{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:30:11Z","timestamp":1760146211541,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T00:00:00Z","timestamp":1728345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Foundations"],"abstract":"<jats:p>After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds.<\/jats:p>","DOI":"10.3390\/foundations4040035","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T12:03:49Z","timestamp":1728389029000},"page":"537-551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Mathematical Examples of Emergent Intuitive Local Time Flow"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4991-7568","authenticated-orcid":false,"given":"Manuel L.","family":"Esqu\u00edvel","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nova School of Science and Technology, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4226-1658","authenticated-orcid":false,"given":"Nadezhda P.","family":"Krasii","sequence":"additional","affiliation":[{"name":"Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Department of Higher Mathematics, Don State Technical University, Gagarin Square 1, Rostov-on-Don 344000, Russia"}]},{"given":"Philippe L.","family":"Didier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nova School of Science and Technology, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Nova Math, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1017\/S0031819100024918","article-title":"Time in Philosophy and in Physics","volume":"54","author":"Dingle","year":"1979","journal-title":"Philosophy"},{"unstructured":"de Ferro, F.M. (1998). Raz\u00e3o e Progresso na Filosofia de Kant, Edi\u00e7\u00f5es Colibri. Forum de Ideias.","key":"ref_2"},{"unstructured":"Guyer, P., and Wood, A.W. (1998). Critique of Pure Reason, Cambridge University Press. The Cambridge Edition of the Works of Immanuel Kant.","key":"ref_3"},{"unstructured":"Zalta, E.N. (2022). Newton\u2019s Views on Space, Time, and Motion. The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University. [Spring 2022 ed.].","key":"ref_4"},{"unstructured":"Baptiste Le Biha, K.M., and Wuthrich, C. (2020). Space and Time in Loop Quantum Gravity. Beyond Spacetime: The Philosophical Foundations of Quantum Gravity, Cambridge University Press.","key":"ref_5"},{"unstructured":"Lawson, R.W. (1920). Relativity: The Special and the General Theory. A Popular Exposition, Methuen & Co., Ltd.","key":"ref_6"},{"unstructured":"Einstein, A. (1954). La Th\u00e9orie de la Relativit\u00e9 Restreinte et G\u00e9n\u00e9rale. Expos\u00e9 \u00c9l\u00e9mentaire. La Relativit\u00e9 et le Probl\u00e8me de L\u2019espace. Traduit de L\u2019allemand par Maurice Solovine, Gauthier-Villars.","key":"ref_7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1002\/andp.19053221004","article-title":"On the electrodynamics of moving bodies","volume":"17","author":"Einstein","year":"1905","journal-title":"Ann. Phys."},{"unstructured":"Stachel, J., Lipscombe, T., Calaprice, A., and Elworthy, S. (1998). (Eds.) Einstein\u2019s Miraculous Year. Five Papers that Changed the Face of Physics, Princeton University Press.","key":"ref_9"},{"doi-asserted-by":"crossref","unstructured":"Nat\u00e1rio, J. (2021). An Introduction to Mathematical Relativity, Springer.","key":"ref_10","DOI":"10.1007\/978-3-030-65683-6"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1023\/A:1001959224682","article-title":"An example of a new type of cosmological solutions of Einstein\u2019s field equations of gravitation","volume":"32","year":"2000","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1119\/1.3086933","article-title":"G\u00f6del, Einstein, Mach, Gamow, and Lanczos: G\u00f6del\u2019s remarkable excursion into cosmology","volume":"77","author":"Rindler","year":"2009","journal-title":"Am. J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"013063","DOI":"10.1088\/1367-2630\/15\/1\/013063","article-title":"Visualization of the G\u00f6del universe","volume":"15","author":"Buser","year":"2013","journal-title":"New J. Phys."