Joseph Woelfel,Edward L. Fink,Deborah A. Cai,Kenton Anderson,Asa Iacobucci,Hua Wang
Inertia in cognitive processes: the case of the COVID-19 vaccine
Inertia,Cognitive processes,Cultural processes,Multidimensional scaling,Attitudes,Behaviors
Developments in factor analysis (Spearman in Am J Psychol 15:201-292, 1904); Thurstone in Multiple factor analysis, University of Chicago Press, Chicago, 1947), multidimensional scaling (Torgerson in Theory and methods of scaling, Wiley Hoboken, New Jersey, 1958; Young and Householder in Psychometrika, 3:19–22, 1938), the Galileo model (Woelfel and Fink in The measurement of communication processes: galileo theory and method, Academic Press Cambridge, Massachusetts, 1980), and, more recently, in computer science, artificial intelligence, computational linguistics, network analysis and other disciplines (Woelfel in Qual Quant 54:263–278, 2020) have shown that human cognitive and cultural beliefs and attitudes can be modeled as movement through a high-dimensional non-Euclidean space. This article demonstrates the theoretical and methodological contribution that multidimensional scaling makes to understand attitude change associated with the COVID-19 vaccine.
Inertia in cognitive processes: the case of the COVID-19 vaccine

Developments in factor analysis (Spearman in Am J Psychol 15:201-292, 1904); Thurstone in Multiple factor analysis, University of Chicago Press, Chicago, 1947), multidimensional scaling (Torgerson in Theory and methods of scaling, Wiley Hoboken, New Jersey, 1958; Young and Householder in Psychometrika, 3:19–22, 1938), the Galileo model (Woelfel and Fink in The measurement of communication processes: galileo theory and method, Academic Press Cambridge, Massachusetts, 1980), and, more recently, in computer science, artificial intelligence, computational linguistics, network analysis and other disciplines (Woelfel in Qual Quant 54:263–278, 2020) have shown that human cognitive and cultural beliefs and attitudes can be modeled as movement through a high-dimensional non-Euclidean space. This article demonstrates the theoretical and methodological contribution that multidimensional scaling makes to understand attitude change associated with the COVID-19 vaccine.

This article reports results from the latest research in a continuing program designed to establish inertial reference frames within a space created for the analysis of cognitive processes. An inertial reference frame is a set of axes that allow us to examine which concepts are stable and which move within a set multidimensional space, or frame, when the concepts are impacted by one or more messages. The concept of an inertial reference frame is not obvious, and it took the physical sciences 2300 years to move from Aristotle’s definition of space as the volume immediately surrounding a body to the concept of an inertial reference frame, whose definition was not fully realized until the late nineteenth century when the term was first coined (Jammer 1997). This article examines the movement of concepts related to the COVID-19 vaccine due to the impact of messages about the vaccine and related concepts within an inertial reference frame.

Summarizing over a decade of research, Woelfel and Fink (1980) published a method for establishing mathematical reference frames for the measurement of cognitive and cultural processes. Briefly, the method required establishing a set of concepts that collectively define a given topic or process and then measuring the mutual distances among those terms using magnitude estimation scales. Following procedures defined by Young and Householder (1938) and Torgerson (1958), measured distances are transformed into their scalar products, and from these products a set of orthogonal basis vectors are extracted as the eigenvectors of the scalar products matrix.1 This transformation provides the basis vectors of a space within which the original concepts can be arrayed while preserving their original interpoint distances. Rotation, translation, and reflection (rigid body transformations) are used to align such reference frames so that the motion of the concepts arrayed in the space can be measured relative to the frame. In 1988, Barnett hypothesized that frequently used concepts should be more massive, and therefore more resistant to acceleration, than concepts less frequently used. For the concept of mass to be meaningful, an inertial reference frame is required; that is, a reference frame in which certain points are held constant—fixed or stable—to serve as a gauge for the relative motion of the moving points. Utilizing the Woelfel and Fink (1980) procedures, Barnett conducted an experiment using four synonyms—pig, hog, boar, and swine. These terms were chosen because they all referred to the same object but occurred with widely different frequencies in English. Participants were randomly assigned to receive a message that used one of the four terms: A credible source presented a persuasive message to the respondents that claimed the specific term was beneficial and attractive. The reference space created by the manipulation was multidimensional and non-Euclidean. Pig was the most massive concept and moved the least, followed by hog and then swine. Although there was a problem with the boar condition, the correlation between the calculated masses of pig, hog, and swine and their frequency of use in the English language was 1.0, based on the Thorndike-Lorge list (1944; Barnett 1988). McIntosh and Woelfel (2017) replicated Barnett’s study in 2016. Some modifications of the original design eliminated some sources of bias (e.g., the source of the message was omitted), but the essence of the experiment was unchanged. The Woelfel and Fink procedures were followed to produce a space in which most of the Barnett concepts were embedded. Four spaces were created: Each space had a set of terms for farm animals that included one of the four porcine terms (pig, hog, boar, swine). Each respondent received a message that the porcine term was beneficial and attractive, as in Barnett’s original message. The McIntosh and Woelfel (2017) experiment reproduced the Barnett space with considerable accuracy, as Fig. 1 shows. Across the 28 years between the two studies, the distances between the concepts used in both studies correlated r = 0.877 (p < 0.0001, df = 53, t = 13.2). The average correlation among the position vectors of these concepts was 0.90, and the average correlation among the dimensions of the space was 0.87, showing that the reference space was quite stable over this period. In the follow-up study, the masses of all four concepts—pig, hog, boar, and swine—were all in the predicted order, and the concepts correlated with their frequency of use in the English language at 0.995, based on the Thorndike-Lorge list (1944). These results demonstrate that the mass of a concept is well predicted by its frequency of use within its relevant language.

