==== Front PLoS One PLoS One plos PLOS ONE 1932-6203 Public Library of Science San Francisco, CA USA 10.1371/journal.pone.0288038 PONE-D-22-30799 Research Article Physical Sciences Mathematics Arithmetic Computer and Information Sciences Data Management Data Visualization Research and Analysis Methods Research Design Experimental Design Computer and Information Sciences Computer Vision Target Detection Biology and Life Sciences Anatomy Body Limbs Arms Hands Medicine and Health Sciences Anatomy Body Limbs Arms Hands Engineering and Technology Signal Processing Speech Signal Processing Physical Sciences Mathematics Discrete Mathematics Combinatorics Permutation Computer and Information Sciences Software Engineering Computer Software Engineering and Technology Software Engineering Computer Software How to not induce SNAs: The insufficiency of directional force Spatial numerical associations and directional force https://orcid.org/0000-0002-0468-4636 Michirev A. Conceptualization Data curation Formal analysis Methodology Validation Visualization Writing – original draft Writing – review & editing 1 https://orcid.org/0000-0002-1334-1563 Kühne K. Conceptualization Data curation Formal analysis Methodology Validation Visualization Writing – original draft Writing – review & editing 2 * Lindemann O. Methodology Writing – review & editing 3 Fischer M. H. Conceptualization Funding acquisition Methodology Project administration Resources Supervision Validation Writing – review & editing 2 ‡ Raab M. Conceptualization Funding acquisition Methodology Project administration Resources Supervision Validation Writing – review & editing 1 4 ‡ 1 Department of Performance Psychology, German Sport University Cologne, Cologne, Germany 2 Division of Cognitive Sciences, University of Potsdam, Potsdam, Germany 3 Department of Psychology, Education and Child Studies Erasmus University, Rotterdam, Netherlands 4 School of Applied Sciences, London South Bank University, London, United Kingdom Tessari Alessia Editor Universita di Bologna, ITALY Competing Interests: The authors have declared that no competing interests exist. ‡ These authors also contributed equally to this work. * E-mail: kkuehne@uni-potsdam.de 29 6 2023 2023 18 6 e02880388 11 2022 19 6 2023 © 2023 Michirev et al 2023 Michirev et al https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. People respond faster to smaller numbers in their left space and to larger numbers in their right space. Here we argue that movements in space contribute to the formation of spatial-numerical associations (SNAs). We studied the impact of continuous isometric forces along the horizontal or vertical cardinal axes on SNAs while participants performed random number production and arithmetic verification tasks. Our results suggest that such isometric directional force do not suffice to induce SNAs. http://dx.doi.org/10.13039/501100001659 Deutsche Forschungsgemeinschaft RA 940/16-2 Raab M. http://dx.doi.org/10.13039/501100001659 Deutsche Forschungsgemeinschaft FI 1915/5-2 Fischer Martin H. This research received by the German Research Foundation (DFG) grant RA 940/16-2 awarded to MR and grant FI 1915/5-2 to MHF. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Data AvailabilityAll the data are uploaded on OSF (https://osf.io/v7dyj/). Data Availability All the data are uploaded on OSF (https://osf.io/v7dyj/). ==== Body pmcIntroduction Cognitive representations of space, time, and number provide a mental structure for how we humans experience our environment [1]. One of those structures is the Spatial-Numerical Associations of Response Codes (SNARC, [2]) effect that describes a representation of magnitude meaning along a horizontal mental number line with smaller numbers represented to the left of larger numbers. Experiments testing the SNARC effect typically rely on bi-manual responses or spatial cueing in a reaction time paradigm across different tasks (e.g., speeded parity judgments, magnitude classifications and simple mental arithmetic; for reviews see [3,4]). The reaction time paradigm is widely accepted and provides evidence for the association of smaller numbers with left space and larger numbers with right space [4]. Originally, the SNARC effect was attributed to reading and writing directions resulting from cultural exposure [2]. However, subsequent research on SNAs produced several alternative hypotheses that consider biological, cognitive, behavioural, and embodied cognition approaches. For example, a biological approach utilizes hemispheric asymmetry of the brain [5–7] whereby the right and left hemispheres are tuned for processing small and large numerosities, respectively. Therefore, the natural spatial scanning direction would go from left to right, which associates left with small and right with large numerosities. A cognitive approach considers positional coding of numbers in serial working memory and utilizes an experienced-based spatial template [8]. This spatial template is then used to arrange information according to the situation and relevant references. For instance, when the number range is between 1 and 9, then the number 5 would provide the reference. In this scenario numbers 1–4 will be placed on the left of the 5 while numbers 6–9 to the right. Together, reading and writing direction, hemispheric asymmetry, and the serial working memory approaches are capable of explaining SNAs along the cardinal horizontal axis. Additionally, a behavioural approach considers stimulus-response associations with polarity correspondence in binary tasks [9,10]. Hereby a stimulus exists in a dimension of polar opposites, such as a number being either small (negative pole) or large (positive pole). In a SNARC paradigm, a large number is paired with the response to the right (i.e., both represent positive poles). Therefore, when the polarities are matched they facilitate the response that can explain the SNARC effect. Note that, within the polarity correspondence account, the SNARC is produced by stimulus-response correspondence and not due to an association of space and numbers. Therefore, SNARC would not be limited to the horizontal axis (but see [11] for empirical evidence against the polarity correspondence account). Linguistic metaphors such as “more is up” are widely found and seem to reflect universal physical laws that explain SNAs on the vertical axis [12,13]. The embodied cognition approach encompasses such linguistic practice and suggests that all concepts, and therefore also SNAs, are bi-directionally related to sensory and motor experiences so that specific activation in one domain (experiential or conceptual) translates to the other [12,13]. Applied to number symbols, the role of the body, especially the systematic use of fingers during number learning, counting [14] and gesturing (e.g., [15]), seems to contribute to the development of numerical cognition (for an overview see [16,17]). The use of fingers seems to be deeply rooted in using the hands to count, which was already evident during Palaeolithic times [18]. In this line of reasoning, the finger-counting hypothesis [14,19] predicts the SNARC effect along the horizontal axes because in Western countries people usually start counting on their left hand and with the left thumb (left-starters), therefore associating small numbers with their left space. Finger-counting then progresses from left to right, just as the typical left-to-right SNARC. Vertical associations instead reflect other sensory and motor experiences, such as the growth of piles during object accumulation or the rise of water in a container, which inspired the linguistic metaphors mentioned above. Notably, there is an asymmetrical relationship between space and numbers, in which the understanding of space is more fundamental. The meaning of numbers is often based on the experiences of space while this is not necessarily the case vice versa [12,13]. Together, the embodied cognition approach predicts SNAs on the horizontal and the vertical axes based on different and independent mechanisms. Recently, more evidence accumulated towards the situatedness and flexibility of finger counting and the starting hand: while finger counting habits seem to be rather stable over time [20] they are also flexible depending on the situation [20,21]. Moreover, even in Western individuals, for example, Hungarians, Germans and Italians, the right hand [22,23] or either hand [24] is used to start counting while in Middle Eastern countries people tend to start counting on their right hand (with their small finger, see [25]). Overall, while the evidence on the direction of counting (starting left or right) is rather mixed and emphasizes situatedness [26], the unequivocal involvement of fingers in SNAs point towards the embodied nature of numerical cognition. Intriguingly, the SNARC effect was recently shown to exist in 3-dimensional space [27] for each of the three cardinal axes (horizontal, vertical and sagittal) whereby left/down/near are associated with smaller and right/up/far with larger numbers (for review see [28]). The authors [27] have interpreted the 3-dimensional SNARC as indicating the existence of three independent mental number lines. These findings are of particular interest because they provide counter-evidence for all the above accounts. For instance, extending the horizontal mental number line to 3-dimensional space opposes the accounts of reading and writing directions [2], hemispheric asymmetry [5–7] and positional coding of numbers in serial working memory [8] because those accounts only explain SNAs along the horizontal cardinal axis. Furthermore, the independence of the three hypothesized mental number lines opposes the polarity correspondence account [9,10] that assumes symmetry of all cardinal axes. Additionally, while the embodied cognition approach can make predictions of SNAs on the horizontal and the vertical axes, it lacks a strong theoretical reason to predict SNAs on the sagittal axis [29]. The goal of the present study is to test the boundary conditions under which the association of numbers and space is manifested in behaviour. We adopted the embodied cognition approach and focused on the horizontal and vertical axes. Examining prototypical experimental conditions under which SNAs are found and not found can help identify a common ingredient across existing experiments (for recent null results for horizontal SNAs see [30,31]). By this rationale, the common ingredient for most studies seems to be the activation of spatial information, namely under the experimental conditions of response space (lateral responses), presentation space (lateralized stimuli), or spatial cueing (arrows and/or instructions) during stimulus presentation (see Fig 1 for a visual overview of prototypical laboratory set-ups). 10.1371/journal.pone.0288038.g001 Fig 1 Prototypical experimental designs reporting SNAs. 1 Lateralized response space with centralized stimuli presentations on the horizontal axis [2]. 2 Lateralized response space with centralized stimuli presentations. The authors wanted to access the vertical axis, however, the spatial alignment of response keys depicts the sagittal axis (e.g., [32–34]). 3 Lateralized response space with centralized stimuli presentations along all three cardinal axes [27]. 4 Lateralized response space with lateralized stimuli presentations (e.g., [35]). 5 Lateralized response space with centralized stimuli presentations on the horizontal or the vertical axis (saccadic movement to the target; e.g., [36,37]). 6 Centralized response space with lateralized target presentations on the horizontal or the vertical axis (visual target detection by key response; e.g., [35,38]). 7 Centralized response space with centralized stimuli presentations but spatial cueing [31]. 8 Centralized response space with centralized stimuli presentations where no SNAs were found [30]. Response space seems to contribute to a range of experiments using bimanual responses (see Fig 1, examples 1 to 4). For example, the original horizontal SNARC [2] was observed during the central presentation of numbers when participants responded bimanually by pressing keys arranged laterally on the left and right side. Additionally, a horizontal SNARC emerges when bimanual response space, as above, is paired with lateralized stimuli presentation (simultaneously displayed digit pairs for magnitude comparison, [35]). Furthermore, response space is not exclusive for the horizontal axis and is present in SNARC research with the vertical axis. However, as a keyboard is commonly used to record responses, this is particularly questionable because the up and down arrows on the keyboard operate the cursor vertically on the computer screen but are located along the sagittal axis (see Fig 1, example 2). In order to have a purer assessment of SNAs along the vertical axis, eye tracking experiments might be more suited. However, even under these experimental conditions spatially aligned targets are typically needed (but see experiments using auditory stimuli [39,40]). For example, when participants are presented with centrally displayed stimuli they are required to respond by saccadic movements to the target that appears inside spatially aligned boxes to either left and right or up and down (see Fig 1, example 5). Finally, response space is also presented in the 3-dimensional SNARC as participants used bimanual responses along the three cardinal axes (see Fig 1, example 3). Very recently, a new study challenged the importance of active responses in the response space. Miklashevsky et al. [41] relied on a paradigm of passive and continuous isometric force readings during numerical tasks. Participants produced more force in the left hand while processing smaller numbers and more force in the right hand while processing larger numbers. These results suggest that active responses might not be required to induce SNAs. However, the authors did not control for spatial effects in their design. Specifically, in the response space, the authors utilized two sensors that were held in the left and in the right hand, thus being laterally displaced. Therefore, spatial information in the response space was apparently sufficient to elicit SNAs in their study. Alternatively to the response space, the presentation space in which magnitudes are displayed can create SNAs. For instance, whenever experiments have centralized their response space in a reaction time paradigm (e.g., by using the space bar) they instead included laterally presented targets (see Fig 1, example 6; target detection paradigm). Such designs remove the response codes; however, they introduce spatial codes in the targets. For example, a stimulus is presented on the screen together with laterally aligned boxes (either left/right or down/up). Then a target appears inside one of these boxes (spatial code) that participants respond to by pressing the space bar (e.g.,[42]). Recently, the target detection paradigm was extended to test the temporal progression of number processing during mental arithmetic (e.g., [38]). During such temporal progression a systematic bias of accepting larger values in addition and smaller values in subtraction problems was observed (Operational Momentum, see [43]) and interpreted as a spatial shift of attention on the mental number line that relies on the spatial nature of number representations. However, the onset time of these SNAs is currently debated in the literature (e.g., [38]). For example, several studies report SNAs after the appearance of single digits [2,42,44]. Others report these SNAs only after both the first operand and the operator of an arithmetic problem had been presented, indicating that spatial associations emerge with arithmetic operators (Operation Sign Space Association; [38,45,46]). Yet another study [47] reported spatially biased eye behaviour only after all problem elements were known to their participants. Therefore, the target detection paradigm provides a useful tool that can help track SNAs on a temporal continuum. However, this paradigm also suffers from spatial confounds as the target is displayed laterally and therefore carries spatial information across either the horizontal or vertical axis. Until now we have presented prototypical examples of experimental conditions under which SNAs emerged, considering both presentation and response space. Next, we consider the SNAs under experimental conditions that have controlled for response and presentation space. These experiments removed spatial information from their SNA assessment by showing non-lateralized numbers and avoiding lateralized responses. If our rationale is correct that spatial information contributes to SNAs, then SNAs should not emerge under such spatially unbiased assessments. Indeed, Pinto et al. [30] (see Fig 1, example 8) deployed such a “purified” assessment of the SNAs under conditions of central presentation space but with varied response space: one task relied on lateralized responses and the second task relied on central responses. The authors showed that only the task that required bi-manual responses (lateralized response space) produced the typical left-to-right SNARC. The other task that aligned response space with the presentation space did not find SNAs (“only press the centralized space bar if the number is larger/smaller than 5, otherwise do not react”). The authors’ interpretation was that the numbers will only spatially align across the mental number line when both, number magnitude is activated and spatial response codes are present (for experimental designs that also centralized the presentation and response space see [48,49]). Another study ([31], see Fig 1, example 7) centralized both presentation and response space. However, this study used spatial cueing with arrows pointing to the left, right, down or up. Despite this spatial cueing the authors found no horizontal SNAs in this version of the parity judgment tasks. Consequently, these findings suggest that the much-studied horizontal SNAs might be ephemeral artefacts of spatial task demands. Together, the above-presented studies point to the conclusion that both, number magnitude and spatial information must be coactivated to produce reliable SNAs [30,31,48,49]. Intriguingly, in the magnitude judgment tasks, the up-to-down cueing succeeded and vertical SNAs persisted and were always stronger than horizontal SNAs. One possible explanation for why vertical SNAs remained stronger than horizontal SNAs could be the hierarchical nature of body-related knowledge representations involved in the task, according to which different mechanisms induce different types of SNAs, referred to as grounded, embodied, or situated cognition [29]. For example, the finger-counting hypothesis considers embodied sensorimotor experiences and cultural influences [14,19]: The hand that is used to start to count defines (or strongly suggests) the direction of horizontal SNAs. Instead, vertical SNAs are grounded in deeply rooted magnitude-space associations that reflect universal physical laws and inspired ubiquitous linguistic metaphors, such as “more is up” [12,13]. Indeed, expressions such as “high number” or “low number” reflect vertical SNAs while no such linguistic connection exists for horizontal SNAs [28]. These linguistic