Results for 'Structure of mathematics'

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  1. Structure in mathematics and logic: A categorical perspective.S. Awodey - 1996 - Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is (...)
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  2.  80
    Structural realism, mathematics, and ontology.Otávio Bueno - 2019 - Studies in History and Philosophy of Science Part A 74:4-9.
  3. Structure in mathematics.Saunders Lane - 1996 - Philosophia Mathematica 4 (2):174-183.
    The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It (...)
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  4.  42
    International Symposium on Structures in Mathematical Theories.Lorenzo PeÑa - 1990 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 5 (1-2):320-324.
  5. Stating structural realism: mathematics‐first approaches to physics and metaphysics.David Wallace - 2022 - Philosophical Perspectives 36 (1):345-378.
    I respond to the frequent objection that structural realism fails to sharply state an alternative to the standard predicate-logic, object / property / relation, way of doing metaphysics. The approach I propose is based on what I call a ‘math-first’ approach to physical theories (close to the so-called ‘semantic view of theories') where the content of a physical theory is to be understood primarily in terms of its mathematical structure and the representational relations it bears to physical systems, rather (...)
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  6.  11
    International Symposium on 'Structures in Mathematical Theories'.Javier Echeverria - 1989 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 4 (2):581-582.
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  7.  75
    H. Garth Dales and W. Hugh Woodin. Super-real fields. Totally ordered fields with additional structure. London Mathematical Society monographs, n.s. no. 14. Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, xv + 357 pp. [REVIEW]M. Dickmann - 2000 - Bulletin of Symbolic Logic 6 (2):218-221.
  8.  68
    Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 pp. [REVIEW]Alessandro Berarducci - 2000 - Bulletin of Symbolic Logic 6 (2):216-218.
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  9.  84
    Mathematical Structure and Empirical Content.Michael E. Miller - unknown - British Journal for the Philosophy of Science 74 (2):511-532.
    Approaches to the interpretation of physical theories provide accounts of how physical meaning accrues to the mathematical structure of a theory. According to many standard approaches to interpretation, meaning relations are captured by maps from the mathematical structure of the theory to statements expressing its empirical content. In this article I argue that while such accounts adequately address meaning relations when exact models are available or perturbation theory converges, they do not fare as well for models that give (...)
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  10. Mathematical anti-realism and explanatory structure.Bruno Whittle - 2021 - Synthese 199 (3-4):6203-6217.
    Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims (...)
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  11.  8
    Philosophy, mathematics and structure.James Franklin - 1995 - Philosopher 1 (2):31-38.
    An early version of the work on mathematics as the science of structure that appeared later as An Aristotelian Realist Philosophy of Mathematics (2014).
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  12. Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1989 - Oxford, England: Oxford University Press.
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
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  13.  10
    Well-structured mathematical logic.Damon Scott - 2013 - Durham, North Carolina: Carolina Academic Press.
    Well-Structured Mathematical Logic does for logic what Structured Programming did for computation: make large-scale work possible. From the work of George Boole onward, traditional logic was made to look like a form of symbolic algebra. In this work, the logic undergirding conventional mathematics resembles well-structured computer programs. A very important feature of the new system is that it structures the expression of mathematics in much the same way that people already do informally. In this way, the new system (...)
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  14.  84
    Mathematical Structural Realism.Christopher Pincock - 2011 - In Alisa Bokulich & Peter Bokulich, Scientific Structuralism. Springer Science+Business Media. pp. 67--79.
    Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the (...)
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  15. Dedekind, structural reasoning, and mathematical understanding.Erich H. Reck - 2009 - In Bart Van Kerkhove, New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics. World Scientific. pp. 150--173.
  16. Structural Analogies Between Mathematical and Empirical Theories.Andoni Ibarra & Thomas Mormann - 1992 - In Javier Echeverría, Andoni Ibarra & Thomas Mormann, The space of mathematics: philosophical, epistemological, and historical explorations. New York: W. de Gruyter.
  17.  39
    Mathematical Logic: On Numbers, Sets, Structures, and Symmetry.Roman Kossak - 2018 - Cham: Springer Verlag.
    This textbook is a second edition of the successful, Mathematical Logic: On Numbers, Sets, Structures, and Symmetry. It retains the original two parts found in the first edition, while presenting new material in the form of an added third part to the textbook. The textbook offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Part I, Logic Sets, and Numbers, shows how mathematical logic is used to (...)
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  18. Mathematical structural realism.Author unknown - manuscript
    Forthcoming in A. Bokulich & P. Bokulich (eds.), Scientific Structuralism, Boston Studies in the Philosophy of Science, Springer. Abstract: Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, (...)
     
