When base rates are learned and used in an experiential manner subjects show better base rate use, perhaps because the implicit learning system is engaged. A causal framework in which base rates are relevant might also be necessary. Humans might thus perform better on more ecologically valid tasks, which are likely to contain those three components.

According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible (...) with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait’s finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories. (shrink)

Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts of its language `implicitly''. Thus whensomeone inquires about the meaning of the set-concept, thestandard response reads that axiomatic set-theory defines itimplicitly and that is the end of it. But can we explainthis assertion in a manner that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong when hesays that implicit definability is ``an obscure notion''''? Doesan explanation of it presuppose any particular view on meaning?Is it not a (...) scandal of the philosophy of mathematics that no answersto these questions are around? We submit affirmative answers to allquestions. We argue that a Wittgensteinian conception of meaninglooms large beneath Hilbert''s conception of implicit definability.Within the specific framework of Horwich''s recent Wittgensteiniantheory of meaning called semantic deflationism, we explain anexplicit conception of implicit definability, and then go on toargue that, indeed, set-theory, defines the set-conceptimplicitly according to this conception. We also defend Horwich''sconception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophicalresources gathered to dissolve all traditional worries about thecoherence of the set-concept, raisedby Frege, Russell and Max Black, and whichrecently have been defended vigorously by Hallett (1984) in hismagisterial monograph Cantorian set-theory and limitationof size. Until this day, scandalously, these worries havebeen ignored too by philosophers of mathematics. (shrink)

The present research seeks to utilize Implicit Theories of Intelligence and Achievement Goal Theory to understand students’ intrinsic motivation and academic performance in mathematics in Singapore. 1,201 lower-progress stream students, ages ranged from 13 to 17 years, from 17 secondary schools in Singapore took part in the study. Using structural equation modeling, results confirmed hypotheses that incremental mindset predicted mastery-approach goals and, in turn, predicted intrinsic motivation and mathematics performance. Entity mindset predicted performance-approach and performance-avoidance goals. Performance-approach goal was (...) positively linked to intrinsic motivation and mathematics performance; performance-avoidance goal, however, negatively predicted intrinsic motivation and mathematics performance. The model accounted for 35.9% of variance in intrinsic motivation and 13.8% in mathematics performance. These findings suggest that intrinsic motivation toward mathematics and achievement scores might be enhanced through interventions that focus on incremental mindset and mastery-approach goal. In addition, performance-approach goal may enhance intrinsic motivation and achievement as well, but to a lesser extent. Finally, the study adds to the literature done in the Asian context and lends support to the contention that culture may affect students’ mindsets and adoption of achievement goals, and their associated impact on motivation and achievement outcomes. (shrink)

Discounting is a manifestation of behavioral impulsivity, which is closely related to self-regulation processes. The decision-making process for intertemporal choices is governed by the inhibition of impulses, which can influence both risk and time-related attitudes. This paper utilizes self-reported measures of risk, impatience, and impulsivity attitudes to examine their impact on the implicit discount rate used when weighing the current purchase cost against future energy savings of appliances. It analyzes and tests the interplay between these attitudes using specific (...) functional forms and causal models. The results highlight the role of risk, impatience and impulsivity on the discount rate and the biases that arise from omitting impulsive attitudes. In addition, other factors such as environmental and social preferences, attitudes, and financial constraints contribute to the implicit discount rate. (shrink)

In the article, questions from the field of philosophy of mathematics are studied. The author is driven by the need to achieve a balance between the philosophy of science and the history of science in formation of concepts of the science development. In this regard, the author justifies the reliance on the methodology of implicit knowledge, combined with the epistemology principle of criticism in studying the development of mathematics as the most expedient and effective. The author expresses the necessity (...) of criticizing the methodology of counterexamples typical for mathematical empiricism declared by I. Lakatos. In the article, an explanation of the concept of implicit knowledge, as well as the examples from mathematics are given. The author believes that application of the concept of implicit knowledge in philosophical and mathematical terms allows us to reveal the unique specificity of mathematical science deeper. The author concludes that it is necessary to recognize the fundamental importance of the methodology of implicit knowledge for the epistemology, as well as for the development of philosophy and methodology of science in general. (shrink)

