El  cero como operador liminal

Andrés E. Vergara Ross

doi:10.48204/2805-1815.11497

Submitted October 20, 2026

Published 2026-10-22

Artículos de investigación y ensayos filosóficos

No. 6 (2026): Analítica

Zero as a liminal operator

Andrés E. Vergara Ross ⁺⁻

DOI https://doi.org/10.48204/2805-1815.11497

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DOI: 10.48204/2805-1815.11497

Published: 2026-10-22

How to Cite

Vergara Ross , A. E. (2026) “ Zero as a liminal operator ”, Analítica, (6), pp. 11–31. doi: 10.48204/2805-1815.11497.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Abstract

This article argues that zero cannot be understood as an element of the number system. Zero is its liminal operator, the internal condition that grants the system positive determination without being a determination itself. The aim is to show that the three dominant traditions in the philosophy of number, logicism, set theory, and structuralism, fail to assimilate zero to their available categories. Each one reproduces, at the formal level, the same aporia. Every definition of zero presupposes a resource that zero itself makes possible. The study proceeds through formal analysis of the Peano axioms, category theory, non-standard analysis, and Gödelian coding, and it draws on recent work on absence perception, referential indeterminacy, and incompleteness. The aporia is not contingent. It recurs across independent frameworks, and that recurrence makes it structural. The category of nihil activum names the ontological status of zero, a trans-systemic function instantiated in diverse ways. Every sufficiently complex system requires a liminal operator that marks its own constitutive void.