practices that describe vertical SNAs differ in their contribution to the embodiment of numbers compared to the horizontal SNAs that seem to rely more on sensorimotor experiences such as finger counting. Moreover, situated factors like interoceptive signals play an additional role in the perception and production of numbers that also might contribute to SNAs ([50]; for review see [51]). Therefore, vertical SNAs might be more general or stable than horizontal SNAs, which could in turn explain why spatial cueing was enough to elicit vertical SNAs but not horizontal SNAs under the exclusion of spatial responses. Overall, the embodiment perspective and its bi-directionality (e.g., [29,52,53]) can shed new light on the association between space and numbers. So far, we have discussed one of the two directions of concept-motor interactions, namely how activations of magnitude representations can affect motor responses and produce spatial biases. The second direction is reversed and predicts how movements can, in turn, affect conceptual representations and produce number magnitude biases. The bi-directional link is part of A Theory of Magnitude (ATOM, see [54,55]) that proposes a generalized magnitude representation system across perception, cognition and action. To test the bi-directionality assumption of the concept-motor interactions, Random Number Generation (RNG) can be utilized. Hereby, the RNG task assesses the cognitive availability of magnitude-related concepts of participants who generate and produce numbers. Specifically, participants are required to produce numbers in specific ranges as randomly as possible [56,57]. RNG can test the concept-motor interactions because number magnitude can produce spatial biases in movement, while movements in space can, in turn, produce magnitude biases. For instance, on the one hand, during a walking task random number generation affects the decision to take either a left or right turn [58], thus producing spatial biases in movement. This is in line with the SNAs research explaining how magnitude activation (small/large numbers) affects movement in space (left/right turns). On the other hand, deciding to take a left/right turn also affects RNG [58]. Other movements through space generally seem to affect RNG as well. For instance, healthy adults generate larger numbers after a right/up head turn and smaller numbers after a left/down head turn ([59]; for RNG across the horizontal axis see also [56]; for review see [28]). Interestingly, such RNG patterns also transfer from active to passive movements in space. For example, passively moving the participants through space produces magnitude biases along the horizontal and vertical axis, however, not along the sagittal axis (passive whole-body movements, see [60]). While the RNG task is well-suited for testing concept-motor interactions, we note that none of the above-presented RNG studies controlled for spatial influences. Therefore, the RNG task is another spatial task producing or relying on movements through space such as a rhythmic head movement to the left/right (e.g., [56]) or a simple button press in a reaction time paradigm representing a location-related action (see Fig 1, examples 1 to 4). Generally, spatial information in the design (e.g., movements through space or responses in space) seems to be essential when investigating SNAs, both as a response producing SNAs and to activate SNAs. One line of the above-described paradigms utilizes button presses, lateral eye movements, or movements that include location-related actions measured by the task. The other line of paradigms rather produces movements in response to number generation. Again, the common ingredient seems to be movement in space. Therefore, it might be movements in space that activate the spatial nature of numbers, thus enabling SNAs. Considering all the presented evidence, we can conclude that spatial information is omnipresent in many experimental paradigms that study SNAs. First, spatial information in the response space plays a major role in accessing SNAs along all three cardinal axes [27]. Second, movement through space can activate SNAs with the RNG task across the horizontal and vertical axes [56,59] but not on the sagittal axes [60]. Third, cognitive spatial cueing induces stronger vertical than horizontal SNAs during centralized stimulus presentation and responses [31]. Fourth, taking a turn to the left/right affects RNG [58]. Based on this we follow the logic of the falsification by elimination approach [61] in which we eliminate spatial task demands from the present experimental design and ask the following question: Will directional but non-spatial movement along the horizontal and vertical axes, thus pressing into a direction of left, right, up, or down, elicit SNAs? In the present experiment we centralized the presentation and the response space. We asked participants to press against a sensor surface, thus recording their isometric force production. Such isometric force was exerted constantly and continuously on the sensor surface in a direction on the horizontal (left or right) and the vertical (up or down) axes during two tasks. Hereby, we utilized the same force sensors as Miklashevsky et al. [41] but altered the paradigm to record the production of isometric force by one-handed presses into a direction instead of grip force (from here on described as “isometric directional force”). Therefore, our isometric force paradigm does not require the execution of a lateralized response after processing a number. Instead, isometric force provides a continuous and implicit measurement of spatial activation during number processing. First, with the RNG task we aimed to test if isometric force, exerted into a direction along the horizontal (left or right) or the vertical (up or down) axis, was sufficient to produce SNAs. Second, we aimed to test if RNG itself can affect isometric force production. This approach allows to test the bi-directionality of the concept-motor interactions. The following hypotheses were not preregistered but formulated a priori and derived from theory and previous empirical work. H1. We hypothesize that isometric directional force activates spatial codes that affect the production of random numbers, thus testing how isometric directional force affects the conceptual representation of numbers. That is, participants are expected to produce more small numbers when pressing in the direction left/down and more large numbers when pressing in the direction right/up [56,59]. H2. We hypothesize that generating numbers by itself affects motor activation [58], thus testing how conceptual representation of numbers affects force production. Generating numbers is assumed to activate within-magnitude associations (ATOM, [54,55]), which in turn, are hypothesized to modulate the amplitude of applied isometric force. Hereby, generating smaller numbers should produce smaller force while generating larger numbers should produce larger force when simultaneously pressing against the surface of a force sensor [54,55]. As an additional research question of the experiment, we aimed to explore SNAs on a temporal continuum during mental arithmetic, thus testing how activation of number concepts affect force production during mental arithmetic. The rationale is similar to that of the target detection paradigm (see Fig 1, example 6). However, we removed spatial information from the presentation and the response space. Therefore, instead of providing spatial information in the form of targets to probe these SNAs, we again relied on central stimuli presentation and isometric directional force. In this task, spatial associations of numbers could become activated in relation to the arithmetic procedure; namely either during the first operand, or the operation sign, or the second operand, or during the answer. Similarly to the reasoning in H2, number meaning related to the task should be expressed by the participants’ spontaneous force production, with smaller numbers generating smaller force and large numbers generating larger force [54,55]. Taken together, this study aims to test SNAs across the horizontal and the vertical axes by applying isometric directional force. This design allows to exclude spatial task demands in the responses by excluding movements through space and in the stimuli by excluding spatially placed targets. Hereby, the RNG task will answer the question if isometric directional force suffices to elicit SNAs while the arithmetic task will help with tracking the dynamics of SNAs across the arithmetical task. General method Participants The total sample size is N = 72 (41 female, mean age 23, median 23, range 17–35 years). All participants were native German speakers and right-handed [the German version of the Lateral Preference Inventory [62] showed a mean of 82,3 and a median of 86.7. The sample was obtained across two cooperating but independent laboratories to ensure the generalizability and replicability of findings across both laboratories and experimental settings. Hereby, Laboratory 1 was allocated at the German Sport University Cologne and Laboratory 2 at the University of Potsdam. Regarding the power analysis, we applied a conservative effect size estimate of dz = .4 for a two-tailed paired t-test (small vs. large number/compatible and incompatible condition) with power .80 and α-level .05 [63], resulting in a required sample size of 52 for each experiment in each laboratory. This is also a general recommendation for sample size estimation in a t-test repeated-measures design [64]. After reaching the sample size of 54 (considering possible drop-out) in Laboratory 1, data collection in Laboratory 2 was also stopped for efficiency reasons since the data fit rather the null hypothesis model. In case of a null effect, for repeated-measures designs a group of at least 60 participants is considered to be sufficient for a Bayesian analysis [64]. Apparatus A customized home-made wooden/metal box was built to enable isometric directional force (i.e., participants’ exerting isometric forces on a fixed force sensor surface) along the vertical and the horizontal axes (two levels each: left/right on a horizontal axis or up/down on the vertical axis). Inside the box a force sensor in the shape of a metallic disc 1.8 cm thick and 4 cm in diameter (FTD-MINI40-E-1.8-M2 from SCHUNK GmbH & Co. KG) was attached to a wall congruent with the experimental force direction condition (see Fig 2, panel 2). Participants placed their right hand inside the box and applied force to the sensor (see Fig 2 for direction left), thus producing isometric directional force measured by the Fz axis of the sensor. With this method, the Fz axis provides most accurate parameters of the force data [65]. The box was adjustable on the vertical and horizontal axes to account for individual hand sizes. 10.1371/journal.pone.0288038.g002 Fig 2 Conceptual drawing of the experimental set-up. Panel 1 shows the general experimental set-up with stimuli (S) presented centrally. The box was centralized relative to the stimulus presentation space, allowing the sensor to be placed into one of the four experimental conditions. Panel 2 shows the Fz axis of the sensor that was used to analyse the force data. Panel 3 depicts the left direction with the sensor attached to the left wall. Force measurements were obtained according to the procedure recommended by Nazir et al. [66]. Two computers were used to conduct the experiment: the first presented stimuli, the second recorded the data. The first computer sent an initial trigger through the parallel port and a terminal trigger to the second computer, then the force signal was recorded using Expyriment software [67] at a sampling rate of 1 kHz through an analog-to-digital converter card. Forces were measured in millinewtons (mN). Experimental tasks 1. Random Number Generation (RNG) task 1.1 Design In order to test the concept-motor interactions of H1 and H2 with the same task, we have utilized two within-subject designs. Both designs utilized one within-subject factor being the force direction (two levels each: left/right on a horizontal axis or up/down on the vertical axis). To test H1, number magnitude constituted the dependent variable with two magnitude levels: small and large. To test H2, number magnitude constituted the independent variable with two magnitude levels: small and large while continuous isometric force was the dependent variable. Additionally to the force directions, there was a control (no-force) condition as a baseline. During the baseline measurement participants were asked to take a relaxed position and rest their hands on their knees to assess unbiased performance. Only the produced numbers were collected and analysed in the baseline condition of H1. 1.2 Procedure Participants were asked to verbally produce a random number from 1 to 9 every 2 seconds. In total participants produced 90 random numbers (cf. [68]) with their eyes closed (for this procedure see as well [50,57]). Closing eyes eliminates possible spatial confounds but maintains isometric directional force as the remaining variable of interest. A metaphor of “drawing numbers from a hat” [50,57,69] was used to instruct participants. A metronome sound, presented with OpenSesame software [70], with a rate of 0.5 Hz was used to indicate the signal for number production every two seconds. Participants responses were recorded with a voice key and noted manually by the experimenter. 1.3 Data analyses of continuous isometric force The following method was not preregistered but closely followed the recommendations of Nazir et al. [66] (for details see S3 File). The selection of time-windows for the force data analysis in the RNG task was based on the trial onset, i.e. the start of the auditory signal in the RNG task after which participants had 2 seconds to state a number. From these 2 seconds we selected the first 700 ms of each trial for calculation because speech production can motorically interfere with manual force [66]. Therefore, we focused our analysis on time-windows prior to speech production, which began on average 576 ms after stimulus onset. We have utilized a cluster permutation analysis (with 5000 permutations and a correction for multiple comparisons) to identify the time-windows of interest during which number magnitude (in operand 1, operand 2, and the result) and the operation (plus, minus) affects force magnitude ([71]; for details see S3 File). In the case of uninformative results of the cluster permutation analysis, five time-windows for each directional condition were selected manually (in ms: 50–150, 150–250, 250–350, 350–450, 450–576). We split the force data into equal chunks of arbitrary length of 100 ms starting from 50 ms to the end of the trial (last time-window being 126 ms long). These time-windows were constructed to capture mainly the effects in the time-windows of 100–250 ms that are associated with semantic activation after a critical stimulus [72]. 2. Single Digits Arithmetic (SDA) task 2.1 Design A within-subject design with three within-subject factors was used for the SDA task, namely: number magnitude (small vs. large), operation (plus/minus), force direction (two levels: left/right on a horizontal axis or up/down on the vertical axis) with isometric force production as the dependent variable (for overview see S1 Table). 2.2 Procedure Participants were asked to watch mathematical equations (from here on, referred to as trials) in which individual stimuli (operand 1, operation sign, operand 2, equal sign, result) were presented sequentially after each other centrally on the computer screen. Some of the proposed results were wrong. In case of a wrong result participant had to reject the result by saying “No“. Each stimulus was shown for 500 ms, then the next stimulus appeared. There was an inter-trial interval of 1500 ms between the trials (see Fig 3). 10.1371/journal.pone.0288038.g003 Fig 3 Time-course of one trial of the SDA task. The task incorporated the Go/No-Go paradigm with wrong mathematical equations (catch trials) to ensure the depth of numerical processing. In GO trials (incorrect result) participants were requested to verbally respond with a “No”. In total, the task contained 116 trials (58 unique trials, presented twice) of which 96 were true-trials (correct results/No-Go condition) and 20 were catch-trials (incorrect results/Go condition). The trials were designed by keeping the proposed result in the single digits and positive numbers, therefore limiting the quantity of unique combinations. These 116 trials were presented in random order for each participant across four blocks, with mandatory self-paced breaks to counter physical fatigue and re-calibrate the force sensor [66]. In total, each participant went through four SDA tasks, one for each force direction (464 total trials per participant). The trials were constructed with single digit operands excluding the number 5 for addition and subtraction (see S2 Table for all stimuli). The digits (size: 100x100 pixel, Calibri font, visual angle of 2.2°) were created with Adobe Photoshop. They were presented one at time at the centre of the PC screen in white on a black background (in OpenSesame software [70]). 2.3 Data analysis of continuous isometric force In the SDA task, the time-windows of interest were the first 500 ms after each stimulus onset. Identically to the RNG task, the time-window selection was based on either a successful cluster permutation analysis (for details see S3 File) or a manual construction of time-windows (in ms: 50–150, 150–250, 250–350, 350–450 and 400–500) based on the same rationale as the construction of the RNG time-windows. Therefore, for each analysis of operand 1, operator, operand 2 or the answer, we only analysed the 500 ms after that stimulus’ onset, respectively. The total time-windows we extracted were the following: operand 1 (0–500 ms), operator (0–1000 ms), operand 2 (0–1500 ms) and answer (0–2500 + 500 ms for the inter-trial interval). This procedure was repeated across all four force-direction conditions. 2.4. General procedure The sequence of force direction conditions and order of tasks was counterbalanced between participants with a Latin square. The experimenter welcomed the participant and seated them in a comfortable chair, at a desk, about 90 cm away from the PC monitor. Participants were informed about the procedure and were asked to sign the informed consent and effector dominance assessment [62]. The experimenter explained the force measurement, as well as the SDA and RNG tasks, using PowerPoint slides supported by a standardized experimental script clarifying any remaining questions if needed. The explanations also covered the exclusion of number 5 and the largest number to be expected to be a 9 (smallest to be 1). Dependent on the force direction condition the participant placed their hand inside the Apparatus. The participant was then instructed to sit as still as possible and to continuously press the force sensor with their index and middle fingers (see Fig 3). The participant then underwent a task-specific familiarization, including force calibration (See S2 File). Importantly, frequent and self-paced breaks were utilized to minimize participants fatigue levels (recommended by [66]). Next, participants took a short break and started the experiment with the predefined force direction condition and task. For example, first, the participant was tested in the SDA task in the right force direction condition. Then, upon completion of the SDA task, the participant proceeded with the RNG task in the same force direction condition. Upon completing one force direction condition the experimenter adjusted the Apparatus to the new force direction condition. In total, each participant completed four force direction conditions. The experiment ended after the completion of all four force direction conditions for both the RNG and SDA tasks. Upon completing the session, the participants filled in the demographics form and were rewarded and debriefed. The typical experiment duration was 90 minutes. The study was reviewed and approved by the Ethics Committee of the University of Potsdam, Germany (approval number 21/2019). The participants provided their informed consent to participate in this study. 