According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (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)

This book examines the philosophical conception of abductive reasoning as developed by Charles S. Peirce, the founder of American pragmatism. It explores the historical and systematic connections of Peirce's original ideas and debates about their interpretations. Abduction is understood in a broad sense which covers the discovery and pursuit of hypotheses and inference to the best explanation. The analysis presents fresh insights into this notion of reasoning, which derives from effects to causes or from surprising observations to explanatory theories. The (...) author outlines some logical and AI approaches to abduction as well as studies various kinds of inverse problems in astronomy, physics, medicine, biology, and human sciences to provide examples of retroductions and abductions. The discussion covers also everyday examples with the implication of this notion in detective stories, one of Peirce’s own favorite themes. The author uses Bayesian probabilities to argue that explanatory abduction is a method of confirmation. He uses his own account of truth approximation to reformulate abduction as inference which leads to the truthlikeness of its conclusion. This allows a powerful abductive defense of scientific realism. This up-to-date survey and defense of the Peircean view of abduction may very well help researchers, students, and philosophers better understand the logic of truth-seeking. (shrink)

After a brief presentation of Feynman diagrams, we criticizise the idea that Feynman diagrams can be considered to be pictures or depictions of actual physical processes. We then show that the best interpretation of the role they play in quantum field theory and quantum electrodynamics is captured by Hughes' Denotation, Deduction and Interpretation theory of models, where “models” are to be interpreted as inferential, non-representational devices constructed in given social contexts by the community of physicists.

In the context of superintelligent AI systems, the term “oracle” has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems (...) and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety. (shrink)

The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery, but also new ways of practicing it in a rational and more systematic way. Ampliative rules, (...) methods, heuristic procedures and even a logic of discovery have been investigated, extracted, reconstructed and refined. The outcome is a ‘scientific discovery revolution’: not only a new way of looking at discovery, but also a construction of tools that can guide us to discover something new. This is a very important contribution of philosophy of science to science, as it puts the former in a position not only to interpret what scientists do, but also to provide and improve tools that they can employ in their activity. (shrink)

In his paper “Explaining Deductive Inference” Prawitz states what he calls «a fundamental problem of logic and the philosophy of logic»: the problem of explaining «Why do certain inferences have the epistemic power to confer evidence on the conclusion when applied to premisses for which there is evidence already?». In this paper I suggest a way of articulating, and partly modifying, the intuitionistic answer to this problem in such a way as to both answer Prawitz’s problem and satisfy a requirement (...) I argue to be crucial for any epistemic theory of the meaning of the logical constants: the requirement that evidence is epistemically transparent. (shrink)

This study applied a method of assisted introspection to investigate the phenomenology of mathematical intuition arousal. The aim was to propose an essential structure for the intuitive experience of mathematics. To achieve an intersubjective comparison of different experiences, several contemporary mathematicians were interviewed in accordance with the elicitation interview method in order to collect pinpoint experiential descriptions. Data collection and analysis was then performed using steps similar to those outlined in the descriptive phenomenological method that led to a generic structure (...) that accounts for the intuition surge in the experience of mathematics which was found to have four irreducible structural moments. The interdependence of these moments shows that a perceptualist view of intuition in mathematics, as defended by Chudnoff, is relevant to the characterization of mathematical intuition. The philosophical consequences of this generic structure and its essential features are discussed in accordance with Husserl’s philosophy of ideal objects and theory of intuition. (shrink)

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...) assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

In this paper we argue that the different positions taken by Dyson and Feynman on Feynman diagrams’ representational role depend on different styles of scientific thinking. We begin by criticizing the idea that Feynman Diagrams can be considered to be pictures or depictions of actual physical processes. We then show that the best interpretation of the role they play in quantum field theory and quantum electrodynamics is captured by Hughes' Denotation, Deduction and Interpretation theory of models (DDI), where “models” are (...) to be interpreted as inferential, non-representational devices constructed in given social contexts by the community of physicists. (shrink)

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is (...) that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...) potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

I DO not have to apologize for entering upon a discussion of intelligence and intelligence tests; it is a field which comes within the purview of philosophy as well as psychology. Any method of testing intelligence is therefore of common interest, especially as the methodology employed is usually based upon some definite theory as to its nature. The very word intelligence covers a wide range of meanings and psychologists seem to select sections of this range at will in accordance with (...) their particular interest. It may include most of the behavioural activities of man, or be narrowed down so that it refers to certain quantitative or relational aspects of experience. To take the case of the factor analyst, after his concept of intelligence has been analysed and classified into g's and s's, it becomes not a description of the mind, but rather a closed cognitive model of it. (shrink)

