Mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century, claims that the philosophy of mathematics cannot concern itself with the making of mathematics, in particular discovery, but only with finished mathematics, namely mathematics presented in finished form. On this basis, mainstream philosophy of mathematics argues that mathematics is theorem proving by the axiomatic method. This, however, is untenable because it is incompatible with Gödel’s incompleteness theorems, and cannot account for many features of mathematics. (...) This book offers an alternative approach, heuristic philosophy of mathematics, according to which the philosophy of mathematics can concern itself with the making of mathematics, in particular discovery. On this basis, the book argues that mathematics is problem solving by the analytic method, and that this can account for all the main features of mathematics: mathematical method, objects, demonstrations, definitions, diagrams, notations, explanations, beauty, applicability, and knowledge. (shrink)

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject develops? (...) What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century. (shrink)

This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical modelling, (...) the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment. (shrink)

Proofs are central to mathematical practice in large part due to the heuristic role that some of them play. Not only do they help establish a result, but often provide new avenues of mathematical research. Jody Azzouni has argued that underlying the practice of creating mathematical proofs there is a very specific norm: to each proof there should be a corresponding algorithmic derivation, a derivation in an algorithmic system. Here a framework is provided to classify and assess (...) mathematical proofs. It is argued that there is a plurality of kinds of proofs in mathematics and a plurality of roles these proofs play. In the end, mathematical practice is far less unified in this respect than it may seem to be. (shrink)

This paper sets out to show how mathematical modelling can serve as a way of ampliating knowledge. To this end, I discuss the mathematical modelling of time in theoretical physics. In particular I examine the construction of the formal treatment of time in classical physics, based on Barrow’s analogy between time and the real number line, and the modelling of time resulting from the Wheeler-DeWitt equation. I will show how mathematics shapes physical concepts, like time, acting as a (...) heuristic means—a discovery tool—, which enables us to construct hypotheses on certain problems that would be hard, and in some cases impossible, to understand otherwise. (shrink)

A mathematical approach to heuristics is proposed, in contrast to Gigerenzer et al.'s assertion that laws of logic and probability are of little importance. Examples are given of effective heuristics in abstract settings. Other short-comings of the text are discussed, including omissions in psychophysics and cognitive science. However, the authors' ecological view is endorsed.

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based (...) on the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science. (shrink)

The philosophical conception of mechanistic explanation is grounded on a limited number of canonical examples. These examples provide an overly narrow view of contemporary scientific practice, because they do not reflect the extent to which the heuristic strategies and descriptive practices that contribute to mechanistic explanation have evolved beyond the well-known methods of decomposition, localization, and pictorial representation. Recent examples from evolutionary robotics and network approaches to biology and neuroscience demonstrate the increasingly important role played by computer simulations and (...) class='Hi'>mathematical representations in the epistemic practices of mechanism discovery and mechanism description. These examples also indicate that the scope of mechanistic explanation must be re-examined: With new and increasingly powerful methods of discovery and description comes the possibility of describing mechanisms far more complex than traditionally assumed. (shrink)

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ (...) approach to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field, that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling. (shrink)

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley. In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and (...) also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures, that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward. (shrink)

Abstract.There is ample evidence that humans (and other primates) possess a knowledge instinct—a biologically driven impulse to make coherent sense of the world at the highest level possible. Yet behavioral decision‐making data suggest a contrary biological drive to minimize cognitive effort by solving problems using simplifying heuristics. Individuals differ, and the same person varies over time, in the strength of the knowledge instinct. Neuroimaging studies suggest which brain regions might mediate the balance between knowledge expansion and heuristic simplification. One (...) region implicated in primary emotional experience is more activated in individuals who use primitive heuristics, whereas two areas of the cortex are more activated in individuals with a strong knowledge drive: one region implicated in detecting risk or conflict and another implicated in generating creative ideas. Knowledge maximization and effort minimization are both evolutionary adaptations, and both are valuable in different contexts. Effort minimization helps us make minor and routine decisions efficiently, whereas knowledge maximization connects us to the beautiful, to the sublime, and to our highest aspirations. We relate the opposition between the knowledge instinct and heuristics to the biblical story of the fall, and argue that the causal scientific worldview is mathematically equivalent to teleological arguments from final causes. Elements of a scientific program are formulated to address unresolved issues. (shrink)