2.5 Forms and questionnaires: Demographic form & lateral preference form The demographic form recorded the age, gender, native language, whether participants have normal or corrected-to-normal vision, or a history of neurological diseases. We also recorded the use of medications that could affect performance. To assess hand dominance, we have relied on the German version of the Lateral Preference Inventory to ensure that all participants are right-handed [62]. 2.6 Pre-processing force data Pre-processing of force data followed the general guidelines proposed by Nazir et al. [66] and is documented in the S3 File. Importantly, due to pre-processing procedures of relevant stimuli in the SDA task (operand 1, the operator, operand 2, and the answer) there was an increased attrition rate for each analysis while progressing through the duration of a trial. Moreover, the attrition rate was not symmetrical across the RNG and the SDA tasks. The accuracy cut-off in the SDA task was set to 90% and was applied for all four directional force conditions separately. In RNG task there was no accuracy cut-off, however two participants were excluded because they failed to comply with the task (they generated numbers outside the range or counted systematically up or down). The experimental protocols of both cooperating but independent laboratories were identical, however, due to possible soft and hardware differences the force data from both laboratories was not collapsed into one data set. This was done to ensure that measuring reliability regarding temporal recordings remains across the individual data sets. Instead, identical analyses were performed on both data sets. Then, the Bayes factors from both analyses were multiplied by each other. This procedure allowed to use Bayes factors as inclusion of prior knowledge [73]. Therefore, the main text includes the combined Bayes tables while the S1 File shows individual analyses from both laboratories. The default width parameter of .707 was used in all analyses since we did not have strong assumptions about the distribution. Moreover, it was shown that the prior width has little impact on the Bayes factor [74]. All analyses were done within each independent axis (horizontal or vertical). Results 1. RNG 1.1 Numbers To test SNAs induction in terms of generated numbers, a Bayesian contingency table with the RNG data was computed by collapsing all data into one data set (Table 1). In total, participants (N = 70) named more small numbers (1–4) than large numbers (6–9; distribution: 14032 vs 13248) with the proportion of 0.51 to 0.49 resulting in a BF10 = 593.26 (very strong evidence) in favour of the alternative hypothesis, indicating a small number bias which is a typical RNG signature [75]. However, there was no effect of direction on number magnitude: neither horizontally (left vs. right) BF01 = 27.05, nor vertically (up vs. down) BF01 = 27.43, nor when comparing each direction to the baseline: Left vs. baseline BF01 = 38.22, right vs. baseline BF01 = 36.58, down vs. baseline BF01 = 30.58, up vs. baseline BF01 = 41.08. All these Bayes factors provide strong to very strong evidence in favour of H0, indicating that the isometric directional force failed to induce SNAs in any direction. 10.1371/journal.pone.0288038.t001 Table 1 Absolute frequencies of randomly produced numbers in each direction (excluding the number five). Direction Numbers Baseline Left Right Down Up Total Large (6–9) 2615 2709 2638 2695 2591 13248 Small (1–4) 2782 2836 2862 2782 2770 14032 Total 5397 5545 5500 5477 5361 27280 1.2 Force during RNG The cluster permutation analysis did not reveal time-windows of interest, therefore five time-windows for each directional condition were selected manually (in ms: 50–150, 150–250, 250–350, 350–450, 450–576). The force data were averaged within the selected time-windows and submitted to Bayesian paired samples t-tests (Student t distribution) with factors small and large within each direction. The tests were conducted in the JASP software 0.16.1 [76], using a Cauchy prior width of .707. The Bayes factors testing small vs. large number magnitudes revealed moderate (BF01 = 2.99) to strong (BF01 = 26.60) evidence for the null hypothesis, depending on the time interval (Table 2), favouring the conclusion that there is no systematic difference between small and large number magnitude regarding force magnitude. Visualized representation of the force data is presented for both laboratories independently in Fig 4A and 4B. The S5 File reports the descriptive statistic of all force data from both laboratories. 10.1371/journal.pone.0288038.g004 Fig 4 A. Visualized Force Data during the RNG Task from Laboratory 1. B. Visualized Force Data during the RNG Task from Laboratory 2. Fig 4A and 4B visualize force profiles of the RNG experiment. The grey lines represent force elicited by a small number, the black lines represent force elicited by a large number, in the respective direction. 10.1371/journal.pone.0288038.t002 Table 2 Bayes factors of Bayesian t-tests of the RNG force data for small vs. large numbers. Time-windows in ms BF01 Left Right Up Down 50–150 24.09 9.33 14.61 8.81 150–250 26.60 2.99 19.56 14.98 250–350 23.86 6.85 22.56 10.38 350–450 20.57 17.19 20.77 14.19 450–576 24.21 19.67 14.19 14.82 The analysis was performed with n = 69, 66, 63, and 67 for the directions left, right, up and down, respectively. 2. SDA 2.1 Force during SDA Symmetrically to the RNG analysis, we also conducted Bayesian paired samples t-tests (Student t distribution) with factors small and large within each direction using a Cauchy prior width of .707. Again, the cluster permutation analysis did not reveal time-windows of interest, therefore six time-windows for each directional condition were selected manually (in ms: 50–150, 150–250, 250–350, 350–450, 400–500). Each trial (Go and No-Go) that remained after data processing was included in the analysis for operand 1, operator and operand 2. However, during the Go trials answers were not analysed because of the substantial noise in the grip force signal created by speech production during the answer. Tables 3 and 4 summarize the results. Visualized representation of the force data is presented for both laboratories independently in Fig 5A and 5B. The S5 File reports the descriptive statistic of all force data from both laboratories. 10.1371/journal.pone.0288038.g005 Fig 5 A. Visualized Force Data during the SDA Task from Laboratory 1. B. Visualized Force Data during the SDA Task from Laboratory 2. Fig 5A and 5B visualize force profiles in the SDA experiment. Representation of a full trial that compares small to large answers. Hereby, there exist only two possible combinations that lead to small and large answers. The grey and black lines represent force elicited in specific conditions (Grey line: Small operand 1 plus small operand 2, and small answer: Black line: Large operand 1 minus small operand 2, and large answer). The vertical dotted line represents the onset of a specific part of the equation (operand 1, operator, operand 2, equals, or the answer, respectively). 10.1371/journal.pone.0288038.t003 Table 3 Bayes factors of Bayesian t-tests of the SDA force data for a small vs large operand 1. Time-windows in ms B01 Operand 1   Left Right Up Down 50–150 1.82 8.02 22.78 15.02 150–250 7.99 9.62 24.39 3.12 250–350 13.87 8.47 20.87 7.54 350–450 18.32 5.73 15.18 5.81 400–500 19.47 8.59 10.39 7.17 The Operand analysis was performed with n = 61, 60, 67, and 65 for the directions left, right, up, and down, respectively. 10.1371/journal.pone.0288038.t004 Table 4 Bayes factors of Bayesian t-tests of the SDA force data for operator (plus vs. minus), operand 2 (small vs. large), and answer (small vs. large).   BF01 Time-windows in ms Small O1 followed by: Large O1 followed by: Operator Operator Left Right Up Down Left Right Up Down 50–150 14.94 1.82 11.91 20.97 16.14 22.40 4.13 2.33 150–250 24.20 3.43 10.94 21.10 16.02 10.40 4.40 3.72 250–350 18.24 1.38 11.45 21.59 16.51 8.88 1.70 7.97 350–450 17.78 0.50 12.33 24.49 24.31 6.21 0.87 6.08 400–500 19.95 0.77 13.40 24.21 24.59 8.78 2.26 4.74   O2 O2   Left Right Up Down Left Right Up Down 50–150 20.99 11.89 17.96 21.58 17.90 15.34 13.98 8.27 150–250 23.52 15.57 17.33 20.41 15.32 12.88 8.72 13.41 250–350 24.21 8.10 18.31 23.78 17.63 16.78 7.49 18.66 350–450 23.44 10.10 14.06 23.00 21.10 17.59 13.62 17.84 400–500 23.60 11.00 12.74 23.92 22.04 16.32 13.68 16.74                     Answer Answer   Left Right Up Down Left Right Up Down 50–150 0.69 15.21 9.72 17.41 14.34 2.65 0.02 3.96 150–250 0.39 15.03 7.77 22.00 13.37 1.03 0.31 4.94 250–350 0.71 16.17 6.62 23.90 16.93 1.15 0.90 9.00 350–450 0.16 17.86 7.65 22.74 15.38 0.62 1.02 8.42 400–500 0.13 17.86 8.56 22.45 20.54 1.10 0.81 7.78 The operator sub table compares plus vs. minus after either a small or a large operand one. O2 and the result sub-tables report the results of small vs. large numbers within referred time-windows after either a small or a large operand 1. The Operator analysis was performed with n = 60, 57, 62, and 63 for the directions left, right, up, and down, respectively. The O2 analysis was performed with n = 60, 52, 55, and 62 for the directions left, right, up and down, respectively. The Answer analysis was performed with n = 59, 43, 46, and 60 for the directions left, right, up, and down, respectively. Discussion In this study, isometric directional force was implemented as an experimental manipulation in two numerical tasks that required producing continuous isometric force. The first objective was to study if isometric directional force would suffice to produce SNAs across the horizontal and the vertical axes during RNG. This question was motivated by prior studies describing how movements through space influence RNG (e.g., [56,58–60]). Notable, our isometric directional force manipulation did not include a movement through space. Instead, isometric directional force utilized a subtler approach of directionally applied pressure attempting to elicit spatial codes. The second objective was to study if number magnitude would affect force production [54,55]. In the RNG task, our experimental manipulation of isometric directional force failed to elicit SNAs along both axes, indicating that the directional force failed to induce SNAs in any direction. Contrary to the prediction of H1, participants did not generate more small numbers in the left/down condition or more large numbers in the right/up condition. In fact, there is strong evidence that the isometric directional force conditions did not differ from the baseline condition in which participants produced numbers in a relaxing position. Nevertheless, participants displayed a typical small number bias [75] indicating that the RNG task, by itself, was carried out appropriately. Overall, this null result is likely due to the design choices of the experiment. Particularly, the experiment was designed to exclude task-specific spatial influences and instead utilized directional (i.e., isometric) force. The central hand position probably eliminated those spatial references, and therefore prevented SNAs [30,31]. The isometric directional force by itself was then insufficient to activate spatial references and, thus, produce SNAs. Excluding spatial task demands is atypical for a study accessing SNAs (see Figs 1 and 2) and follows the logic of the falsification by elimination approach [61]. In this experiment, we therefore systematically excluded known spatial factors under which prototypical SNAs studies are conducted. Our finding is in contrast to other RNG studies that found SNAs along the horizontal and vertical axes by using either active movement ([59]; see also [56] for RNG across the horizontal axis; for review see [28]) or passive movement through space [60]. Therefore, it is likely that activating a spatial code in the response space, it requires a movement with a spatial component, such as movement through space or movement allocated in space (e.g., contacting spatially aligned response buttons), to create SNAs. Interestingly, theoretical accounts described in the Introduction, such as the reading and writing directions [2], hemispheric asymmetry, serial working memory [8], the polarity correspondence [9,10], and the embodied cognition accounts (e.g., [13,29]) all make no predictions for cases where movement in space is not present. Importantly, movement in space is often measured as a response (e.g., manual response, gaze shift) or executed before the response (RNG paradigm, e.g., [56]). Therefore, spatial information (e.g., movement through space) seems to be deeply rooted into SNAs assessment. To investigate the role of movement in space, future experiments could contrast designs utilizing movement in space against designs that utilize static and central versus static and lateral responses (for static and lateral responses see [41]). While isometric directional force failed to induce SNAs, H2 predicted a reverse influence such that number magnitude affects force magnitude [54,55,58]. However, the average Bayes factors provide moderate to strong evidence that number magnitude had no effect on force magnitude. Specifically, in the RNG task, the intention of producing either small or large numbers had no effect on force magnitude. Apparently, the mechanisms involved in random production of numbers either did not induce changes in isometric force or we could not capture them under the current experimental design. This being said, we should bear in mind that we only analysed the force data before voice onset; this was necessary to prevent artefact contamination by the speech act. However, given that we relied on the mean voice onset, it cannot be ruled out that the last 100–200 ms were already in many trials part of the speech production phase [77] and did not only reflect numerical processing. Another research question of the experiment was to explore SNAs on a temporal continuum during mental arithmetic. Hereby, number magnitude also did not affect force magnitude in the SDA task systematically at any time point during sequential presentation of both operands, the operator, and during the answer. Notably, there are two exceptions during the processing of the answer. First, there is moderate evidence of an effect (BF01 = 0.16 and 0.13 translating to BF10 = 6.25 and 7.69) while processing the answer (350–450 and 400–500 ms after the onset of the answer) following a small operand 1 in the direction left. However, inspecting the visual data of the individual laboratories (see Fig 5A and 5B) shows an opposite effect of number magnitude on force magnitude. Laboratory 1 shows larger forces for larger answers while Laboratory 2 shows larger forces for smaller answer. Second, similar evidence is found while processing an answer following a large operand 1 (50–150 and 150–250 ms) in the direction up. Here, we report strong to moderate BF01 = 0.02 and 0.31 (translating to BF10 = 50 and 3.33), respectively. Again, inspecting the visual data of the individual laboratories (see Fig 5A and 5B) shows an opposite effect of number magnitude on force magnitude with Laboratory 1 showing larger forces for larger answers and Laboratory 2 showing larger forces for smaller answer. Both of these asymmetrical and inconsistent effects could have been the result of different sample sizes. The data collection of Laboratory 2 was stopped before fulfilling the a priori calculated sample size requirement of N = 52. Apart from these exceptions, there was no main effect of numbers and operator: neither large numbers/nor the operational sign affected force production. Overall, Bayes factors during the SDA task were not as consistent as during the RNG task (SDA task: BF01 range: 0.02–24.59 and RNG task: BF01 range: 2.99–26.6). One plausible explanation would be that the mental arithmetic procedure in itself activated some spatial information [78], however, not strong enough to elicit reliable magnitude activation in isometric force. This could be attributed to the habitual reading of arithmetical tasks on paper which are read from left to right. Theoretically, the justification of a null effect is challenging because no such conclusion is logically permitted. There always will be the possibility that there is an effect which we failed to find. Therefore, and to better describe the null results, the Bayesian framework was chosen over the Frequentist approach. Specifically, there are two main reasons to apply the Bayesian framework. First, it allows for the quantification of the amount of evidence in favour or either the null or the alternative hypothesis. This approach does not force us to decide whether there is an effect or not as the Frequentist framework does. Instead, the Bayes factors indicate how much evidence there is in favour of a hypothesis. Second, the practical reason is how Bayes factors allow to combine data sets. For this experiment, we have collected data across two laboratories with similar set-ups, identical code and study protocols. Our objective was to provide converging evidence by replicating findings across laboratories. Yet, combining the datasets might be a suboptimal procedure due to small technical differences between laboratories. For example, the soft- and hardware of the computers could produce timing differences that systematically apply to one specific laboratory. Therefore, we chose to produce two individual result tables (see S1 File) and then sum them up to one by multiplying the Bayes factors. This logic is based on Bayesian knowledge updating and how data sets are used to inform each other using prior probability [73]. Limitations In this study, directional force generation was ensured by measuring isometric force production. For the experimental manipulation to induce SNAs by directional force, it was therefore sufficient to press the force sensor into an instructed direction (see Fig 4A and 4B). Indeed, as previously shown, an isometric force manipulation (but not measurement) during a lateral movement was successfully applied to induce spatial biases across the horizontal axis in a mathematical problem-solving task (Experiment 2 in [79]). Participants generated more addition solutions following a right-sided movement