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References

Ahlfors, L. V. (1979). Complex analysis (3.ª ed.). McGraw-Hill.
Aristóteles. (1995). Física (G. R. de Echandía, Trad.). Gredos. (Obra original ca. siglo IV a. C.)
Awodey, S. (2010). Category theory (2.ª ed.). Oxford University Press.
Badiou, A. (1999). El ser y el acontecimiento (R. Cerdeiras et al., Trad.). Manantial. (Obra original publicada en 1988)
Baire, R.-L. (1899). Sur les fonctions de variables réelles. Annali di Matematica Pura e Applicata, 3, 1-123.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford University Press.
Barton, N. (2019). Absence perception and the philosophy of zero. Synthese, 197(9), 3823-3850. https://doi.org/10.1007/s11229-019-02220-x
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74(1), 47-73. https://doi.org/10.2307/2183530
Berkeley, G. (2013). El analista (M. Vázquez Campos, Trad.). Tirant Humanidades. (Obra original publicada en 1734)
Boccuni, F. (2019). Realismo e indeterminatezza nei fondamenti della matematica. Istituto Lombardo. Accademia di Scienze e Lettere, Incontri di Studio. https://doi.org/10.4081/incontri.2019.464
Button, T., & Walsh, S. (2016). Structure and categoricity: Determinacy of reference and truth value in the philosophy of mathematics. Philosophia Mathematica, 24(3), 283-307. https://doi.org/10.1093/philmat/nkw007
Chaitin, G. (2007). The halting probability omega: Irreducible complexity in pure mathematics. Milan Journal of Mathematics, 75(1), 291-304. https://doi.org/10.1007/s00032-006-0064-2
Cheng, Y. (2021). Current research on Gödel's incompleteness theorems. The Bulletin of Symbolic Logic, 27(2), 113-167. https://doi.org/10.1017/bsl.2020.44
Cohen-Tannoudji, C., Diu, B., & Laloë, F. (1977). Mecánica cuántica (2 vols.). Reverté.
Dauben, J. W. (1995). Abraham Robinson: The creation of nonstandard analysis. Princeton University Press.
Derrida, J. (2000). De la gramatología (O. del Barco y C. Ceretti, Trad.). Siglo XXI. (Obra original publicada en 1967)
Diehl, C. (2008). The empty space in structure: Theories of the zero from Gauthiot to Deleuze. Diacritics, 38(3), 93-119. https://doi.org/10.1353/dia.0.0059
Dirac, P. A. M. (1990). Los principios de la mecánica cuántica (J. Pérez Mercader, Trad.). Reverté. (Obra original publicada en 1958)
Dummett, M. (1991a). Frege: Philosophy of mathematics. Duckworth.
Dummett, M. (1991b). The logical basis of metaphysics. Harvard University Press.
Eckhart, M. (2014). El fruto de la nada y otros escritos (A. Vega Esquerra, Trad.). Siruela.
Edgerton, S. Y. (2009). The mirror, the window, and the telescope. Cornell University Press.
Field, H. (2016). Science without numbers (2.ª ed.). Oxford University Press.
Franzén, T. (2005). Gödel's theorem: An incomplete guide to its use and abuse. A K Peters.
Frege, G. (2016). Los fundamentos de la aritmética (C. U. Moulines, Trad.). UNAM. (Obra original publicada en 1884)
Gödel, K. (1986). Collected works (Vol. I, S. Feferman et al., Eds.). Oxford University Press.
Goldstein, L. (2002). The indefinability of "one". Journal of Philosophical Logic, 31(1), 29-42. https://doi.org/10.1023/A:1015077403773
Grabiner, J. V. (1981). The origins of Cauchy's rigorous calculus. MIT Press.
Grattan-Guinness, I. (2000). The search for mathematical roots, 1870-1940. Princeton University Press.
Guicciardini, N. (1999). Reading the Principia. Cambridge University Press.
Hallett, M. (1984). Cantorian set theory and limitation of size. Clarendon Press.
Heidegger, M. (2003). Ser y tiempo (J. E. Rivera, Trad.). Trotta. (Obra original publicada en 1927)
Hellman, G., & Shapiro, S. (2019). Mathematical structuralism. Cambridge University Press.
Hemmo, M., & Shenker, O. (2013). Probability zero in Bohm's theory. Philosophy of Science, 80(5), 1148-1158. https://doi.org/10.1086/673891
Jackson, J. D. (1999). Classical electrodynamics (3.ª ed.). Wiley.
Kant, I. (2005). Crítica de la razón pura (P. Ribas, Trad.). Taurus. (Obra original publicada en 1781)
Kant, I. (2010). Ensayo para introducir en la filosofía el concepto de magnitudes negativas. En Obras (P. Ribas, Trad., Vol. I). Gredos. (Obra original publicada en 1763)
Kleene, S. C. (1952). Introduction to metamathematics. North-Holland.
Kosmann-Schwarzbach, Y. (2011). The Noether theorems (B. E. Schwarzbach, Trad.). Springer.
Kunen, K. (2011). Set theory. College Publications.
Lawvere, F. W., & Schanuel, S. (2009). Conceptual mathematics (2.ª ed.). Cambridge University Press.
Leibniz, G. W. (2015). Nuevo método para el máximo y el mínimo. En Obras filosóficas y científicas (C. Álvarez Pereira et al., Eds., Vol. 7a). Comares. (Obra original publicada en 1684)
Levey, S. (1998). Leibniz on mathematics and the actually infinite division of matter. The Philosophical Review, 107(1), 49-96. https://doi.org/10.2307/2998315
Linnebo, ??. (2017). Philosophy of mathematics. Princeton University Press.
Liu, C. (1996). Gauge invariance, Cauchy problem, indeterminism, and symmetry breaking. Philosophy of Science, 63(S3), S71-S79. https://doi.org/10.1086/289938
Llorente Cardo, J. (2018). El no-ser se da de muchas maneras: La tabla kantiana de la nada ante la fenomenología de la donación. Ingenium, 12. https://doi.org/10.5209/inge.62420
Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic. Springer.
Maddy, P. (1990). Realism in mathematics. Clarendon Press.
McGinn, B. (2005). The harvest of mysticism in medieval Germany. Crossroad.
McLarty, C. (1992). Elementary categories, elementary toposes. Clarendon Press.
Meinong, A. (2008). Teoría del objeto y presentación personal (J. Rivera de Rosales, Trad.). Encuentro. (Obra original publicada en 1904)
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (2017). Gravitation. Princeton University Press.
Ni, L. (2007). Zero and metaphysics: Thoughts about being and nothingness from mathematics, Buddhism, Daoism to phenomenology. Frontiers of Philosophy in China, 2(4), 547-556. https://doi.org/10.1007/s11466-007-0035-x
Noether, E. (1918). Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235-257.
O'Meara, D. J. (1993). Plotinus: An introduction to the Enneads. Clarendon Press.
Oxtoby, J. C. (1980). Measure and category (2.ª ed.). Springer.
Parménides. (2007). Poema: Fragmentos y tradición textual (A. Bernabé, Trad.). Istmo. (Obra original ca. siglo V a. C.)
Parsons, C. (2008). Mathematical thought and its objects. Cambridge University Press.
Peirce, C. S. (1974). La ciencia de la semiótica (A. Sercovich, Trad.). Nueva Visión.
Platón. (2000). Sofista (N. L. Cordero, Trad.). Gredos. (Obra original ca. siglo IV a. C.)
Plotino. (2005). Enéadas (J. Igal, Trad., 3 vols.). Gredos.
Priest, G. (2016). Towards non-being (2.ª ed.). Clarendon Press.
Rivera de Rosales, J. (2011). El punto de partida de la Metafísica Transcendental. UNED.
Robinson, A. (1974). Non-standard analysis (ed. rev.). North-Holland.
Rotman, B. (1993). Signifying nothing: The semiotics of zero. Stanford University Press. (Obra original publicada en 1987)
Russell, B., & Whitehead, A. N. (1963). Principia mathematica (2.ª ed., 3 vols.). Cambridge University Press. (Obra original publicada 1925-1927)
Ryskin, G. (2020). Vanishing vacuum energy. Astroparticle Physics, 115, Artículo 102387. https://doi.org/10.1016/j.astropartphys.2019.102387
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford University Press.
Sutherland, D. (2017). Kant's conception of number. The Philosophical Review, 126(2), 147-190. https://doi.org/10.1215/00318108-3771988
Vermeiren, F. (2021). Bergson and the Kantian concept of intensive magnitude. Journal of the British Society for Phenomenology, 53(1), 91-104. https://doi.org/10.1080/00071773.2021.1910854
Volovik, G. E. (2006). Vacuum energy: Myths and reality. International Journal of Modern Physics D, 15(12), 1987-2010. https://doi.org/10.1142/S0218271806009431
Walsh, S. (2014). Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, 7(1), 84-119. https://doi.org/10.1017/S1755020313000397
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281.
Zinkin, M. (2012). Kant on negative magnitudes. Kant-Studien, 103(4). https://doi.org/10.1515/kant-2012-0030