The Polish philosophy of mathematics in the 19th century had its origins in the Romantic period under the influence of the then-predominant idealist philosophies. The decline of Romantic philosophy precipitated changes in general philosophy, but what is less well known is how it triggered changes in the philosophy of mathematics. In this paper, we discuss how the Polish philosophy of mathematics evolved from the metaphysical approach that had been formed during the Romantic era to the more modern positivistic paradigm. These (...) evolutionary changes are attributed to the philosophers Henryk Struve, Antoni Molicki and Julian Ochorowicz, and mathematicians Karol Hertz and Samuel Dickstein. We also show how implicit ideas (i.e., those not declared openly) from the area between the philosophy of science and general philosophy played a crucial role in the paradigm shift in the Polish philosophy of mathematics. (shrink)

The implicit commitment thesis is the claim that believing in a mathematical theory S carries an implicit commitment to further sentences not deductively entailed by the theory, such as the consistency sentence Con(S). I provide a new argument for this thesis based on the notion of mathematical certainty. I also reply to a recent argument by Walter Dean against the implicit commitment thesis, showing that my formulation of the thesis avoids the difficulties he raises.

Although implicit tests of positive and negative affect exist, implicit measures of distinct emotional states are scarce. Three experiments examined whether a novel implicit emotion-assessment task, the rating of emotion expressed in abstract images, would reveal distinct emotional states. In Experiment 1, participants exposed to a sadness-inducing story inferred more sadness, and less happiness, in abstract images. In Experiment 2, an anger-provoking interaction increased anger ratings. In Experiment 3, compared to neutral images, spider images increased fear ratings (...) in spider-fearful participants but not in controls. In each experiment, the implicit task indicated elevated levels of the target emotion and did not indicate elevated levels of non-target negative emotions; the task thus differentiated among emotional states of the same valence. Correlations also supported the convergent and discriminant validity of the implicit task. Supporting the possibility that heuristic processes underlie the ratings, group differences were stronger among those who responded relatively quickly. (shrink)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains so far an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection (...) principle is still lacking. We provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one's implicit commitments, and sheds light on the reason why this is so. We argue that the theory has interesting epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent proposals for the analysis of implicit commitment based on truth or epistemic notions. (shrink)

Leibniz’s universal characteristic is a fundamental aspect of his theory of cognition. Without symbols or characters it would be difficult for the human mind to define several concepts and to achieve many demonstrations. In most disciplines, and particularly in mathematics, the mind must then focus on symbols and their combinatorial rules rather than on mental contents. For Leibniz, mental perception is most of the time too confused for attaining distinct notions and valid deductions. In this paper, I argue that the (...) functions of symbolization differ depending upon the kind of concepts that are replaced with characters. In my view, most commentators did not sufficiently underline the distinction between two main functions of formal substitution in Leibniz’s characteristic: either increasing our knowledge or simply structuring it. In the first case, we complete our knowledge because formal substitution makes sensible and imaginary concepts more distinct. In the second case, symbolization helps to organize contents that are already conceived of by reason. Thus the process of substitution is not always identically applicable, for symbols replace different types of concepts. (shrink)

In this article my primary intention is to engage in a discussion on the inherent constraints of models, taken as models of theories, that reaches beyond the epistemological level. Naturally the paper takes into account the ongoing debate between proponents of the syntactic and the semantic view of theories and that between proponents of the various versions of scientific realism, reaching down to the most fundamental, subjective level of discourse. In this approach, while allowing for a limited discussion of physical (...) and positive science models, I am primarily focused on the structure and ontology of mathematical models, in particular Cohen’s forcing models and to a lesser extent Gödel’s constructible universe, to the extent that these were designed to answer questions bearing on the scope, the capacity and ultimately the ontology of models themselves, therefore influencing in one or the other way the status of models in general. This status, it is argued, is largely defined by the way models subsume a set-theoretical structure whose constraints, reducible to an extra-linguistic level of discourse, may implicitly condition the epistemic status of models as representations of axiomatic theories. In the last section I deal extensively with the inner constraints of theories as subjectively originated in a less technical philosophically oriented discussion with certain prompts from phenomenology. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This (...) new meta-methodological concept is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

The article deals with historical dynamics of implicit and intuitive elements of mathematical knowledge. The author describes historical dynamics of implicit and intuitive elements and discloses a historical and evolutionary mechanism of building up mathematical knowledge. Each requirement to increase the level of theoretical rigor in mathematics is historically realized as a three-stage process. The first stage considers some general conditions of valid mathematical knowledge recognized by the mathematical community. The second one reveals the (...) level of theoretical rigor increasing, while the third one is characterized by explication of the hidden lemmas. A detailed discussion of historical substantiation of the basic algebra theorem is conducted according to the proposed technique. (shrink)

