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Mathematics embodied: Merleau-Ponty on geometry and algebra as fields of motor enaction

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Abstract

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity.

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Notes

  1. For example, Cassou-Noguès (1998) is mostly critical, while Hass and Hass (2000) and Matherne (2018) are appreciative. See also Besmer (2007) and Baldwin (2013), whose positions are mixed.

  2. I note that Hutto does not explicitly connect idealism with Husserl. However, among the authors discussed by Gallagher, Husserl seems to be one of those who may be labelled an idealist.

  3. In connection to this discussion, see also Hohol (2020, 89–108), who provides an analogical criticism of the idea of neural simulation of sensory experiences based on his review of literature that includes the works discussed by Gallagher and Hutto. However, Hohol adopts a moderately embodied stance and does not entirely reject the notion of representation.

  4. Husserl’s thoughts on this matter developed over time. While in his late discussions on the role of language and writing, Husserl (1989) comes close to the idea that expression plays a constitutive role with regard to ideal meaning, he explicitly rejects this in his earlier works (e.g., 1982, 296).

  5. Speech “is a praxis,” Merleau-Ponty claims, and the mathematical ideality appears “at the edge of speech” (2002, 56, 46). Regarding speech as operative intentionality implicated in abstract thought, including mathematics, see also Merleau-Ponty (1968b, 155, 188). For analyzes of Merleau-Ponty’s notion of speaking speech in a broader context, see Baldwin (2007); Kee (2018); Kiverstein and Rietveld (2021).

  6. While Merleau-Ponty emphasizes the role of perceptual and bodily expression from early on, he properly grasped the social and linguistic aspects only after 1947, when he adopted elements of structural linguistics. See, for example, Merleau-Ponty’s description of the process through which we express our thoughts within a dialogue (1973, 133–146). For interpretations of Merleau-Ponty’s theory of expression, see in particular Fóti (2013), Hass (2008, 146–192), Kee (2018), and Landes (2013).

  7. Merleau-Ponty is alluding here to Bergson’s (1946, 7–31) idea of the “retrograde movement of the true.”.

  8. For a more elaborate discussion of this point, see Sect. 4.3.

  9. Regarding this point, see Merleau-Ponty’s discussion of the idea of Stiftung (1970, 161–67; 2002, 16–65; 2010, 50–61). For interpretations of Merleau-Ponty’s relationship to Husserl (1989), see Baldwin (2013); Besmer (2007); Hass and Hass (2000), Hass (2008, 146–169); Lawlor (2002); Robert (2000); Vallier (2005); Watson (2016).

  10. In contrast to this view, Besmer’s (2007) reading of Merleau-Ponty, for example, emphasizes his continuity with Husserl and is correlatively origin-nostalgic, past-oriented, and teleologically objectivistic (see, e.g., 135–136). However, I believe that since Merleau-Ponty ultimately defined our relation to reality as “interrogative” and thus open-ended, he would not accept a teleologically objectivistic view (see, e.g., 1996, 375).

  11. The available English translations of Merleau-Ponty’s Résumés de cours (1968a) and La prose du monde (1969) contain many inaccuracies. I have modified the passages cited from these texts as noted and I invite the reader to compare the French originals by following the pagination indicated after the slash in the citations.

  12. Cassou-Noguès (1998, 398) and Baldwin (2013, 305, 325) criticize Merleau-Ponty for not acknowledging formalization as a specifically mathematical procedure and as an evidence-providing instrument. Yet neither Cassou-Noguès nor Baldwin discusses Merleau-Ponty’s own interpretation of formalization and his explicit criticism of this procedure. Moreover, Cassou-Noguès’ and Baldwin’s interpretations seem to lack an accurate account of the presumed counterpart of formal thought, namely, “intuitive” thought (a term Merleau-Ponty uses only very rarely, e.g., 2012, 405). Baldwin (2013, 326) claims that Merleau-Ponty rejects formal thought in favor of “informal” thought, but this is unconvincing. Unlike Husserl, Merleau-Ponty does not proceed with the idea of an intuitive insight, since he considers all experience dependent on habitual bodily schematizations and, more strongly, on socio-cultural acquisitions, which always formalize our experience at least to some degree. In contrast to Cassou-Noguès and Baldwin, Hass and Hass (2000, 182) explicitly analyze Merleau-Ponty’s account of geometry in relation to formalism in mathematics and find Merleau-Ponty’s position persuasive. In their view, a transition from premises to conclusions is never a purely formal but rather an “expressive” or structurally productive operation (cf. Watson, 2007, 536–37; Gallagher, 2017, 208). Moreover, as Hohol (2020, 135) explains, the contemporary formal approach to proving practice in geometry was completely unknown to the inventors of Euclidian geometry.

  13. Similarly, Hohol (2020, chapter 4) recently argued that Euclidian geometry is a cognitive artefact that cannot be explained universalistically.