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[1] Ahlfors, L. V. (1979). Complex analysis (3.ª ed.). McGraw-Hill.

[2] Aristóteles. (1995). Física (G. R. de Echandía, Trad.). Gredos. (Obra original ca. siglo IV a. C.)

[3] Awodey, S. (2010). Category theory (2.ª ed.). Oxford University Press.

[4] Badiou, A. (1999). El ser y el acontecimiento (R. Cerdeiras et al., Trad.). Manantial. (Obra original publicada en 1988)

[5] Baire, R.-L. (1899). Sur les fonctions de variables réelles. Annali di Matematica Pura e Applicata, 3, 1-123.

[6] Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford University Press.

[7] Barton, N. (2019). Absence perception and the philosophy of zero. Synthese, 197(9), 3823-3850. https://doi.org/10.1007/s11229-019-02220-x

[8] Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74(1), 47-73. https://doi.org/10.2307/2183530

[9] Berkeley, G. (2013). El analista (M. Vázquez Campos, Trad.). Tirant Humanidades. (Obra original publicada en 1734)

[10] Boccuni, F. (2019). Realismo e indeterminatezza nei fondamenti della matematica. Istituto Lombardo. Accademia di Scienze e Lettere, Incontri di Studio. https://doi.org/10.4081/incontri.2019.464

[11] Button, T., & Walsh, S. (2016). Structure and categoricity: Determinacy of reference and truth value in the philosophy of mathematics. Philosophia Mathematica, 24(3), 283-307. https://doi.org/10.1093/philmat/nkw007

[12] Chaitin, G. (2007). The halting probability omega: Irreducible complexity in pure mathematics. Milan Journal of Mathematics, 75(1), 291-304. https://doi.org/10.1007/s00032-006-0064-2

[13] Cheng, Y. (2021). Current research on Gödel's incompleteness theorems. The Bulletin of Symbolic Logic, 27(2), 113-167. https://doi.org/10.1017/bsl.2020.44

[14] Cohen-Tannoudji, C., Diu, B., & Laloë, F. (1977). Mecánica cuántica (2 vols.). Reverté.

