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  1. Changes of language in the development of mathematics.Ladislav Kvasz - 2000 - Philosophia Mathematica 8 (1):47-83.
    The nature of changes in mathematics was discussed recently in Revolutions in Mathematics. The discussion was dominated by historical and sociological arguments. An obstacle to a philosophical analysis of this question lies in a discrepancy between our approach to formulas and to pictures. While formulas are understood as constituents of mathematical theories, pictures are viewed only as heuristic tools. Our idea is to consider the pictures contained in mathematical text, as expressions of a specific language. Thus we get formulas and (...)
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  • Proofs and pictures.James Robert Brown - 1997 - British Journal for the Philosophy of Science 48 (2):161-180.
    Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.
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  • On ensuring that physicalism is not a dual attribute theory in sheep's clothing.Frank Jackson - 2006 - Philsophical Studies 131 (1):227-249.
    Physicalists are committed to the determination without remainder of the psychological by the physical, but are they committed to this determination being a priori? This paper distinguishes this question understood de dicto from this question understood de re, argues that understood de re the answer is yes in a way that leaves open the answer to the question understood de dicto.
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  • Structuring a Philosophical Approach.Richard Startup - 2019 - Open Journal of Philosophy 9 (4):452-469.
    A framework is developed for understanding what is “taken for granted” both in philosophy and in life generally, which may serve to orient philosophical inquiry and make it more effective. The framework takes in language and its development, as well as mathematics, logic, and the empirical sphere with particular reference to the exigencies of life. It is evaluated through consideration of seven philosophical issues concerned with such topics as solipsism, sense data as the route to knowledge, the possible reduction of (...)
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  • Experimental Mathematics.Alan Baker - 2008 - Erkenntnis 68 (3):331-344.
    The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is (...)
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  • Spatial visualization and sex-related differences in mathematical problem solving.Julia A. Sherman - 1996 - Behavioral and Brain Sciences 19 (2):262-263.
    Spatial visualization as a key variable in sex-related differences in mathematical problem solving and spatial aspects of geometry is traced to the 1960s. More recent relevant data are presented. The variability debate is traced to the latter part of the nineteenth century and an explanation for it is suggested. An idea is presented for further research to clarify sex-related brain laterality differences in solving spatial problems.
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  • On the relationship between plane and solid geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
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  • Explanation, independence and realism in mathematics.Michael D. Resnik & David Kushner - 1987 - British Journal for the Philosophy of Science 38 (2):141-158.
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  • Some Remarks on Wittgenstein’s Philosophy of Mathematics.Richard Startup - 2020 - Open Journal of Philosophy 10 (1):45-65.
    Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem (...)
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  • (1 other version)Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction.Bart Van Kerkhove & Bendegem - 2005 - Philosophia Scientiae 9 (2):61-78.
    In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.
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  • We are far from understanding sex-related differences in spatial-mathematical abilities despite the theory of sexual selection.Üner Tan - 1996 - Behavioral and Brain Sciences 19 (2):264-264.
    I have provided evidence that Geary's model does not explain male dominance in spatial abilities by sexual selection. The current literature concerning the relations of nonverbal IQ to testosterone, hand preference, and right- and left-hand skill, as well as the organizing effects of testosterone on cerebral lateralization during the perinatal period, does not support Geary's arguments.
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  • Sexual-selection accounts of human characteristics: Just So Stories or scientific hypotheses?Nora Newcombe & Mary Ann Baenninger - 1996 - Behavioral and Brain Sciences 19 (2):259-260.
    We evaluate three of Geary's claims, finding that there is little evidence for sex differences in object- vs. person-orientation; sex differences in competition, even if biologically caused, lead to sex differences in mathematics only given a certain style of teaching; and sex differences in mental rotation, though real, are not well explained in a sociobiological framework or by the proximate biological variables assumed by Geary.
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  • Is there a comparative psychology of implicit mathematical knowledge?Hank Davis - 1996 - Behavioral and Brain Sciences 19 (2):250-250.
    Geary suggests that implicit mathematical principles exist across human cultures and transcend sex differences. Is such knowledge present in animals as well, and is it sufficient to account for performance in all species, including our own? I attempt to trace the implications of Gearys target article for comparative psychology, questioning the exclusion of “subitizing” in describing human mathematical performance, and asking whether human researchers function as cultural agents with animals, elevating their implicit knowledge to secondary domains of numerical performance.
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  • All sex differences in cognitive ability may be explained by an X-Y homologous gene determining degrees of cerebral asymmetry.T. J. Crow - 1996 - Behavioral and Brain Sciences 19 (2):249-250.
    Male superiority in mathematical ability (along with female superiority in verbal fluency) may reflect the operation of an X-Y homologous gene (the right-shift-factor) influencing the relative rates of development of the cerebral hemispheres. Alleles at the locus on the Y chromosome will be selected at a later mean age than alleles on the X, and only by females.
