The Ideal in mathematics

A Spinozist Marxian Elaboration and Revision of the Theory of Knowledge Objectification


  • Wolff-Michael Roth University of Victoria



Spinoza; Marx; idealism; materialism; consciousness; society; praxis; mediation; individualism; abstraction; mathematics education


The theory of knowledge objectification, initially presented and developed by Luis Radford, has gained some traction in the field of mathematics education. As with any developing theory, its presentation contains statements that may contradict its stated intents; and these problems are exacerbated in its uptake into the work of other scholars. The purpose of this study is to articulate a Spinozist-Marxian approach, in which the objectification exists not in things—semiotic means that mediate interactions—but as real relation between people. As a consequence, the (problematic) concept of mediation is unnecessary and can be abandoned. A concrete classroom example from Radford’s own studies is used to exemplify and develop pertinent issues. In particular, the societal nature of the ideal—a synecdoche of relations between objects that reflect relations between people—should be added and the notion of (sign) mediation no longer is required.


Arievitch, I., & Stetsenko, A. (2014). The “magic of signs”: Developmental trajectory of cultural mediation. In A. Yasnitsky, R. van der Veer, & M. Ferrari (Eds.), The Cambridge handbook of cultural-historical psychology (pp. 217–244). Cambridge: Cambridge University Press.
Arzarello, F., Robutti, O., & Thomas, M. (2015). Growth point and gestures: looking inside mathematical meanings. Educational Studies in Mathematics, 90, 19–37.
Bartolini Bussi, M., & Mariotti, M., A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English (Ed.), Handbook of international research in mathematics education (2nd Edition) (pp. 746–783). New York: Routledge.
Bateson, G. (1979). Mind and nature: A necessary unity. New York: E. P. Dutton.
Carlsen, M. (2010). Appropriating geometric series as a cultural tool: a study of student collaborative learning. Educational Studies in Mathematics, 74, 95–116.
Chahine, I. C. (2013). The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures. Journal of Mathematical Behavior, 32, 434–449.
Cole, M., & Engeström, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.) Distributed cognitions (pp. 1–46). Cambridge: Cambridge University Press.
El'konin, B. D. (1994). Vvedenie v psixologiju razvitija: B tradicii kul’turno-istoričeskoj teorii L. S. Vygotskogo [Introduction to the psychology of development: In the tradition of the cultural-historical theory of L. S. Vygotsky]. Moscow: Trivola.
Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82, 97–124.
Ilyenkov, E. V. (1977). Dialectical logic: Essays on its history and theory. Moscow: Progress.
Ilyenkov, E. (1982). Dialectics of the abstract and the concrete in Marx’s Capital. Moscow: Progress.
Ilyenkov, E. V. (2012). Dialectics of the ideal. Historical Materialism 20 (2), 149–193.
Kaput, J. (2009). Building intellectual infrastructure to expose and understand ever-increasing complexity. Educational Studies in Mathematics, 70, 211–215.
LaCroix, L. (2014). Learning to see pipes mathematically: preapprentices’ mathematical activity in pipe trades training. Educational Studies in Mathematics, 86, 157–176.
Leont'ev, A. N. (1978). Activity, consciousness, and personality. New Jersey: Prentice-Hall.
Leont'ev, A. N. (1994). Filosopfija psixologii: iz naučnogo nasledija [Philosophy of psychology: of scientific heritage]. Moskva: Izdatel’stvo Moskovskogo universiteta.
Mamardašvili, M. (1986). Analysis of consciousness in the works of Marx. Studies in Soviet Thought, 32, 101–120.
Marx, K., & Engels, F. (1962). Werke Band 23 [Works vol. 23]. Berlin: Dietz.
Marx, K., & Engels, F. (1978). Werke Band 3 [Works vol. 3]. Berlin: Dietz.
Marx, K., & Engels, F. (1983). Werke Band 42 [Works vol. 42]. Berlin: Dietz.
Mead, G. H. (1938). Philosophy of the act. Chicago: University of Chicago Press.