[15] Dauben, J. W. (1995). Abraham Robinson: The creation of nonstandard analysis. Princeton University Press.

[16] Derrida, J. (2000). De la gramatología (O. del Barco y C. Ceretti, Trad.). Siglo XXI. (Obra original publicada en 1967)

[17] Diehl, C. (2008). The empty space in structure: Theories of the zero from Gauthiot to Deleuze. Diacritics, 38(3), 93-119. https://doi.org/10.1353/dia.0.0059

[18] Dirac, P. A. M. (1990). Los principios de la mecánica cuántica (J. Pérez Mercader, Trad.). Reverté. (Obra original publicada en 1958)

[19] Dummett, M. (1991a). Frege: Philosophy of mathematics. Duckworth.

[20] Dummett, M. (1991b). The logical basis of metaphysics. Harvard University Press.

[21] Eckhart, M. (2014). El fruto de la nada y otros escritos (A. Vega Esquerra, Trad.). Siruela.

[22] Edgerton, S. Y. (2009). The mirror, the window, and the telescope. Cornell University Press.

[23] Field, H. (2016). Science without numbers (2.ª ed.). Oxford University Press.

[24] Franzén, T. (2005). Gödel's theorem: An incomplete guide to its use and abuse. A K Peters.

[25] Frege, G. (2016). Los fundamentos de la aritmética (C. U. Moulines, Trad.). UNAM. (Obra original publicada en 1884)

[26] Gödel, K. (1986). Collected works (Vol. I, S. Feferman et al., Eds.). Oxford University Press.

[27] Goldstein, L. (2002). The indefinability of "one". Journal of Philosophical Logic, 31(1), 29-42. https://doi.org/10.1023/A:1015077403773

[28] Grabiner, J. V. (1981). The origins of Cauchy's rigorous calculus. MIT Press.

[29] Grattan-Guinness, I. (2000). The search for mathematical roots, 1870-1940. Princeton University Press.

[30] Guicciardini, N. (1999). Reading the Principia. Cambridge University Press.

[31] Hallett, M. (1984). Cantorian set theory and limitation of size. Clarendon Press.

[32] Heidegger, M. (2003). Ser y tiempo (J. E. Rivera, Trad.). Trotta. (Obra original publicada en 1927)

[33] Hellman, G., & Shapiro, S. (2019). Mathematical structuralism. Cambridge University Press.

[34] Hemmo, M., & Shenker, O. (2013). Probability zero in Bohm's theory. Philosophy of Science, 80(5), 1148-1158. https://doi.org/10.1086/673891

[35] Jackson, J. D. (1999). Classical electrodynamics (3.ª ed.). Wiley.

[36] Kant, I. (2005). Crítica de la razón pura (P. Ribas, Trad.). Taurus. (Obra original publicada en 1781)

[37] Kant, I. (2010). Ensayo para introducir en la filosofía el concepto de magnitudes negativas. En Obras (P. Ribas, Trad., Vol. I). Gredos. (Obra original publicada en 1763)

[38] Kleene, S. C. (1952). Introduction to metamathematics. North-Holland.

[39] Kosmann-Schwarzbach, Y. (2011). The Noether theorems (B. E. Schwarzbach, Trad.). Springer.

[40] Kunen, K. (2011). Set theory. College Publications.

[41] Lawvere, F. W., & Schanuel, S. (2009). Conceptual mathematics (2.ª ed.). Cambridge University Press.