Despite the increasing recognition that heuristics may be involved in myriad scientific activities, much about how to use them prudently remains obscure. As typically defined, heuristics are efficient rules or procedures for converting complex problems into simpler ones. But this increased efficiency and problem-solving power comes at the cost of a systematic bias. As Wimsatt showed, biased modelling heuristics can conceal errors, leading to poor decisions or inaccurate models. This liability to produce errors presents a fundamental challenge (...) to the philosophical value of heuristic analyses. Heuristics may be powerful and efficient procedures, but their practical value for science and epistemology is significantly mitigated if we do not have a principled methodology for knowing how to use them wisely. In this essay, I extend Wimsatt’s analyses to argue that this challenge for a heuristic methodology can be met by appealing to second-order, or meta-, heuristics—that is, practical guidelines that prescribe the appropriate conditions for a first-order heuristic’s use. 1 Introduction2 Background3 Heuristics and Meta-heuristics4 A Hierarchical Model and Three Applications4.1 Decision making in emergency medicine4.2 Mathematical explanation in physics4.3 Resilience in ecological management5 Conclusion. (shrink)

Some viewpoints on the foundations of mathematics and its philosophy are more connected to scientific practice and its heuristics, mainly with the construction of physical theories and the search for the best explanations of physical phenomena by means of abduction or the solution of problems by the analytical method. Some researchers have introduced the importance of human cultural activities into the cognitive aspects of the mental processes of scientists, proposing an embodied approach in the bridge between mathematics and reality. (...) Fluid mechanics is an interesting area in this sense due to its position if the network of mathematics. By means of an historical example on vortex motion by Helmholtz, we show that the intuitive idea of eddy contains cognitive properties of a mental schema and that it gives many heuristic options for an embodied mathematical explanation about vortex dynamics. (shrink)

. Heuristic is a central concept of Lakatos' philosophy both in his early works and in his later work, the methodology of scientific research programs. The term itself, however, went through significant change of meaning. In this paper I study this change and the ‘metaphysical’ commitments behind it. In order to do so, I turn to his mathematical heuristic elaborated in Proofs and Refutations. I aim to show the dialogical character of mathematical knowledge in his account, which can (...) open a door to hermeneutic studies of mathematical practice. (shrink)

There is ample evidence that humans (and other primates) possess a knowledge instinct—a biologically driven impulse to make coherent sense of the world at the highest level possible. Yet behavioral decision-making data suggest a contrary biological drive to minimize cognitive effort by solving problems using simplifying heuristics. Individuals differ, and the same person varies over time, in the strength of the knowledge instinct. Neuroimaging studies suggest which brain regions might mediate the balance between knowledge expansion and heuristic simplification. One (...) region implicated in primary emotional experience is more activated in individuals who use primitive heuristics, whereas two areas of the cortex are more activated in individuals with a strong knowledge drive: one region implicated in detecting risk or conflict and another implicated in generating creative ideas. Knowledge maximization and effort minimization are both evolutionary adaptations, and both are valuable in different contexts. Effort minimization helps us make minor and routine decisions efficiently, whereas knowledge maximization connects us to the beautiful, to the sublime, and to our highest aspirations. We relate the opposition between the knowledge instinct and heuristics to the biblical story of the fall, and argue that the causal scientific worldview is mathematically equivalent to teleological arguments from final causes. Elements of a scientific program are formulated to address unresolved issues. (shrink)

