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.
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Notes
Proper noun: the name of a particular person, place, or object that is spelled with a capital letter.
Common noun: a noun that is the name of a group of similar things, such as “table” or “book”, and not of a single person, place, or thing. (Online Cambridge Dictionary).
There are other properties that challenge the classification of “one” as a proper noun: «Proper nouns are not normally preceded by an article or other limiting modifier, as any or some. Nor are they usually pluralized. But the language allows for exceptions» (Dictionary.com) One exception proves the rule...
Many books have been written about the story of numbers, this is a classical one. Other interesting ones are (Dantzig 1930; Ifrah 1981) and more recently (Corry 2015). The literal translation of the original French title of Ifrah’s book would be The universal story of numbers with subtitle The intelligence of human beings narrated through numbers and counting. It has been creatively translated in English as From One to Zero: A Universal History of Numbers. This book is an interesting best seller but weak points have been pointed out, especially regarding the last part about the development of computation, see e.g. Dauben’s critcisms (Dauben 2002). Joseph Dauben (1944-) is the editor of an annotated bibliography about the history of mathematics (Dauben 1985) where the reader will find indications about many other books concerning numbers. And classical books about the history of mathematics are (Rouse Ball 1888; Bell 1940; Dieudonné 1987).
To avoid confusion the glyph for the number zero is sometimes slashed, but this convention has been used the other way round by IBM and in mathematics this is similar to the empty set; the glyph for number 1 has not yet been slashed. The capital letter “I” can also be confused with symbols for the number one. “I” incarnates the first person, not any one.
From this point of view, one is the starting point of Geometry.
Du Pasquier, a former student of ETH Zurich, was professor at the University of Neuchâtel and responsible for the edition of part of Leonhard Euler’s Opera Omnia.
This is an untranslatable French word, which, punnily enough, has a double meaning: the primary one meaning taking out, the secondary one meaning counting the votes.
The phallic aspect of the “1” contrasts with the feminine aspect of the “0”. Both are present in mathematics, but like Adam, 1 was the first. They can be seen as the two main characters of the story of mathematics, from which everything is springing.
This part of the work of Leibniz was mainly rediscovered by Couturat (1901). In this seminal book he gives a good analysis both of lingua universalis and calculus raciocinator. Couturat is also the author of a comprehensive book of 700 pages on the history of the universal language (joint work with Leau in 1903) and an interesting paper on the logical definition of number (Couturat 1900). The correspondence between Couturat and Russell which was frozen during many years in La Chaux-de-Fonds, Switzerland, was published in 2001 by Anne-Françoise Schmid.
As we have pointed out in a paper on identity (Beziau 2015), Recorde’s symbol is, like the balance of justice, a double symbolization: a pictogram representing an object which is a prototypical specimen of the thing it signifies. About the process of symbolization see our book La pointure du symbole (2014), related to a congress we have organized at the University of Neuchâtel in 2005 on symbolic thinking, including two papers on mathematical symbolism: (Robert 2014; Pont 2014).
Practiced by Rudolf Carnap (1891-1970), a member of the abortive project of a unified science including an International Encyclopedia; see Carnap (1934), Morris (1960), Salmi (2012) and Otto Neurath and the unity of science by Symons et al. (2011) published in the book series Logic, Epistemology and the Unity of Sciences (Springer, Dordrecht) launched by Shahid Rahman as a way to take up the torch.
“Qui dit mathématiques, dit démonstration”; this is the famous opening of The Element of Mathematics by the General Bourbaki (1934-). This has been reinforced by one of his disciples, Jean Dieudonné (1906-1992).
See the recent work by Guo (2014) for a comparative study of Greek rationality and Chinese thinking.
See e.g. the paper by Tsuji et al. (1998) stating in particular negative results about Nash equilibrium.
In a sense 1 was not considered as a number in Ancient Greece, but this is from the point of view of its cardinal nature–the Greeks emphasizing the dichotomy between unity and multiplicity, see next section (The Only One). From the point of view of numeration it stands in the same line as the others.
It is also possible to attribute a sign to zero, cf. the IEEE 754 for floating-point arithmetic.