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  • The role of inversion in the genesis, development and the structure of scientific knowledge.Nagarjuna G. - manuscript
    The main thrust of the argument of this thesis is to show the possibility of articulating a method of construction or of synthesis--as against the most common method of analysis or division--which has always been (so we shall argue) a necessary component of scientific theorization. This method will be shown to be based on a fundamental synthetic logical relation of thought, that we shall call inversion--to be understood as a species of logical opposition, and as one of the basic monadic (...)
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  • The three arrows of Zeno.Craig Harrison - 1996 - Synthese 107 (2):271 - 292.
    We explore the better known paradoxes of Zeno including modern variants based on infinite processes, from the point of view of standard, classical analysis, from which there is still much to learn (especially concerning the paradox of division), and then from the viewpoints of non-standard and non-classical analysis (the logic of the latter being intuitionist).The standard, classical or Cantorian notion of the continuum, modeled on the real number line, is well known, as is the definition of motion as the time (...)
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  • Beltrami's model and the independence of the parallel postulate.J. Scanlan Michael - 1988 - History and Philosophy of Logic 9 (1):13-34.
    E. Beltrami in 1868 did not intend to prove the consistency of non-euclidean plane geometry nor the independence of the euclidean parallel postulate. His approach would have been unsuccessful if so intended. J. Hoüel in 1870 described the relevance of Beltrami's work to the issue of the independence of the euclidean parallel postulate. Hoüel's method is different from the independence proofs using reinterpretation of terms deployed by Peano about 1890, chiefly in using a fixed interpretation for non-logical terms. Comparing the (...)
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  • Between-sex differences are often averaging artifacts.Hoben Thomas - 1996 - Behavioral and Brain Sciences 19 (2):265-265.
    The central problem in Geary's theory is how differences are conceptualized. Recent research has shown that between-sex differences on certain tasks are a consequence of averaging within sex differences. A mixture distribution models between-sex differences on several tasks well and does not appear congenial to a sexual-selection perspective.
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  • Omissions relevant to gender-linked mathematical abilities.Herman T. Epstein - 1996 - Behavioral and Brain Sciences 19 (2):251-252.
    Analyses of bodies of data usually omit some relevant studies. Geary omits some studies looking at functional correlates of basic biological data, studies of developmental implications for functioning, and the recent achievement of acceleration of cognitive development.
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  • (1 other version)An argument for P=NP.Selmer Bringsjord - manuscript
    Selmer Bringsjord & Joshua Taylor∗ Department of Cognitive Science Department of Computer Science The Rensselaer AI & Reasoning (RAIR) Lab Rensselaer Polytechnic Institute (RPI) Troy NY 12180 USA http://www.rpi.edu/∼brings {selmer,tayloj}@rpi.edu..
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  • (1 other version)An Argument for P = NP.Selmer Bringsjord - 2017 - Minds and Machines 27 (4):663-672.
    I articulate a novel modal argument for P=NP.
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  • Why do mathematicians re-prove theorems?John W. Dawson Jr - 2006 - Philosophia Mathematica 14 (3):269-286.
    From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)
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  • Differences in male and female cognitive abilities: Sexual selection or division of labor?Michael T. Ghiselin - 1996 - Behavioral and Brain Sciences 19 (2):254-255.
    In Darwinian terminology, “sexual selection” refers to purely reproductive competition and is conceptually distinct from natural selection as it affects reproduction generally. As natural selection may favor the evolution of sexual dimorphism by virtue of the division of labor between males and females, this possibility needs to be taken very seriously.
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  • Still far too sexy a topic.Susan F. Chipman - 1996 - Behavioral and Brain Sciences 19 (2):248-249.
    Geary is highly selective in his use of the literature on gender differences. His assumption of consistent female inferiority in mathematics is not necessarily supported by the facts.
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  • Abstract Explanations in Science.Christopher Pincock - 2014 - British Journal for the Philosophy of Science 66 (4):857-882.
    This article focuses on a case that expert practitioners count as an explanation: a mathematical account of Plateau’s laws for soap films. I argue that this example falls into a class of explanations that I call abstract explanations.explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need not be causally relevant to the explanandum for its features to be explanatorily relevant. However, it remains unclear how to unify (...)
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  • Able youths and achievement tests.Julian C. Stanley & Heinrich Stumpf - 1996 - Behavioral and Brain Sciences 19 (2):263-264.
    Achievement test differences between boys and girls and between young men and young women, mostly favoring males, extend far beyond mathematics. Such pervasive differences, illustrated here, may require an explanatory theory broader than Geary's.
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  • On Norton’s dome.Jon Pérez Laraudogoitia - 2013 - Synthese 190 (14):2925-2941.
