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MANY 1

A Transversal Imaginative Journey across the Realm of Mathematics
  • Jean-Yves Beziau
Article
  • 42 Downloads

Abstract

We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.

Keywords

One Philosophy of mathematics Logic Truth-value Boole Universal logic 

Notes

Acknowledgements

Although this paper is self-contained, it is the continuation of many previous ones in particular the one I just wrote before this one: “Is the principle of contradiction a consequence of x 2 = x?” (2016), related to a plenary talk I gave at the University of St Petersburg for the congress The 12th International Conference Logic Today: Developments and Perspectives in June 2016. Thanks again to my Russian colleagues for the invitation, in particular to Elena Lisanyuk and Ivan Mikirtumov.

I had the original idea of the present paper in July 2016 when in the Island of Santorini in Greece, formerly known as Kallíste (Kαλλιστη, “the most beautiful one”) and fictionally as Atlantis, (Ἀτλαντὶς νῆσος, “the island of Atlas”). I was invited on the Island by Ioannis Vandoulakis, organizer of the event The Logics of Image: Visualization, Iconicity, Imagination and Human Creativity. Following the idea of this congress, I made extensive use in this paper of images, similarly as in recent papers, in particular “Possibility, Imagination and Conception” (2016a, b, c) that I presented at this event. This is related to a project I am developing to promote the use of images in philosophy, The World Journal of Pictorial Philosophy: www.wjpp.org.

I did not expound the present “MANY 1” paper at the event but had the occasion to discuss some of its contents with Ioannis who is a specialist of history of mathematics, and with other participants of the event, in particular Dénes Nagy the president of the The International Society for the Interdisciplinary Study of Symmetry, former student of the great Hungarian historian of mathematics, Árpád Szabó (whose work I know since my youth; Szabo 1969, 1984), and the plastic artist Catherine Chantilly.

I would like also to thank Mihir Chakraborty, founder of the Kolkata Logic Circle, who invited me to write this paper for a volume dedicated to pluralism in mathematics, a volume he has prepared with Michele Friend, author of the book Pluralism in mathematics (Friend 2014). I know Mihir since a couple of years. He was an invited speaker at the 3rd Congress on the Square of Opposition we organized in Beirut in June 2012, and he gave a tutorial at the 4th World Congress and School on Universal Logic we organized in Rio de Janeiro in April 2013. After that I have organized with him the 5th World Congress on Paraconsistency in February 2014, at the Indian Statistical Institute in Kolkata, whose motto is “Unity in diversity”. Mihir also invited me to take part to another event in Kolkata just after this one: International Congress on History and Philosophy of Mathematics—Tribute to SIR Ashutosh Mookherjee where I presented the talk “Bourbaki and Modern Mathematics”, which I never transformed into a paper but some things I said there are included in the present paper.

I was immersed in a Bourba-très-chic atmosphere since my youth but I deepened my knowledge about the history and philosophy of Bourbaki when in São Paulo, Brazil, in 1991–1992 working with Newton da Costa who used to take me to the house of his former teacher Edison Farah (1911–2006), the host of André Weil, Jean Dieudonné and Alexandre Grothendieck during their frequent visits to the University of São Paulo in the 1940s and 1950s. Farah proved a conjecture that Weil thought was false: general distributivity of conjunction relatively to disjunction is equivalent to the Axiom of Choice (Farah 1954). Therefore, one more formulation of AC among many ones.

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Authors and Affiliations

  1. 1.University of BrazilRio de JaneiroBrazil

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