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A constructive sequence algebra for the calculus of indications

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Abstract

In this paper, we investigate some aspects of Spencer–Brown’s Calculus of Indications. Drawing from earlier work by Kauffman and Varela, we present a new categorical framework that allows to characterize the construction of infinite arithmetic expressions as sequences taking values in grossone.

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Notes

  1. This is akin to Peirce’s \(\alpha \) graphs (Zeman 1974; Gangle et al. 2020), although with a basic but immaterial difference between the representations of True and False in the two formalisms.

  2. The introduction of the numeral ① may require, in some contexts, the augmentation of the set of natural \({\mathbb {N}}\) to a larger set \(\hat{{\mathbb {N}}}\), where

    For the purpose of this paper we will not need to make use of \(\hat{{\mathbb {N}}}\).

  3. Evidence of the efficacy of the grossone approach is highlighted by its successful application to many fields of applied mathematics, including optimization (see Cococcioni et al. 2018, 2020; De Cosmis and De Leone 2012; Sergeyev et al. 2018; Iavernaro et al. 2020; Iudin et al. 2012; Sergeyev 2007, 2013b; Zhigljavsky 2012), fractals and cellular automata (see Caldarola 2018; D’Alotto 2012, 2015, 2013) as well as infinite decision-making processes, game theory, and probability (see Fiaschi and Cococcioni 2018; Rizza 2018, 2019), whereas the formal logical foundation of grossone has been investigated in Lolli (2012); Margenstern (2011); Montagna et al. (2015). The approach presented in this paper bears similarities to the application of grossone to Turing machines as described in Sergeyev and Garro (2010). Further investigation of these analogies remains open.

  4. The two systems are slightly different in terms of their rules of deduction (or rewriting). We do not examine this point any further in this paper.

  5. The relevance of grossone to this construction will appear in the subsequent sections.

  6. Our approach is similar to that of Varela and Goguen (1978) in that it involves limits of sequences. But by calculating values of the sequences themselves, and not only their limits, the (complex algebra) waveform interpretation of Kauffman and Varela (1980) is partially maintained as well.

  7. Here the objects are the natural numbers, and there is a morphism between two natural numbers \(m\rightarrow n\) if and only if \(m\ge n\). The choice of reversing the natural order is made in order for the functor F to be contravariant.

  8. The condition given here guarantees that all nested cuts are of at most depth \(n-1\).

  9. A monic arrow in any category is defined as follows. Given any morphism \(f: A\longrightarrow B\), f is said to be monic if, for any two morphisms \(g,h: Z\longrightarrow A\) such that \(f\circ g=f\circ h\), it is the case that \(g=h\).

  10. \(\coprod \) is well defined in \({\mathcal {CI}}\) since it is a topos.

  11. A formal categorical characterization of a variant of this procedure, applied to cuts-only \(\alpha \) graphs can found in Gangle et al. (2020).

  12. Note that it is important to distinguish this operator (as an operation on \(\alpha \)) from the circle drawn around 1 in the grossone numeral ① .

References

  • Barwise J, Etchemendy J (1989) The liar: an essay on truth and circularity. Oxford University Press, New York

    MATH  Google Scholar 

  • Caldarola F (2018) The Sierpinski curve viewed by numerical computations with infinities and infinitesimals. Appl Math Comput 318:321–328

    MATH  Google Scholar 

  • Caterina G, Gangle R (2013) Iconicity and abduction: a categorical approach to creative hypothesis-formation in Peirce’s existential graphs. Logic J IGPL 21(6):1028–1043

    Article  MathSciNet  Google Scholar 

  • Caterina G, Gangle R (2016) Iconicity and abduction. Springer, New York

    Book  Google Scholar 

  • Cococcioni M, Pappalardo M, Sergeyev YD (2018) Lexicographic multi-objective linear programming using grossone methodology: theory and algorithm. Appl Math Comput 318:298–311

    MATH  Google Scholar 

  • Cococcioni M, Cudazzo A, Pappalardo M, Sergeyev YD (2020) Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology. Commun Nonlinear Sci Numer Simul 84:1–20

    Article  MathSciNet  Google Scholar 

  • D’Alotto L (2012) Cellular automata using infinite computations. Appl Math Comput 218(16):8077–8082

    MathSciNet  MATH  Google Scholar 

  • D’Alotto L (2013) A classification of two-dimensional cellular automata using infinite computations. Indian J Math 55:143–158

    MathSciNet  MATH  Google Scholar 

  • D’Alotto L (2015) A Classification of One-Dimensional Cellular Automata using Infinite Computations. Applied Mathematics and Computation 255:15–24

    Article  MathSciNet  Google Scholar 

  • De Cosmis S, De Leone R (2012) The use of grossone in mathematical programming and operations research. Appl Math Comput 218(16):8029–8038

    MathSciNet  MATH  Google Scholar 

  • Fiaschi L, Cococcioni M (2018) Numerical asymptotic results in game theory using Sergeyev’s infinity computing. Int J Unconvent Comput 14(1):1–25