A nationwide study conducted by the Gallup organization (Riffkin 2014) showed that nurses were the most trusted occupational group in the U.S., and members of Congress were the least trusted. These rankings have not changed much since then (Saad 2022). The notion that credible sources produce more attitude change than noncredible sources has been supported in a good deal of prior research (e.g., Bochner and Insko 1966). However, even more compelling evidence from neuroscience has shown that the formation of connections among neurons is not under voluntary control (Hebb 1949): One cannot choose not to be influenced by incoming stimuli. Thus, one’s belief about an incoming message will not be relevant to the message’s effect on that individual’s neural network. When several stimuli are attended to simultaneously, the process of neural connection begins.2 This relationship between neurons that are synoptically connected, in which a stimulus that excites one neuron results in changes in another neuron and brings about what we refer to as learning, is known as the Hebb Rule (Hebb 1949). To test the role of credibility and neural networks on attitude change, we used the information about nurses and Congress to conduct an experiment among 150 undergraduate students at a university in the Northeast of the United States. In this experiment, approximately one-third of the students read the following high-credibility message:This questionnaire will ask your opinion about the Health Care Recovery Act (HCRA), which a committee of nurses said was beneficial and attractive. Another third received the following low-credibility message:This questionnaire will ask your opinion about the Health Care Recovery Act (HCRA), which a committee of members of Congress said was beneficial and attractive. The final third was a control group, which received the following message: This questionnaire will ask your opinion about the Health Care Recovery Act (HCRA). The words beneficial and attractive were taken from the original Barnett study about barnyard animals and its replication (McIntosh and Woelfel 2017). A manipulation check in our experiment showed that nurses were considered more credible than members of Congress, with nurses 27.1 units away from credible, while members of Congress were twice as far from credible at 55.0 units (p < 0.01). To help interpret the magnitude of these differences, McIntosh and Woelfel (2019) specified the metric as follows:Please estimate how different or “far apart” each of the following words or phrases are from each of the others. The more different, or further apart they seem to be, the larger the number you should write. To help you know what size number to write, remember reliable and unreliable are 100 units apart. If two words or phrases are not different at all, please write zero (0). If you have no idea, just leave the space blank. Respondents then compared the differences [(12 × 11)/2 = 66] among the 12 concepts beneficial, attractive, trustworthy, credible, nurses, members of congress, good, HCRA, unreliable, untrustworthy, health, and yourself. The largest difference was 76.6279 between beneficial and unreliable, whereas the smallest difference measured was 14.0926 between health and nurses. These are ratio scales, so ratio comparisons are appropriate. Overall results showed that the space was reliably multidimensional and non-Euclidean with good precision of measurement. As the Hebb Rule hypothesized, and contrary to the historical record of most attitude change research, the more credible source did not produce more change than the less credible source. Instead, the change produced by the less credible source was in a different direction than that produced by the more credible source. The movement produced by the more credible source made the Health Care Recovery Act (HCRA) closer to the mean self-point, whereas the low credibility source moved HCRA orthogonally, neither closer to nor further from the self-point. The reason for greater or lesser attitude change is not the credibility or ethos of the message’s source, but rather the source’s position in the space. If the source is located close to the subjects’ self-point, averaging the source’s distance from the self-point with the other message concepts will improve the effectiveness of the message; if the source is located far from the self-point, including its distance into the average will reduce the effect. Because nurses were close to the self, adding nurses into the message helped move the HCRA closer to the self; because Congress was far from the self, adding it into the message did not help move the HCRA closer to the self. Observing the multidimensional space makes it possible to see this difference. Had typical unidimensional scaling methods been used, it would have appeared that the credible source produced considerably more change. Figure 2 shows the change produced in the high credibility condition (nurses). Rather than considering nurses to be only the source of the message, it is reasonable to consider nurses to be part of the message. The message terms form the triangle nurses, beneficial, and attractive on the lefthand side of Fig. 2. The movement of HCRA is along the vector from its position in the control group (all upper case) through its new location (all lower case).The figure shows movement toward the middle of the message triangle. Note that this movement is in the direction of yourself, which creates a more favorable motion of HCRA, because it moves HCRA toward the subjects’ self-concept. Figure 3 shows a similar set of relationships, with the low credibility source (Congress) forming a part of the message triangle between Congress, beneficial, and attractive. In this case, however, the source (Congress) is located toward the right edge of Fig. 3, near the concepts unreliable and untrustworthy. Again, the HCRA moves toward the center of the message triangle, but this movement vector is orthogonal to the self-concept vector (yourself), so the result is not an improvement in favorability. If favorability were measured unidimensionally, examining HCRA and yourself without observing the relative positions would make it appear that the less credible source produced less attitude change, but this conclusion would be incorrect. It is the movement of HCRA in relation to the self-concept vector that provides a more accurate conclusion about the role of the less credible source.