and more subtraction solutions following a left-sided movement. Notably, these authors relied on lateral movements along the surface of a touch-screen, meaning that participants pressed the touch-screen while conducting a lateral movement. However, there can be the case that isometric force as a measurement lacks sensitivity to reflect cognitive processes. There are two counterarguments to this statement. First, visually observing our own force data of the SDA task (see Fig 5A and 5B) indicates a sudden drop after the presentation of the answer (2000 ms). We suggest that this drop in isometric force represents a decrease in cognitive load [80] after the arithmetic procedure is completed. Second, as stated in the introduction, Miklashevsky et al. [41] found SNAs in numerical tasks during bimanual and passive isometric force recordings. The authors employed an experimental design utilizing two lateralized force sensors in a task that did not require active responses and instead recorded passive forces applied to the sensors. The authors found that smaller numbers induced more force in the left hand and larger numbers induced more force in the right hand. Such a finding is particularly intriguing because SNAs were produced without an active response and therefore not relying on a movement through space. This finding suggests that movements through space are not necessary in order to elicit SNAs. Instead, spatial information about effector positions by itself might be the sole contributor to SNAs formation (cf. [81]). Indeed, the authors utilized two spatially aligned sensors and did not control for spatial information in the response space. Additionally, keep in mind that Miklashevsky et al. [41] employed two different tasks compared to our study. The authors themselves suggested that the task itself is critical in finding SNAs as they report weaker SNAs in their surface numerical decision task (“decide if this a number or a letter”) than in the classical magnitude classification task (“is this number larger or smaller than 5”). Therefore, it is an open empirical question how their finding will translate to other experimental tasks such as random number generation and mental arithmetic as was the case in the current study. Together, Miklashevky et al.’s [41] study and our own can be interpreted as suggesting that active responses (and movements) might not be per se necessary to elicit SNAs, while spatial information might play a critical role. Future experiments could test more systematically if SNAs can exist in an experimental design space that excludes both, spatial information in the presentation and response space, as well as movements in the response space. Therefore, the current study contributes to the overarching discussion on design space and effect specification [82] necessary to elicit SNAs. Another limitation is a substantial difference between Miklashevsky et al.’s [41] and the current study is how the other authors employed the force sensor, namely in a manual precision grip. Adjusting one’s precision grip so the object does not slip, reflects exquisite fine motor control [83]. However, the precision grip is motorically different from the rather coarse directional isometric conditions employed in the current study. In contrast to the precision grip, the current study utilized the crude motor capabilities of participant’s fingers, especially the index and the middle fingers. It is likely that these fingers do not produce such finely tuned force patterns as is typical for the precision grip. Indeed, recently, it has been shown that the magnitude of unintentional force drift is large for individual fingers, especially the index finger [84]. Additionally, visually, we can observe the pattern of force in different directional conditions and how it changes drastically between directions (see Figs 4A, 4B, 5A and 5B). We attribute this to the asymmetrical motor difficulty between force directional conditions of left/down and right/up. Especially the right/up conditions, in contrast to the left/down conditions, are suffering from a larger attrition rate due to motor difficulty. This is unfortunate as the right/up directions should be associated with larger numbers. However, both left/ down directions suffer less from artefact rejection and yet present average Bayes factors pointing at moderate evidence in favour of a null effect regarding numerical magnitude processing. At the very least, isometric force served as a manipulation in any direction and ensured that participants deployed directional force (see limitations of Experiment 2 in [79]). Yet, in our independent assessment of H1, we do not find any SNAs in any direction during the RNG task. Alternatively, the precision grip can be associated with purpose and affordances [85] as it is real-life relevant and frequently used for fine-motor activities such as holding a pen. This is also true for the field of numerical cognition where multiple studies found that small numerical values facilitate a precision grip, while large numerical values facilitate a power grip [86–88]. In the current study, the experimental task by itself might have not been relevant to activate affordances for the directional force that resulted in the absence of SNAs. This is also in line with the embodiment approach according to which movements become only meaningful when they are goal-directed (e.g., [89]). Another limitation can be found in the SDA task itself. The process of mental arithmetic presupposed showing several subsequent numbers connected by an operator. This chain of stimuli might have resulted in accumulation of response biases to separate stimuli, making it difficult to disentangle the effects of each component. Further studies could display stimuli with a longer duration to increase the visibility of overlapping effects as a possible solution to the problem. Conclusion Studies that find spatial-numerical associations usually include spatial information in their experimental design. Such spatial information can be found in lateralized presentation of stimuli, visual directional cueing, laterally arranged response buttons and movement in space. The current study has eliminated such spatial information by centrally aligning the space of presentation and response. The aim was to produce SNAs with isometric force applied in the directions along the horizontal and vertical axes, thus removing movement and lateralized spatial information. Such isometric directional force was insufficient to produce SNAs along the horizontal and the vertical axes during the RNG task. Therefore, our motor task did not interact with the concept of magnitude. Additionally, we explored whether number magnitude affects force magnitude in such a design. Again, we report null results for such magnitude and force interactions. Therefore, participants’ concept of magnitude did not interact with the motor task. We conclude that applying pressure into the direction of left, right, up, and down does not suffice in eliciting SNAs along the horizontal and the vertical axes. Supporting information S1 Table Experimental factors of the SDA task. S1 Table shows the experimental factors of the SDA task. (DOCX) Click here for additional data file. S2 Table Experimental stimuli of the SDA experiment. S2 Table shows the experimental stimuli of the SDA task. In total, 58 math problems (48 true and 10 distractors) were used as stimuli. The stimuli were presented twice across four blocks. Number magnitude was defined as small (numbers 1,2,3,4) and large (6,7,8,9). Number 5 was excluded from both operands and the result. (DOCX) Click here for additional data file. S1 File Data from individual laboratories. (DOCX) Click here for additional data file. S2 File Calibration procedure. (DOCX) Click here for additional data file. S3 File Additional information for the analysis. (DOCX) Click here for additional data file. S4 File Pilot studies. (DOCX) Click here for additional data file. S5 File Descriptive statistics of force data from both laboratories. (DOCX) Click here for additional data file. We would like to thank Andre’ Dähne for his graphical support. We also would like to thank Jan Kaminski, Dominik von Hertlein, Tobias Nolte, and David Voß for their contributions in data collection. 10.1371/journal.pone.0288038.r001 Decision Letter 0 Tessari Alessia Academic Editor © 2023 Alessia Tessari 2023 Alessia Tessari https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version0 7 Feb 2023 PONE-D-22-30799How to not induce SNAs: the insufficiency of covert directional movementPLOS ONE Dear Dr. Kühne, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Mar 24 2023 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. 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Additional Editor Comments: Dear Dr Kühne, We have now received two reviews. After reading the paper myself, I found it interesting for PLOS ONE, but before considering it for publication, you have to arrange the manuscript following the reviewers' requests. I look forward to receiving your revised version of the manuscript. Sincerely, Alessia Tessari [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: Partly ********** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: Yes Reviewer #2: Yes ********** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: No Reviewer #2: No ********** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ********** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: The present paper reports a study in which participants are asked to maintain a constant force with a single hand on a disk plate in one of four directions: upwards, downwards, leftwards or rightwards, while they carry out a random number generation task and an arithmetic verification task. No movement was produced, only isometric force. In the number generation task they measured the magnitud of the generated numbers as a function of instructed force direction, and they also measured the magnitud of the force generated as a function of the magnitud of the number generated. In the arithmetic verification task they measured the magnitud of force produced as a function of the magnitud of each operand, the type of operation (addition vs. subtraction), and the magnitud of the result. The goal was to detect Space-Number Associations (SNAs), such that left or up space is associated to small numbers and right or low space is associated to large numbers. In all cases, they failed to find any significant SNA. Bayesian analyses showed that there was clear support for the null hypothesis. The authors interpret that it is the lack of outer directional movement what is causing the absence of SNAs. The paper is very well written, the topic is relevant, the methods are sound, the data are clear and clearly support the null hypothesis and the interpretation follows from the context and the data. All in all, I have very few concerns or suggestions for improvement, and I do recommend its publication after improving on those points. My questions and suggestions are the following: - PLoS ONE requests that the data be openly available and the authors claim that they will be uploaded to an open repository after the acceptance of the manuscript. However, in the Data Availability Statement at the end of the manuscript the authors say that data are “available upon reasonable request from the corresponding author”. In my opinion, this goes against the open data policy of the journal. In my experience, asking authors for data of published studies succeeds in very rare occasions. The data should be deposited in a open repository such as OSF, where they can be directly accessed in the future without any intervention from the authors. - I don’t understand why the hypotheses on the random number generation task are posed as confirmatory and those on the arithmetic verification task are posed as exploratory. Lacking a preregistration of the study (as it is the case), all of them should be considered exploratory. - I also don’t understand why, if the authors have examined the interaction between number magnitud and the four force directions, they assert that their design includes the factor force direction with two levels “(left/right, up/down)”. If they carried out independent analyses for the two levels within each axis (horizontal and vertical), they should say that there were two independent designs. If they included all four directions into a single design, the factor has four levels. - The preprocessing and analysis of force data is complex and implies a high number of choices. As a suggestion for future research, I would be more convinced if every choice would be pre-registered. - I would like the authors to justify their choice of prior in the Bayesian analyses (line 506). - My MAIN SUGGESTION for improvement is to include in the Discussion the recent study by Miklashevsky et al. (2022), who found SNAs using grip force and without any overt movements. This study seems to clash with their conclusion that it is the lack of overt movements what is responsible from the absence of SNAs, and should be taken into consideration and provide some speculation as to why the contrast between the studies. - Is citation number 73 correct? (line 655). The cited study does not measure any forces. - In the paragraph starting in line 667, the authors entertain a possible explanation of the absence of effects in the present study that strikes me as undermining the whole study: they say that perhaps the kind of force that they measured (pushing with two fingers in a given direction) may not be sensitive to high level cognitive processes in the way that force in a precision grip has been already shown to be. If this is so, the authors are saying that they are using a measure that is invalid as an index of the cognitive processes of interest. They should strive to rule out this possibility, either with good arguments or by carrying out an experimental construct validation of the measure. If the authors finally accept that they cannot be sure that the measure is a valid one, then they should change their conclusions: the SNAs are absent either because there is no overt directional movement or because the measure is not valid. References: Miklashevsky, A., Fischer, M. H., & Lindemann, O. (2022). Spatial-numerical associations without a motor response? Grip force says ‘Yes’. Acta Psychologica, 231, 103791. https://doi.org/10.1016/j.actpsy.2022.103791 Reviewer #2: There are several unique merits to the paper, not the least of which is its creative methodological approach. The paper is mostly clear (with one exception, see below), and I see no problems with the design or analysis. My main concern(s) stem(s) from the conclusions that are reached based on the results. Below, I elaborate on what I see as the central concern(s). 1. It seems that the central argument relies on the interpretation of a null effect. Of course, there are some instances where null effects are valuable, and the statistics were appropriate given the sort of conclusion the authors wished to make. Nevertheless, I had trouble making sense of this particular null effect, because I have no standard upon which to evaluate it. Are there cases where we *should* expect this method to reveal SNAs? If I'm understanding correctly, this approach has only been used successfully in the study of language. I'm not an expert on those studies or this methodology, but it seems that even the authors agree their approach is somewhat different from what is used in prior work. Therefore, I would be uncomfortable concluding that covert directional movements *cannot* induce SNAs if we do not have strong reason to believe that this design is sufficiently sensitive to detect covert directional movements to the relevant extent. In other words: I feel that we'd need a comparable positive effect in order to evaluate the lack of an effect here. Is there anyway to demonstrate that this method *is* sensitive enough to detect some effect that we'd have strong reason to predict? 2. Is there something circular about the central argument? The authors argue that explicit movement is required to induce SNAs. But isn't it also possible that this measure is just not a measure of SNAs? In other words: Am I correct to understand that there is no independent evidence of SNAs in this task? If that's true, is it not possible that participants simply exhibited no SNAs, but that, if they had, this measure may have correctly detected them? In other words, there are two possible conclusions one could draw from the null effect. One conclusion is that subjects are not exhibiting SNAs. But another equally valid conclusion is that the subjects are exhibiting SNAs, but that this measure isn't sufficient to detect it. Perhaps, for instance, eye movements could have revealed SNAs even if these motor movements did not. 3. In the arithmetic task, how are the equations appearing? The language in the Methods is not clear. It says the stimuli were presented "sequentially", but where? Were these stimuli arranged spatially in any way, or did they all appear in the same location one after the other? If the latter, what was the timing of that like? Ultimately, I think it could be valuable to have this result published. Perhaps others would be inspired to take a similar approach, which could prove valuable. However, I feel that the results should not be published without, at the very least, clearly qualifying the results. I'm just not sure that we could take from a single sort of null effect, in a very untested paradigm, that anything certainly is or certainly is not true. Then again, the authors' may have some very strong argument as to why they think these results do merit the conclusions that they made. Or it may be that I am misunderstanding something about the study. If either of these things were true, I would be able to more enthusiastically recommend publication. Other: 1. I found it surprisingly difficult to understand the key manipulation. I spent much time wondering whether the experiment involved applying pressure to the subjects' hands, or whether the authors were measuring covert movements -- or, both. I'm still not sure. I think this stems from the ambiguity in phrases like "We studied continuous isometric forces..." It just wasn't clear to me whether the 'forces' referred to a force that was being applied by the device, or by the participant. I think this problem can be easily solved by just revisiting this language through the manuscript, to ensure that it would be clear to a reader that has no prior knowledge of the procedure or design. 2. I had trouble understanding what counts as 'overt' directional movement vs. 