    Google Scholar 

  • Gangle R, Caterina G, Tohme F (2020) A generic figures reconstruction of Peirce’s Existential Graphs (Alpha). Erkenntnis. https://doi.org/10.1007/s10670-019-00211-5

  • Iavernaro F, Mazzia F, Mukhametzhanov MS, Sergeyev YD (2020) Conjugate-symplecticity properties of Euler–Maclaurin methods and their implementation on the infinity computer. Appl Numer Math 155:58–72

    Article  MathSciNet  Google Scholar 

  • Iudin D, Sergeyev YD, Hayakawa M (2012) Interpretation of percolation in terms of infinity computations. Appl Math Comput 218(16):8099–8111

    MathSciNet  MATH  Google Scholar 

  • Kauffman LH (2012) Laws of Form: an Exploration in Mathematics and Foundations. http://homepages.math.uic.edu/~kauffman/Laws.pdf

  • Kauffman LH, Varela FJ (1980) Form dynamics. J Soc Biol Struct 3:171–206

    Article  Google Scholar 

  • Lolli G (2012) Infinitesimals and infinites in the history of mathematics: a brief survey. Appl Math Comput 218(16):7979–7988

    MathSciNet  MATH  Google Scholar 

  • Luhmann N (1995) Social systems. Stanford University Press, Stanford

    Google Scholar 

  • Mac Lane S (1998) Categories for the working mathematician. Springer, Berlin

    MATH  Google Scholar 

  • Margenstern M (2011) Using Grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers. Ultrametric Anal Appl 3(3):196–204

    Article  MathSciNet  Google Scholar 

  • Montagna F, Simi G, Sorbi A (2015) Taking the Pirahã seriously. Commun Nonlinear Sci Numer Simul 21(1–3):52–69

    Article  MathSciNet  Google Scholar 

  • Reyes M, Reyes G, Zolfaghari H (2004) Generic figures and their glueings. Polimetrica, Milan

    MATH  Google Scholar 

  • Rizza D (2018) A study of mathematical determination through Bertrand’s Paradox. Philosophia Mathematica 26(3):375–395

    Article  MathSciNet  Google Scholar 

  • Rizza D (2019) Numerical methods for infinite decision-making processes. Int J Unconvent Comput 14(2):139–158

    Google Scholar 

  • Sergeyev YD (2013a). Arithmetic of infinity. Edizioni Orizzonti Meridionali, 2nd edition

  • Sergeyev YD (2007) Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers. Chaos, Solitons Fract 33:50–75

    Article  Google Scholar 

  • Sergeyev YD (2010) Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. USA Patent 7(860):914

    Google Scholar 

  • Sergeyev YD (2013b) Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer. Appl Math Comput 219(22):10668–10681

  • Sergeyev YD (2017) Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv Math Sci 4(2):219–320

    Article  MathSciNet  Google Scholar 

  • Sergeyev YD, Garro A (2010) Observability of turing machines: a refinement of the theory of computation. Informatica 21(3):425–454

    Article  MathSciNet  Google Scholar 

  • Sergeyev YD, Kvasov DE, Mukhametzhanov MS (2018) On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Commun Nonlinear Sci Numer Simul 59:319–330

    Article  MathSciNet  Google Scholar 

  • Spencer-Brown G (1969) Laws of form. Allen & Unwin, London

    MATH  Google Scholar 

  • Tohmé F, Caterina G, Gangle Rocco (2020) Computing Truth Values in the Topos of \(\alpha \)-Infinite Peirce’s Existential Graphs. In press, Applied Mathematics and Computation

  • Varela F, Goguen JA (1978) The arithmetic of closure. J Cybern 8:291–324

    Article  MathSciNet  Google Scholar 

  • Zalamea F (2010) Towards a complex variable interpretation of Peirce’s existential graphs. In: Bergman M et al (eds) Ideas in action. Proceedings of the applying peirce conference, Nordic Pragmatism Network, Helsinki, pp 254–264

  • Zeman J (1974) Peirce’s logical graphs. Semiotica 12:239–256

    Google Scholar 

  • Zhigljavsky A (2012) Computing sums of conditionally convergent and divergent series using the concept of Grossone. Appl Math Comput 218:8064–8076

    MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Center for Diagrammatic and Computational Philosophy, Endicott College, Beverly, MA, USA

    Rocco Gangle & Gianluca Caterina

  2. Universidad Nacional del Sur/CONICET, Bahía Blanca, Buenos Aires, Argentina

    Fernando Tohmé

Authors
  1. Rocco Gangle
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  2. Gianluca Caterina
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  3. Fernando Tohmé
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Correspondence to Gianluca Caterina.

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Communicated by Yaroslav D. Sergeyev.

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Gangle, R., Caterina, G. & Tohmé, F. A constructive sequence algebra for the calculus of indications. Soft Comput 24, 17621–17629 (2020). https://doi.org/10.1007/s00500-020-05121-1

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