8 results
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2. Naturalising Mathematics? A Wittgensteinian Perspective.
- Author
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Stam, Jan, Stokhof, Martin, and Van Lambalgen, Michiel
- Subjects
MENTAL work ,MATHEMATICS - Abstract
There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Naturalising Mathematics? A Wittgensteinian Perspective
- Author
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Jan Stam, Martin Stokhof, and Michiel Van Lambalgen
- Subjects
neuroscience ,mathematics ,Wittgenstein ,naturalism ,Logic ,BC1-199 ,Philosophy (General) ,B1-5802 - Abstract
There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy.
- Published
- 2022
- Full Text
- View/download PDF
4. A New Model of Mathematics Education: Flat Curriculum with Self-Contained Micro Topics
- Author
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Miklós Hoffmann and Attila Egri-Nagy
- Subjects
mathematics ,education ,philosophy of education ,flat curriculum ,micro topics ,deconstruction ,Logic ,BC1-199 ,Philosophy (General) ,B1-5802 - Abstract
The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative.
- Published
- 2021
- Full Text
- View/download PDF
5. A New Model of Mathematics Education: Flat Curriculum with Self-Contained Micro Topics
- Author
-
Attila Egri-Nagy and Miklós Hoffmann
- Subjects
Structure (mathematical logic) ,education ,flat curriculum ,Deductive reasoning ,BC1-199 ,Logic ,mathematics ,Teaching method ,media_common.quotation_subject ,B1-5802 ,dependency graph ,philosophy of education ,Philosophy ,Dependency graph ,History and Philosophy of Science ,deconstruction ,Mathematics education ,Conversation ,micro topics ,Philosophy of education ,Philosophy (General) ,Curriculum ,Parallels ,media_common - Abstract
The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative.
- Published
- 2021
- Full Text
- View/download PDF
6. Operators in Nature, Science, Technology, and Society: Mathematical, Logical, and Philosophical Issues
- Author
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Mark Burgin and Joseph Brenner
- Subjects
function ,information ,logic ,mathematics ,ontology ,epistemology ,nature ,operator ,science ,society ,system ,theory ,Logic ,BC1-199 ,Philosophy (General) ,B1-5802 - Abstract
The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators.
- Published
- 2017
- Full Text
- View/download PDF
7. Should Computability Be Epistemic? a Logical and Physical Point of View
- Author
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Florent Franchette
- Subjects
computability ,physical process ,Physical point ,Computability ,Church–Turing thesis ,Church-Turing thesis ,Epistemology ,usability ,Set (abstract data type) ,physical computability ,Philosophy ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Computable function ,History and Philosophy of Science ,Logical conjunction ,Computability logic ,Effective method ,physical Church-Turing thesis ,Mathematics - Abstract
Although the formalizations of computability provided in the 1930s have proven to be equivalent, two different accounts of computability may be distinguished regarding computability as an epistemic concept. While computability, according to the epistemic account, should be based on epistemic constraints related to the capacities of human computers, the non-epistemic account considers computability as based on manipulations of symbols that require no human capacities other than the capacity of manipulating symbols according to a set of rules. In this paper, I shall evaluate, both from a logical and physical point of view, whether computability should be regarded as an epistemic concept, i.e., whether epistemic constraints should be added on (physical) computability for considering functions as (physically) computable. Specifically, I shall argue that the introduction of epistemic constraints have deep implications for the set of computable functions, for the logical and physical Church-Turing thesis—cornerstones of logical and physical computability respectively—might turn out to be false according to which epistemic constraints are accepted.
- Published
- 2015
8. Operators in Nature, Science, Technology, and Society: Mathematical, Logical, and Philosophical Issues
- Author
-
Joseph E. Brenner and Mark Burgin
- Subjects
Computer science ,media_common.quotation_subject ,Functionalism (philosophy of mind) ,02 engineering and technology ,system ,050105 experimental psychology ,information ,Operator (computer programming) ,History and Philosophy of Science ,Logical conjunction ,0202 electrical engineering, electronic engineering, information engineering ,0501 psychology and cognitive sciences ,ontology ,lcsh:BC1-199 ,Function (engineering) ,theory ,lcsh:B1-5802 ,science ,media_common ,function ,logic ,mathematics ,lcsh:Philosophy (General) ,05 social sciences ,epistemology ,nature ,lcsh:Logic ,Variety (cybernetics) ,Epistemology ,Philosophy ,Range (mathematics) ,society ,operator ,Ontology ,020201 artificial intelligence & image processing ,Mathematical structure - Abstract
The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators.
- Published
- 2017
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