    Norton’s very simple case of indeterminism in classical mechanics has given rise to a literature critical of his result. I am interested here in posing a new objection different from the ones made to date. The first section of the paper expounds the essence of Norton’s model and my criticism of it. I then propose a specific modification in the absence of gravitational interaction. The final section takes into consideration a surprising consequence for classical mechanics from the new model introduced (...)
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  • Genetic influences on sex differences in outstanding mathematical reasoning ability.Ada H. Zohar - 1996 - Behavioral and Brain Sciences 19 (2):266-267.
    Sexual selection provides an adequate partial explanation for the difference in means between the distributions, but fails to explain the difference in variance, that is, the overrepresentation of both boys with outstanding mathematical reasoning ability and boys with mental retardation. Other genetic factors are probably at work. While spatial ability is correlated with OMRA, so are other cognitive abilities. OMRA is not reducible to spatial ability; hence selection for navigational skill is unlikely to be the only mechanism by which males (...)
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  • Sex differences and evolutionary by-products.Thomas Wynn, Forrest Tierson & Craig Palmer - 1996 - Behavioral and Brain Sciences 19 (2):265-266.
    From the perspective of evolutionary theory, we believe it makes more sense to view the sex differences in spatial cognition as being an evolutionary by-product of selection for optimal rates of fetal development. Geary does not convince us that his proposed selective factors operated with “sufficient precision, economy, and efficiency.” Moreover, the archaeological evidence does not support his proposed evolutionary scenario.
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  • The twain shall meet: Uniting the analysis of sex differences and within-sex variation.David C. Rowe - 1996 - Behavioral and Brain Sciences 19 (2):262-262.
    Spatial and mathematical abilities may be “sex-limited” traits. A sex-limited trait has the same determinants of variation within the sexes, but the genetic or environmental effects would be differentially expressed in males and females. New advances in structural equation modeling allow means and variation to be estimated simultaneously. When these statistical methods are combined with a genetically informative research design, it should be possible to demonstrate that the genes influencing spatial and mathematical abilities are sex-limited in their expression. This approach (...)
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  • Abstraction and Intuition in Peano's Axiomatizations of Geometry.Davide Rizza - 2009 - History and Philosophy of Logic 30 (4):349-368.
    Peano's axiomatizations of geometry are abstract and non-intuitive in character, whereas Peano stresses his appeal to concrete spatial intuition in the choice of the axioms. This poses the problem of understanding the interrelationship between abstraction and intuition in his geometrical works. In this article I argue that axiomatization is, for Peano, a methodology to restructure geometry and isolate its organizing principles. The restructuring produces a more abstract presentation of geometry, which does not contradict its intuitive content but only puts it (...)
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  • The logic of the sociobiological model Geary-style.Diane Proudfoot - 1996 - Behavioral and Brain Sciences 19 (2):261-261.
    Geary's is the traditional view of the sexes. Yet each part of his argument – the move from sex differences in spatial ability and social preferences to a sex difference in mathematical ability, the claim that the former are biologically primary, and the sociobiological explanation of these differences – requires considerable further work. The notion of a biologically secondary ability is itself problematic.
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  • A Voyage of Mathematical and Cultural Awareness for Students of Upper Secondary School.Evangelos N. Panagiotou - 2014 - Science & Education 23 (1):79-123.
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  • Explanation, Existence and Natural Properties in Mathematics – A Case Study: Desargues’ Theorem.Marc Lange - 2015 - Dialectica 69 (4):435-472.
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  • Resources dimorphism sexual selection and mathematics achievement.Diana Eugenie Kornbrot - 1996 - Behavioral and Brain Sciences 19 (2):259-259.
    Geary's model is a worthy effort, but ambiguous on important issues. It ignores differential resource allocation, although this follows directly from sexual selection via differential parental investment. Dimorphism in primary traits is arbitrarily attributed to sexual selection via intramale competition, rather than direct evolutionary pressures. Dubious predictions are made about the consequences of raising mathematics achievement.
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  • Some problematic links between hunting and geometry.Meredith M. Kimball - 1996 - Behavioral and Brain Sciences 19 (2):258-259.
    Geary's emphasis on hunting ignores the possible importance of other human activities, such as scavenging and gathering, in the evolution of spatial abilities. In addition, there is little evidence that links spatial abilities and math skills. Furthermore, such links have little practical importance given the small size of most differences and girls' superior performance in mathematics classrooms.
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  • A critic with a different perspective.Lloyd G. Humphreys - 1996 - Behavioral and Brain Sciences 19 (2):257-258.
    To the extent that Geary's theory concerning biologically primary and secondary behaviors depends on factor analytic methods and findings, it is woefully weak. Factors have been mistakenly called primary mental abilities, but the adjective “primary” represents reification of a mathematical dimension defined by correlations. Fleshing out a factor beyond its mathematical properties requires much additional quantitative experimental and correlational research that goes far beyond mere factoring.
