Axioms as Definitions: Revisiting Poincaré and Hilbert
A fondamental problem in the discussion on the foundations of mathematics is to clarify what an axiom is. This is especially important in the light of the most recent advances in set theory where new axioms have been proposed whose legitimacy is highly controversial (for example, large cardinal axio...
| Main Author: | |
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| Format: | Article |
| Language: | deu |
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Éditions Kimé
2019-02-01
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| Series: | Philosophia Scientiæ |
| Online Access: | http://journals.openedition.org/philosophiascientiae/1827 |
| _version_ | 1797402582147137536 |
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| author | Laura Fontanella |
| author_facet | Laura Fontanella |
| author_sort | Laura Fontanella |
| collection | DOAJ |
| description | A fondamental problem in the discussion on the foundations of mathematics is to clarify what an axiom is. This is especially important in the light of the most recent advances in set theory where new axioms have been proposed whose legitimacy is highly controversial (for example, large cardinal axioms); this paper is a contribution to this discussion. By analysing the view of Poincaré and Hilbert on axioms, we observe that, despite the deep differences in their philosophical thinking, the two logicians came to the same conception of the axioms of geometry as definitions in disguise. We revisit and generalise this view by arguing that any axiomatic system (set theory in particular) is the definition of some concepts. |
| first_indexed | 2024-03-09T02:26:22Z |
| format | Article |
| id | doaj.art-15feec70c75f471385924c05839016ea |
| institution | Directory Open Access Journal |
| issn | 1281-2463 1775-4283 |
| language | deu |
| last_indexed | 2024-03-09T02:26:22Z |
| publishDate | 2019-02-01 |
| publisher | Éditions Kimé |
| record_format | Article |
| series | Philosophia Scientiæ |
| spelling | doaj.art-15feec70c75f471385924c05839016ea2023-12-06T15:53:38ZdeuÉditions KiméPhilosophia Scientiæ1281-24631775-42832019-02-0123116718310.4000/philosophiascientiae.1827Axioms as Definitions: Revisiting Poincaré and HilbertLaura FontanellaA fondamental problem in the discussion on the foundations of mathematics is to clarify what an axiom is. This is especially important in the light of the most recent advances in set theory where new axioms have been proposed whose legitimacy is highly controversial (for example, large cardinal axioms); this paper is a contribution to this discussion. By analysing the view of Poincaré and Hilbert on axioms, we observe that, despite the deep differences in their philosophical thinking, the two logicians came to the same conception of the axioms of geometry as definitions in disguise. We revisit and generalise this view by arguing that any axiomatic system (set theory in particular) is the definition of some concepts.http://journals.openedition.org/philosophiascientiae/1827 |
| spellingShingle | Laura Fontanella Axioms as Definitions: Revisiting Poincaré and Hilbert Philosophia Scientiæ |
| title | Axioms as Definitions: Revisiting Poincaré and Hilbert |
| title_full | Axioms as Definitions: Revisiting Poincaré and Hilbert |
| title_fullStr | Axioms as Definitions: Revisiting Poincaré and Hilbert |
| title_full_unstemmed | Axioms as Definitions: Revisiting Poincaré and Hilbert |
| title_short | Axioms as Definitions: Revisiting Poincaré and Hilbert |
| title_sort | axioms as definitions revisiting poincare and hilbert |
| url | http://journals.openedition.org/philosophiascientiae/1827 |
| work_keys_str_mv | AT laurafontanella axiomsasdefinitionsrevisitingpoincareandhilbert |