The theory that emerges from this research begins by assuming that concepts are clusters of interconnected neurons, and, therefore, they act as physical entities subject to the laws of physics and biology. Concepts may be modeled as point masses in a multidimensional space from which emanate meaning fields that vary with distance in the space; that is, the closer concepts are to a meaning field, the more they participate in determining the meaning of that field. The human brain comprises approximately 85 billion neurons, each of which is connected, on average, to fewer than about 10,000 other neurons. Therefore, the matrix of interconnections is expected to be sparse, and the space between the clusters is expected to be discontinuous. Instead, there exist smaller clusters of concepts, or neighborhoods, consistent with Mead’s (1934) notion of the situation. Thus, concepts designated by the same symbolic names may have different meanings in different subspaces. For example, mustang can refer to a horse or a car, and it may possibly have other meanings. Thus, meanings of concepts are understood by their context. Associating concepts in messages creates forces that move them toward or away from each other. The frequency of use strengthens concepts’ connections, which makes the concepts more massive, and, subsequently, results in greater inertia, so they are more resistant to acceleration.

Based on this theory we can generate the following hypotheses, which we examine in this study:

The neural processes underlying cognitive and cultural processes may be represented by a multidimensional spatial manifold. We believe that H1 is a hypothesis that needs to be tested. It is easily falsified; if the predictions generated from Hebb’s rule are not supported by these data, this hypothesis should clearly be false. Similarly, because all the dimensions were calculated from the data, failure to find multiple dimensions, or failure of the dimensional structure to match across time, would also falsify the hypothesis.

The cognitive spatial manifold will generally be non-Euclidean.

The Woelfel-Fink procedures provide precise and reliable measurements of the cognitive spatial manifold.

Points in the cognitive spatial manifold will exhibit inertial properties. In other words, more massive points will be accelerated less by any given force than less massive points.

This study further investigates the role of inertia in cognitive and cultural processes. As with previous studies, the concept used in this research, the HCRA, is a fictional entity with no history. Therefore, this concept should have little or no mass, so it is expected that HCRA is likely to be moved by a very brief one-sentence message. In contrast with the previous study, for this experiment a more massive concept, COVID Vaccine, was used. COVID Vaccine has been a constant focus of media attention during the recent pandemic, and, even before COVID, vaccination has been a hotly disputed topic. Except for replacing HCRA with COVID Vaccine, the same paired comparison magnitude estimation procedure, as described by Woelfel and Fink (1980), was used as in the earlier study. Data analysis was performed using Galileo v.5.7 software.