},{"doi-asserted-by":"crossref","unstructured":"Passon, O., Benzm\u00fcller, C., and Falkenburg, B. (2023). The Philosophical Meaning of the G\u00f6del Universe. On G\u00f6del and the Nonexistence of Time \u2013 G\u00f6del und die Nichtexistenz der Zeit: Kurt G\u00f6del Essay Competition 2021\u2014Kurt-G\u00f6del-Preis 2021, Springer.","key":"ref_14","DOI":"10.1007\/978-3-662-67045-3"},{"doi-asserted-by":"crossref","unstructured":"G\u00f6del, K., and Feferman, S. (1995). Some observations about the relationship between theory of relativity and Kantian philosophy. Collected Works of Kurt G\u00f6del Vol. III, Oxford University Press. Drafts from G\u00f6del Nachlass for the Article G\u00f6del, 1949a.","key":"ref_15","DOI":"10.1093\/oso\/9780195072556.001.0001"},{"unstructured":"Anscombe, G.E. (1961). Notebooks 1914\u20131916, Harper & Brothers Publishers.","key":"ref_16"},{"unstructured":"Wittgenstein, L. (1974). Tractatus Logico-Philosophicus, Routledge Classics, Routledge & Kegan Paul.","key":"ref_17"},{"unstructured":"Rhees, R. (1978). Philosophical Grammar, Philosophical Grammar, University of California Press.","key":"ref_18"},{"doi-asserted-by":"crossref","unstructured":"Stadler, F., and St\u00f6ltzner, M. (2006). Wittgenstein on Time (1929\u20131933). Time and History, Proceedings of the 28. International Ludwig Wittgenstein Symposium, Kirchberg am Wechsel, Austria 2005, De Gruyter.","key":"ref_19","DOI":"10.1515\/9783110333213"},{"unstructured":"Rovelli, C. (2018). The Order of Time, Penguin Random House. [1st ed.].","key":"ref_20"},{"unstructured":"Connes, A. (1994). Noncommutative Geometry, Academic Press, Inc.","key":"ref_21"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1088\/0264-9381\/11\/12\/007","article-title":"Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories","volume":"11","author":"Connes","year":"1994","journal-title":"Class. Quantum Gravity"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1515\/cmb-2020-0110","article-title":"From ODE to Open Markov Chains, via SDE: An application to models for infections in individuals and populations","volume":"8","author":"Guerreiro","year":"2020","journal-title":"Comput. Math. Biophys."},{"unstructured":"Hoffman, D. (1978). A Course in Differential Geometry, Springer. Graduate Texts in Mathematics.","key":"ref_24"},{"doi-asserted-by":"crossref","unstructured":"Thorpe, J.A. (1979). Elementary Topics in Differential Geometry, Springer. Undergraduate Texts in Mathematics.","key":"ref_25","DOI":"10.1007\/978-1-4612-6153-7"},{"unstructured":"Braemer, J.M., and Kerbrat, Y. (1976). G\u00e9om\u00e9trie Des Courbes et Des Surfaces, Hermann. Collection M\u00e9thodes. [Methods Collection].","key":"ref_26"},{"unstructured":"Folland, G.B. (1999). Real Analysis, John Wiley & Sons, Inc.. [2nd ed.]. Modern Techniques and Their Applications, A Wiley-Interscience Publication.","key":"ref_27"},{"unstructured":"Gray, A., Abbena, E., and Salamon, S. (2006). Modern Differential Geometry of Curves and Surfaces with Mathematica\u00ae, Chapman & Hall, CRC. [3rd ed.]. Studies in Advanced Mathematics.","key":"ref_28"},{"doi-asserted-by":"crossref","unstructured":"Jost, J. (2011). Riemannian Geometry and Geometric Analysis, Springer. [6th ed.]. Universitext.","key":"ref_29","DOI":"10.1007\/978-3-642-21298-7"},{"unstructured":"Embarek, D. (1980). Chapitres Suppl\u00e9mentaires de la Th\u00e9orie Des \u00c9quations Diff\u00e9rentielles Ordinaires, \u201cMir\u201d.","key":"ref_30"},{"unstructured":"Lang, S. (1983). Real Analysis, Addison-Wesley Publishing Company. [2nd ed.].","key":"ref_31"},{"unstructured":"Arnol\u2019 d, V. (1974). \u00c9quations Diff\u00e9rentielles Ordinaires, \u00c9ditions Mir. Traduit du russe par Djilali Embarek.","key":"ref_32"},{"doi-asserted-by":"crossref","unstructured":"Barndorff-Nielsen, O.E., and Shiryaev, A. (2010). Change of Time and Change of Measure, World Scientific Publishing Co., Pte., Ltd.","key":"ref_33","DOI":"10.1142\/7928"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1080\/03610918.2014.901352","article-title":"Brownian bridge and other path-dependent Gaussian processes vectorial simulation","volume":"44","author":"Sousa","year":"2015","journal-title":"Comm. Statist. Simul. Comput."},{"unstructured":"do Ros\u00e1rio Grossinho, M., Shiryaev, A., Oliveira, P.E., and Esqu\u00edvel, M.L. (2004). Dynamical Value-at-Risk via Ito line integrals. Stochastic Finance 2004, CIM\u2014Centro Internacional de Matem\u00e1tica.","key":"ref_35"}],"container-title":["Foundations"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2673-9321\/4\/4\/35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:09:13Z","timestamp":1760112553000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2673-9321\/4\/4\/35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,8]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["foundations4040035"],"URL":"https:\/\/doi.org\/10.3390\/foundations4040035","relation":{},"ISSN":["2673-9321"],"issn-type":[{"type":"electronic","value":"2673-9321"}],"subject":[],"published":{"date-parts":[[2024,10,8]]}}}