This book argues for the need to put into practice a profound and comprehensive intellectual revolution, affecting to a greater or lesser extent all branches of scientific and technological research, scholarship and education. This intellectual revolution differs, however, from the now familiar kind of scientific revolution described by Kuhn. It does not primarily involve a radical change in what we take to be knowledge about some aspect of the world, a change of paradigm. Rather it involves a radical change in (...) the fundamental, overall intellectual aims and methods of inquiry. At present inquiry is devoted to the enhancement of knowledge. This needs to be transformed into a kind of rational inquiry having as its basic aim to enhance personal and social wisdom. This new kind of inquiry gives intellectual priority to the personal and social problems we encounter in our lives as we strive to realize what is desirable and of value – problems of knowledge and technology being intellectually subordinate and secondary. For this new kind of inquiry, it is what we do and what we are that ultimately matters: our knowledge is but an aspect of our life and being. (shrink)

It is proposed that the collective identification of objects is a necessary phase in the evolution of language. A procedure of identification based on the first-person perspective is assumed that is guided by object constancy, the naming insight, and other more obvious features of language production. The mental image is the basic element of this procedure. The comparison of mental images is used to define recognition, and the identification and naming of macroscopic objects by a group of persons is formulated. (...) This is shown to effect a binary relation of the order type on the set of objects that reduces to an equivalence relation in the case of observer independence. (shrink)

Mental imagery (varieties of which are sometimes colloquially refered to as “visualizing,” “seeing in the mind's eye,” “hearing in the head,” “imagining the feel of,” etc.) is quasi-perceptual experience; it resembles perceptual experience, but occurs in the absence of the appropriate external stimuli. It is also generally understood to bear intentionality (i.e., mental images are always images of something or other), and thereby to function as a form of mental representation. Traditionally, visual mental imagery, the most discussed variety, was thought (...) to be caused by the presence of picturelike representations (mental images) in the mind, soul, or brain, but this is no longer universally accepted. (shrink)

Prior to the twentieth century, theories of knowledge were inherently perceptual. Since then, developments in logic, statis- tics, and programming languages have inspired amodal theories that rest on principles fundamentally different from those underlying perception. In addition, perceptual approaches have become widely viewed as untenable because they are assumed to implement record- ing systems, not conceptual systems. A perceptual theory of knowledge is developed here in the context of current cognitive science and neuroscience. During perceptual experience, association areas in the (...) brain capture bottom-up patterns of activation in sensory-motor areas. Later, in a top-down manner, association areas partially reactivate sensory-motor areas to implement perceptual symbols. The stor- age and reactivation of perceptual symbols operates at the level of perceptual components – not at the level of holistic perceptual expe- riences. Through the use of selective attention, schematic representations of perceptual components are extracted from experience and stored in memory (e.g., individual memories of green, purr, hot). As memories of the same component become organized around a com- mon frame, they implement a simulator that produces limitless simulations of the component (e.g., simulations of purr). Not only do such simulators develop for aspects of sensory experience, they also develop for aspects of proprioception (e.g., lift, run) and introspec- tion (e.g., compare, memory, happy, hungry). Once established, these simulators implement a basic conceptual system that represents types, supports categorization, and produces categorical inferences. These simulators further support productivity, propositions, and ab- stract concepts, thereby implementing a fully functional conceptual system. Productivity results from integrating simulators combinato- rially and recursively to produce complex simulations. Propositions result from binding simulators to perceived individuals to represent type-token relations. Abstract concepts are grounded in complex simulations of combined physical and introspective events. Thus, a per- ceptual theory of knowledge can implement a fully functional conceptual system while avoiding problems associated with amodal sym- bol systems. Implications for cognition, neuroscience, evolution, development, and artificial intelligence are explored. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

We discuss Einstein’s knowledge of projective geometry. We show that two pages of Einstein’s Scratch Notebook from around 1912 with geometrical sketches can directly be associated with similar sketches in manuscript pages dating from his Princeton years. By this correspondence, we show that the sketches are all related to a common theme, the discussion of involution in a projective geometry setting with particular emphasis on the infinite point. We offer a conjecture as to the probable purpose of these geometric considerations.