'covert' directional movement. In the classic SNARC design, is there overt directional movement? Subjects are just keeping their hands in one place. Why would this be overt? And here, if forces are being applied in one direction or the other, isn't that quite 'overt'? It's an explicit manipulation that the subjects would ostensibly be aware of. There are several other border cases that I'm just not sure about. For instance, there are a few papers from Stella Lourenco's lab that seem to fall somewhere in between: Holmes, K. J., Ayzenberg, V., & Lourenco, S. F. (2016). Gamble on gaze: Eye movements reflect the numerical value of blackjack hands. Psychonomic Bulletin & Review, 23, 1974-1981. Aulet, L. S., Yousif, S. R., & Lourenco, S. F. (2021). Spatial–numerical associations from a novel paradigm support the mental number line account. Quarterly Journal of Experimental Psychology, 74(10), 1829-1840. Aulet, L. S., & Lourenco, S. F. (2018). The developing mental number line: Does its directionality relate to 5-to 7-year-old children’s mathematical abilities?. Frontiers in Psychology, 9, 1142. In the first, eye movements alone were used as a measure of SNAs. In the second and third, placement errors in a spatial task were used as a measure of SNAs. The latter certainly required movement, but of a very different kind that in many SNARC paradigms. The former obviously involves a kind of movement, but surely the present study also involves eye movements -- so shouldn't SNAs be induced by that, regardless of the other manipulation? I don't have any strong opinions about this, except that, having read the paper several times, I'm not sure what counts as covert vs. overt. 3. Related to both points above: If the manipulation involves instructing participants to apply forces in different directions, how is that not overt? Sorry to be difficult about all of this; I find myself very confused. Minor: 1. The first sentence of the abstract doesn't stand on its own. What does it mean that people "respond faster to the left", for instance? Respond to what? ********** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: Yes: Julio Santiago Reviewer #2: No ********** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. 10.1371/journal.pone.0288038.r002 Author response to Decision Letter 0 Submission Version1 22 Mar 2023 Reviewer #1: 1. PLoS ONE requests that the data be openly available and the authors claim that they will be uploaded to an open repository after the acceptance of the manuscript. However, in the Data Availability Statement at the end of the manuscript, the authors say that data are “available upon reasonable request from the corresponding author”. In my opinion, this goes against the open data policy of the journal. In my experience, asking authors for data from published studies succeeds on very rare occasions. Reply: Thank you for this reasonable suggestion. We have now deposited our data and all the analyses scripts in an open repository (OSF, https://osf.io/v7dyj/) and corrected our Data Availability Statement correspondingly (lines 887-888). 2. I don’t understand why the hypotheses on the random number generation task are posed as confirmatory and those on the arithmetic verification task are posed as exploratory. Lacking preregistration of the study (as is the case), all of them should be considered exploratory. Reply: Thank you for this comment. Indeed, we did not preregister any of the hypotheses of this experiment. We explicitly state this now in lines 325-327: “The following hypotheses were not preregistered but formulated a priori and derived from theory and previous empirical work.”. Additionally, we removed Hypothesis 3 and reframed it as a research question exploring potential effects. In lines 340-343 it now reads “As an additional research question of the experiment, we aimed to explore SNAs on a temporal continuum during mental arithmetic, thus testing how activation of number concepts affects force production during mental arithmetic.”. We have also explicitly stated that the analysis method also was not preregistered but generally followed the guidelines proposed by Nazir et al. (2017). In lines 441-442 it now reads “The following method was not preregistered but closely followed the recommendations of Nazir et al. [61, for details see supplementary file S5]. ”. As a general comment, we want to clarify that our RNG hypotheses are both strongly derived from and predicted by empirical and theoretical work. Therefore, we had clear hypotheses we put to test. To make it clearer, we have highlighted the bi-directionality of concept-motor interaction before introducing the first two hypotheses. In line 324-325 that now reads “This approach allows to test the bi-directionality of the concept-motor interactions.”. Additionally, half sentences were added to H1 and H2 that specify the concept-motor directions and now reads as (lines 328-332) “H1. We hypothesize that covert directional movement activates spatial codes that affect the production of random numbers, thus testing how (covert) movement affects the conceptual representation of numbers. That is, participants are expected to produce more small numbers when pressing in the direction left/down and more large numbers when pressing in the direction right/up [51,54].” And (lines 333-339) “H2. We hypothesize that generating numbers by itself affects motor activation [53], thus testing how conceptual representation of numbers affects force production during covert movement. Generating numbers is assumed to activate within-magnitude associations [ATOM, 49,50], which in turn, are hypothesized to modulate the amplitude of applied isometric force. Hereby, generating smaller numbers should produce smaller force while generating larger numbers should produce larger force when simultaneously pressing against the surface of a force sensor [49,50].”. Similarly, the research question (original H3) is also based on the theoretical predictions of A Theory of Magnitude (Walsh, 2003; 2015). However, the important difference is that the timing of the effect is theoretically unclear and the empirical evidence is rather mixed. Therefore, we did not specify it. We have improved the section that now reads (lines 340-353): “As an additional research question of the experiment, we aimed to explore SNAs on a temporal continuum during mental arithmetic, thus testing how activation of number concepts affects force production during mental arithmetic. The rationale is similar to that of the target detection paradigm (see Fig 1, example 6). However, we removed spatial information from the presentation and the response space. Therefore, instead of providing spatial information in the form of targets to probe these SNAs, we again relied on central stimuli presentation and covert directional force. In this task, spatial associations of numbers could become activated in relation to the arithmetic procedure; namely either during the first operand, or the operation sign, or the second operand, or during the answer. Similarly to the reasoning in H2, number meaning related to the task should be expressed by the participants’ spontaneous force production, with smaller numbers generating smaller force and large numbers generating larger force [49,50].”. Further, we fully agree that pre-registering is a better scientific practice. We have not preregistered this experiment because it is the first in a larger project that helps us to accumulate knowledge about our experimental design and method. Further experiments within this project will be preregistered. 3. I also don’t understand why, if the authors have examined the interaction between the number magnitude and the four force directions, they assert that their design includes the factor force direction with two levels “(left/right, up/down)”. If they carried out independent analyses for the two levels within each axis (horizontal and vertical), they should say that there were two independent designs. If they included all four directions into a single design, the factor has four levels. Reply: Thank you for this valuable notice. We have changed it accordingly and specified that our design presupposed two independent axes (horizontal and vertical). Changes to this effect appear in the General Method section in lines 388-389: “A customized home-made wooden/metal box was built to enable covert directional movements (i.e., participants’ exerting isometric forces on a fixed force sensor surface) along the vertical and the horizontal axes (two levels each: left/right on a horizontal axis or up/down on the vertical axis).” Also edited in lines 416-418:” We have utilized a within-subject design with one within-subject factor being the force direction (two levels each: left/right on a horizontal axis or up/down on the vertical axis).”. We also made in clearer in the Introduction why we expect the horizontal and the vertical axes to be independent of each other. In lines 100-104 it now reads: ”Vertical associations instead reflect other sensory and motor experiences, such as the growth of piles during object accumulation or the rise of water in a container, which inspired the linguistic metaphors mentioned above. Together, the embodied cognition approach predicts SNAs on the horizontal and the vertical axes based on different and independent mechanisms. ”. We also extended this argument in lines 240-243 that now reads “One possible explanation for why vertical SNAs remained while horizontal SNAs disappeared could be the hierarchical nature of body-related knowledge representations involved in the task, according to which different mechanisms induce different types of SNAs, referred to as grounded, embodied, or situated [28]. ”. 4. The preprocessing and analysis of force data is complex and implies a high number of choices. As a suggestion for future research, I would be more convinced if every choice would be pre-registered. Reply: Thank you for the comment. We agree that in future research we will pre-register the method. To clarify, we have rigorously followed the general guidelines of an existing methodological paper describing the same method we have used (Nazir et al., 2017). We have now made it clearer by adding the reference in the “pre-processing force data” section (lines 542-543). Now it says “Pre-processing of force data followed the general guidelines proposed by Nazir et al. [61] and is documented in the S5 file.” Additionally, we made changes to the Supplemental File (S5, lines 13-16) to clarify that there was one parameter (amplitude range) that we adjusted. We adjusted amplitude range from ±200 to ±400 mN because we utilized longer trials than Nazir et al. [1] that naturally have higher force fluctuation. Specifically, Nazir et al. [1] utilized 1 second trials while we had 2 seconds (RNG) and 2.5 seconds (SDA).” References: Nazir, T. A., Hrycyk, L., Moreau, Q., Frak, V., Cheylus, A., Ott, L., Lindemann, O., Fischer, M. H., Paulignan, Y., & Delevoye-Turrell, Y. (2017). A simple technique to study embodied language processes: the grip force sensor. Behavior Research Methods, 49(1), 61–73. https://doi.org/10.3758/s13428-015-0696-7 5. I would like the authors to justify their choice of prior in the Bayesian analyses (line 506). Reply: Thank you for your suggestion. We have now explicitly justified the choice of the Cauchy prior in the Bayesian analyses (lines 562-565). It now reads: “The default width parameter of .707 was used in all analyses since we did not have strong assumptions about the distribution. Moreover, it was shown that the prior width has little impact on the Bayes factor [70]. All analyses were done within each independent axis (horizontal or vertical).”. References: van Ravenzwaaij, D., & Wagenmakers, E-J. (2022). Advantages masquerading as "issues" in Bayesian hypothesis testing: A commentary on Tendeiro and Kiers (2019). Psychological Methods, 27(3), 451-465. https://doi.org/10.1037/met0000415 6. My MAIN SUGGESTION for improvement is to include in the Discussion the recent study by Miklashevsky et al. (2022), who found SNAs using grip force and without any overt movements. This study seems to clash with their conclusion that it is the lack of overt movements that are responsible for the absence of SNAs, and should be taken into consideration and provide some speculation as to why the contrast between the studies. Reply: Thank you for the valuable suggestion. Indeed, that paper is highly relevant. We have fully integrated it in our Introduction and Discussion and it enormously helped extending our argument and the limits of the current study and also the validity of our force measure. It also helped us to change and adapt our conclusion on several occasions throughout the manuscript. The main changes in the introduction are in lines 180-189 which now reads: “Very recently, a new study challenged the importance of active responses in the response space. Miklashevsky et al. [40] relied on a paradigm of passive and continuous isometric force readings during numerical tasks. Participants produced more force in the left hand while processing smaller numbers and more force in the right hand while processing larger numbers. These results suggest that active responses might not be required to induce SNAs. However, the authors did not control for spatial effects in their design. Specifically, in the response space, the authors utilized two sensors that were held in the left and in the right hand, thus being laterally displaced. Therefore, spatial information in the response space was apparently sufficient to elicit SNAs in their study.“. In the Discussion, in lines 785-811 it now reads: “Second, as stated in the Introduction, Miklashevsky et al. [40] found SNAs in numerical tasks during passive isometric force recordings utilizing the same method as in our study. The authors employed an experimental design utilizing two lateralized force sensors in a task that did not require active responses and instead recorded passive forces applied to the sensors. The authors found that smaller numbers induced more force in the left hand and larger numbers induced more force in the right hand. Such a finding is particularly intriguing because SNAs were produced without an active response and therefore not relying on an overt movement through space. This finding suggests that overt movements through space are not necessary in order to elicit SNAs. Instead, spatial information about effector positions by itself might be the sole contributor to SNAs formation [cf. 78]. Indeed, the authors utilized two spatially aligned sensors and did not control for spatial information in the response space. Additionally, keep in mind that the authors employed two different tasks compared to our study. The authors themselves suggested that the task itself is critical in finding SNAs as they report weaker SNAs in their surface numerical decision task (“decide if this a number or a letter”) than in the classical magnitude classification task (“is this number larger or smaller than 5”). Therefore, it remains to be seen how their finding will translate to other experimental tasks such as random number generation and mental arithmetic as was the case in the current study. Together, Miklashevky et al.’s [40] study and our own can be interpreted as suggesting that active responses (and overt movements) might not be per se necessary to elicit SNAs, while spatial information might play a critical role. Future experiments could test more systematically if SNAs can exist in an experimental design space that excludes both, spatial information in the presentation and response space, as well as overt movements in the response space. Therefore, the current study contributes to the overarching discussion on design space and effector specification [79] necessary to elicit SNAs.”. Other adaptations of the conclusion: 1. Lines 31-33 “Here we argue that overt movements in space contribute to the formation of spatial-numerical associations (SNAs).”. 2. Lines 36-37 we have deleted the final sentence. It no longer reads: “We conclude that overt movements through space enable SNAs.”. 3. Lines 291-293:” Generally, spatial information in the design (e.g., movements through space or responses in space) seems to be essential when investigating SNAs, both as a response producing SNAs and to activate SNAs.”. 4. Lines 300-302:” Considering all the presented evidence, we can conclude that spatial information is omnipresent in many experimental paradigms that study SNAs.”. 5. Lines 697-698: “Therefore, spatial information (e.g., movement through space) seems to be deeply rooted into SNAs assessment. To investigate the role of movement in space, future experiments could contrast designs utilizing movement in space against designs that utilize static and central versus static and lateral responses [for static and lateral responses see 40].”. 6. Lines 866-867: “We conclude that lateralized spatial information in the design space plays a crucial role in eliciting SNAs.”. References: Almaatouq, A., Griffiths, T. L., Suchow, J. W., Whiting, M. E., Evans, J., & Watts, D. J. (2022). Beyond Playing 20 Questions with Nature: Integrative Experiment Design in the Social and Behavioral Sciences. Behavioral and Brain Sciences, 1-55. https://doi.org/10.1017/S0140525X22002874 Miklashevsky, A., Fischer, M. H., & Lindemann, O. (2022). Spatial-numerical associations without a motor response? Grip force says ‘Yes’. Acta Psychologica, 231, 103791. https://doi.org/10.1016/j.actpsy.2022.103791 Rizzolatti, G., Riggio, L., & Sheliga, B. (1994). Space and Selective Attention. In Attention and Performance XV. https://doi.org/10.7551/mitpress/1478.003.0016 7. Is citation number 73 correct? (line 655). The cited study does not measure any forces. Reply: Thank you for the comment. Indeed, the authors do not report any force measurements. We have now clarified that the study (now not citation number 73 but 76) only manipulated the movement direction and isometric force but did not measure isometric force. In lines 771-779 it now reads: “Indeed, as previously shown, an isometric force manipulation (but not measurement) during a lateral movement was successfully applied to induce spatial biases across the horizontal axis in a mathematical problem-solving task [Experiment 2 in 76]. Participants generated more addition solutions following a right-sided movement and more subtraction solutions following a left-sided movement. Notably, these authors relied on lateral movements along the surface of a touch-screen, meaning that participants pressed the touch-screen while conducting a lateral movement.”. References: Werner, K., Raab, M., & Fischer, M. H. (2019). Moving arms: the effects of sensorimotor information on the problem-solving process. Thinking & Reasoning, 25(2), 171–191. https://doi.org/10.1080/13546783.2018.1494630 8. In the paragraph starting in line 667, the authors entertain a possible explanation of the absence of effects in the present study that strikes me as undermining the whole study: they say that perhaps the kind of force that they measured (pushing with two fingers in a given direction) may not be sensitive to high-level cognitive processes in the way that force in a precision grip has been already shown to be. If this is so, the authors are saying that they are using a measure that is invalid as an index of the cognitive processes of interest. They should strive to rule out this possibility, either with good arguments or by carrying out an experimental construct validation of the measure. If the authors finally accept that they cannot be sure that the measure is a valid one, then they should change their conclusions: the SNAs are absent either because there is no overt directional movement or because the measure is not valid. Reply: Thank you for your valuable feedback. We think that we have a valid measurement based on previous studies (Aravena et al., 2012; Aravena et al., 2014, Frak et al., 2010; Nazir et al., 2017) and recent evidence from the Miklashevsky et al. (2022) study that you suggested. In this study, it cannot be fully ruled out that this exact measure might have not been sensitive enough to pinpoint the SNAs in force data. However, we did show evidence against SNAs in our design in the RNG task with an independent measure: In all directions, no interaction between the small number bias and the direction of the force application was detected. Namely, our participants produced equally more small numbers in all directions, which is a classical signature of RNG. Due to the bidirectional nature of the embodiment perspective, such an interaction was to be expected in the case of reliable SNAs in our design. We have added an argument in lines 834-837 that now reads: “At the very least, isometric force served as a manipulation in any direction and ensured that participants deployed directional force [see limitations of Experiment 2 in 76]. Yet, in our independent assessment of H1, we do not find any SNAs in any direction during the RNG task.”