{"title":"数学中逻辑与美学的边界空间","authors":"Gerhard Heinzmann","doi":"10.1007/s10516-024-09720-7","DOIUrl":null,"url":null,"abstract":"<p>The main thesis defended in this paper is that, interpreted in the light of reflections of Peirce and Poincaré, one can found in mathematical reasoning a non-logical symptom that may be aesthetic in Goodman’s sense. This symptom is called exemplification and serves to distinguish between only logically correct and even explanatory proofs. It broadens the scope of aesthetics to include all activities involving symbolic systems and blurs the boundaries between logic and aesthetics in mathematics. It gives a better understanding of Poincaré’s thesis that to affect aesthetic value to certain properties is not simply an added value, a bonus that somehow rewards the mathematician’s mechanical labor, but on the contrary, taking the aesthetic value into account can be helpful to mathematical practice. As an example, three proofs of the irrationality of √2 are compared for their aesthetic functioning.</p>","PeriodicalId":44799,"journal":{"name":"Axiomathes","volume":"2 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Border Space between Logic and Aesthetics in Mathematics\",\"authors\":\"Gerhard Heinzmann\",\"doi\":\"10.1007/s10516-024-09720-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The main thesis defended in this paper is that, interpreted in the light of reflections of Peirce and Poincaré, one can found in mathematical reasoning a non-logical symptom that may be aesthetic in Goodman’s sense. This symptom is called exemplification and serves to distinguish between only logically correct and even explanatory proofs. It broadens the scope of aesthetics to include all activities involving symbolic systems and blurs the boundaries between logic and aesthetics in mathematics. It gives a better understanding of Poincaré’s thesis that to affect aesthetic value to certain properties is not simply an added value, a bonus that somehow rewards the mathematician’s mechanical labor, but on the contrary, taking the aesthetic value into account can be helpful to mathematical practice. As an example, three proofs of the irrationality of √2 are compared for their aesthetic functioning.</p>\",\"PeriodicalId\":44799,\"journal\":{\"name\":\"Axiomathes\",\"volume\":\"2 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-09-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Axiomathes\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s10516-024-09720-7\",\"RegionNum\":3,\"RegionCategory\":\"哲学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"0\",\"JCRName\":\"PHILOSOPHY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Axiomathes","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s10516-024-09720-7","RegionNum":3,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"0","JCRName":"PHILOSOPHY","Score":null,"Total":0}
The Border Space between Logic and Aesthetics in Mathematics
The main thesis defended in this paper is that, interpreted in the light of reflections of Peirce and Poincaré, one can found in mathematical reasoning a non-logical symptom that may be aesthetic in Goodman’s sense. This symptom is called exemplification and serves to distinguish between only logically correct and even explanatory proofs. It broadens the scope of aesthetics to include all activities involving symbolic systems and blurs the boundaries between logic and aesthetics in mathematics. It gives a better understanding of Poincaré’s thesis that to affect aesthetic value to certain properties is not simply an added value, a bonus that somehow rewards the mathematician’s mechanical labor, but on the contrary, taking the aesthetic value into account can be helpful to mathematical practice. As an example, three proofs of the irrationality of √2 are compared for their aesthetic functioning.
期刊介绍：
Axiomathes: Where Science Meets PhilosophyResearch in many fields confirms that science is changing its nature. Natural science, cognitive and social sciences, mathematics and philosophy (i.e., the set of tools developed to understand and model reality) exceed the conceptual framework introduced by Galileo and Descartes. Complexity and chaos; network dynamics; anticipatory systems; qualitative aspects of experience (intentionality, for example); emergent properties and objects; forward, upward, and downward causation: all portend a new scientific agenda.Axiomathes publishes studies of evolving ideas, perspectives, and methods in science, mathematics, and philosophy. Many aspects of this dawning are unknown: there will be startlingly good ideas, and many blind-alleys. We welcome this ferment. While Axiomathes’ scope is left open, scholarly depth, quality and precision of presentation remain prerequisites for publication.Axiomathes welcomes submissions, regardless of the tradition, school of thought, or disciplinary background from which they derive. The members of the journal’s editorial board reflect this approach in the diversity of their affiliations and interests. Axiomathes includes one issue per year under the title Epistemologia. Please see the tab on your right for more information about this joint publication.All submissions are subjected to double-blind peer review, the average peer review time is 3 months.Axiomathes publishes:· Research articles, presenting original ideas and results.· Review articles, which comprehensively synthesize and critically assess recent, original works or a selected collection of thematically related books.· Commentaries, brief articles that comment on articles published previously.· Book symposia, in which commentators are invited to debate an influential book with the author, who answers with a concluding reply.· Special issues, in which an expert collaborates with the journal as a guest editor, in order to identify an interesting topic in science, mathematics or philosophy, and interacts with the selected contributors, being in charge of a whole issue of the journal. Axiomathes invites potential guest-editors, who might be interested in collecting and editing such special issue, to contact the Editor in order to discuss the feasibility of the project.· Focused debates, collecting submissions and invited articles around a particular theme, as part of a normal issue of the journal.· Authors wishing to submit a reply article, or a proposal for a review article, a book symposium, a special issue or a focused debate, are invited to contact the Editor for further information.
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