  14. The English translation (1973, 105) of this passage is incorrect.

  15. In my opinion, Cassou-Noguès fails to recognize this point (see, e.g., 1998, 382).

  16. It is important to note that in the passages interpreted here, Merleau-Ponty speaks of drawing in a very narrow sense. However, in his other writings, he considers drawing and painting as expressive operations in their own right and acknowledges their capacity to bring forth a specific type of generality. Therefore, the difference between the generality of an expressive drawing and a geometric figure should be understood as of a degree and not of a kind.

  17. For example, the development of a non-Euclidian geometry enables a transition from a still naïve Euclidian expression of space to a less naïve alternative, not to an absolute truth (cf. Merleau-Ponty 2012, 414; 1973, 100/141, 103/146, 127–28/178–79). Euclidian space is by no means a priori for Merleau-Ponty, and even if it might be considered privileged in relation to other expressions of space, it is a historical invention and its privileged position is not absolute (1968b, 213; cf. Hass, 2008, 166). As Hass and Hass (2000, 180–81) note, “mathematical truths are historically and geographically located” and the demonstrative value of mathematical proofs comes from the fact that the mathematical object perseveres throughout the structural transformations involved in the proof (cf. Merleau-Ponty 2012, 405). In accordance with this view, Hohol (2020, chapter 4.4.) argues that the main source of generality and necessity of Euclid’s proofs is the intersubjective repeatability of reasonings “scaffolded on the consistent use of [cognitive] artifacts” such as geometric diagrams and formulas (2020, 137).

  18. One may understand this phenomenon by analogy with threshold phenomena in perception (cf., e.g., Merleau-Ponty 2012, 9).

  19. Merleau-Ponty later briefly interpreted an analogous geometric problem, namely, the calculation of the area of a parallelogram (2010, 55; based on Wertheimer, 2020, 14–78).

  20. In his interpretation of geometric demonstration, Merleau-Ponty does not address the differences between the actual visual field and the imaginary visual field. Such an explanation is, however, necessary if Merleau-Ponty’s argument is to be made entirely plausible. Generally, Merleau-Ponty interprets the imaginary field as founded on some elements of the perceptual world and refuses to conceive of the imaginary world as a purely mental domain (cf., e.g., 2010, 46–50; 1970, 48, 68–69). From this point of view, a transition to the field of imagination does not affect the fundamental aspects of Merleau-Ponty’s argument concerning geometry. For an analogical contemporary attempt to ground imagination in embodied action, see Rucińska and Gallagher (2021).

  21. Building on Merleau-Ponty, Irwin (2017) has developed an analogical interpretation of abstract words.

  22. For a detailed explanation of how the body schema provides the ground for perceptual figures according to Merleau-Ponty, see Halák (2021a, 35–38).

  23. Cf. Wertheimer’s detailed discussions of analogic geometrical examples (e.g., 2020, 18).

  24. Hohol (2020, 46), for example, does not seem to distinguish between perceptual figures and geometric properties. In contrast, see Zahidi and Myin’s (2016) critique of this widespread cognitive bias.

  25. Alternatively, it is possible that painters such as Cézanne were capable of liberating their perception from the influence of previously learned geometry. Cf. Merleau-Ponty’s analysis of “perspectival distortions” in Cézanne’s paintings (1964c, 13–15) or his thoughts on topological space (1968b, 212–13).

  26. The English translation (1970, 105) of this passage is incorrect.

  27. Cf. Kiverstein and Rietveld’s (2021, 178–180) critique of Gibson on this point.

  28. Merleau-Ponty would therefore satisfy the requirement for a relative independence of the two systems as presented by Hohol (2020), based on his review of recent empirical experiments.

  29. Merleau-Ponty (2020a, 123) makes a similar point regarding the system of speech.

  30. See Merleau-Ponty (1973, 105–106/149–150; 125–26/176–177; cf. 2010, 55–56).

  31. In contrast to that, Schneider cannot dispense with the operation itself. See below, Sect. 4.3.

  32. Husserl (1989) considers “sedimentation” the process through which an originally experienced ideal meaning becomes stabilized in language or other symbolic systems and thus, communicable across time and space (cf. Blomberg 2019).

  33. See Sect. 3.4. for an analogical argument regarding perceptual “physiognomies” and geometrical properties. For discussion on this relationship from the perspective of embodied cognition, see Zahidi (2021), Zahidi and Myin (2016), and Fabry (2018, 796–98).

  34. See above, note 7.

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Acknowledgements

I would like to thank the two anonymous reviewers for their valuable comments on the manuscript. I am also thankful to Raymond Tallis for his interest and critique, and to Sunwoo Lee for her continous support.

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Work on this study was supported by the project “The Dynamics of Corporeal Intentionality,” Palacký University Olomouc, Reg. No. JG_2019_006.

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    Jan Halák

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Halák, J. Mathematics embodied: Merleau-Ponty on geometry and algebra as fields of motor enaction. Synthese 200, 34 (2022). https://doi.org/10.1007/s11229-022-03526-z

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