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  • Brain differences, anthropological stories, and educational implications.Christy Hammer & R. Valentine Dusek - 1996 - Behavioral and Brain Sciences 19 (2):257-257.
    Criticism of sex differences in mathematical ability and sex roles in sociobiology and the pernicious influence of these ideas on education.
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  • Mating, math achievement, and other multiple relationships.Diane F. Halpern - 1996 - Behavioral and Brain Sciences 19 (2):256-256.
    Although Geary's partitioning of mathematical abilities into those that are biologically primary and secondary is an advance over most sociobiological theories of cognitive sex differences, it remains untestable and ignores the spatial nature of women's traditional work. An alternative model based on underlying cognitive processes offers other advantages.
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  • Sex differences in mathematical abllity: Genes, environment, and evolution.Jeffrey W. Gillger - 1996 - Behavioral and Brain Sciences 19 (2):255-256.
    Geary proposes a sociobiological hypothesis of how sex differences in math and spatial skills might have jointly arisen. His distinction between primary and secondary math skills is noteworthy, and in some ways analogous to the closed versus open systems postulated to exist for language. In this commentary issues concerning how genes might affect complex cognitive skills, the interpretation of heritability estimates, and prior research abilites are discussed.
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  • Sexual selection and sex differences in mathematical abilities.David C. Geary - 1996 - Behavioral and Brain Sciences 19 (2):229-247.
    The principles of sexual selection were used as an organizing framework for interpreting cross-national patterns of sex differences in mathematical abilities. Cross-national studies suggest that there are no sex differences in biologically primary mathematical abilities, that is, for those mathematical abilities that are found in all cultures as well as in nonhuman primates, and show moderate heritability estimates. Sex differences in several biologically secondary mathematical domains are found throughout the industrialized world. In particular, males consistently outperform females in the solving (...)
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  • On the biology and politics of cognitive sex differences.David C. Geary - 1996 - Behavioral and Brain Sciences 19 (2):267-284.
    The male advantage in certain mathematical domains contributes to the difference in the numbers of males and females that enter math-intensive occupations, which in turn contributes to the sex difference in earnings. Understanding the nature and development of the sex difference in mathematical abilities is accordingly of social as well as scientific concern. A more complete understanding of the biological contributions to these differences can guide research on educational techniques that might someday produce more equal educational outcomes in mathematics and (...)
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  • Mary has more: Sex differences, autism, coherence, and theory of mind.Uta Frith & Francesca Happé - 1996 - Behavioral and Brain Sciences 19 (2):253-254.
    We challenge the notion that differences in spatial ability are the best or only explanation for observed sex differences in mathematical word problems. We suggest two ideas from the study of autism: sex differences in theory of mind and in central coherence.
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  • Arithmetic and old lace.Jeffrey Foss - 1996 - Behavioral and Brain Sciences 19 (2):252-253.
    Geary's project faces the severe methodological difficulty of tracing the biological effects of gender on mathematical ability in a system that is massively open. Two methodological stratagems he uses are considered. The first is that pancultural sex differences are biological in nature, which is dubious in the domain of mathematics, since it is completely culture-bound. The second is that sociosexual differences are partly caused by biosexual differences, which renders his thesis unfalsifiable and empirically empty.
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  • On an evolutionary model of sex differences in mathematics: Do the data support the theory?Alan Feingold - 1996 - Behavioral and Brain Sciences 19 (2):252-252.
    The target article draws on evolutionary theory to formulate a biosocial model of sex differences in quantitative abilities. Unfortunately, the data do not support some of the crucial hypotheses. The male advantage in geometry is not appreciably greater than the male advantagi in algebra, and the greater male variability in mathematics cited by Gear is not cross-culturally invariant.
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  • How important is spatial ability to mathematics?Ann Dowker - 1996 - Behavioral and Brain Sciences 19 (2):251-251.
    This commentary focuses on one of the many issues raised in Geary's target article: the importance of gender differences in spatial ability to gender differences in mathematics. I argue that the evidence for the central role of spatial ability in mathematical ability, or in gender differences in it, is tenuous at best.
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  • How real are real numbers?Gregory Chaitin - 2011 - Manuscrito 34 (1):115-141.
    We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel.
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  • Do gender differences in spatial skills mediate gender differences in mathematics among high-ability students?M. Beth Casey - 1996 - Behavioral and Brain Sciences 19 (2):247-248.
    Based on Geary's theory, intelligence may determine which males utilize innate spatial knowledge to inform their mathematical solutions. This may explain why math gender differences occur mainly with higher abilities. In support, we found that mental rotation ability served as a mediator of gender differences on the math Scholastic Assessment Test for two high-ability samples. Our research suggests, however, that environment and biology interact to influence mental rotation abilities.
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