The present experiment investigated the development of implicit and explicit knowledge during concept learning. According to Cleeremans and Jiménez , the content of a representation can be conscious only when the representation is of a sufficiently good quality; on this theory, increasing explicit and decreasing implicit knowledge might be expected with training. The view that implicit knowledge arises from compilation of explicit knowledge makes the opposite prediction. The present research tested these possibilities using subjective measures based on (...) confidence ratings. One group of participants was presented with blocks of category exemplars that activated prior knowledge, whereas another group was presented with blocks of categories that did not elicit any useful prior knowledge. The results showed that, irrespective of the knowledge group participants were allocated to, explicit knowledge increased over the course of learning, whereas implicit knowledge either stayed the same or decreased, consistent with Cleeremans and Jiménez’s prediction. (shrink)

We have been oversold on the base rate fallacy in probabilistic judgment from an empirical, normative, and methodological standpoint. At the empirical level, a thorough examination of the base rate literature (including the famous lawyer–engineer problem) does not support the conventional wisdom that people routinely ignore base rates. Quite the contrary, the literature shows that base rates are almost always used and that their degree of use depends on task structure and representation. Specifically, base rates play a relatively (...) larger role in tasks where base rates are implicitly learned or can be represented in frequentist terms. Base rates are also used more when they are reliable and relatively more diagnostic than available individuating information. At the normative level, the base rate fallacy should be rejected because few tasks map unambiguously into the narrow framework that is held up as the standard of good decision making. Mechanical applications of Bayes's theorem to identify performance errors are inappropriate when (1) key assumptions of the model are either unchecked or grossly violated, and (2) no attempt is made to identify the decision maker's goals, values, and task assumptions. Methodologically, the current approach is criticized for its failure to consider how the ambiguous, unreliable, and unstable base rates of the real world are and should be used. Where decision makers' assumptions and goals vary, and where performance criteria are complex, the traditional Bayesian standard is insufficient. Even where predictive accuracy is the goal in commonly defined problems, there may be situations (e.g., informationally redundant environments) in which base rates can be ignored with impunity. A more ecologically valid research program is called for. This program should emphasize the development of prescriptive theory in rich, realistic decision environments. (shrink)

Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations between autonomous (...) and intentional objects. In particular we develop a phenomenology of mathematical works, which has the stratified intentional structure discovered by Ingarden in his study of the literary work. (shrink)

Translator's introduction -- Fundamental questions of geometry -- The decidability requirement -- The origin of the concept of number -- Implicit definition and the proper grounding of mathematics -- Rigid bodies in geometry -- Prelude to geometry : the essential ideas -- Physical and mathematical geometry -- Natural geometry -- The concept of the differential -- Reflections on the proper grounding of mathematics I -- Concepts and proofs in mathematics -- Dimension and space in mathematics -- Reflections on (...) the proper grounding of mathematics II -- The axiomatic method in modern mathematics. (shrink)

The study of intellectual humility is still in its early stages and issues of definition and measurement are only now being explored. To inform and guide the process of defining and measuring this important intellectual virtue, we conducted a series of studies into the implicit theory – or ‘folk’ understanding – of an intellectually humble person, a wise person, and an intellectually arrogant person. In Study 1, 350 adults used a free-listing procedure to generate a list of descriptors, one (...) for each person-concept. In Study 2, 335 adults rated the previously generated descriptors by how characteristic each was of the target person-concept. In Study 3, 344 adults sorted the descriptors by similarity for each person-concept. By comparing and contrasting the three person-concepts, a complex portrait of an intellectually humble person emerges with particular epistemic, self-oriented, and other-oriented dimensions. (shrink)

In one of his contributions to The Nation, Peirce claims that ―Crystal clearness, such as we justly require in mathematics, in law, in economics, is in philosophy the characteristic of the second-rates.‖ The statement might seem paradoxical enough: isn't Peirce the author of How to Make our Ideas Clear (hereafter: HMIC), the seminal paper for the pragmatist tradition, a paper that is sure to be included in each and every anthology of American thought? How can clearness then be ―the characteristic (...) of the second-rates‖? I take it that these perplexities arise only in a superficial reading of HMIC and that the present quote tells quite a lot about Peirce's doctrine and style. My first section concerns the method one should adopt in philosophy according to Peirce and explains in part why belief and doubt play such an important role in the Illustrations series. The second one concerns the kind of problems Peirce felt attracted to and how we can make sense of what he says about the ―"strongest" thinkers. (shrink)