Music is at the centre of what it means to be human – it is the sounds of human bodies and minds moving in creative, story-making ways. We argue that music comes from the way in which knowing bodies (Merleau-Ponty) prospectively explore the environment using habitual 'patterns of action' which we have identified as our innate ‘communicative musicality’. To support our argument, we present short case studies of infant interactions using micro analyses of video and audio recordings to show the (...) timings and shapes of intersubjective vocalisations and body movements of adult and child while they improvise shared narratives of meaning. Following a survey of the history of discoveries of infant abilities, we propose that the gestural narrative structures of voice and body seen as infants communicate with loving caregivers are the building blocks of what become particular cultural instances of the art of music, and of dance, theatre and other temporal arts. Children enter into a musical culture where their innate communicative musicality can be encouraged and strengthened through sensitive, respectful, playful, culturally informed teaching in companionship. The central importance of our abilities for music as part of what sustains our well-being is supported by evidence that communicative musicality strengthens emotions of social resilience to aid recovery from mental stress and illness. Drawing on the experience of the first author as a counsellor, we argue that the strength of one person’s communicative musicality can support the vitality of another’s through the application of skilful techniques that encourage an intimate, supportive, therapeutic, spirited companionship. Turning to brain science, we focus on hemispheric differences and the affective neuroscience of Jaak Panksepp. We emphasise that the psychobiological purpose of our innate musicality grows from the integrated rhythms of energy in the brain for prospective, sensation-seeking affective guidance of vitality of movement. We conclude with a Coda that recalls the philosophy of the Scottish Enlightenment, which built on the work of Heraclitus and Spinoza. This view places the shared experience of sensations of living – our communicative musicality – as inspiration for rules of logic formulated in symbols of language. (shrink)

This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and assessing (...) the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a case study of (...) ... (shrink)

The necessary but not sufficient conditions for biological informational concepts like signs, symbols, memories, instructions, and messages are (1) an object or referent that the information is about, (2) a physical embodiment or vehicle that stands for what the information is about (the object), and (3) an interpreter or agent that separates the referent information from the vehicle’s material structure, and that establishes the stands-for relation. This separation is named the epistemic cut, and explaining clearly how the stands-for relation is (...) realized is named the symbol-matter problem. (4) A necessary physical condition is that all informational vehicles are material boundary conditions or constraints acting on the lawful dynamics of local systems. It is useful to define a dependency hierarchy of information types: (1) syntactic information (i.e., communication theory), (2) heritable information acquired by variation and natural selection, (3) non-heritable learned or creative information, and (4) measured physical information in the context of natural laws. High information storage capacity is most reliably implemented by discrete linear sequences of non-dynamic vehicles, while the execution of information for control and construction is a non-holonomic dynamic process. The first epistemic cut occurs in self-replication. The first interpretation of base sequence information is by protein folding; the last interpretation of base sequence information is by natural selection. Evolution has evolved senses and nervous systems that acquire non-heritable information, and only very recently after billions of years, the competence for human language. Genetic and human languages are the only known complete general purpose languages. They have fundamental properties in common, but are entirely different in their acquisition, storage and interpretation. (shrink)

Reading through Mechanica1 Intelligence, volume III of Alan Turing's Collected Works, one begins to appreciate just how propitious Turing's timing was. If Turing's major accomplishment in ‘On Computable Numbers’ was to expose the epistemological premises built into formalism, his main achievement in the 1940s was to recognize the extent to which this outlook both harmonized with and extended contemporary psychological thought. Turing sought to synthesize these diverse mathematical and psychological elements so as to forge a union between ‘embodied rules’ and (...) ‘learning programs’. Through their joint service in the Mechanist Thesis each would validate the other: and the frameworks from whence each derived. In this paper I will try to show how Turing's psychological thesis forces us to reassess the consequences of establishing AI on the epistemological foundation that underlies behaviourism. (shrink)

