. No particular feature of the space of informational states suggests that such a codomain should be selected. It is therefore meaningful to look for alternative numerical codomains, on which utility functions may exist. In other words, it is reasonable to conjecture that a lack of fit exists not between utility functions and spaces of information states, but between these spaces and the ordered reals

Chromogranin A: From Laboratory to Clinical Aspects of Patients with Neuroendocrine Tumors

Background. Neuroendocrine tumors (NETs) are characterized by having behavior and prognosis that depend upon tumor histology, primary site, staging, and proliferative index. The symptoms associated with carcinoid syndrome and vasoactive intestinal peptide tumors are treated with octreotide acetate. The PROMID trial assesses the effect of octreotide LAR on the tumor growth in patients with well-differentiated metastatic midgut NETs. The CLARINET trial evaluates the effects of lanreotide in patients with nonfunctional, well-, or moderately differentiated metastatic enteropancreatic NETs. Everolimus has been approved for the treatment of advanced pancreatic NETs (pNETs) based on positive PFS effects, obtained in the treated group. Sunitinib is approved for the treatment of patients with progressive gastrointestinal stromal tumor or intolerance to imatinib, because a randomized study demonstrated that it improves PFS and overall survival in patients with advanced well-differentiated pNETs. In a phase II trial, pasireotide shows efficacy and tolerability in the treatment of patients with advanced NETs, whose symptoms of carcinoid syndrome were resistant to octreotide LAR. An open-label, phase II trial assesses the clinical activity of long-acting repeatable pasireotide in treatment-naive patients with metastatic grade 1 or 2 NETs. Even if the growth of the neoplasm was significantly inhibited, it is still unclear whether its antiproliferative action is greater than that of octreotide and lanreotide. Because new therapeutic options are needed to counter the natural behavior of neuroendocrine tumors, it would also be useful to have a biochemical marker that can be addressed better in the management of these patients. Chromogranin A is currently the most useful biomarker to establish diagnosis and has some utility in predicting disease recurrence, outcome, and efficacy of therapy

A Study of Mathematical Determination through Bertrand’s Paradox

Certain mathematical problems prove very hard to solve because some of their intuitive features have not been assimilated or cannot be assimilated by the available mathematical resources. This state of affairs triggers an interesting dynamic whereby the introduction of novel conceptual resources converts the intuitive features into further mathematical determinations in light of which a solution to the original problem is made accessible. I illustrate this phenomenon through a study of Bertrand’s parado