Humans recognise reddish-coloured faces as angry. However, does facial colour also affect “implicit” facial expression perception of which humans are not explicitly aware? In this study, we investigated the effects of facial colour on implicit facial expression perception. The experimental stimuli were “hybrid faces”, in which the low-frequency component of the neutral facial expression image was replaced with the low-frequency component of the facial expression image of happiness or anger. In Experiment 1, we confirmed that the hybrid face (...) stimuli were perceived as neutral and, therefore, supported implicit facial expression perception. In Experiment 2, the hybrid face stimuli were adjusted to natural and reddish facial colours, and their friendliness ratings were compared. The results showed that the expression of happiness was rated as more friendly than the expression of anger. In addition, the expression of happiness was rated as friendlier when the low-frequency happy component was red, but the friendliness rating of the expression of anger did not change when it was presented in red. In Experiment 3, we affirmed the implicit facial expression perception even in reddish colours. These results suggest that facial colour modulates the perception of implicit facial expressions in hybrid facial stimuli. (shrink)

This new volume on logic follows a recognizable format that deals in turn with the topics of mathematical logic, moving from concepts, via definitions and inferences, to theories and axioms. However, this fresh work offers a key innovation in its ‘pyramidal’ graph system for the logical formalization of all these items. The author has developed this new methodology on the basis of original research, traditional logical instruments such as Porphyrian trees, and modern concepts of classification, in which pyramids are (...) the central organizing concept. The pyramidal schema enables both the content of concepts and the relations between the concept positions in the pyramid to be read off from the graph. Logical connectors are analyzed in terms of the direction in which they connect within the pyramid. Additionally, the author shows that logical connectors are of fundamentally different types: only one sort generates propositions with truth values, while the other yields conceptual expressions or complex concepts. On this basis, strong arguments are developed against adopting the non-discriminating connector definitions implicit in Wittgensteinian truth-value tables. Special consideration is given to mathematical connectors so as to illuminate the formation of concepts in the natural sciences. To show what the pyramidal method can contribute to science, a pyramid of the number concepts prevalent in mathematics is constructed. The book also counters the logical dogma of ‘false’ contradictory propositions and sheds new light on the logical characteristics of probable propositions, as well as on syllogistic and other inferences. (shrink)

The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...) small apertures. (shrink)

In four experiments using an artificial grammar learning procedure, the authors examined the links between the “classic” mere exposure effect [heightened affect for previously encountered stimulus items ] and the “structural” mere exposure effect [greater hedonic appreciation for novel stimuli that conform to an implicitly acquired underlying rule system ]. After learning, participants: classified stimuli according to whether they conformed to the principles of the grammar and, rated them in terms of how much they liked them. In some experiments unusual (...) and unfamiliar symbols were used to instantiate the AG, in others highly familiar characters were used. In all cases participants showed standard AG learning. However, whether the two exposure effects emerged was dependent on symbol familiarity. Symbols with high a priori familiarity produced a structural mere exposure effect. Moderately familiar symbols produced only the classic, but not the structural, mere exposure effect. Highly unfamiliar symbols produced neither exposure effect. Results are discussed in the context of implicit learning theory and implications for a general theory of aesthetics are presented. (shrink)

We introduce the Relational Responding Task (RRT) as a tool for capturing beliefs at the implicit level. Flemish participants were asked to respond as if they believed that Flemish people are more intelligent than immigrants (e.g., respond “true” to the statement “Flemish people are wiser than immigrants”) or to respond as if they believed that immigrants are more intelligent than Flemish people (e.g., respond “true” to the statement “Flemish people are dumber than immigrants”). The difference in performance between these (...) two tasks correlated with ratings of the extent to which participants explicitly endorsed the belief that Flemish people are more intelligent than immigrants and with questionnaire measures of subtle and blatant racism. The current study provides a first step towards validating RRT effects as a viable measure of implicit beliefs. (shrink)