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  19. Structural Analogies Between Mathematical and Empirical Theories.Andoni Ibarra & Thomas Mormann - 1992 - In Javier Echeverría, Andoni Ibarra & Thomas Mormann, The space of mathematics: philosophical, epistemological, and historical explorations. New York: W. de Gruyter.
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  20.  29
    Finite Mathematical Structures.John G. Kemeny, Hazleton Mirkill, J. Laurie Snell & Gerald L. Thompson - 1959 - Journal of Symbolic Logic 24 (3):221-222.
  21.  14
    Mathematical Structures and Physical Necessity.Roberto Torretti - 1992 - In Javier Echeverría, Andoni Ibarra & Thomas Mormann, The space of mathematics: philosophical, epistemological, and historical explorations. New York: W. de Gruyter. pp. 132.
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  22. Mathematical Structural Realism.Chris Pincock - 2011 - In Alisa Bokulich & Peter Bokulich, Scientific Structuralism. Springer Science+Business Media.
    Epistemic structural realists have argued that we are in a better epistemic position with respect to the structural claims made by our theories than the non-structural claims. Critics have objected that we cannot make the structure/non-structure distinction precise. I respond that a focus on mathematical structure leads to a clearer understanding of this debate. Unfortunately for the structural realist, however, the contribution that mathematics makes to scientific representation undermines any general confidence we might have in the (...)
     
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  23. Structure and applied mathematics.Travis McKenna - 2022 - Synthese 200 (5):1-31.
    ‘Mapping accounts’ of applied mathematics hold that the application of mathematics in physical science is best understood in terms of ‘mappings’ between mathematical structures and physical structures. In this paper, I suggest that mapping accounts rely on the assumption that the mathematics relevant to any application of mathematics in empirical science can be captured in an appropriate mathematical structure. If we are interested in assessing the plausibility of mapping accounts, we must ask ourselves: how plausible (...)
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  24.  61
    Mathematical structures, universals, and singular terms.Bahram Assadian - 2022 - In B. Assadian N. Kürbis, Knowledge, Number, and Reality; Encounters with the Work of Keith Hossack. pp. 203-216.
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  25. Mathematical reasoning vs. abductive reasoning: A structural approach.Atocha Aliseda - 2003 - Synthese 134 (1-2):25 - 44.
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  26. Interdisciplinary Connections between Radical Constructivist Approaches in Mathematical Problem Solving and Structural Design in Architecture.V. Sevim - 2014 - Constructivist Foundations 9 (3):411-412.
    Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: In the target article, Christiane Herr offers an insightful characterization of how von Glasersfeld’s radical constructivism can be implemented in structural design education in architecture. In this commentary, I articulate possible connections between research on problem solving and problem posing in mathematics education and design processes in structural design education as described in the target article.
     
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  27.  47
    Introduction to mathematics: number, space, and structure.Scott A. Taylor - 2023 - Providence, Rhode Island: American Mathematical Society.
    This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history (...)
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  28.  60
    Societal, Structural, and Conceptual Changes in Mathematics Teaching: Reform Processes in France and Germany over the Twentieth Century and the International Dynamics.Hélène Gispert & Gert Schubring - 2011 - Science in Context 24 (1):73-106.
    ArgumentThis paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics education or (...)
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  29.  21
    The Constructivism, Structuralism and Structure-Constructivism in Mathematical Philosophy. 문장수 - 2022 - Journal of the New Korean Philosophical Association 109:225-262.
    본 연구는 수학철학의 핵심적인 두 유파는 구성주의와 구조주의라는 것을 논증하면서, 양자는 다시 구조-구성주의라는 하나의 유파로 종합될 수 있으며 종합되어야 한다는 것을 논증하고자 한다. 이를 위해 역사-비판적 검토를 수행한다. 역사-비판적 관점에서 볼 때, 수학철학은 광의의 관점에서 10여개의 유파로 구분되지만, 세부적으로는 20여개 이상으로 세분화된다. 그러나 가장 큰 흐름은 수학적 대상들은 인간의 정신으로부터 독립적이고 물질적 대상들로부터도 독립적으로 존재하는 “추상적 실체”라고 주장하는 플라톤주의이고, 이것에 대립적인 것은 심리학적 구성주의이다. 후자에 따르면, 수학적 대상들은 인간정신의 구성의 산물이다. 그러나 여기서 말하는 구성의 근거가 논리적 장치인가 아니면 기호적 상징체계인가 (...)
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  30. Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction.Bart Van Kerkhove & Jean Van 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 (...)
     