. Additionally, the discussion of Miklashevskys et al. (2022) paper (your point 6) should have addressed this reviewer’s concerns about the validity of the method and the implications for SNA research. In lines 318-320 we directly state that we utilize the same method as Miklashevsky et al. (2022). It now reads: ” Hereby, we utilized the same isometric force paradigm as Miklashevsky et al. [40]. Therefore, our isometric force paradigm does not require the execution of a lateralized response after processing a number.”. References: Aravena, P., Courson, M., Frak, V., Cheylus, A., Paulignan, Y., Deprez, V., & Nazir, T. A. (2014). Action relevance in linguistic context drives word-induced motor activity. Frontiers in Human Neuroscience, 8(1), 163. https://doi.org/10.3389/fnhum.2014.00163 Aravena, P., Delevoye-Turrell, Y., Deprez, V., Cheylus, A., Paulignan, Y., Frak, V., & Nazir, T. (2012). Grip Force Reveals the Context Sensitivity of Language-Induced Motor Activity during “Action Words” Processing: Evidence from Sentential Negation. PLoS ONE, 7(12), e50287. https://doi.org/10.1371/journal.pone.0050287 Frak, V., Nazir, T., Goyette, M., Cohen, H., & Jeannerod, M. (2010). Grip Force Is Part of the Semantic Representation of Manual Action Verbs. PLoS ONE, 5(3), e9728. https://doi.org/10.1371/journal.pone.0009728 Miklashevsky, A., Fischer, M. H., & Lindemann, O. (2022). Spatial-numerical associations without a motor response? Grip force says ‘Yes’. Acta Psychologica, 231, 103791. https://doi.org/10.1016/j.actpsy.2022.103791 Nazir, T. A., Hrycyk, L., Moreau, Q., Frak, V., Cheylus, A., Ott, L., Lindemann, O., Fischer, M. H., Paulignan, Y., & Delevoye-Turrell, Y. (2017). A simple technique to study embodied language processes: the grip force sensor. Behavior Research Methods, 49(1), 61–73. https://doi.org/10.3758/s13428-015-0696-7 Reviewer #2: 1. It seems that the central argument relies on the interpretation of a null effect. Of course, there are some instances where null effects are valuable, and the statistics were appropriate given the sort of conclusion the authors wished to make. Nevertheless, I had trouble making sense of this particular null effect, because I have no standard upon which to evaluate it. Are there cases where we *should* expect this method to reveal SNAs? If I'm understanding correctly, this approach has only been used successfully in the study of language. I'm not an expert on those studies or this methodology, but it seems that even the authors agree their approach is somewhat different from what is used in prior work. Therefore, I would be uncomfortable concluding that covert directional movements *cannot* induce SNAs if we do not have strong reason to believe that this design is sufficiently sensitive to detect covert directional movements to the relevant extent. In other words: I feel that we'd need a comparable positive effect to evaluate the lack of an effect here. Is there any way to demonstrate that this method *is* sensitive enough to detect some effect that we'd have strong reason to predict? Reply: Thank you for pointing this out. Indeed, we have argued that our measure is valid based on the grip force studies from linguistics. Now we have additional evidence for its validity based on a study published right after the submission of this manuscript, utilizing the same method (Miklashevsky et al., 2022). Miklashevsky et al. (2022) have utilized the same force measures as in the current study and demonstrated that it can be employed for studying embodied numerical processing. The authors found SNAs in their study, namely smaller numbers inducing more force in the left hand and larger numbers inducing more force in the right hand. This finding shows that SNAs can be found outside overt movements and with continuously employed isometric force. This has two major implications for our current study. First, the finding helps us extend the methodology to numerical cognition. Second, and at the same time, it challenges, as you rightfully point out, our conclusion that overt movements are necessary to induce SNAs. We have now integrated this new study in our revised Introduction and Discussion. This new study has enormously helped extending our argument and the limits of the current study and also the validity of our force measure. It also helped to change and adopt our conclusion on several occasions through the manuscript. The main changes in the Introduction are in lines 180-189 which now reads: “Very recently, a new study challenged the importance of active responses in the response space. Miklashevsky et al. [40] relied on a paradigm of passive and continuous isometric force readings during numerical tasks. Participants produced more force in the left hand while processing smaller numbers and more force in the right hand while processing larger numbers. These results suggest that active responses might not be required to induce SNAs. However, the authors did not control for spatial effects in their design. Specifically, in the response space, the authors utilized two sensors that were held in the left and in the right hand, thus being laterally displaced. Therefore, spatial information in the response space was apparently sufficient to elicit SNAs in their study. “ Further, in lines 318-320 we directly state that we utilize the same method as Miklashevsky et al. (2022) to show that the method is valid. It now reads: ” Hereby, we utilized the same isometric force paradigm as Miklashevsky et al. [40]. Therefore, our isometric force paradigm does not require the execution of a lateralized response after processing a number.”. In the Discussion the changes are in lines 785-811 which now reads: “Second, as stated in the Introduction, Miklashevsky et al. [40] found SNAs in numerical tasks during passive isometric force recordings utilizing the same method as in our study. The authors employed an experimental design utilizing two lateralized force sensors in a task that did not require active responses and instead recorded passive forces applied to the sensors. The authors found that smaller numbers induced more force in the left hand and larger numbers induced more force in the right hand. Such a finding is particularly intriguing because SNAs were produced without an active response and therefore not relying on an overt movement through space. This finding suggests that overt movements through space are not necessary in order to elicit SNAs. Instead, spatial information about effector positions by itself might be the sole contributor to SNAs formation [cf. 78]. Indeed, the authors utilized two spatially aligned sensors and did not control for spatial information in the response space. Additionally, keep in mind that the authors employed two different tasks compared to our study. The authors themselves suggested that the task itself is critical in finding SNAs as they report weaker SNAs in their surface numerical decision task (“decide if this a number or a letter”) than in the classical magnitude classification task (“is this number larger or smaller than 5”). Therefore, it remains to be seen how their finding will translate to other experimental tasks such as random number generation and mental arithmetic as was the case in the current study. Together, Miklashevky et al.’s [40] study and our own can be interpreted as suggesting that active responses (and overt movements) might not be per se necessary to elicit SNAs, while spatial information might play a critical role. Future experiments could test more systematically if SNAs can exist in an experimental design space that excludes both, spatial information in the presentation and response space, as well as overt movements in the response space. Therefore, the current study contributes to the overarching discussion on design space and effector specification [79] necessary to elicit SNAs.”. Additionally, we have further addressed your point by specifying isometric force in a different example in lines 771-779. Now it reads: “Indeed, as previously shown, an isometric force manipulation (but not measurement) during a lateral movement was successfully applied to induce spatial biases across the horizontal axis in a mathematical problem-solving task [Experiment 2 in 76]. Participants generated more addition solutions following a right-sided movement and more subtraction solutions following a left-sided movement. Notably, these authors relied on lateral movements along the surface of a touch-screen, meaning that participants pressed the touch-screen while conducting a lateral movement.”. This finding is also a positive finding that an isometric force manipulation (but not measurement) should be sufficient to induce SNAs. However, in our study this was not the case as our directional isometric force manipulation did not induce SNAs in the RNG task (see lines 668-676). We have further developed the argument in lines 833-836 that now reads: “At the very least, isometric force served as a manipulation in any direction and ensured that participants deployed directional force [see limitations of Experiment 2 in 76]. Yet, in our independent assessment of H1, we do not find any SNAs in any direction during the RNG task.”. Other adaptations of the conclusion: 1. Lines 31-33 “Here we argue that overt movements in space contribute to the formation of spatial-numerical associations (SNAs).”. 2. Lines 36-37 we have deleted the final sentence. It no longer reads: “We conclude that overt movements through space enable SNAs.”. 3. Lines 291-293:” Generally, spatial information in the design (e.g., movements through space or responses in space) seems to be essential when investigating SNAs, both as a response producing SNAs and to activate SNAs.”. 4. Lines 300-302:” Considering all the presented evidence, we can conclude that spatial information is omnipresent in many experimental paradigms that study SNAs.”. 5. Lines 697-698: “Therefore, spatial information (e.g., movement through space) seems to be deeply rooted into SNAs assessment. To investigate the role of movement in space, future experiments could contrast designs utilizing movement in space against designs that utilize static and central versus static and lateral responses [for static and lateral responses see 40].”. 6. Lines 866-867: “We conclude that lateralized spatial information in the design space plays a crucial role in eliciting SNAs.”. References: Almaatouq, A., Griffiths, T. L., Suchow, J. W., Whiting, M. E., Evans, J., & Watts, D. J. (2022). Beyond Playing 20 Questions with Nature: Integrative Experiment Design in the Social and Behavioral Sciences. Behavioral and Brain Sciences, 1-55. https://doi.org/10.1017/S0140525X22002874 Werner, K., Raab, M., & Fischer, M. H. (2019). Moving arms: the effects of sensorimotor information on the problem-solving process. Thinking & Reasoning, 25(2), 171–191. https://doi.org/10.1080/13546783.2018.1494630 Miklashevsky, A., Fischer, M. H., & Lindemann, O. (2022). Spatial-numerical associations without a motor response? Grip force says ‘Yes’. Acta Psychologica, 231, 103791. https://doi.org/10.1016/j.actpsy.2022.103791 Rizzolatti, G., Riggio, L., & Sheliga, B. (1994). Space and Selective Attention. In Attention and Performance XV. https://doi.org/10.7551/mitpress/1478.003.0016 2. Is there something circular about the central argument? The authors argue that explicit movement is required to induce SNAs. But isn't it also possible that this measure is just not a measure of SNAs? In other words: Am I correct to understand that there is no independent evidence of SNAs in this task? If that's true, is it not possible that participants simply exhibited no SNAs, but that, if they had, this measure may have correctly detected them? In other words, there are two possible conclusions one could draw from the null effect. One conclusion is that subjects are not exhibiting SNAs. But another equally valid conclusion is that the subjects are exhibiting SNAs, but that this measure isn't sufficient to detect it. Perhaps, for instance, eye movements could have revealed SNAs even if these motor movements did not. Reply: Thank you for the valid comment and the concern. We have thoroughly addressed this issue in your point 1 and replying to the other reviewer. For clarification, there is independent evidence in the RNG task that no SNAs were activated in the study (see lines 668-676). Additionally, we have extended the argument. In lines 833-836 it now reads: “At the very least, isometric force served as a manipulation in any direction and ensured that participants deployed directional force [see limitations of Experiment 2 in 76]. Yet, in our independent measure of H1, we do not find any SNAs in any direction during the RNG task.”. References: Werner, K., Raab, M., & Fischer, M. H. (2019). Moving arms: the effects of sensorimotor information on the problem-solving process. Thinking & Reasoning, 25(2), 171–191. https://doi.org/10.1080/13546783.2018.1494630 3. In the arithmetic task, how are the equations appearing? The language in the Methods is not clear. It says the stimuli were presented "sequentially", but where? Were these stimuli arranged spatially in any way, or did they all appear in the same location one after the other? If the latter, what was the timing of that like? Ultimately, I think it could be valuable to have this result published. Perhaps others would be inspired to take a similar approach, which could prove valuable. However, I feel that the results should not be published without, at the very least, clearly qualifying the results. I'm just not sure that we could take from a single sort of null effect, in a very untested paradigm, that anything certainly is or certainly is not true. Then again, the authors may have some very strong arguments as to why they think these results do merit the conclusions that they made. Or it may be that I am misunderstanding something about the study. If either of these things were true, I would be able to more enthusiastically recommend publication. Reply Part a: Thank you for your suggestion. We have elaborated on the spatial information in the introduction in lines 344-347 that now reads: “However, we removed spatial information from the presentation and the response space. Therefore, instead of providing spatial information in the form of targets to probe these SNAs, we again relied on central stimuli presentation and covert directional force.”. Further, we have adjusted in the General Method section (lines 468-474) both, spatial and temporal information that now reads:” Participants were asked to watch mathematical equations (from here on, referred to as trials) in which individual stimuli (operand 1, operation sign, operand 2, equal sign, result) were presented sequentially after each other centrally on the computer screen. Some of the proposed results were wrong. In case of a wrong result participant had to reject the result by saying “No“. Each stimulus was shown for 500 ms, then the next stimulus appeared. There was an inter-trial interval of 1500 ms between the trials (Fig 3).”. Reply Part b (“Ultimately…”): Thank you for this comment. As you will read, we have implemented your comments in our manuscript and feel that it has now substantially improved. In your point 2 we have discussed why we think our method is valid in this current study and how our study contributes to the field of numerical cognition and the boundary conditions of SNAs. On the general level, point 1 that you have raised is especially important in understanding how our results inform the field of numerical cognition. For this, we have implemented Miklashevsky et al. (2022) study in our Introduction and Discussion and contrasted their positive results to ours. We feel that with the integration of that study we found a strong argument on how to interpret boundary conditions of SNAs. We also suggest future directions on how to obtain more evidence on the relevance of overt movements and spatial information in the design space. On the general level, our manuscript and the findings strongly contribute to the overarching discussion on how to specify, rather than quantify psychological effects ensuring replicability and generalizability of experimental phenomena (Almaatouq et al., 2022). In lines 804-811 in now reads: “Together, Miklashevky et al.’s [40] study and our own can be interpreted as suggesting that active responses (and overt movements) might not be per se necessary to elicit SNAs, while spatial information might play a critical role. Future experiments could test more systematically if SNAs can exist in an experimental design space that excludes both, spatial information in the presentation and response space, as well as overt movements in the response space. Therefore, the current study contributes to the overarching discussion on design space and effector specification [79] necessary to elicit SNAs”. References: Almaatouq A, Griffiths TL, Suchow JW, Whiting ME, Evans J, Watts DJ. Beyond Playing 20 Questions with Nature: Integrative Experiment Design in the Social and Behavioral Sciences. Behav Brain Sci. 2022; 1–55. doi:10.1017/S0140525X22002874 Miklashevsky, A., Fischer, M. H., & Lindemann, O. (2022). Spatial-numerical associations without a motor response? Grip force says ‘Yes’. Acta Psychologica, 231, 103791. https://doi.org/10.1016/j.actpsy.2022.103791 Other: 1. I found it surprisingly difficult to understand the key manipulation. I spent much time wondering whether the experiment involved applying pressure to the subjects' hands, or whether the authors were measuring covert movements -- or, both. I'm still not sure. I think this stems from the ambiguity in phrases like "We studied continuous isometric forces..." It just wasn't clear to me whether the 'forces' referred to a force that was being applied by the device, or by the participant. I think this problem can be easily solved by just revisiting this language through the manuscript, to ensure that it would be clear to a reader that has no prior knowledge of the procedure or design. Reply: Thank you for the comment. We have written clarifying sentences in multiple sections. 