For the definition of contrariety see for example our recent paper “Disentangling contradiction from contrariety via incompatibility” (2016b), and for other work on the square of opposition, see in the bibliography of the present paper the volumes we have edited on the topic.
Both ∀ and ∃ are generally used, although it is well known that, in classical logic, it is enough to take one of them as primitive.
This has been emphasized in particular in Section 4 of Chapter 1 of Dantzig’s 1930 classical book on number.
The remarkable correspondence between Cantor and Dedekind has been published by Emmy Noether and Jean Cavaillès in 1937(see the reference in the name Cantor in our Bibliography).
Since we are taking examples of commutative operations, we are not making here the distinction between left and right identities.
The expression “imaginary number” is due to Descartes, the sign “i” to Euler; another connection between the number 1 and the 9th letter of the alphabet. About this use of the word “imagination” see our recent paper “Possibility, Imagination and Conception” (2016a).
About this notion, see our 2010b paper: “What is a logic?—Towards axiomatic emptiness”.
This second disadvantage does not manifest in Portuguese or in German where the first capitalized letter of the word for truth is not “\({ \top }\)” but “V” and “W” respectively. In English sometimes people use the lower case “t”, in this case this second confusion is also avoided.
The notion of tautology was essentially promoted by Wittgenstein (1889–1951), mainly in the Tractatus, but he did not reach the symbolization “T” for it and/or the corresponding top notion. Wittgenstein was against the Fregean notion represented by Frege (1848–1925) with “\({ \vdash }\)” (see Rombout 2011). So the notation “\({ \vdash } { \top }\)” is completely anti-Wittgensteinian. Frege on the other hand did not make the difference between truth and logical truth, the symbol “\({ \vdash }\)” he introduced means for him truth, not logical truth, as used later on. So for him the notation “\({ \vdash } { \top }\)” would also be meaningless. Frege introduced in logic the expression “truth-value” and the corresponding two truth-values but did not represent them by 0 and 1, moreover although he considered them as objects, it was by contrast to the notion of function (see Heck and May 2016). Furthermore we can distinguish in Frege the truth-value he calls “True” from truth which is not for him an object (see Greimann 2007).
Initially Łukasiewicz (1920) used the symbol “2” for the third value, which is a bit absurd. He then shifted to ½. Asenjo, who was the first to develop paraconsistent logic based on three-valued truth-tables similar to the ones of Łukasiewicz, also used “2” for the third value, in paraconsistent logic it can make sense.
Post (1921) originally denoted truth and falsity by “+” and “−”. Our notation naturally allows the reading of this diagram as a bi-lattice taking into consideration this double reading.
about the background of this lecture, see (Rowe 2013).
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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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This paper has been written in such a way that it can be understood and/or tasted by any gentleman or gentlewoman with an average IQ but is not recommended for people with an emotional intelligence of less than ℵ1. This is a revised version taking in account the reflections of many ones including I, me, mine. I am grateful to Jonathan Westphal, Michele Friend and Srećko Kovač who carefully read the original version and indicated in particular (but no only) English incorrectnesses and/or topys (like this 1). My style of writing is inspired in particular by Fredric Brown (see e.g. Brown 1958). Thanks also to Jasmin Blanchette, Catherine Chantilly, Newton da Costa, Luis Estrada-González, Yvon Gauthier, Paul Healey, Wilfrid Hodges, Jean-Louis Hudry, Colin James III, Jens Lemanski, Dominique Luzeaux, Daniel Parrochia, Arnaud Plagnol, Andrei Rodin, Denis Saveliev, Matthias Schirn, Sergey Sudoplatov, José Veríssimo, Gereon Wolters and John Woods for their useful comments and/or feedback. I would like to dedicate this paper to my mother who gave me birth, introduced me to modern mathematics through the work of Georges Papy (who passed away on 11.11.11) and provided me access to the pathless land.
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Beziau, JY. MANY 1. J. Indian Counc. Philos. Res. 34, 259–287 (2017). https://doi.org/10.1007/s40961-016-0081-7
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DOI: https://doi.org/10.1007/s40961-016-0081-7