Three hundred and fifty-three respondents from three different locations (locations not provided to maintain confidentiality) responded to the paper and pencil instrument, yielding 23,177 paired comparison measurements. (This was the last use of paper and pencil responses the researchers employed.) Because the criterion standard was a large distance (“reliable and unreliable are 100 units apart”), a maximum value filter was set at 101, as it was in the earlier study (MacIntosh and Woelfel 2019). Error detection software eliminated all values above 100 as well as any coding errors and missing data, which resulted in an overall rejection rate of 5.88%, leaving 21,638 paired comparisons for analysis. Based on the twelve concepts reported in Table 1, the resulting space matches the multidimensional non-Euclidean space from the earlier study very precisely. Correlations among the eleven highly correlated pairs of position vectors are between 0.78 and 0.99, with a median correlation of 0.97 and a mean correlation of 0.90. The one notable exception is the correlation for HCRA with COVID Vaccine; r = 0.08, which correlates HCRA in the previous study with COVID Vaccine in the present study, so HCRA should not be expected to correlate with other concepts that are reported in Table 1. Table 2 shows that the dimensional structure of the space is also very stable over the five-year period between studies, with very high and statistically significant correlations between all eleven corresponding dimensions (any set of N points can always be projected onto N-1 dimensions). Results show that the present study is more precise than the previous study, with standard errors around the paired comparisons generally between 2 and 3 units within the 100-point range, which gives about a 3% rate of error compared to the 5–6% error rate in the previous study. This is nowhere as precise as we would hope for a physical science experiment, but far more precise than a typical social science study is likely to have, and close enough for substantive analyses. The increase in precision reflects that the second study had 233% more cases than the first study. Because the last three dimensions of the space are imaginary and statistically significant, the space is reliably both multidimensional and non-Euclidean. As a result, Hypotheses 1 and 2 are both supported. Attitudes are measured in these procedures by the distances between the self-point (yourself) and the remaining concepts. Overall, results show very high levels of precision; correlations between attitudes in the control group and the randomly assigned treatment groups are 0.99 between the control group and the Congress group, and 0.99 between the control group and the nurses group. The goodness of fit of the Congress and the nurses spaces as compared to the control space and to the Congress and nurses spaces in the prior McIntosh and Woelfel (2019) experiment, along with the very high correlations between attitudes in the control group and treatment groups, provide strong support for Hypothesis 3.

The complete paired comparison measurement instrument allows for an internal manipulation check. As Table 3 shows, nurses is a substantially more credible source than members of Congress. Figure 4 shows the first three dimensions of the two spaces plotted on joint coordinates (i.e., the spaces have been rotated to least squares best fit, excluding HCRA and COVID-19). This figure provides a helpful visualization of the information provided in Tables 1 and 2. Notice that the concepts from the two studies are quite close to each other on the map, except for COVID Vaccine, which is much closer to yourself in the present study than HCRA was in the previous study. This result is likely because all 353 members of the sample were vaccinated (see Barnett et al. 1976).

The Hebb Rule predicts that “neurons wire together if they fire together” (Löwel and Singer 1992, p. 211), although one must fire before the other (Hebb 1949). Therefore, we can hypothesize that associating two or more concepts in a message should generate forces toward convergence, because the synapses between the concepts are strengthened. Concepts that are more closely connected should appear more similar and hence closer together in the space. Each assertion, such as “the COVID Vaccine is beneficial,” should generate forces moving the COVID Vaccine and beneficial closer together in the space. The forces are represented mathematically by vectors in the space. With compound messages, such as “the COVID Vaccine is beneficial and attractive,” the resultant force is given by the vector average of the two individual force vectors. It follows from the Hebb Rule that a message source should not be considered differently from the other concepts in the message. When a source is involved, it should be assumed that it is part of the message. For the message “Members of Congress say that the COVID Vaccine is beneficial and attractive,” the resultant force is given by the vector average of all these three individual force vectors, beneficial, attractive, and members of Congress. All these effects were observed in the McIntosh and Woelfel (2019) experiment. Because the matrix of interpoint distances is, in general and in this specific case, non-Euclidean, multidimensional, and locally Riemannian, these vectors are calculated by the extended theory of Pythagoras for n dimensions. The ith dimension is labeled pi.where sij = distance between concepts i and j, summed over all dimensions (from 1 to n) and square-rooted, and where pi = dimension i. When the space is non-Euclidean, as it is here, some of the dimensions will be imaginary. Therefore, their squared values will be negative and the quantity under the radical will be reduced. These calculations were performed in the present experiment by the computer program Buffalo, which has a free-standing algorithm that calculates the vector average of the concepts. Each message concept is represented by a position vector, which points from the origin to the concept given by its coordinates. The average of the coordinates is the geometric center of the position vectors. The result of these calculations is a point at the vector average of the message concepts that indicates the hypothetical equilibrium point at which the forces of the message exerted on the affected concepts sum to zero. When the mass of the concepts in the message is small, as was the case for the fictional HCRA, we expect the concepts to move substantially toward the equilibrium point. On the other hand, for more massive concepts, such as the COVID Vaccine, we expect a considerably smaller effect.3 Table 3 shows the distances moved by the nascent (low-mass) concept HCRA in the prior study, along with the distance moved by the COVID Vaccine (high mass) concept in the present experiment.