1. Lines 312-318: “In the present experiment we centralized the presentation and the response space utilizing covert directional movement. We asked participants to push against a sensor surface, thus generating a covert movement. Here, we define a covert movement as an application of constant and continuous isometric force to a surface. Such a covert movement was exerted by the participant to a surface in a direction on the horizontal (left or right) and the vertical (up or down) axes during two tasks.”. 2. Lines 344-347: “However, we removed spatial information from the presentation and the response space. Therefore, instead of providing spatial information in the form of targets to probe these SNAs, we again relied on central stimuli presentation and covert directional force.”. 3. Lines 386-390: “A customized home-made wooden/metal box was built to enable covert directional movements (i.e., participants’ exerting isometric forces on a fixed force sensor surface) along the vertical and the horizontal axes (two levels each: left/right on a horizontal axis or up/down on the vertical axis).”. 2. I had trouble understanding what counts as 'overt' directional movement vs. 'covert' directional movement. In the classic SNARC design, is there overt directional movement? Subjects are just keeping their hands in one place. Why would this be overt? And here, if forces are being applied in one direction or the other, isn't that quite 'overt'? It's an explicit manipulation that the subjects would ostensibly be aware of. There are several other border cases that I'm just not sure about. For instance, there are a few papers from Stella Lourenco's lab that seem to fall somewhere in between: Holmes, K. J., Ayzenberg, V., & Lourenco, S. F. (2016). Gamble on gaze: Eye movements reflect the numerical value of blackjack hands. Psychonomic Bulletin & Review, 23, 1974-1981. Aulet, L. S., Yousif, S. R., & Lourenco, S. F. (2021). Spatial–numerical associations from a novel paradigm support the mental number line account. Quarterly Journal of Experimental Psychology, 74(10), 1829-1840. Aulet, L. S., & Lourenco, S. F. (2018). The developing mental number line: Does its directionality relate to 5-to 7-year-old children’s mathematical abilities?. Frontiers in Psychology, 9, 1142. In the first, eye movements alone were used as a measure of SNAs. In the second and third, placement errors in a spatial task were used as a measure of SNAs. The latter certainly required movement, but of a very different kind that in many SNARC paradigms. The former obviously involves a kind of movement, but surely the present study also involves eye movements -- so shouldn't SNAs be induced by that, regardless of the other manipulation? I don't have any strong opinions about this, except that, having read the paper several times, I'm not sure what counts as covert vs. overt. Reply: Thank you for raising this question and introducing this interesting literature. Under an overt movement, we understand a movement through space that has a starting point and an endpoint. This also includes movements of response keys. A covert movement we define a movement that has no spatial component. We expanded the definition in the manuscript. 1. Overt movement: Lines 286-290: “Hereby, an overt movement is defined as a movement through space that has a starting point and an endpoint. For instance, an overt movement can be a rhythmic head movement to the left/right/up/down [e.g., 51] or a simple button press in a reaction time paradigm representing a location-related action (Fig 1, examples 1 to 4).“ 2. Covert movement: Lines 312-318: “In the present experiment we centralized the presentation and the response space utilizing covert directional movement. We asked participants to push against a sensor surface, thus generating a covert movement. Here, we define a covert movement as an application of constant and continuous isometric force to a surface. Such a covert movement was exerted by the participant to a surface in a direction on the horizontal (left or right) and the vertical (up or down) axes during two tasks.”. With this definition, it is the case that the literature you suggest falls into the category of overt movements. As for Holmes et al. (2016), in our Fig 1 (examples 5 and 6, see also lines 174-179) we specifically address eye movements as overt movements. Also, eye movements should have played no role in our current study. For instance, the RNG task was performed with the eyes closed (lines 430-432). During the SDA task, stimuli were presented centrally and sequentially (lines 468-474). Therefore, eyes movements during the SDA task would have been task irrelevant (see also argument in the Discussion that only goal-directed behaviour might be detectable, lines 844-846). As for Aulet & Lourenco (2018) and Aulet et al. (2021), both designs have movements in the response space either by moving a mouse and clicking or moving the hand to touch a screen. Therefore, both would also fall into the category of an overt movement. 3. Related to both points above: If the manipulation involves instructing participants to apply forces in different directions, how is that not overt? Sorry to be difficult about all of this; I find myself very confused. Reply: Thank you for the valid comment on the necessity to provide thorough definitions. We tried to make it more accessible and addressed it in response to your comment 2 above. Minor: 1. The first sentence of the abstract doesn't stand on its own. What does it mean that people "respond faster to the left", for instance? Respond to what? Reply: Thank you for pointing out this unclarity. We have paraphrased the sentence to „ People respond faster to smaller numbers in their left space and to larger numbers in their right space.” (Lines 30-31) 10.1371/journal.pone.0288038.r003 Decision Letter 1 Tessari Alessia Academic Editor © 2023 Alessia Tessari 2023 Alessia Tessari https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version1 2 May 2023 PONE-D-22-30799R1How to not induce SNAs: the insufficiency of covert directional movementPLOS ONE Dear Dr. Kühne, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. 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However, one of the reviewers still asks for minor revisions. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #1: (No Response) Reviewer #2: (No Response) ********** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. 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Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ********** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: The manuscript has been improved in many ways. Still, I found a number of inadequacies and also have some suggestions for further improvement. I will mention them in the same order as they appear in the manuscript. - Title: I am not convinced by the label “covert movement”. In my opinion, a movement is covert if it is planned and mentally simulated, but without it reaching the muscles. In this study the planned action does reach the muscles, and the participant does not plan to perform any movement, just a permanent pressure. It would be better to use a different label. “Isometric force” maybe a good alternative, although when I first read it I had difficulty to parse it. Maybe “static directional pressure” or some other option. - line 54: “produces” is not adequate. Options: “supports”, “suggests”... - lines 72-80 (polarity correspondence account): It would be important to cite the work by Santiago and Lakens (2015) that failed to support the polarity correspondence account. In general, I don’t think there is much point of describing and discussing all these theoretical approaches in the introduction of the paper, as the data are not going to be relevant to any of them and they are not taken up again in the discussion. - line 84: not all embodied approaches presume that the relation between the abstract and the concrete domain must be bidirectional. Conceptual metaphor theory is an embodied theory and it suggests an asymmetrical relation. Moreover, not all abstract concepts are the same: some abstract concepts seem to be of a different kind to numbers, being more based on interoceptive experiences and language (see the recent review by Borghi, Shaki and Fischer, 2022). - line 126: “become associated with space” does not seem to me to be the best way to put it. It is clear that numbers have an association with space, and this association is built because of the accummulation of experiences. I think what the authors mean is that they want to study the conditions under which the association between numbers and space is manifested in behavior. - line 131: the authors claim that the “common ingredient” of studies showing SNAs is physical space. However, this is not always the case. For Shaki and Fischer (2018) the key ingredient is that either number or space is part of the explicit definition of the task. - line 150: under these conditions, Shaki and Fischer (2018) DID find a vertical SNA in one experiment (but none in the other experiment). - line 181: the authors try and explain the results by Miklashevsky by having spatial information in the response space. However, other studies such as Shaki and Fischer (2018) did not have any spatial information in the response space and found lateral SNAs in magnitud comparison and a vertical SNA in parity judgement in one out of two experiments. Moreover, there are a number of studies that just present a number and find SNAs in the pattern of eye movements over a blank screen. The authors should try and give a coherent explanation of the whole pattern of findings currently available. - line 214: the cite to Pinto [24] does not seem to be correct, as it refers to the study with neglect patients. Actually, it would be important to include all the studies by Pinto (2019 a y b, and 2021) into the set of findings that the authors try and integrate in their review of the literature. - line 221: the conclusion that follows from Pinto’s studies is not that left and right spatial codes are necessary for SNAs. Instead it is that both numbers and space must be present AND linked in the definition of the task. - line 225-226: incorrect. Shaki and Fischer (2018) found absolutely no horizontal SNARC in the parity task. - lines 228-230: not completely correct: the vertical SNARC in the parity task was found in one experiment but not in the other. - line 293: it is not correct to say that spatial cueing induces vertical but not horizontal SNAs with centralized stimuli and responses (see the last two comments). - line 306: it is asserted that the present study uses “the same isometric force paradigm as Miklashevsky”. Even though the same sensor is used, the fact that Miklashevsky used a two-handed precision grip is a potentially important difference in paradigm (as discussed by the authors at the end of the paper). Therefore, it is misleading to say here that the paradigm in the two studies was the same. - line 395: this is two designs, not one. - line 399-400: exchange H1 and H2. - line 438: this is two designs, not one. - lines 474-477: the time windows overlap. - line 654: as I have pointed out above, “spatial information seems to be deeply rooted into SNAs assessment” fails to take into account Pinto’s studies. - line 705: I can’t see how the argument in the prior lines (703-704) “fits the reasoning of “Absence of evidence is not evidence of absence”. Bayesian statistics are indeed able to quantify evidence for absence. - line 738: again, the authors claim that Miklashevsky used the same method as in the present study. Although they also measured isometric force, the differences in methods are substantial. - line 813-814: From the present study it does not follow that “lateralized spatial information in the design space plays a crucial role in eliciting SNAs”. This is because in the present design there is lateralized spatial information, as the participants are instructed to press the sensor in lateral (and vertical) directions, and because, in general, the study cannot conclude anything about what is essential for SNAs to arise. It can only conclude that directional static pressures are not able by themselves to make SNAs to arise. Reviewer #2: The authors responded admirably to the feedback they received. I see no reason that the paper should not be accepted at this point. Still, I want to comment on one issue. Previously, I'd found myself confused about the key distinction in the paper between 'overt' and 'covert' movement. The authors' reply about this point left me more confused, I think. Ordinarily I wouldn't care so much about this difference, but it is central to the paper. I'm not sure that (1) this distinction is so clear-cut as the authors imply, that (2) the distinction here is one that maps onto how most people would naturally think of the difference, nor that (3) this distinction is the relevant one. In addition, I felt that grouping all three of the studies I mentioned as 'overt' was very surprising to me. I genuinely wouldn't have expected that. For that reason, I was hoping the authors would have said slightly more in the paper about exactly what things are and are not overt, referring to more specific examples like these. (I found the information in Figure 1 to be valuable but hard to fully process, even with extensive knowledge of these studies; a non-expert might have even more trouble.) I don't find this disqualifying, however, because the results can be understood and interpreted without respect to that distinction. For this reason, I only wish to suggest that the authors should once more reconsider whether this language is apt, or whether it is being explained in the best way. Maybe it is; I'm not sure. It stood out to me, so I wanted to make a note of it. ********** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: Yes: Julio Santiago Reviewer #2: No ********** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. 10.1371/journal.pone.0288038.r004 Author response to Decision Letter 1 Submission Version2 17 May 2023 Reviewer #1: The manuscript has been improved in many ways. Still, I found a number of inadequacies and also have some suggestions for further improvement. I will mention them in the same order as they appear in the manuscript. - Title: I am not convinced by the label “covert movement”. In my opinion, a movement is covert if it is planned and mentally simulated, but without it reaching the muscles. In this study the planned action does reach the muscles, and the participant does not plan to perform any movement, just a permanent pressure. It would be better to use a different label. “Isometric force” maybe a good alternative, although when I first read it I had difficulty to parse it. Maybe “static directional pressure” or some other option. Reply: Thanks to both reviewers for pointing this out. We have now completely removed the words “overt” and “covert” from the manuscript as both reviewers made similar points regarding the wording and how it led to confusion. We now use “isometric force” and “isometric directional force” and already applied them in the title. Additionally, we have reformulated the sentence with the aim of the study to make the idea more accessible before introducing “isometric directional force”. We have also contrasted movement through space with our isometric directional force manipulation early in the Discussion. We have renamed the manuscript to “How to not induce SNAs: the insufficiency of directional force”. In lines 324-326, it now reads: “Will directional but non-spatial movement along the horizontal and vertical axes, thus pressing into a direction of left, right, up, or down, elicit SNAs?”. In lines 328-338, we follow up with: “We asked participants to press against a sensor surface, thus recording their isometric force production. Such isometric force was exerted constantly and continuously on the sensor surface in a direction on the horizontal (left or right) and the vertical (up or down) axes during two tasks. Hereby, we utilized the same force sensors as Miklashevsky et al. [41] but altered the paradigm to record the production of isometric force by one-handed presses into a direction instead of grip force (from here on described as “isometric directional force”).” In lines 695-699, it now reads: “Notable, our isometric directional force manipulation did not include a movement through space. Instead, isometric directional force utilized a subtler approach of directionally applied pressure attempting to elicit spatial codes. - line 54: “produces” is not adequate. Options: “supports”, “suggests”... Reply: Corrected. In lines 54-56, it now reads: “The reaction time paradigm is widely accepted and provides evidence for the association of smaller numbers with left space and larger numbers with right space [4].” - lines 72-80 (polarity correspondence account): It would be important to cite the work by Santiago and Lakens (2015) that failed to support the polarity correspondence account. In general, I don’t think there is much point of describing and discussing all these theoretical approaches in the introduction of the paper, as the data are not going to be relevant to any of them and they are not taken up again in the discussion. Reply: We have referenced the Santiago and Lakens (2015) paper to address existing counterevidence for the polarity correspondence account. In lines 80-82, it now reads: “Therefore, SNARC would not be limited to the horizontal axis (but see [11] for empirical evidence against the polarity correspondence account).” Regarding the Introduction. The theoretical approaches discussed in the Introduction seem important to keep allowing us to introduce the context in which the experiment is based on the existing literature. This approach reflects our planning in the methodology and facilitates the comprehension of our study for the readers. Thus, we tried to be concise in our Introduction but respectfully decline the request to eliminate our theoretical context. Moreover, to establish better theoretical coherence between Introduction and Discussion, we now briefly discuss that the theories may need to be extended to allow predictions in cases of an absent movement (see lines 725-735). References: Santiago, J., & Lakens, D. (2015). Can conceptual congruency effects between number, time, and space be accounted for by polarity correspondence? Acta Psychologica, 156, 179–191. https://doi.org/10.1016/j.actpsy.2014.09.016 - line 84: not all embodied approaches presume that the relation between the abstract and the concrete domain must be bidirectional. Conceptual metaphor theory is an embodied theory and it suggests an asymmetrical relation. Moreover, not all abstract concepts are the same: some abstract concepts seem to be of a different kind to numbers, being more based on interoceptive experiences and language (see the recent review by Borghi, Shaki, and Fischer, 2022). Reply: Thank you for the suggestions. We now acknowledge the asymmetrical relation of numbers and space. In lines 102-105, it now reads: “Notably, there is an asymmetrical relationship between space and numbers, in which the understanding of space is more fundamental. The meaning of numbers is often based on the experiences of space while this is not necessarily the case vice versa [12,13]. Additionally, we have integrated the review by Borghi, Shaki, and Fischer (2022) into an existing section where we compared the mechanisms behind the vertical and the horizontal SNAs (lines 249-268). Concretely, in lines 261-265, it now reads: “These linguistic practices that describe vertical SNAs differ in their contribution to the embodiment of numbers compared to the horizontal SNAs that seem to rely more on sensorimotor experiences such as finger counting. Moreover, situated factors like interoceptive signals play an additional role in the perception and production of numbers that also might contribute to SNAs ([50]; for review see [51]). References: Belli, F., Felisatti, A., & Fischer, M. H. (2021). “BreaThink”: breathing affects production and perception of quantities. Experimental Brain Research, 239(8), 2489–2499. https://doi.org/10.1007/s00221-021-06147-z Borghi, A. M., Shaki, S., & Fischer, M. H. (2023). Abstract concepts: external influences, internal constraints, and methodological issues. Psychological Research, 86(8), 2370–2388. https://doi.org/10.1007/s00426-022-01698-4 - line 126: “Become associated with space” does not seem to me to be the best way to put it. It is clear that numbers have an association with space, and this association is built because of the accummulation of experiences. I think what the authors mean is that they want to study the conditions under which the association between numbers and space is manifested in behavior. Reply: Corrected. In lines 133-135, it now reads: “The goal of the present study is to test the boundary conditions under which the association of numbers and space is manifested in behavior.”