I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the inferential microstructures that shape this construction, that is, the study of both the very first, ampliative inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations. I discuss how this paradigmatic case study supports the (...) recent approaches to problem-solving and philosophy of mathematics, and how it suggests refinements of them. In more detail, I argue that the inferential micro-structures enable us to shed more light on the informal, heuristic side of mathematical practice, and its inferential and rational procedures. I show how they enable the generation of a problem, the construction of its conditions of solvability, the search for a hypothesis to solve it, and how these processes are representation-sensitive. On this base, I argue that: 1.the recent development of the philosophy of mathematics was right in moving from Lakatos’ initial investigation of the formal side of a mathematical proof to the investigation of the semi-formal, heuristic side of the mathematical practice as a way of understanding mathematical knowledge and its advancement. 2.The investigation of mathematical practice and discovery can be improved by a finer-grained study of the inferential micro-structures that are built during mathematical problem-solving. (shrink)

Several philosophers of science have claimed that the correspondence principle can be generalized from quantum physics to all of (particularly physical) science and that in fact it constitutes one of the major heuristical rules for the construction of new theories. In order to evaluate these claims, first the use of the correspondence principle in (the genesis of) quantum mechanics will be examined in detail. It is concluded from this and from other examples in the history of science that the principle (...) should be qualified with respect to its nature and relativized with respect to its scope of application. At the same time this conclusion implies a qualification and a relativization of the heuristic power of the principle. Generally speaking, intertheoretical correspondence is primarily of a formal-mathematical and empirical but not of a conceptual nature. Moreover, it only applies to certain parts of the theories involved. Finally, a number of philosophical justifications of the principle are discussed and some conclusions are drawn concerning the debates on theory reduction and on the discovery-justification distinction. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance (...) for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical (...) ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

This book introduces a new form of argumentative analysis: rhetorical heuremes. The method applies the concepts of heuristic thinking, probability, and contingency in order to develop a better understanding of complex arguments in classical oratory. A new theory is required because Greek and Roman rhetoric cannot provide detailed answers to problems of strategic argumentation in the analysis of speeches. Building on scholarship in Ciceronian oratory, this book moves beyond the extant terminology and employs a concept of heuristic reasoning derived from (...) the psychology of decision making and mathematical problem solving. The author analyses selected passages from Cicero's forensic speeches where arguments of probability are deployed, and shows that the Sophistic concept of probability can link ancient rhetoric and modern theories of argumentation. Six groups of heuremes are identified, each of which represents a form of probabilistic reasoning by which the orator plays upon the perception of the jurors. (shrink)

: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting (...) to clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the (...) potential to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science with the activities of presenting scientific results. It combines the heuristic and the deductive side of science. The heuristic (...) side is represented, e.g., by the creative methods to find the 'best' hypotheses, to design experimental systems for empirical research in order to formulate general laws, or to create axiomatic systems. The other side consists of the production of deductive knowledge. This combination leads to a clear hierarchy: the heuristic side provides the basic presuppositions from which the deductive side takes off. The former is used to make deductions possible. The deductive method can be presented as an analysis-synthesis scheme as it can be found, e.g., in the tradition of Kant, Jakob Friedrich Fries, and Leonard Nelson. Nelson's critical philosophy can be seen as a key for understanding the philosophy behind David Hilbert's early axiomatic method. This axiomatic method is usually restricted to a non-philosophical approach to pure mathematics. But Hilbert was not an exclusive formalist; he proposed a mathesis universalis in the Cartesian-Leibnizian sense according to which mathematics is the syntactical tool for a general philosophy of science, applicable to all scientific disciplines. In this function, mathematics takes its problems from the sciences. Hilbert did not deny that mathematics should play a role in explaining the world. The analysis-synthesis scheme helps to provide a consistent interpretation of these two sides of Hilbert's attitude towards his working field. (shrink)

With democracy in mind, promoting students’ cognitive, personal, and social development can inform and shape the mathematics curriculum and classroom practice with the goal of their becoming more capable, self-reflective, and socially aware human beings. Toward that realization, their mathematics experience could include: heuristics, as it provides a natural language for problem solving; habits of mind, so students can think and act with a more developed “reflective intelligence”; and multiple-centers investigations, where collaborations based on shared mathematical interest can (...) be pursued. In this way, their education would be “a fostering, a nurturing, a cultivating process” that supports the development of democratic society. (shrink)

The systematic mapping of the interplay of ontology and epistemology in the context of present day philosophy of mathematics constitutes an important heuristic goal. In order to achieve it, we must analyze and reinterpret the position of mathematics in philosophy." -- Back cover.