This study aims to improve the effectiveness of English grammar teaching in high school. Firstly, the Science, Technology, Engineering, Art, and Mathematics educational model is comprehensively discussed. Then, the current situation and difficulties of English grammar teaching in high school are analyzed. Finally, based on the traditional and the STEAM teaching mode, a comprehensive study on the psychological motivation of students is carried out in high school English grammar teaching. The traditional teaching model had little effect on students’ psychological motivation (...) in the process of English grammar teaching. Therefore, students’ satisfaction with English grammar teaching under the traditional teaching model is very low, especially in innovation and practice. The STEAM teaching model has greatly improved the psychological motivation of high school students in the English grammar teaching process. The overall satisfaction rate of the students with the English grammar teaching in the STEAM teaching model is 70%. This teaching model can greatly improve students’ requirements for innovation and practice. This study not only provides a reference for the improvement of English teaching in high schools but also contributes to the reform and development of education. (shrink)

In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of his examples. This inadequacy (...) is remediable in each case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (shrink)

Education and the collective construction of knowledge. An overview -- Teaching modality : a corpus analysis of the use of must by university Spanish learners of English -- Motivation and anxiety towards mathematical learning in primary education -- Criteria for the sequencing of contemporary music in education -- Sustainability training evaluation of Spanish university students -- Art as a means of fostering peace, communication and freedom of expression from childhood -- Culturemes vs. humoremes as translation problems in practice exemplified (...) in the German Cabaret of Marc-Uwe Kling and his Kangaroo stories -- Is there homogeneity among business students regarding the level of concern towards the sustainable development goals? A cluster analysis -- Use trends and methodological pertinence of Delphi within educational research in IberoAmerica : a systematic review in Scopus -- Gamification in the improvement of speaking competences in the English as a foreign language classroom : a case study -- Quality management in the Spanish university : an analysis of the success and performance rates in communication degree programs -- Teacher stories from the future : technology for the language and literature classroom. (shrink)

In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...) method of analysis, which are intended to shed light on his view about the status of mathematical axioms. I describe the position Russell develops in consequence as “immanent logicism,” in contrast to what Irving (1989) describes as “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament, and to propose a method for discovering structural relationships of dependence within mathematical theories. (shrink)

Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to (...) the classical tradition shape prominent debates in philosophy of mathematics, and I initiate a project of reconstruction within this field. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain strongly algebraizable logics, all (...) connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

One influential interpretation of Newton's formulation of his calculus has regarded his work as an organized, cohesive presentation, shaped primarily by technical issues and implicitly motivated by a knowledge of the form which a "finished" calculus should take. Offered as an alternative to this view is a less systematic and more realistic picture, in which both philosophical and technical considerations played a part in influencing the structure and interpretation of the calculus throughout Newton's mathematical career. This analysis sees the (...) development of Newton's calculus not principally as a calculated movement toward "rigorous" justification (in the sense of Cauchy, Weierstrass, and their 19th century contemporaries) and refinement of techniques, but as an evolution in the light of his involvement in debates over philosophical and scientific issues central to the rise of modern philosophy and science in the 17th and early 18th centuries. (shrink)

A modified version of the mere exposure effect paradigm was utilized in an implicit artificial grammar learning task in an attempt to develop a procedure that would be more sensitive in assesing nonconscious learning processes than the methods currently utilized within the field of implicit learning. Subjects were presented with stimuli generated from a finite-state artificial grammar and then had to either decide if novel items conformed to the rule structure of the grammar or rate the degree (...) to which they liked novel items. Because the latter of the task was more indirect of the two procedures, subjects′ availability to discriminate between well-formed and ill-formed items on this liking task was taken as a more sensitive piece of evidence of implicit learning, compared to performance on the rule conformity task. A subsequent test of subjects′ explicit knowledge of the rules of the grammar showed that subjects do have some conscious knowledge of the artificial grammar, but subjects making liking decisions exhibited significantly less conscious rule knowledge than subjects initially making rule-conformity judgements. The findings are discussed in relation to synergistic relationships among implicit perception, implicit memory, and implicit learning proceses. (shrink)

Research shows that goal-relevant objects are rated positively, which results from their functionality towards the aim. In previous studies these objects were always external to the agent. However, relevant knowledge of self is also potentially accessible during goal pursuit, as self-esteem is an indicator of aim’s feasibility. In two experimental studies we tested whether goal activation affects temporal changes in automatic evaluations of personality traits related to the dimensions of agency and communion. We administered affect misattribution procedure where participants rated (...) neutral Chinese hexagrams proceeded by words describing traits. The list of words comprised agentic and communal traits. The rating took place twice - before and after introducing a manual task. In the first study, goal activation led to slightly more positive implicit evaluations of agentic and more negative evaluation of communal traits, which is consistent with empirical data on self-perception depending on agentic knowledge rather than communal one. In the second study we showed that goal activation led to changes only for promotion-, but not prevention-oriented individuals, which is explained by motivation strength. The results indicate that valuation of traits changes temporarily along with goal pursuit. (shrink)

