Testimonial accounts of mathematical problem choice typically rely on intrinsic constraints. They focus on the worth of the problem and feelings of beauty. These are often developed as both descriptive and normative constraints on problem choice. In this paper, I aim to add an extrinsic constraint of no less importance: the assurance of contact of minds with a desired audience. A number of elements for the relationship between mathematician and his audience make up this contact. This constraint stems from (...) the mathematician’s role as an arguer, as one of the pre-requisites to argumentation is contact of minds. I examine two exceptional cases which fail to be explained by intrinsic constraints on motivation and posit how this contact could influence usual cases. While not the only constraint or drive in problem choice, establishing contact of minds plays an important role worth further examination. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

Testimonial accounts of mathematical problem choice typically rely on intrinsic constraints. They focus on the worth of the problem and feelings of beauty. These are often developed as both descriptive and normative constraints on problem choice. In this paper, I aim to add an extrinsic constraint of no less importance: the assurance of contact of minds with a desired audience. A number of elements for the relationship between mathematician and his audience make up this contact. This constraint stems from (...) the mathematician’s role as an arguer, as one of the pre-requisites to argumentation is contact of minds. I examine two exceptional cases which fail to be explained by intrinsic constraints on motivation and posit how this contact could influence usual cases. While not the only constraint or drive in problem choice, establishing contact of minds plays an important role worth further examination. (shrink)

Testimonial accounts of mathematical problem choice typically rely on intrinsic constraints. They focus on the worth of the problem and feelings of beauty. These are often developed as both descriptive and normative constraints on problem choice. In this paper, I aim to add an extrinsic constraint of no less importance: the assurance of contact of minds with a desired audience. A number of elements for the relationship between mathematician and his audience make up this contact. This constraint stems from (...) the mathematician’s role as an arguer, as one of the pre-requisites to argumentation is contact of minds. I examine two exceptional cases which fail to be explained by intrinsic constraints on motivation and posit how this contact could influence usual cases. While not the only constraint or drive in problem choice, establishing contact of minds plays an important role worth further examination. (shrink)

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.

This paper concerns the role of mathematical problems in the epistemology of Jean Cavaillès. Most occurrences of the term “problem” in his texts refer to mathematical problems, in the sense in which mathematicians themselves use the term: for an unsolved question which they hope to solve. Mathematical problems appear as breaking points in the succession of mathematical theories, both giving a continuity to the history of mathematics and illuminating the way in which the (...) history of mathematics breaks up into successive theories with different kinds of operations and, in a sense, different kinds of a prioris. We briefly compare mathematical becoming to the succession of episteme in Foucault’s Les Mots et les choses. We then come back to the necessity that Cavaillès attributes to mathematical becoming, and which the position of mathematical problems illustrates, in order to discuss its various consequences in Cavaillès’ later works but also in Canguilhem’s discussion of Cavaillès’ role in the Resistance. Finally, we study other types of problems, in Cavaillès’ writings: philosophical problems and what we will call “questions” rather than “problems,” and which contrary to mathematical problems, as Cavaillès uses the term, cannot be solved but pervade the whole of the history of mathematics. We will put these “questions” in relation to Lautman’s Ideas. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods (...) from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) class='Hi'>mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)

In this paper the reader is asked to engage in some simple problem-solving in classical pure number theory and to then describe, on the basis of a series of questions, what it is like to solve the problems. In the recent philosophy of mind this “what is it like” question is one way of signaling a turn to phenomenological description. The description of what it is like to solve the problems in this paper, it is argued, leads to (...) several morals about the epistemology and ontology of classical pure mathematical practice. Instead of simply making philosophical judgments about the subject matter in advance, the exercise asks the reader to briefly engage in a mathematical practice and to then reflect on the practice. (shrink)

Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and thus engage in cognitive processes (...) that make their problem solving algorithms computationally suboptimal, in contrast with the optimal algorithms studied in the computational approach. Therefore, in order to accurately model the human cognitive tasks involved in mathematical problem solving, we need to expand our methodology to also include aspects relevant to the algorithmic level. This allows us to study algorithms that are cognitively optimal for human problem solvers. Since problem solving methods are not universal, I propose that they should be studied in the framework of enculturation, which can explain the expected cultural variance in the humanly optimal algorithms. While mathematical problem solving is used as the case study, the considerations in this paper concern modeling of cognitive tasks in general. (shrink)

Isaac Newton's (1642-1727) contribution to the quantitative aspects of mathematics are well known compared to his views on it's qualitative aspect. In this paper, the author attempts to examine Newton.s position with regard to the orientation of mathematical problems based on some of his own writings on the subject.

