{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:40:12Z","timestamp":1777516812274,"version":"3.51.4"},"reference-count":51,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2019,11,15]],"date-time":"2019-11-15T00:00:00Z","timestamp":1573776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2020,2,26]]},"abstract":"<jats:p>In La G\u00e9om\u00e9trie, Descartes proposed a \u201cbalance\u201d between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.<\/jats:p>\n                  <jats:p>Descartes\u2019 geometric foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has been extended beyond the limits of algebraic polynomials in two different periods: by late 17th century tractional motion and by early 20th century differential algebra. This paper proves that, adopting these extensions, it is possible to define a new convergence of machines (analog computation), algebra (symbolic manipulations) and a well determined class of mathematical objects that gives scope for a constructive foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. To establish this balance, a clear definition of the constructive limits of tractional motion is provided by a differential universality theorem.<\/jats:p>","DOI":"10.3233\/com-180208","type":"journal-article","created":{"date-parts":[[2019,11,15]],"date-time":"2019-11-15T11:50:06Z","timestamp":1573818606000},"page":"51-83","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":5,"title":["A differential extension of Descartes\u2019 foundational approach: A\u00a0new\u00a0balance between symbolic and analog computation"],"prefix":"10.1177","volume":"9","author":[{"given":"Pietro","family":"Milici","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Bretagne Occidentale, Brest, France."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2019,11,15]]},"reference":[{"key":"ref001","first-page":"21","volume":"11","author":"Adda F.B.","year":"1997","journal-title":"Journal of Fractional Calculus"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-33099-2"},{"key":"ref003","unstructured":"V.\u00a0Blasjo, Transcendental Curves in the Leibnizian Calculus, Academic Press, 2017."},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1111\/j.1600-0498.1988.tb00714.x"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0087-8"},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1515\/9783110922752.109"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-24021-3_8"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2011.12.022"},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-51082-6_73"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021105915386"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.1007\/s00283-019-09895-7"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735172"},{"key":"ref013","first-page":"1","volume":"12","author":"Di Paola B.","year":"2012","journal-title":"Acta Didactica Universitatis Comenianae \u2013 Mathematics"},{"key":"ref014","unstructured":"M.P.\u00a0Do Carmo, Differential Geometry of Curves and Surfaces, Vol.\u00a02, Prentice-Hall, Englewood Cliffs, 1976."},{"key":"ref015","doi-asserted-by":"publisher","DOI":"10.1007\/s10958-009-9689-3"},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.2478\/s13540-014-0174-4"},{"key":"ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF02430507"},{"key":"ref018","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45084-X_2"},{"key":"ref019","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(89)90120-5"},{"key":"ref020","doi-asserted-by":"publisher","DOI":"10.1016\/S0040-9383(01)00034-9"},{"key":"ref021","first-page":"213","volume":"7","author":"Kempe A.B.","year":"1876","journal-title":"Proceedings of the London Mathematical Society"},{"key":"ref022","unstructured":"E.R.\u00a0Kolchin, Differential Algebra & Algebraic Groups, Pure and Applied Mathematics, Vol.\u00a054, Academic Press, 1973."},{"key":"ref023","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8840-9"},{"key":"ref024","unstructured":"S.\u00a0Lang, Undergraduate Algebra, Springer, 2005."},{"issue":"3","key":"ref025","first-page":"221","volume":"8","author":"Milici P.","year":"2012","journal-title":"International Journal of Unconventional Computing"},{"key":"ref026","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-32894-7_16"},{"key":"ref027","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-10434-8_1"},{"issue":"4","key":"ref028","first-page":"3520","volume":"14","author":"Milici P.","year":"2017","journal-title":"Oberwolfach Reports"},{"key":"ref029","doi-asserted-by":"publisher","DOI":"10.1007\/s00283-012-9308-x"},{"key":"ref030","first-page":"486","volume":"4","author":"Milici P.","year":"2012","journal-title":"International Journal for Mathematics in Education, HMS i JME"},{"key":"ref031","unstructured":"K.S.\u00a0Miller and B.\u00a0Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993."},{"key":"ref032","doi-asserted-by":"publisher","DOI":"10.1111\/j.1600-0498.1963.tb00546.x"},{"key":"ref033","doi-asserted-by":"publisher","DOI":"10.1098\/rstl.1706.0017"},{"key":"ref034","doi-asserted-by":"publisher","DOI":"10.1098\/rstl.1714.0038"},{"issue":"4","key":"ref035","first-page":"367","volume":"5","author":"Podlubny I.","year":"2002","journal-title":"Fractional Calculus & Applied Analysis"},{"key":"ref036","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2953-5"},{"key":"ref037","unstructured":"J.\u00a0Poleni, Epistolarum Mathematicarum Fasciculus, Typogr. 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