. - line 131: the authors claim that the “common ingredient” of studies showing SNAs is physical space. However, this is not always the case. For Shaki and Fischer (2018) the key ingredient is that either number or space is part of the explicit definition of the task. Reply: Thank you for pointing this out. We have now removed the term “physical space” and corrected it to “spatial information”. Spatial information is a broader term that covers the response space, presentation space, and cueing. In lines 140-145, it now reads: “By this rationale, the common ingredient for most studies seems to be the activation of spatial information, namely under the experimental conditions of response space (lateral responses), presentation space (lateralized stimuli), or spatial cueing (arrows and/or instructions) during stimulus presentation (see Fig 1 for a visual overview of prototypical laboratory set-ups).” - line 150: under these conditions, Shaki and Fischer (2018) DID find a vertical SNA in one experiment (but none in the other experiment). Reply: This is correct; however, the reviewer seems to have mixed up the order of the references. Line 150 (162-163 in the revised marked manuscript) refers to the study by Pinto et al. (2021; reference number 30) where no SNAs were found in experiment 2. Reference number 31(lines 160-161) is for Shaki and Fischer (2018) only describing the experimental design with no description of the result. - line 181: the authors try and explain the results by Miklashevsky by having spatial information in the response space. However, other studies such as Shaki and Fischer (2018) did not have any spatial information in the response space and found lateral SNAs in magnitud comparison and a vertical SNA in parity judgement in one out of two experiments. Moreover, there are a number of studies that just present a number and find SNAs in the pattern of eye movements over a blank screen. The authors should try and give a coherent explanation of the whole pattern of findings currently available. Reply: Thank you for the comment. The studies that we refer to in the Introduction point to the inconsistency of the literature and how we use it to motivate our own research. For this, we chose the best-fitting examples. However, based on the latter comments of the reviewer, we have also extended the literature by integrating some new studies suggested by the reviewer (lines 226-248). Together, these studies motivate our research and we do not think that adding even more examples would benefit the paper. We think that from lines 244-246 we can already draw the necessary conclusion: “Together, the above-presented studies point to the conclusion that both, number magnitude and spatial information must be coactivated to produce reliable SNAs [30,31,48,49].”. References to the additional studies included: Pinto, M., Pellegrino, M., Marson, F., Lasaponara, S., Cestari, V., D’Onofrio, M., & Doricchi, F. (2021). How to trigger and keep stable directional Space–Number Associations (SNAs). Cortex, 134, 253–264. https://doi.org/10.1016/j.cortex.2020.10.020 Pinto, M., Pellegrino, M., Marson, F., Lasaponara, S., & Doricchi, F. (2019). Reconstructing the origins of the space-number association: spatial and number-magnitude codes must be used jointly to elicit spatially organised mental number lines. Cognition, 190, 143–156. https://doi.org/10.1016/j.cognition.2019.04.032 - line 214: the cite to Pinto [24] does not seem to be correct, as it refers to the study with neglect patients. Actually, it would be important to include all the studies by Pinto (2019 a y b, and 2021) into the set of findings that the authors try and integrate into their review of the literature. Reply: Thank you for pointing this out. We have verified the citation (Pinto et al. 2021) and can confirm its accuracy (reference number 30 in the revised manuscript). This was a single but important citation showing an experimental design of a centralized presentation and response space. Based on your suggestions and for the interested readers, we have now referenced other studies by Pinto that fulfill the same criteria and further strengthen our argument. In lines 233-237, it now reads: “The authors’ interpretation was that the numbers will only spatially align across the mental number line when both, number magnitude is activated and spatial response codes are present (for experimental designs that also centralized the presentation and response space see [48,49]).” References: Pinto, M., Pellegrino, M., Lasaponara, S., Scozia, G., D’Onofrio, M., Raffa, G., Nigro, S., Arnaud, C. R., Tomaiuolo, F., & Doricchi, F. (2021). Number space is made by response space: Evidence from left spatial neglect. Neuropsychologia, 154. https://doi.org/10.1016/j.neuropsychologia.2021.107773 Pinto, M., Pellegrino, M., Marson, F., Lasaponara, S., Cestari, V., D’Onofrio, M., & Doricchi, F. (2021). How to trigger and keep stable directional Space–Number Associations (SNAs). Cortex, 134, 253–264. https://doi.org/10.1016/j.cortex.2020.10.020 Pinto, M., Pellegrino, M., Marson, F., Lasaponara, S., & Doricchi, F. (2019). Reconstructing the origins of the space-number association: spatial and number-magnitude codes must be used jointly to elicit spatially organised mental number lines. Cognition, 190, 143–156. https://doi.org/10.1016/j.cognition.2019.04.032 - line 221: the conclusion that follows from Pinto’s studies is not that left and right spatial codes are necessary for SNAs. Instead, it is that both numbers and space must be present AND linked in the definition of the task. Reply: Thank you for pointing this out. We have now added multiple sentences to incorporate this. In lines 233-237, it now reads: “The authors’ interpretation was that the numbers will only spatially align across the mental number line when both, number magnitude is activated and spatial response codes are present (for experimental designs that also centralized the presentation and response space see [48,49]).” In lines, 244-246: “Together, the above-presented studies point to the conclusion that in which both, number magnitude and spatial information must be coactivated to produce reliable SNAs [30,31,48,49].” - line 225-226: incorrect. Shaki and Fischer (2018) found absolutely no horizontal SNARC in the parity task. Reply: Corrected (line 241). - lines 228-230: not completely correct: the vertical SNARC in the parity task was found in one experiment but not in the other. Reply: Thank you for the comment. We agree that our report was not complete as we do not present the full findings of the experiment because this goes beyond the scope of the paper. However, based on the reviewer’s other suggestions the section leads to the conclusion we needed to motivate our research. In lines 244-246, it now reads: “Together, the above-presented studies point to the conclusion that both, number magnitude and spatial information must be coactivated to produce reliable SNAs [30,31,48,49].”. - line 293: it is not correct to say that spatial cueing induces vertical but not horizontal SNAs with centralized stimuli and responses (see the last two comments). Reply: Corrected. In lines 319-321, it now reads: “Third, cognitive spatial cueing induces stronger vertical than horizontal SNAs during centralized stimulus presentation and responses [31].” Additionally, we have corrected a sentence in lines 249-253 that now reads: “One possible explanation for why vertical SNAs remained stronger than horizontal SNAs could be the hierarchical nature of body-related knowledge representations involved in the task, according to which different mechanisms induce different types of SNAs, referred to as grounded, embodied, or situated cognition [29].”. - line 306: it is asserted that the present study uses “the same isometric force paradigm as Miklashevsky”. Even though the same sensor is used, the fact that Miklashevsky used a two-handed precision grip is a potentially important difference in paradigm (as discussed by the authors at the end of the paper). Therefore, it is misleading to say here that the paradigm in the two studies was the same. Reply: We appreciate your helpful suggestion and corrected the sentence. In lines 334-338, it now reads: “Hereby, we utilized the same force sensors as Miklashevsky et al. [41] but altered the paradigm to record the production of isometric force by one-handed presses into a direction instead of grip force (from here on described as “isometric directional force”).“ - line 395: this is two designs, not one. - line 399-400: exchange H1 and H2. Reply to both points: Thank you for pointing this out. In lines 441-450, now reads: “In order to test the concept-motor interactions of H1 and H2 with the same task, we have utilized two within-subject designs. Both designs utilized one within-subject factor being the force direction (two levels each: left/right on a horizontal axis or up/down on the vertical axis). To test H1, number magnitude constituted the dependent variable with two magnitude levels: small and large. To test H2, number magnitude constituted the independent variable with two magnitude levels: small and large while continuous isometric force was the dependent variable.”. Regarding the second point, we cannot exchange H1 and H2 as this would be incorrect. - line 438: this is two designs, not one. In lines 490-494, we describe the design of the SDA task. This task does not test the bi-directionality of motor-concept interactions but only the effect of number magnitudes on force magnitudes. Therefore, it is one design. - lines 474-477: the time windows overlap. Reply: The time windows were constructed manually after the uninformative cluster permutation analysis of both tasks. They were constructed to capture mainly the effects in the time windows of 100-250 ms. However, we covered the duration until the end of the trial to explore the data in more detail. Mathematically, the time windows had to overlap during the last fragment of the SDA trials. In lines 480-487, we describe the full reasoning. Concretely, in lines 485-487 it reads: “These time-windows were constructed to capture mainly the effects in the time-windows of 100-250 ms that are associated with semantic activation after a critical stimulus [72].”. - line 654: as I have pointed out above, “spatial information seems to be deeply rooted into SNAs assessment” fails to take into account Pinto’s studies. Reply: Thank you for the comment. Earlier, we have acknowledged the position by Pinto et al. (2019, 2021a,2021b) in lines 244-246. However, in that specific paragraph we emphasize on the role of spatial information in the movement itself. We have rewritten the section to make the point clear. In lines 719-725, it now reads: “Our finding is in contrast to other RNG studies that found SNAs along the horizontal and vertical axes by using either active movement ([59]; see also [56] for RNG across the horizontal axis; for review see [28]) or passive movement through space [60]. Therefore, it is likely that activating a spatial code in the response space, requires a movement with a spatial component, such as movement through space or movement allocated in space (e.g., contacting spatially aligned response buttons), to create SNAs.”. - line 705: I can’t see how the argument in the prior lines (703-704) “fits the reasoning of “Absence of evidence is not evidence of absence”. Bayesian statistics are indeed able to quantify evidence for absence. Reply: Thank you for pointing this out. We have removed the sentence (lines 787-788). - line 738: again, the authors claim that Miklashevsky used the same method as in the present study. Although they also measured isometric force, the differences in methods are substantial. Reply: Thank you for pointing this out. We have removed the comparison to our study. Further differences from our study to Miklashevksy’s are discussed in the Limitations section. In lines 821-823, it now reads: “Second, as stated in the introduction, Miklashevsky et al. [41] found SNAs in numerical tasks during bimanual and passive isometric force recordings.”. - line 813-814: From the present study it does not follow that “lateralized spatial information in the design space plays a crucial role in eliciting SNAs”. This is because in the present design there is lateralized spatial information, as the participants are instructed to press the sensor in lateral (and vertical) directions, and because, in general, the study cannot conclude anything about what is essential for SNAs to arise. It can only conclude that directional static pressures are not able by themselves to make SNAs arise. Reply: Thank you for pointing this out. We have now changed the concluding sentence in lines 900-901 that now reads as: “We conclude that applying pressure into the direction of left, right, up, and down does not suffice in eliciting SNAs along the horizontal and the vertical axes.”. Reviewer #2: The authors responded admirably to the feedback they received. I see no reason that the paper should not be accepted at this point. Still, I want to comment on one issue. Previously, I'd found myself confused about the key distinction in the paper between 'overt' and 'covert' movement. The authors' reply about this point left me more confused, I think. Ordinarily I wouldn't care so much about this difference, but it is central to the paper. I'm not sure that (1) this distinction is so clear-cut as the authors imply, that (2) the distinction here is one that maps onto how most people would naturally think of the difference, nor that (3) this distinction is the relevant one. In addition, I felt that grouping all three of the studies I mentioned as 'overt' was very surprising to me. I genuinely wouldn't have expected that. For that reason, I was hoping the authors would have said slightly more in the paper about exactly what things are and are not overt, referring to more specific examples like these. (I found the information in Figure 1 to be valuable but hard to fully process, even with extensive knowledge of these studies; a non-expert might have even more trouble.) I don't find this disqualifying, however, because the results can be understood and interpreted without respect to that distinction. For this reason, I only wish to suggest that the authors should once more reconsider whether this language is apt, or whether it is being explained in the best way. Maybe it is; I'm not sure. It stood out to me, so I wanted to make a note of it. Reply: Thanks to both reviewers for pointing this out. We have now completely removed the words “overt” and “covert” from the manuscript as both reviewers made similar points regarding the wording and how it led to confusion. We now use “isometric force” and “isometric directional force” and already applied it in the title. Additionally, we have reformulated the sentence with the aim of the study to make the idea more accessible before introducing” isometric directional force”. We have also contrasted movement through space with our isometric directional force manipulation early in the discussion. We have renamed the manuscript to “How to not induce SNAs: the insufficiency of directional force”. In lines 324-326, it now reads: “Will directional but non-spatial movement along the horizontal and vertical axes, thus pressing into a direction of left, right, up, or down, elicit SNAs?”. In lines 328-338, we follow up with: “We asked participants to press against a sensor surface, thus recording their isometric force production. Such isometric force was exerted constantly and continuously on the sensor surface in a direction on the horizontal (left or right) and the vertical (up or down) axes during two tasks. Hereby, we utilized the same force sensors as Miklashevsky et al. [41] but altered the paradigm to record the production of isometric force by one-handed presses into a direction instead of grip force (from here on described as “isometric directional force”). ”. In lines 695-699, it now reads: “Notable, our isometric directional force manipulation did not include a movement through space. Instead, isometric directional force utilized a subtler approach of directionally applied pressure attempting to elicit spatial codes. Changes to the reference list The reference list changed from the original submission to the current version. Beneath, the updated reference list is displayed. Red-coloured references are new additions while the orange references were removed from the reference list. 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Behav Brain Sci. 2001;24: 849–878. doi:10.1017/S0140525X01000103 Attachment Submitted filename: Response to reviewers_Revision2.pdf Click here for additional data file. 10.1371/journal.pone.0288038.r005 Decision Letter 2 Tessari Alessia Academic Editor © 2023 Alessia Tessari 2023 Alessia Tessari https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version2 19 Jun 2023 How to not induce SNAs: the insufficiency of directional force PONE-D-22-30799R2 Dear Dr. Kühne, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. 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If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org. If we can help with anything else, please email us at plosone@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Professor Alessia Tessari Academic Editor PLOS ONE ==== Refs References 1 Dehaene S , Brannon EM . Space, time, and number: a Kantian research program. 2010. doi: 10.1016/j.tics.2010.09.009 20980194 2 Dehaene S , Bossini S , Giraux P . The mental representation of parity and number magnitude. J Exp Psychol Gen. 1993;122 : 371–396. doi: 10.1037/0096-3445.122.3.371 3 Fischer MH , Shaki S . 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