Some of the most relevant problems today both in Science and practical problems involves Coupled Socio-ecological Systems, which are some of the best examples of Complex Systems. In this work we discuss groundwater-management as an example of these Coupled Socio-ecological System, also known as Coupled Human and Natural Systems. We argue that it is possible and even necessary to construct a contextual statistical theory of groundwater management. Contextuality implies some very different statistical features as entanglement and complementarity. We discuss some (...) interpretation about statistical entanglement and statistical complementarity in terms of groundwater flow theory and extraction equation. To this end, we propose a non-Kolmogorobian approach following the Växjö school of thought. One of the most straightforward conclusions is that in this way, a statistical-contextual treatment of groundwater management may not only give place to new flow equations, but it could have a profound impact in resource management. Then the new basic unity of analysis should be the indivisible pair. We should not talk any more about aquifer dynamics or resource management plans by themselves. Physical and social systems are coupled in a statistical-contextual fundamental way, and a socio-hydrogeological theory should be developed. (shrink)

This book explores new findings on the long-neglected topic of theory construction and discovery, and challenges the orthodox, current division of scientific development into discrete stages: the stage of generation of new hypotheses; the stage of collection of relevant data; the stage of justification of possible theories; and the final stage of selection from among equally confirmed theories. The chapters, written by leading researchers, offer an interdisciplinary perspective on various aspects of the processes by which theories rationally should, and descriptively (...) are, built. They address issues such as the role of problem-solving and heuristic reasoning in theory-building; how inferences and models shape the pursuit of scientific knowledge; the relation between problem-solving and scientific discovery; the relative values of the syntactic, semantic, and pragmatic view of theories in understanding theory construction; and the relation between ampliative inferences, heuristic reasoning, and models as a means for building new theories and knowledge. Through detailed arguments and examinations, the volume collectively challenges the orthodox view’s main tenets by characterizing the ways in which the different “stages” are logically, temporally, and psychologically intertwined. As a group, the chapters provide several attempts to answer long-standing questions about the possibility of a unified conceptual framework for building theories and formulating hypotheses. (shrink)

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

How do we get new knowledge? Following the maverick tradition in the philosophy of science, Carlo Cellucci gradually came to the conclusion that logic can only fulfill its role in mathematics, science and philosophy if it helps us to answer this question. He argues that mathematical logic is inadequate and that we need a new logic, framed in a naturalistic conception of knowledge and philosophy - the heuristic conception. This path from logic to a naturalistic conception of knowledge and (...) philosophy explains the title, From a Heuristic Point of View, which recalls the celebrated collection of essays, From a Logical Point of View, by Willard Van Orman Quine, the father of modern naturalized epistemology. The word 'heuristic' points to Cellucci's favorite theme and the main difference between him and Quine: the emphasis on discovery and building a 'logic' for generating new knowledge. This book is a collection of essays from leading figures in this field who discuss, criticize, or expand on the main topics in Cellucci's work, dealing with some of the most challenging questions in logic, science and philosophy. (shrink)

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does (...) a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha. (shrink)

This contribution defends two claims. The first is about why thought experiments are so relevant and powerful in mathematics. Heuristics and proof are not strictly and, therefore, the relevance of thought experiments is not contained to heuristics. The main argument is based on a semiotic analysis of how mathematics works with signs. Seen in this way, formal symbols do not eliminate thought experiments (replacing them by something rigorous), but rather provide a new stage for them. The formal world (...) resembles the empirical world in that it calls for exploration and offers surprises. This presents a major reason why thought experiments occur both in empirical sciences and in mathematics. The second claim is about a looming aporia that signals the limitation of thought experiments. This aporia arises when mathematical arguments cease to be fully accessible, thus violating a precondition for experimenting in thought. The contribution focuses on the work of Vladimir Voevodsky (1966–2017, Fields medalist in 2002) who argued that even very pure branches of mathematics cannot avoid inaccessibility of proof. Furthermore, he suggested that computer verification is a feasible path forward, but only if proof is not modeled in terms of formal logic. (shrink)