Three competing ‘methods’ have been endorsed for inferring phylogenetic hypotheses: parsimony, likelihood, and Bayesianism. The latter two have been claimed superior because they take into account rates of sequence substitution. Can rates of substitution be justified on its own accord in inferences of explanatory hypotheses? Answering this question requires addressing four issues: (1) the aim of scientific inquiry, (2) the nature of why-questions, (3) explanatory hypotheses as answers to why-questions, and (4) acknowledging that neither parsimony, likelihood, nor Bayesianism are inferential (...) actions leading to explanatory hypotheses. The aim of scientific inquiry is to acquire causal understanding of effects. Observation statements of organismal characters lead to implicit or explicit why-questions. Those questions, conveyed in data matrices, assume the truth of observation statements, which is contrary to subsequently invoking substitution rates within inferences to phylogenetic hypotheses. Inferences of explanatory hypotheses are abductive in form, such that some version of an evolutionary theory(ies) is/are included or implied. If rates of sequence evolution are to be considered, it must be done prior to, rather than within abduction, which requires renaming those putatively-shared nucleotides subject to substitution rates. There are, however, no epistemic grounds for renaming characters to accommodate rates, calling into question the legitimacy of causally accounting for sequence data. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain strongly algebraizable logics, all (...) connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

This book discusses both the philosophy of mathematics and of mathematics education. The first part is a critique of existing approaches and a new philosophy of mathematics. Chapters include: (1) "A Critique of Absolutist Philosophies of Mathematics," (2) "The Philosophy of Mathematics Reconceptualized," (3) "Social Constructivism as a Philosophy of Mathematics," (4) "Social Constructivism and Subjective Knowledge," and (5) "The Parallels of Social Constructivism." The second part of the book explores the philosophy of mathematics education and shows that many aspects (...) of mathematics education rest on underlying, usually implicit, philosophical assumptions. Making these assumptions explicit puts a critical tool into the hands of teachers and researchers. Chapters in this section are: (6) "Aims and Ideologies of Mathematics Education," (7) "Groups with Utilitarian Ideologies," (8) "Groups with Purist Ideologies," (9) "The Social Change Ideology of the Public Educators," (10) "Critical Review of Cockcroft and the National Curriculum," (11) "Hierarchy in Mathematics, Learning, Ability and Society," (12) "Mathematics, Values and Equal Opportunities," and (13) "Investigation, Problem Solving and Pedagogy." Each chapter contains notes. (Contains 654 references.) (MKR). (shrink)

The implicit activities of the living body in sports (such as heart rate, involuntary gestures, stress, reflex, emotional regulation and interaction expressions) emerge in the consciousness of the lived body without our voluntary control. We demonstrate physiological emersion, and how, including in dramaturgical perception, physiological flows and processes collide with the image of a whole body. In this paper, we introduce corporeal non-property as the missing (?) link between phenomenology and neuroscience, renewed by research on the cerebral unconscious (...) and the perception of their lived body. Rather than mindfulness, physical activity obliges the subject to learn the unconscious and his motor skills from his living body: a new psychoanalysis of sport is thus established by studying what emerges from the activity of the living body up to the level of the conscious description of motor action. In this paper, we explain why that awareness occurs, and why it matters for the understanding of unconscious activity of the athlete’s living body. This is potentially interesting, although it needs, as a theory, to overcome the obvious reliance of the body on unconscious or habitual/automated physiological processes. But the incorporation of body techniques forms a habitus. (shrink)

A traditional problem of ethics in mathematics is the denial of social responsibility. Pure mathematics is viewed as neutral and value free, and therefore free of ethical responsibility. Applications of mathematics are seen as employing a neutral set of tools which, of themselves, are free from social responsibility. However, mathematicians are convinced they know what constitutes good mathematics. Furthermore many pure mathematicians are committed to purism, the ideology that values purity above applications in mathematics, and some historical reasons for this (...) are discussed. MacIntyre’s virtue ethics accommodates both the good mathematician and the ethics of the social practice of mathematics. It demonstrates that purism is compatible with acknowledging the social responsibility of mathematics. Four aspects of this responsibility are mentioned, two concerning the impact of mathematics via education, and two concerning explicit and implicit applications of mathematics. The last of these opens up the performativity of mathematical and measurement applications in society, which change the very processes they are supposed to measure. Although these applications are not explored in detail, they illustrate the importance of considering the ethics and social responsibility of mathematics in society. MacIntyre’s virtue theory opens a broad approach to the controversial topic of the ethics of mathematics encompassing purism, and absolutist and social constructivist philosophies of mathematics, but still enabling ethical critiques of the impact of mathematics on society. (shrink)

