Identifying the borders of mathematical knowledge

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Identifying the borders of mathematical knowledge
Authors: Filipi Nascimento Silva, Bruno A. N. Travençolo, Matheus P. Viana, Luciano da Fontoura Costa [edit item]
Citation: Journal of Physics A: Mathematical and Theoretical 43 (32): . 2010.
Publication type: Journal article
Peer-reviewed: Yes
Database(s):
DOI: 10.1088/1751-8113/43/32/325202.
Google Scholar cites: Citations
Link(s): Paper link
Added by Wikilit team: Added on initial load
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Identifying the borders of mathematical knowledge is a publication by Filipi Nascimento Silva, Bruno A. N. Travençolo, Matheus P. Viana, Luciano da Fontoura Costa.


[edit] Abstract

Based on a divide and conquer approach, knowledge about nature has been organized into a set of interrelated facts, allowing a natural representation in terms of graphs: each `chunk' of knowledge corresponds to a node, while relationships between such chunks are expressed as edges. This organization becomes particularly clear in the case of mathematical theorems, with their intense cross-implications and relationships. We have derived a web of mathematical theorems from Wikipedia and, thanks to the powerful concept of entropy, identified its more central and frontier elements. Our results also suggest that the central nodes are the oldest theorems, while the frontier nodes are those recently added to the network. The network communities have also been identified, allowing further insights about the organization of this network, such as its highly modular structure.

[edit] Research questions

"In this paper, we set out to investigate how scientific knowledge, specifically mathematical theorems, is organized and interrelated. Thanks to Wikipedia, a comprehensive network of theorems has been compiled and investigated for the first time using complex networks."

Research details

Topics: Other corpus topics [edit item]
Domains: Mathematics, Physics [edit item]
Theory type: Analysis [edit item]
Wikipedia coverage: Sample data [edit item]
Theories: "Undetermined" [edit item]
Research design: Statistical analysis [edit item]
Data source: Wikipedia pages [edit item]
Collected data time dimension: Cross-sectional [edit item]
Unit of analysis: Article, Subject [edit item]
Wikipedia data extraction: Live Wikipedia [edit item]
Wikipedia page type: Article [edit item]
Wikipedia language: English [edit item]

[edit] Conclusion

"The question of how knowledge is organized represents a theme of broad interest, equally appealing to researchers from the most diverse areas as well as to lay people. In this paper, we set out to investigate how scientific knowledge, specifically mathematical theorems, is organized and interrelated. Thanks to Wikipedia, a comprehensive network of theorems has been compiled and investigated for the first time using complex networks. In particular, we identified the centrality of each theorem with respect to the whole network, which also paved the way to the identification of the frontier theorems—both classifications were possible because of the discriminative power of the diversity entropy, which is, for this network, correlated with the betweenness centrality. This approach is particularly interesting because it can provide insights about the development and organization of mathematical knowledge, as was confirmed by the strong negative correlation found between the diversity and the century of the theorem proof. According to our results, the oldest theorems tend to be the most important ones, in the sense that they have higher values of diversity entropy, in the average. On the other hand, the frontiers theorems are those recently added to the network. Additional structural information have been provided by the identification of the communities of theorems. It has been found that some of the most important mathematical theorems from different fields (e.g. Algebra, Calculus and Geometry) are tightly interconnected and tend to belong to the same community. As a future work, we intend to analyze Wikipedia in more depth, considering other scientific areas and respective relationships between them."

[edit] Comments


Further notes[edit]

Facts about "Identifying the borders of mathematical knowledge"RDF feed
AbstractBased on a divide and conquer approach, knBased on a divide and conquer approach, knowledge about nature has been organized into a set of interrelated facts, allowing a natural representation in terms of graphs: each `chunk' of knowledge corresponds to a node, while relationships between such chunks are expressed as edges. This organization becomes particularly clear in the case of mathematical theorems, with their intense cross-implications and relationships. We have derived a web of mathematical theorems from Wikipedia and, thanks to the powerful concept of entropy, identified its more central and frontier elements. Our results also suggest that the central nodes are the oldest theorems, while the frontier nodes are those recently added to the network. The network communities have also been identified, allowing further insights about the organization of this network, such as its highly modular structure.ork, such as its highly modular structure.
Added by wikilit teamAdded on initial load +
Collected data time dimensionCross-sectional +
ConclusionThe question of how knowledge is organizedThe question of how knowledge is organized represents a theme of broad interest, equally

appealing to researchers from the most diverse areas as well as to lay people. In this paper, we set out to investigate how scientific knowledge, specifically mathematical theorems, is organized and interrelated. Thanks to Wikipedia, a comprehensive network of theorems has been compiled and investigated for the first time using complex networks. In particular, we identified the centrality of each theorem with respect to the whole network, which also paved the way to the identification of the frontier theorems—both classifications were possible because of the discriminative power of the diversity entropy, which is, for this network, correlated with the betweenness centrality. This approach is particularly interesting because it can provide insights about the development and organization of mathematical knowledge, as was confirmed by the strong negative correlation found between the diversity and the century of the theorem proof. According to our results, the oldest theorems tend to be the most important ones, in the sense that they have higher values of diversity entropy, in the average. On the other hand, the frontiers theorems are those recently added to the network. Additional structural information have been provided by the identification of the communities of theorems. It has been found that some of the most important mathematical theorems from different fields (e.g. Algebra, Calculus and Geometry) are tightly interconnected and tend to belong to the same community. As a future work, we intend to analyze Wikipedia in more depth, considering

other scientific areas and respective relationships between them.
and respective relationships between them.
Data sourceWikipedia pages +
Doi10.1088/1751-8113/43/32/325202 +
Google scholar urlhttp://scholar.google.com/scholar?ie=UTF-8&q=%22Identifying%2Bthe%2Bborders%2Bof%2Bmathematical%2Bknowledge%22 +
Has authorFilipi Nascimento Silva +, Bruno A. N. Travençolo +, Matheus P. Viana + and Luciano da Fontoura Costa +
Has domainMathematics + and Physics +
Has topicOther corpus topics +
Issue32 +
Peer reviewedYes +
Publication typeJournal article +
Published inJournal of Physics A: Mathematical and Theoretical +
Research designStatistical analysis +
Research questionsIn this paper, we set out to investigate hIn this paper, we set out to investigate how scientific knowledge, specifically mathematical theorems, is

organized and interrelated. Thanks to Wikipedia, a comprehensive network of theorems has

been compiled and investigated for the first time using complex networks.
for the first time using complex networks.
Revid10,810 +
TheoriesUndetermined
Theory typeAnalysis +
TitleIdentifying the borders of mathematical knowledge
Unit of analysisArticle + and Subject +
Urlhttp://dx.doi.org/10.1088/1751-8113/43/32/325202 +
Volume43 +
Wikipedia coverageSample data +
Wikipedia data extractionLive Wikipedia +
Wikipedia languageEnglish +
Wikipedia page typeArticle +
Year2010 +