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  31.  17
    Structures Mères: Semantics, Mathematics, and Cognitive Science.Silvano Zipoli Caiani & Alberto Peruzzi (eds.) - 2020 - Springer.
    This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information and inspiration for (...)
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  32. A Tree-Structured List in a Mathematical Series Text from Mesopotamia.Christine Proust - 2015 - In Karine Chemla & Jacques Virbel, Texts, Textual Acts and the History of Science. Springer International Publishing.
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  33.  62
    Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction.Bart Van Kerkhove & Jean Paul Van Bendegem - 2005 - Philosophia Scientiae 9-2 (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 (...)
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  34. Naturalizing Badiou: mathematical ontology and structural realism.Fabio Gironi - 2014 - New York: Palgrave-Macmillan.
    This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical (...)
  35.  36
    (1 other version)A Metaphysics for Mathematical and Structural Realism.Adam InTae Gerard - 2009 - Stance 2:76-89.
    The goal of this paper is to preserve realism in both ontology and truth for the philosophy of mathematics and science. It begins by arguing that scientific realism can only be attained given mathematical realism due to the indispensable nature of the latter to the prior. Ultimately, the paper argues for a position combining both Ontic Structural Realism and Ante Rem Structuralism, or what the author refers to as Strong Ontic Structural Realism, which has the potential to reconcile realism (...)
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  36.  65
    Do Ante Rem Mathematical Structures Instantiate Themselves?Scott Normand - 2019 - Australasian Journal of Philosophy 97 (1):167-177.
    ABSTRACTAnte rem structuralists claim that mathematical objects are places in ante rem structural universals. They also hold that the places in these structural universals instantiate themselves. This paper is an investigation of this self-instantiation thesis. I begin by pointing out that this thesis is of central importance: unless the places of a mathematical structure, such as the places of the natural number structure, themselves instantiate the structure, they cannot have any arithmetical properties. But if places do not (...)
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  37.  10
    Mathematical Metaphors Presuppose Common Logico-Mathematical Structures.Anderson Norton & Vladislav Kokushkin - 2021 - Constructivist Foundations 16 (3):285-287.
    Constructivist and embodied theories of learning each focus on action as the basis for cognition. However, in restricting action to sensorimotor activity, some embodied perspectives eschew ….
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  38. A logico-mathematic, structural methodology. Part II: Experimental design and epistemological issues.Robert E. Haskell - 2003 - Journal of Mind and Behavior 24 (3-4):401-422.
    In this first of two companion papers to a logico-mathematic, structural methodology , a meta-level analysis of the non metric structure is presented in relation to critiques based on standard experimental, statistical, and computational methods of contemporary psychology and cognitive science. The concept of a non metric methodology is examined as it relates to the epistemological and scientific goals of experimental, statistical, and computational methods. While sharing in these goals, differences and similarities between the two methodological approaches are outlined. (...)
     