Introduction Codex Upsaliensis Graecus 8 contains a miscellany of Greek texts, mostly from the Byzantine period, ranging all the way from Stephanites et Ichnelates to botanical lexica. Among these varied texts is a collection of mathematical problems on ff. 324r–331r. We might compare it to other known Byzantine textbooks of mathematical problems, such as the following: the mathematical epigrams attributed to Metrodorus in the Greek Anthology (14:116–146); the papyrus found at Akhmim from the 7th or (...) 8th century; Nicholas Rhabdas' letter on mathematics to Theodore Tzabouches; an anonymous collection found in the 14th century cod. Par. Suppl. Gr. 387 (Vogel 1968); and another anonymous collection from the latter half of the 15th century found in cod. Vind. Phil. Gr. 65 (Hunger-Vogel). The collection in Uppsala most resembles these last two textbooks. (shrink)

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in (...) the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed. (shrink)

The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of (...) the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

In this article some intriguing aspects of electromagnetic theory and its relation to mathematics and reality are discussed, in particular those related to the suppositions needed to obtain the wave equations from Maxwell equations and from there Helmholtz equation. The following questions are discussed. How is that equations obtained with so many irreal or fictitious assumptions may provide a description that is in a high degree verifiable? Must everything that is possible to deduce from a theoretical mathematical model occur (...) in the world? Does everything that takes place in the world have a mathematical description? (shrink)

I build on Heras-Escribano’s ontological characterization to address issues of affordances related to mathematics education, particularly about how it can enable fruitful conceptualizations for ….

Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account (...) of this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)

Preservice mathematics teachers’ accurate understanding of mathematical problem solving and its teaching is key to the performance of their professional quality. This study aims to investigate preservice mathematics teachers’ understanding of problem solving and its teaching and compares it with the understanding of in-service mathematics teachers. After surveying 326 in-service mathematics teachers, this study constructs a reliable and valid tool for the cognition of mathematical problem solving and its teaching and conducts a questionnaire survey on 26 preservice mathematics (...) teachers. Survey results reveal that preservice mathematics teachers have a good understanding of mathematical problem solving and its teaching and are more confident in the transfer value of problem solving ability. By contrast, in-service teachers are more optimistic that problem solving requires exploration, continuous thinking, and the participation of metacognition. This article concludes that preservice mathematics teachers should focus more on the initiative and creativity of students and put students at the center of education. In addition, teacher educators should provide more teaching practice opportunities for preservice teachers. The findings also show that in-service teachers’ understanding of problem solving and its teaching is inferior to that of preservice teachers on some indicators, implying the importance of post-service training for in-service teachers. (shrink)

Spatial visualization as a key variable in sex-related differences in mathematical problem solving and spatial aspects of geometry is traced to the 1960s. More recent relevant data are presented. The variability debate is traced to the latter part of the nineteenth century and an explanation for it is suggested. An idea is presented for further research to clarify sex-related brain laterality differences in solving spatial problems.

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, and (...) creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: In the target article, Christiane Herr offers an insightful characterization of how von Glasersfeld’s radical constructivism can be implemented in structural design education in architecture. In this commentary, I articulate possible connections between research on problem solving and problem posing in mathematics education and design processes in structural design education as described in the target article.

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis (...) of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

In previous papers (see Van Bendegem [1993], [1996], [1998], [2000], [2004], [2005], and jointly with Van Kerkhove [2005]) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if (...) you like, to construct a proof. (shrink)

Upshot: In this response to the open peer commentaries on our target article, we address two emerging themes: the need to explicate further the nature of learning processes from a radical constructivist perspective, and the need to investigate further the implications of our research for classroom teaching.

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Problem solving is an enormous field of study, where so-called “problems” can end up having very little in common. One of the least studied categories of problems is open-ended mathematical modeling research. Cifarelli and Sevim’s framework - although not developed for this purpose - may be (...) a useful lens for studying the development of mathematical modelers and researchers in applied mathematics. (shrink)

In the mathematical documents which have come down to us from these peoples, there are no theorems or demonstrations, and the fundamental concepts of ...

This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a (...) hitherto unnoticed problem for any Orthodox Realist who attempts to explain the a priori. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of (...) Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential (...) reading for anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

It is often supposed that we can use mathematics to capture the time evolution of any physical system. By this, I mean that we can capture the basic truths about the time evolution of a physical system with a set of mathematical assertions, which can then be used as premises in arbitrary mathematical arguments to deduce more complex properties of the system. ;I would like to argue that this picture of the role of mathematics in physics is incorrect. (...) Specifically, I shall assert: ;The Deduction Failure Thesis: Bodies of knowledge in physics are generally not closed under otherwise valid mathematical argument forms. ;The Representation Failure Thesis: We cannot assume that the state of any system, together with its fundamental laws, can be captured by some set of mathematical assertions or equations. In fact, it is more likely that the world is not representable by a set of mathematical assertions or equations than that it is. ;The dissertation largely consists of arguments for these two theses. (shrink)