This paper attempts to define Exploratory Data Analysis (EDA) more precisely than usual, and to produce the beginnings of a philosophy of this topical and somewhat novel branch of statistics. A data set is, roughly speaking, a collection of k-tuples for some k. In both descriptive statistics and in EDA, these k-tuples, or functions of them, are represented in a manner matched to human and computer abilities with a view to finding patterns that are not "kinkera". A kinkus is a (...) pattern that has a negligible probability of being even partly potentially explicable. A potentially explicable pattern is one for which there probably exists a hypothesis of adequate "explicativity", which is another technical probabilistic concept. A pattern can be judged to be probably potentially explicable even if we cannot find an explanation. The theory of probability understood here is one of partially ordered (interval-valued), subjective (personal) probabilities. Among other topics relevant to a philosophy of EDA are the "reduction" of data; Francis Bacon's philosophy of science; the automatic formulation of hypotheses; successive deepening of hypotheses; neurophysiology; and rationality of type II. (shrink)

Mr. A. M. Turing was quoted in The Times about a year ago as saying it would be interesting to discover the degree of intellectual activity of which a machine was capable and to what extent it could think for itself. He has now pressed this suggestion further and given the results of his researches in an article called “Computing Machines and Intelligence,” together with a brief account of a “child-machine” which he has attempted to educate . I intend to (...) discuss this article in some detail and examine his claim that “machines can think.”. (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)

In our culture, intelligence, emotions and embodied intuitions continue to be seen as separate categories. The body is regarded as a medium of identity as well as social and sexual appeal, but neglected as the ground of embodied existence and silent knowledge, or the full understanding of the human condition. Prevailing educational and pedagogic practices also still separate the mental and intellectual capacities from emotions and the senses, and the multifarious dimensions of human embodiment.

For a variety of reasons, learning should be studied as a cultural phenomenon. The task of the first part of this article is to clear up the terminological questions about various ideal types of learning cultures, and how ideal type analysis may be used to study value and knowledge transfer and knowledge acquirement in various types of organisations. The important task of the second part is to analyse how implementation of environmental management systems, like BS-7750, contribute to a certain learning (...) practice. Moreover the task is to evaluate and discuss supplementing or alternative modes of learning to just formalisation of knowledge transfer. (shrink)

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, (...) the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis. (shrink)

In this paper, I present a framework where possible relations between learning and mental models are explored. In particular, I’ll be concerned with non-symbolic gestures accompanying discourse and their role in inducing the construction of models and therefore deep comprehension and learning in the listener. Also, I’ll be concerned with cognitive and socio-cognitive conflicts and their roles in inducing construction of alternative models of a problem and therefore in learning to reason. Human ability to learn is of great importance for (...) individuals interested in change. Indeed, to learn both declarative and procedural knowledge means to change, and in order to be able to intervene on change in a desired way it is necessary to have a theory of the mental representations and processes involved in learning and a theory of the communication and contexts that favour learning. (shrink)

The for-profit sector is an active, viable and financially successful piece of the landscape of education and assumed to continue growing. “Edupreneurs” or private, for-profit education companies provide desirable and affordable educational products and services for students, or better, customers. At the tertiary level, for-profit higher education is defined “private institution[s] in which the individual or agency in control receives compensation other than wages, rent, or other expenses for the assumption of risk”. In other words, public higher education and private (...) not-for-profit colleges and universities on the one hand are not entitled to benefit private interests and net earnings cannot be distributed to owners or shareholders. On the other hand, for-profit institutions set their goal to make a profit for their owners or shareholders. According to John Sperling, For-profit universities offer several advantages over non-profit institutions, among which are the for-profit’s accountability for educational effectiveness, operational efficiency, cost benefits, and the time it takes them to respond to changes in the education needs. Fueled by the trends of internationalization, globalization, commercialization, and privatization in the education sector, for-profit education expands worldwide. This research intends to feature the Chinese echoes to the trend of For-profit education. The purpose of the study is three-fold. To begin with, the author aims to portray the scope and size of Chinese for-profit education sector, and make a tentative classification for “Edupreneurs” operating in Chinese education and training market. Next, the author aims to show the panorama of Chinese for-profit education, looking into the yesterday, today, and tomorrow of Chinese “Edupreneurs”. Last but not least, the researcher proposes to promote educational cooperation between Germany and China. Germany is blessed with excellent educational resources and services, and among one of the most popular destinations for international student mobility. Nevertheless, Germany has been avoiding the private surge, and thus a for-profit surge so far, even when faced with severe budget cuts and funding problems. Is this a voluntary or reluctant rejection, under the current educational system lacking self-management and autonomy? A quest for combining educational provision and consumption between Germany and China will then be incorporated in this study. Qualitative research methods are used to collect data. The primary source of data comes from semi-structured interviews with middle or senior administrators of selected for-profit educational companies. Other sources include direct observation made by the researcher during the periods of visiting the interviewees and companies; official documents ; internal documents; company fact book; company website; journalism, and others. (shrink)