[42] Leibniz, G. W. (2015). Nuevo método para el máximo y el mínimo. En Obras filosóficas y científicas (C. Álvarez Pereira et al., Eds., Vol. 7a). Comares. (Obra original publicada en 1684)

[43] Levey, S. (1998). Leibniz on mathematics and the actually infinite division of matter. The Philosophical Review, 107(1), 49-96. https://doi.org/10.2307/2998315

[44] Linnebo, ??. (2017). Philosophy of mathematics. Princeton University Press.

[45] Liu, C. (1996). Gauge invariance, Cauchy problem, indeterminism, and symmetry breaking. Philosophy of Science, 63(S3), S71-S79. https://doi.org/10.1086/289938

[46] Llorente Cardo, J. (2018). El no-ser se da de muchas maneras: La tabla kantiana de la nada ante la fenomenología de la donación. Ingenium, 12. https://doi.org/10.5209/inge.62420

[47] Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic. Springer.

[48] Maddy, P. (1990). Realism in mathematics. Clarendon Press.

[49] McGinn, B. (2005). The harvest of mysticism in medieval Germany. Crossroad.

[50] McLarty, C. (1992). Elementary categories, elementary toposes. Clarendon Press.

[51] Meinong, A. (2008). Teoría del objeto y presentación personal (J. Rivera de Rosales, Trad.). Encuentro. (Obra original publicada en 1904)

[52] Misner, C. W., Thorne, K. S., & Wheeler, J. A. (2017). Gravitation. Princeton University Press.

[53] Ni, L. (2007). Zero and metaphysics: Thoughts about being and nothingness from mathematics, Buddhism, Daoism to phenomenology. Frontiers of Philosophy in China, 2(4), 547-556. https://doi.org/10.1007/s11466-007-0035-x

[54] Noether, E. (1918). Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235-257.

[55] O'Meara, D. J. (1993). Plotinus: An introduction to the Enneads. Clarendon Press.

[56] Oxtoby, J. C. (1980). Measure and category (2.ª ed.). Springer.

[57] Parménides. (2007). Poema: Fragmentos y tradición textual (A. Bernabé, Trad.). Istmo. (Obra original ca. siglo V a. C.)

[58] Parsons, C. (2008). Mathematical thought and its objects. Cambridge University Press.

[59] Peirce, C. S. (1974). La ciencia de la semiótica (A. Sercovich, Trad.). Nueva Visión.

[60] Platón. (2000). Sofista (N. L. Cordero, Trad.). Gredos. (Obra original ca. siglo IV a. C.)

[61] Plotino. (2005). Enéadas (J. Igal, Trad., 3 vols.). Gredos.

[62] Priest, G. (2016). Towards non-being (2.ª ed.). Clarendon Press.

[63] Rivera de Rosales, J. (2011). El punto de partida de la Metafísica Transcendental. UNED.

[64] Robinson, A. (1974). Non-standard analysis (ed. rev.). North-Holland.

[65] Rotman, B. (1993). Signifying nothing: The semiotics of zero. Stanford University Press. (Obra original publicada en 1987)

[66] Russell, B., & Whitehead, A. N. (1963). Principia mathematica (2.ª ed., 3 vols.). Cambridge University Press. (Obra original publicada 1925-1927)

[67] Ryskin, G. (2020). Vanishing vacuum energy. Astroparticle Physics, 115, Artículo 102387. https://doi.org/10.1016/j.astropartphys.2019.102387

[68] Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford University Press.

[69] Sutherland, D. (2017). Kant's conception of number. The Philosophical Review, 126(2), 147-190. https://doi.org/10.1215/00318108-3771988

[70] Vermeiren, F. (2021). Bergson and the Kantian concept of intensive magnitude. Journal of the British Society for Phenomenology, 53(1), 91-104. https://doi.org/10.1080/00071773.2021.1910854

[71] Volovik, G. E. (2006). Vacuum energy: Myths and reality. International Journal of Modern Physics D, 15(12), 1987-2010. https://doi.org/10.1142/S0218271806009431

[72] Walsh, S. (2014). Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, 7(1), 84-119. https://doi.org/10.1017/S1755020313000397

[73] Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281.

[74] Zinkin, M. (2012). Kant on negative magnitudes. Kant-Studien, 103(4). https://doi.org/10.1515/kant-2012-0030

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