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  39.  75
    Fundamental physical theories: mathematical structures grounded on a primitive ontology.Valia Allori - 2007 - Dissertation, Rutgers
    In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a (...)
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  40.  34
    Mathematical Structure Applied to Metaethical Dialectics.Deborah C. Arangno & Lorraine Marie Arangno - 2022 - Philosophia 50 (4):1563-1577.
    This paper seeks to utilize mathematical methods to formally define and analyze the metaethical theory that is ethical reductionism. In contemporary metaethics, realist-antirealist debates center on the ontology of moral properties. Our research reflects an innovative methodology using methods from Graph Theory to clarify a debated position of Meta-Ethics, previously encumbered by intrinsic vagueness and ambiguity. We employ rigorous mathematical formalism to symbolize, parse, and thus disambiguate, particular philosophical questions regarding ethical ontological materialism of the reductionist variety. In this paper, (...)
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  41.  78
    Applying Mathematics: Immersion, Inference, Interpretation.Otávio Bueno & Steven French - 2018 - Oxford, England: Oxford University Press. Edited by Steven French.
    How is that when scientists need some piece of mathematics through which to frame their theory, it is there to hand? What has been called 'the unreasonable effectiveness of mathematics' sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character, whereas others have pointed to the range of structures that mathematics offers. Otavio Bueno and Steven French offer a middle way, which focuses on the moves that (...)
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  42. Mathematical Thought and its Objects.Charles Parsons - 2007 - New York: Cambridge University Press.
    Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept (...)
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  43.  57
    Geoffrey Hellman. Mathematics without numbers. Towards a modal-structural interpretation. Clarendon Press, Oxford University Press, Oxford and New York1989, xi + 154 pp. [REVIEW]Peter Clark - 1995 - Journal of Symbolic Logic 60 (4):1310-1312.
  44.  24
    Factors influencing microgame adoption among secondary school mathematics teachers supported by structural equation modelling-based research.Tommy Tanu Wijaya, Yiming Cao, Martin Bernard, Imam Fitri Rahmadi, Zsolt Lavicza & Herman Dwi Surjono - 2022 - Frontiers in Psychology 13.
    Microgames are rapidly gaining increased attention and are highly being considered because of the technology-based media that enhances students’ learning interests and educational activities. Therefore, this study aims to develop a new construct through confirmatory factor analysis, to comprehensively understand the factors influencing the use of microgames in mathematics class. Participants of the study were the secondary school teachers in West Java, Indonesia, which had a 1-year training in microgames development. We applied a quantitative approach to collect the data (...)
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  45.  10
    The Phenomenon of Life.Christopher Alexander & Center for Environmental Structure - 2002
    Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to students, (...)
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  46. Towards collaborative content management and version control for structured mathematical knowledge.Michael Kohlhase - unknown
    We propose an infrastructure for collaborative content management and version control for structured mathematical knowledge. This will enable multiple users to work jointly on mathematical theories with minimal interference. We describe the API and the functionality needed to realize a cvs-like version control and distribution model. This architecture extends the cvs architecture in two ways, motivated by the specific needs of distributed management of structured mathematical knowledge on the Internet. On the one hand the one-level client/server model of cvs is (...)
     
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  47.  70
    Unifying scientific theories: physical concepts and mathematical structures.Talel A. Debs - 2003 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34 (1):151-153.
  48. Unifying Scientific Theories: Physical Concepts and Mathematical Structures.Margaret Morrison - 2000 - Cambridge University Press.
    This book is about the methods used for unifying different scientific theories under one all-embracing theory. The process has characterized much of the history of science and is prominent in contemporary physics; the search for a 'theory of everything' involves the same attempt at unification. Margaret Morrison argues that, contrary to popular philosophical views, unification and explanation often have little to do with each other. The mechanisms that facilitate unification are not those that enable us to explain how or why (...)
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  49. Can we dissolve physical entities into mathematical structures?Tian Yu Cao - 2003 - Synthese 136 (1):57 - 71.
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  50.  61
    Model Organisms and Mathematical and Synthetic Models to Explore Gene Regulation Mechanisms.Andrea Loettgers - 2007 - Biological Theory 2 (2):134-142.
    Gene regulatory networks are intensively studied in biology. One of the main aims of these studies is to gain an understanding of how the structure of genetic networks relates to specific functions such as chemotaxis and the circadian clock. Scientists have examined this question by using model organisms such as Drosophila and mathematical models. In the last years, synthetic models—engineered genetic networks—have become more and more important in the exploration of gene regulation. What is the potential of this new (...)
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