Many scientific discoveries have depended on external diagrams or visualizations. Many scientists also report to use an internal mental representation or mental imagery to help them solve problems and reason. How do scientists connect these internal and external representations? We examined working scientists as they worked on external scientific visualizations. We coded the number and type of spatial transformations (mental operations that scientists used on internal or external representations or images) and found that there were a very large number of (...) comparisons, either between different visualizations or between a visualization and the scientists’ internal mental representation. We found that when scientists compared visualization to visualization, the comparisons were based primarily on features. However, when scientists compared a visualization to their mental representation, they were attempting to align the two representations. We suggest that this alignment process is how scientists connect internal and external representations. (shrink)

Chance and accidents play important roles in scientific discoveries, but they are not blind luck. Serendipity is not merely stumbling on things unsought for, it is the ability to see significances and find values in the things stumbled upon. Without this ability, accidents do not lead to discoveries, as Pasteur observed: “Chance favors the prepared mind.” What are the characteristics of a prepared mind in science? How do chance and serendipity work in scientific research in general and drug discovery in (...) particular? (shrink)

Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given (...) axioms. Moreover, all the evidence suggests that all practitioners work with the same ontology. (My number 7 is exactly the same as yours.) How can we reconcile the notion that people construct mathematics, with this apparent choice-free, predetermined objectivity? I believe the answer is to be found by examining what mathematical thinking is (as a mental activity) and the way the human brain acquired the capacity for mathematical thinking. (shrink)

Ernst Mach was the first to write on the question of the implications on human knowledge, scientific knowledge and science education resulting from the evolutionary theory (i.e. the ideas by Lamarck, Erasmus Darwin, Spencer or Wallace as synthesised by Charles Darwin 1859 in his Origin of the Species). The ideas made a consistent genetic world view possible, which develops all scientific ideas (from all sciences) and experiences from the senses. Mach described this as the “adaptation of the thoughts to the (...) facts and the facts to each other” (Mach 1905). For Mach, science education is the key to identify the necessary changes as well as implementing them. The question is how can we as humans adapt(Adapt and adaptation is meant here as a reciprocal process between the “world” and the “self”. The individual and the world are not separate, but the individual and human thoughts are part of “the world”, as well as “the world” is a necessary part of human perception and thinking. This is Mach’s monism. For Mach (and for Richard Avenarius) the separation of the “world” and the “self” is only the result of a pragmatic adaptation of the individual to the fact that some clusters of sensual impressions (one’s own body perception) are more consistent over time than others. Initially for the child there is no such separation. For instance, the concept of the “others” is constructed before a consistent concept of self.) our thoughts to a consistent world view, which is plastic, i.e. itself continuously adapting and transforming. (shrink)

Umerez’s analysis made me aware of the fundamental differences in the culture of physics and molecular biology and the culture of semiotics from which the new field of biosemiotics arose. These cultures also view histories differently. Considering the evolutionary span and the many hierarchical levels of organization that their models must cover, models at different levels will require different observables and different meanings for common words, like symbol, interpretation, and language. These models as well as their histories should be viewed (...) as complementary rather than competitive. The relation of genetic language and human language is the central issue. They are separated by 4 billion years and require entirely different models. Nevertheless, these languages have in common a unique unlimited expressive power that allows open-ended evolution and creative thought. Understanding the nature of this expressive power and how it arises remains a basic unsolved problem of biosemiotics. (shrink)

The mind maps symbols and the extra-symbolic relationships amongst them to specific meanings. When symbols of various levels are placed in a hierarchical ordering, one may look at such ordered classes as distinct worlds where one class represents objects and the other represents the objects’ corresponding meanings. However, such an explanation can only be partial because the number of potential levels in such an ordering is infinite and, therefore, it engenders problems of recursion and infinite regress. There are also logical (...) problems in the form of paradoxes that emanate from the consideration of sets of sets. Given that most prior studies only consider symbols that are classical objects in associative relationships, we argue that there is a need to also consider objects with shifting boundaries and quantum objects. We believe that objects belonging to each of these three classes—that is classical objects, objects with shifting boundaries, and quantum objects—play a role in the workings of the mind. (shrink)

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.