Lee and Schwarz (L&S) argue that “much like cognitive capacities in general are grounded in sensorimotor ones, […] mental procedures in particular are grounded in physical procedures […]” (sect. 3.1, para. 5). Their grounded procedures are generalizable because brains are incorporated in bodies that interact with the environment, so embodied cognition integrates physical properties of the world, sensorimotor constraints of our body, and contextual factors. Despite broad agreement, Matheson and Barsalou (Reference Matheson, Barsalou, Wixted and Thompson-Schill2018) recently diagnosed that an “overarching theory of embodiment and grounded cognition” is impeded by rather heterogeneous contributions from different disciplines. A hierarchical distinction between grounded, embodied, and situated cognition, initially proposed for the domain of numerical cognition (Fischer, Reference Fischer2012), can accomplish the desired theoretical integration.

The proposed hierarchy (see also Myachykov, Scheepers, Fischer, & Kessler, Reference Myachykov, Scheepers, Fischer and Kessler2014; Pezzulo et al., Reference Pezzulo, Barsalou, Cangelosi, Fischer, McRae and Spivey2013) first considers universal physical constraints on cognition, resulting from our environment (grounding through physical laws that shaped our nervous systems). A second level establishes embodiment of cognition through our sensory-motor history, including learned procedures. Finally, representing specific task instructions situates cognition and explains flexible performance signatures. Importantly, our conceptual distinction explains performance biases across domains, as we now illustrate.

Consider grounding first. L&S wrote about “causal links between physical cleansing and various psychological variables” and asked: “Empirically, how robust are they?” (Abstract). Because of its evolutionary origin, physical cleansing grounded on basic emotions will be most robust and hold universally for all separation/connection procedures based on approach and avoidance, for example, disgust towards rotten food (Ekman, Sorenson, & Friesen, Reference Ekman, Sorenson and Friesen1969). In fact, the feeling of disgust is triggered not only by physical contaminants, but also by moral impurities and “form(s) part of a behavioural loss aversion system aimed at protecting valuable resources, including the integrity of one's body” (Schnall, Reference Schnall2017, p. 50).

Equally grounded is the fact that object accumulations produce higher piles because physical laws prevent objects from penetrating each other. The universal association “more is up” consequently informs metaphorical language (Lakoff & Johnson, Reference Lakoff and Johnson1980) and arithmetic intuitions (Lakoff & Núñez, Reference Lakoff and Núñez2000) and also induces judgment biases when accepting results that exceed the correct sum (“addition is more”; Shaki, Pinhas, & Fischer, Reference Shaki, Pinhas and Fischer2018).

Consider embodiment next. L&S describe how cleansing separates the self from failure (sect. 3.1, para. 6) and how physical and mental connecting procedures are related (sect. 5). Again, we find similar embodiment signatures in mathematical cognition, where physical procedures of separation and connection prime subtraction and addition solutions, respectively (Werner & Raab, Reference Werner and Raab2013; Werner, Raab, & Fischer, Reference Werner, Raab and Fischer2019). Conceptualization of addition/subtraction as connection/separation is also revealed by semantic priming between linguistic expressions defining commonly related entities and additions (Bassok, Pedigo, & Oskarsson, Reference Bassok, Pedigo and Oskarsson2008) and by the importance of gestures in math education (Sinclair & Heyd-Metzuyanim, Reference Sinclair and Heyd-Metzuyanim2014).

By distinguishing grounding from embodiment we understand cleansing behaviors both as universally grounded mechanisms of connection and separation, and as culturally learned and experienced embodied metaphors of morality or guilt. These descriptions are hierarchically organized and complementary and explain cross-domain interactions, such as increased prosocial (Ding et al., Reference Ding, Xie, Sun, Li, Wang and Zhen2016; Liao, Yam, Johnson, Liu, & Song, Reference Liao, Yam, Johnson, Liu and Song2018) or self-punishing behavior (Schei, Sheikh, & Schnall, Reference Schei, Sheikh and Schnall2019) to compensate for moral transgression.

Having transferred the hierarchical distinction between grounded and embodied cognition from calculation to cleansing, we now wish to show how mental arithmetic can equally benefit from L&S's study of separation and connection procedures. In line with a hierarchical approach to cognition, they suggested that “grounded procedures of separation can be a proximate mechanism underlying cleansing effects” (Introduction, para. 4) and “once activated, whether physically or mentally, a procedure can be applied across content domains, even in unrelated situations” (sect. 3.1, para. 5). Doing just this, we realize that “at the core of grounded procedures are physical actions, which move through space” (sect. 7, para. 4), although at the core of mathematical learning are physical manipulation of quantity across space. Indeed, abstract concepts are typically understood in terms of concrete concepts: Counting numbers originated from piling up pebbles to quantify sheep (Keranen, Reference Keranen2016, p. 12); children acquire basic arithmetic during putting objects into and out of containers, thus constructing metaphors grounded in everyday experiences (Lakoff & Núñez, Reference Lakoff and Núñez2000).

The fact that separation/connection procedures and subtraction/addition procedures are grounded on similar physical mechanisms, predicts that they may also activate emotions similarly because bodily states and perceptions form an integral part of emotional experiences (Winkielman, Niedenthal, Wielgosz, Eelen, & Kavanagh, Reference Winkielman, Niedenthal, Wielgosz, Eelen, Kavanagh, Mikulincer, Shaver, Borgida and Bargh2015). Indeed, “acts of separation are more likely to be triggered by negative entities, acts of connection are more likely to be triggered by positive entities” (sect. 5, para. 2). Confirming this extension across domains, several studies showed that separation procedures affect emotion perception by reducing negative moral emotions (Lee, Tang, Wan, Mai, & Liu, Reference Lee, Tang, Wan, Mai and Liu2015; Zhong & Liljenquist, Reference Zhong and Liljenquist2006), decreasing negative (guilt and shame) and increasing positive emotions (happiness; Tang et al., Reference Tang, Lu, Su, Liang, Mai and Liu2017) or even resetting them (clean slate effect), which consequently reduced the strictness of moral judgments (Kaspar, Krapp, & König, Reference Kaspar, Krapp and König2015). Similarly, studies demonstrating effects of emotion on arithmetic problem-solving further support this extension (Fabre & Lemaire, Reference Fabre and Lemaire2019; Schimmack & Derryberry, Reference Schimmack and Derryberry2005).

If we assume that cleansing behaviors and mathematical thinking share mechanisms of separation versus connection, then cross-domain priming paradigms can test this prediction. Thus, in the framework of embodied mathematics, subtraction presupposes separation (Lakoff & Núñez, Reference Lakoff and Núñez2000), parallel to cleansing behaviors, such as hand washing. Will participants solve subtraction tasks faster after washing their hands? Or will they slow down because they already “separated” themselves and a greater separation would mean more effort? Answering these questions will clarify the mechanisms connecting everyday behaviors.

We end this comparison with a challenge pointing towards situated influences on cognition. Decision-making, as well as mathematical reasoning, features heuristics and biases (Kahneman, Reference Kahneman2011; Shaki et al., Reference Shaki, Pinhas and Fischer2018). While L&S (sect. 7, para. 3) predicted that anchoring bias should increase through acts of connection, anchoring is instead more prevalent in subtraction than in addition (Shaki et al., Reference Shaki, Pinhas and Fischer2018). How can we explain this wrong prediction? Although the hierarchical understanding of grounded procedures invites a multi-layered analysis of behavior, pervasive heuristics/biases signal additional, context-dependent influences.