Using analogical representations for mathematical concept formation

Alison Pease; Simon Colton; Ramin Ramezani; Alan Smaill; Markus Guhe

doi:10.1007/978-3-642-15223-8_17

Using analogical representations for mathematical concept formation

Alison Pease (Lead / Corresponding author), Simon Colton (Lead / Corresponding author), Ramin Ramezani, Alan Smaill, Markus Guhe

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

Abstract

We argue that visual, analogical representations of mathematical concepts can be used by automated theory formation systems to develop further concepts and conjectures in mathematics. We consider the role of visual reasoning in human development of mathematics, and consider some aspects of the relationship between mathematics and the visual, including artists using mathematics as inspiration for their art (which may then feed back into mathematical development), the idea of using visual beauty to evaluate mathematics, mathematics which is visually pleasing, and ways of using the visual to develop mathematical concepts. We motivate an analogical representation of number types with examples of “visual” concepts and conjectures, and present an automated case study in which we enable an automated theory formation program to read this type of visual, analogical representation.

Original language	English
Title of host publication	Model-Based Reasoning in Science and Technology
Subtitle of host publication	Abduction, Logic, and Computational Discovery
Editors	Lorenzo Magnani, Walter Carnielli, Claudio Pizzi
Place of Publication	Berlin
Publisher	Springer
Pages	301-314
Number of pages	14
ISBN (Electronic)	9783642152238
ISBN (Print)	9783642152238
DOIs	https://doi.org/10.1007/978-3-642-15223-8_17
Publication status	Published - 2010
Event	Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery - State University of Campinas , Campinas, Brazil Duration: 17 Dec 2009 → 19 Dec 2009 http://www-3.unipv.it/webphilos_lab/courses/progra4.html

Publication series

Name	Studies in Computational Intelligence
Volume	314

Conference

Conference	Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery
Abbreviated title	MBR'09_Brazil
Country/Territory	Brazil
City	Campinas
Period	17/12/09 → 19/12/09
Internet address	http://www-3.unipv.it/webphilos_lab/courses/progra4.html

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10.1007/978-3-642-15223-8_17

http://copac.ac.uk/search?&isn=9783642152221

Cite this

Pease, A., Colton, S., Ramezani, R., Smaill, A., & Guhe, M. (2010). Using analogical representations for mathematical concept formation. In L. Magnani, W. Carnielli, & C. Pizzi (Eds.), Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery (pp. 301-314). (Studies in Computational Intelligence; Vol. 314). Springer . https://doi.org/10.1007/978-3-642-15223-8_17

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title = "Using analogical representations for mathematical concept formation",

abstract = "We argue that visual, analogical representations of mathematical concepts can be used by automated theory formation systems to develop further concepts and conjectures in mathematics. We consider the role of visual reasoning in human development of mathematics, and consider some aspects of the relationship between mathematics and the visual, including artists using mathematics as inspiration for their art (which may then feed back into mathematical development), the idea of using visual beauty to evaluate mathematics, mathematics which is visually pleasing, and ways of using the visual to develop mathematical concepts. We motivate an analogical representation of number types with examples of “visual” concepts and conjectures, and present an automated case study in which we enable an automated theory formation program to read this type of visual, analogical representation.",

author = "Alison Pease and Simon Colton and Ramin Ramezani and Alan Smaill and Markus Guhe",

year = "2010",

doi = "10.1007/978-3-642-15223-8_17",

language = "English",

isbn = "9783642152238",

series = "Studies in Computational Intelligence",

publisher = "Springer ",

pages = "301--314",

editor = "Magnani, {Lorenzo } and Carnielli, {Walter } and {Pizzi }, {Claudio }",

booktitle = "Model-Based Reasoning in Science and Technology",

note = "Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery, MBR'09_Brazil ; Conference date: 17-12-2009 Through 19-12-2009",

url = "http://www-3.unipv.it/webphilos_lab/courses/progra4.html",

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Pease, A, Colton, S, Ramezani, R, Smaill, A & Guhe, M 2010, Using analogical representations for mathematical concept formation. in L Magnani, W Carnielli & C Pizzi (eds), Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery. Studies in Computational Intelligence, vol. 314, Springer , Berlin, pp. 301-314, Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery, Campinas, Brazil, 17/12/09. https://doi.org/10.1007/978-3-642-15223-8_17

Using analogical representations for mathematical concept formation. / Pease, Alison (Lead / Corresponding author); Colton, Simon (Lead / Corresponding author); Ramezani, Ramin et al.
Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery. ed. / Lorenzo Magnani; Walter Carnielli; Claudio Pizzi . Berlin: Springer , 2010. p. 301-314 (Studies in Computational Intelligence; Vol. 314).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Using analogical representations for mathematical concept formation

AU - Pease, Alison

AU - Colton, Simon

AU - Ramezani, Ramin

AU - Smaill, Alan

AU - Guhe, Markus

PY - 2010

Y1 - 2010

N2 - We argue that visual, analogical representations of mathematical concepts can be used by automated theory formation systems to develop further concepts and conjectures in mathematics. We consider the role of visual reasoning in human development of mathematics, and consider some aspects of the relationship between mathematics and the visual, including artists using mathematics as inspiration for their art (which may then feed back into mathematical development), the idea of using visual beauty to evaluate mathematics, mathematics which is visually pleasing, and ways of using the visual to develop mathematical concepts. We motivate an analogical representation of number types with examples of “visual” concepts and conjectures, and present an automated case study in which we enable an automated theory formation program to read this type of visual, analogical representation.

AB - We argue that visual, analogical representations of mathematical concepts can be used by automated theory formation systems to develop further concepts and conjectures in mathematics. We consider the role of visual reasoning in human development of mathematics, and consider some aspects of the relationship between mathematics and the visual, including artists using mathematics as inspiration for their art (which may then feed back into mathematical development), the idea of using visual beauty to evaluate mathematics, mathematics which is visually pleasing, and ways of using the visual to develop mathematical concepts. We motivate an analogical representation of number types with examples of “visual” concepts and conjectures, and present an automated case study in which we enable an automated theory formation program to read this type of visual, analogical representation.

U2 - 10.1007/978-3-642-15223-8_17

DO - 10.1007/978-3-642-15223-8_17

M3 - Conference contribution

SN - 9783642152238

T3 - Studies in Computational Intelligence

SP - 301

EP - 314

BT - Model-Based Reasoning in Science and Technology

A2 - Magnani, Lorenzo

A2 - Carnielli, Walter

A2 - Pizzi , Claudio

PB - Springer

CY - Berlin

T2 - Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery

Y2 - 17 December 2009 through 19 December 2009

ER -

Pease A, Colton S, Ramezani R, Smaill A, Guhe M. Using analogical representations for mathematical concept formation. In Magnani L, Carnielli W, Pizzi C, editors, Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery. Berlin: Springer . 2010. p. 301-314. (Studies in Computational Intelligence). doi: 10.1007/978-3-642-15223-8_17

Using analogical representations for mathematical concept formation

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