Mikhailov, F. T. (2001). The “Other Within” for the psychologist. Journal of Russian and East European Psychology, 39(1), 6–31.
Mikhailov, F. T. (2004). Object-oriented activity—Whose? Journal of Russian and East European Psychology, 42(3), 6–34.
Mikhailov, F. T. (2006). Problems of the method of cultural-historical psychology. Journal of Russian and East European Psychology, 44 (1), 21–54.
Moutsios-Rentzos, A., Spyrou, P., & Peteinara, A. (2014). The objectification of the right-angled triangle in the teaching of the Pythagorean Triangle: an empirical investigation. Educational Studies in Mathematics, 85, 29–51.
Negri, A. (1991). The savage anomaly: The power of Spinoza’s metaphysics and politics. Minneapolis, MN: University of Minnesota Press.
Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.
Radford, L. (2003). Gestures, speech and the sprouting of signs. Mathematical Thinking and Learning, 5, 37–70.
Radford, L. (2006). The anthropology of meaning. Educational Studies in Mathematics, 61, 39–65.
Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215–234). Rotterdam, The Netherlands: Sense Publishers.
Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70, 111–126.
Radford, L. (2012). Education and the illusions of emancipation. Educational Studies in Mathematics, 80, 101–118.
Radford, L. (2013). Three key concepts of the theory of objectification: knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2, 7–44.
Radford, L. (2015). The epistemological foundations of the theory of objectification. Isonomia—Epistemologia, 7, 127–149.
Radford, L., Bardini, C., & Sabena, C. (2006). Rhythm and the grasping of the general. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 393–400). Prague: PME.
Radford, L., Edwards, L., & Arzarello, F. (2009). Introduction: beyond words. Educational Studies in Mathematics, 70, 91–95.
Radford, L., & Roth, W.-M. (2011). Beyond Kantian individualism: An activity perspective on classroom interaction. Educational Studies in Mathematics, 77, 227–245.
Roth, W.-M. (2007). On mediation: Toward a cultural-historical understanding of the concept. Theory & Psychology, 15, 655–680.
Roth, W.-M. (2016). On the social nature of mathematical reasoning. For the Learning of Mathematics, 26(2), 34–39.
Roth, W.-M. (2017). The mathematics of mathematics: Thinking with the late, Spinozist Vygotsky. Rotterdam: Sense Publishers.
Roth, W.-M. (2018). Birth of signs: A (Spinozist-Marxian) materialist approach. In N. Presmeg, L. Radford, W.-M. Roth, G. Kadunz, & L. Puig (Eds.), Signs of signification: Semiotics in mathematics education (pp. 37–53). Dordrecht: Springer.
Roth, W.-M., & Jornet, A. (2019). Theorizing with/out “mediators.” Integrative Psychological and Behavioral Science, 53, 323–343.
Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Rotterdam: Sense Publishers.
Siebert, B. (2014). Prospects for a cultural-historical psychology of intelligence. In A. Levant & V. Oittinen (Eds.), Dialectics of the ideal: Evald Ilyenkov and creative Soviet Marxism (pp. 97–105). Leiden: Brill.
Vygotskij, L. S. (2005). Psixologija razvitija čeloveka [Psychology of human development]. Moscow: Eksmo.
Vygotsky, L. S. (1987). The collected works of L. S. Vygotsky, vol. 1: Problems of general psychology. New York: Springer.
Vygotsky, L. S. (1989). Concrete human psychology. Soviet Psychology, 27(2), 53–77.
Vygotsky, L. S. (1997). The collected works of L. S. Vygotsky, vol. 4: The history of the development of higher mental functions. New York: Springer.
Zavershneva, E. Iu. (2010). The way to freedom. Journal of Russian and East European Psychology, 48(1), 61–90.
Zavershneva, E. Iu. (2016). Vygotsky the unpublished: an overview of the personal archive (1912–1934). In A. Yasnitsky & R. van der Veer (Eds.), Revisionist revolution in Vygotsky studies (pp. 94–126). London: Routledge.
Zurina, H., & Williams, J. (2011). Gesturing for oneself. Educational Studies in Mathematics, 77, 175–188.




How to Cite

Roth, W.-M. (2020). The Ideal in mathematics: A Spinozist Marxian Elaboration and Revision of the Theory of Knowledge Objectification. Outlines. Critical Practice Studies, 21(02), 60–88.