Arguments for the effectiveness, and even the indispensability, of mathematics in scientific explanation rely on the claim that mathematics is an effective or even a necessary component in successful scientific predictions and explanations. Well-known accounts of successful mathematical explanation in physical science appeals to scientists’ ability to solve equations directly in key domains. But there are spectacular physical theories, including general relativity and fluid dynamics, in which the equations of the theory cannot be solved directly in target domains, and (...) yet scientists and mathematicians do effective work there (McLarty 2023, Elder 2023). Building on extant accounts of structural scientific explanation (Bokulich 2011, Leng 2021), I argue that philosophical accounts of the role of equations in scientific explanation need not rely on scientists’ ability to solve equations independently of their understanding of the empirical or experimental context. For instance, the process of formulating solutions to equations can involve significant appeal to information about experimental contexts (Curiel 2010) or of physically similar systems (Sterrett 2023). Working from a close analysis of work in fluid mechanics by Martin Bazant and Keith Moffatt (2005), I propose an account of heuristic structural explanation in mathematics (Einstein 1921, Pincock 2021), which explains how physical explanations can be constructed even in domains where basic equations cannot be solved directly. (shrink)

There are presently two leading foreign policy decision-making paradigms in vogue. The first is based on the classical or rational model originally posited by von Neumann and Morgenstern to explain microeconomic decisions. The second is based on the cybernetic perspective whose groundwork was laid by Herbert Simon in his early research on bounded rationality. In this paper we introduce a third perspective — thepoliheuristic theory of decision-making — as an alternative to the rational actor and cybernetic paradigms in international relations. (...) This theory is drawn in large part from research on heuristics done in experimental cognitive psychology. According to the poliheuristic theory, policy makers use poly (many) heuristics while focusing on a very narrow range of options and dimensions when making decisions. Among them, the political dimension is noncompensatory. The paper also delineates the mathematical formulations of the three decision-making models. (shrink)

There are presently two leading foreign policy decision-making paradigms in vogue. The first is based on the classical or rational model originally posited by von Neumann and Morgenstern to explain microeconomic decisions. The second is based on the cybernetic perspective whose groundwork was laid by Herbert Simon in his early research on bounded rationality. In this paper we introduce a third perspective -- the poliheuristic theory of decision-making -- as an alternative to the rational actor and cybernetic paradigms in international (...) relations. This theory is drawn in large part from research on heuristics done in experimental cognitive psychology. According to the poliheuristic theory, policy makers use poly (many) heuristics while focusing on a very narrow range of options and dimensions when making decisions. Among them, the political dimension is noncompensatory. The paper also delineates the mathematical formulations of the three decision-making models. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

This paper is an essay-review of J. Yoder's "Unrolling Time: Christian Huygens and the Mathematization of Nature" (Cambridge, 1989). Highlighting the scholarly thoroughness and mathematical competence of Yoder's reconstruction of Huygens's heuristic path to his ground-breaking results on centrifugal force, cycloidal motion and evolutes, the essay also deals with Yoder's attempts to characterize Huygens's way of using mathematics in physical problems. In opposition to Yoder's thesis, this paper argues that evidence internal to Huygens's work as well as the contemporary (...) reaction to it suggest the existence of substantial methodological differences between Huygens's mathematization of nature and Newton's. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

In considering a domain of knowledge – language, music, mathematics, or morality – it is necessary to derive principles that can describe the mature state and explain how an individual reaches this state. Although Sunstein's heuristics go some way toward a description of our moral sense, it is not clear that they are at the right level of description, and as stated, they provide no guidelines for looking at the acquisition process – the problem of explanatory adequacy.

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)