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

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

Mathematics stand in a privileged relationship with aesthetics: a relationship that follows two main directions. The first concerns the introduction of mathematical considerations into aesthetic discourse. For instance, it is common to mention the mathematical architecture of certain artistic productions. The second leads from aesthetics to mathematics. In this case, the question is that of the role and meaning that aesthetic considerations may assume in mathematics. It is indeed a widely held view among mathematicians, of whatever socio-historical context, (...) not only to see their discipline as presenting a strong aesthetic dimension, but also to consider that this dimension plays a fundamental role in the process of developing and understanding mathematics. The main ambition of this paper is to show how Nelson Goodman’s aesthetics can be used to justify this point of view and to propose a thesis concerning the aesthetic functioning of mathematics. This first result allows to resituate Goodman’s aesthetics within a very classical tradition that will be described. Finally, the underlying ambition is to show the keys provided by Goodman’s theory for the philosophy of mathematics. (shrink)

This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, analysis, geometry, etc., (...) including numbers, functions, sets, points, lines, spaces, etc., whether it be such objects themselves or the possibilities of such, as postulated by modal and modal structural accounts?2 What real difference does it make, whether it be the existence of such things or the mathematical possibility of such things? Even fictionalist views seem to confront analogous problems. After all, we rely on mathematics for myriad scientific applications; so such mathematics had better at least be coherent, even if not true. But coherence requires at least formal consistency; so we seem implicitly to be committed to things like formal derivations, viz. the absence of any such having contradictory consequences, framed within the appropriate system of axioms and rules. But derivations are themselves akin to mathematical objects, as derivations are strings of strings of symbols. Moreover, derivations provided by axioms and rules form an infinite class, such that, at any future time, infinitely many will not have been written down. Moreover, such strings, as Gödel famously showed, can be coded as natural numbers. Thus, even fictionalist accounts (such as that of Field in his Science without Numbers [2016]) seem to confront problems very much like those plaguing platonist accounts. (shrink)

Implicit definitions have been much discussed in the history and philosophy of science in relation to logical positivism. Not only have the logical positivists been influential in establishing this notion, but they have addressed the main problems connected with the use of such definitions, in particular the question whether there can be such definitions, and the problem of delimiting their scope. This paper aims to draw further insights on implicit definitions from the development of this notion from its (...) first occurrence in German language in Enriques’s “Principles of Geometry” (1907) to Schlick’s General Theory of Knowledge (1918). Enriques was one of the first to acknowledge that implicit definitions in mathematics are possible only for higher-order entities or structures, which can have infinitely many interpretations in terms of physical objects. While Schlick introduced coordinating principles to account for the scientific interpretations of implicit definitions, Enriques addressed the problem of bridging the gap between abstract and concrete terms in a different way: He identified, within mathematics, structural patterns that provide a clarification of conceptual relations, and so also serve (indirectly) the purposes of applied mathematics. My suggestion is that Enriques’s analysis of these patterns deserves deeper consideration also from a contemporary perspective on mathematical concept formation. (shrink)

In contemporary historical studies, Peano is usually included in the logical tradition pioneered by Frege. In this paper, I shall first demonstrate that Frege and Peano independently developed a similar way of using logic for the rigorous expression and proof of mathematical laws. However, I shall then suggest that Peano also used his mathematical logic in such a way that anticipated a formalisation of mathematical theories which was incompatible with Frege’s conception of logic.

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the (...) exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth, and he claimed that alternate deflationary formalizations are unable to explain some central uses of the truth predicate in mathematics. He also used Beth’s definability theorem to argue that, contrary to deflationists’ claims, the T-schema cannot provide an ‘implicit definition’ of truth. In this article, I want to (...) challenge this final argument. Whatever other faults deflationism may have, the T-schema does provide an implicit definition of the truth predicate. Or so, at any rate, I shall argue. (shrink)