McIntosh and Woelfel (2019) calculated inertial mass as the ratio of distances moved by.the concepts. The mass of the concepts is calculated as the inverse of the amount the concepts moved (Woelfel and Fink 1980, pp. 148–149). As Table 4 shows, the differences in motion between the HCRA and the COVID Vaccine are large and highly significant. Taking these ratios of the movements for the two conditions (nurses and Congress), we find that COVID Vaccine is 23.02 times as massive as the HCRA in the nurses condition and 5.35 times as massive as HCRA in the Congress condition. Averaging these two estimates gives a value of 14.19 for the relative mass of COVID-19 as compared to HCRA. These values are valid because HCRA and the COVID Vaccine were both affected by the same message. It is clear that impacting different objects with the same message (i.e., the same amount of change advocated, the same source credibility, and the same duration of exposure), and using identical messages in the identical context, should bring about the same outcome. The subjects in the high credibility condition received exactly the same message in this experiment as the subjects did in the HCRA study:This questionnaire will ask your opinion about the Health Care Recovery Act (HCRA), which a committee of nurses said was beneficial and attractive. Similarly, subjects in the low credibility condition and in the control condition received the exact same wording as did subjects in the HCRA study. But the meaning of the messages depends ultimately not on the dictionary definition of the words, but rather on the relative locations of each of the terms in the message in the recipients’ cognitive space. Analysis with the Buffalo software indicates that the distance between COVID Vaccine and the equilibrium point of the message for the Congress condition is 39.8 units, and for the nurses condition it is 70.4 units. These distances are a measure of the amount of change advocated by each message. It has been well established that, when higher levels of change are advocated in a message, greater attitude change is likely to occur (see, e.g., Bochner and Insko 1966); therefore, we use change advocated as a measure of message force.4 Figure 5 shows the equilibrium point for the nurses and Congress conditions. In both conditions, the equilibrium point lies further from the self-point than from COVID Vaccine, so neither of them will result in improving the attitude of the respondents to the vaccine. In physics, the relationship between force, mass, and distance is well understood, as described in the following5:where F = force, m = mass, a = acceleration, andwhere s = distance moved and t = time. From these equations we can derive If we assume that time is about the same in both conditions, we can set t to 1, to derive Using Eq. 5, we get two estimates of the mass of HCRA and two estimates of the mass of COVID Vaccine. The resulting mean estimate of the masses of HCRA and COVID Vaccine are not very different from the ratio of our movements’ measures (i.e., the masses that are established by the movement of the concepts). Averaging the two measures gives the following: (14.19 + 10.22)/2 = 12.21, which is probably as precise a value of the masses as we can derive from these data. Any of these estimates, however, show that the COVID Vaccine is considerably more massive than HCRA, which supports Hypothesis 4.

This research is consistent with the Hebb Rule, that the near-simultaneous activation of neurons enhances the synaptic connections among the active neurons. It also provides support for a mathematical model of this process, in which concepts are modeled as points in a multidimensional non-Euclidean space. In this model, the co-occurrence in consciousness (i.e., neurons firing together) is modeled as forces moving the concurrent concepts toward each other in the space. The increasing physical size of synaptic connections among frequently co-firing neurons is modeled as the inertial mass of the points (Table 5). This model provides a possible explanation for why unidimensional studies of attitude change often conclude that more credible sources result in more attitude change. More credible sources are more likely to be closer to the self-concept in typical meaning spaces. This relationship suggests that the source is more strongly connected to the neural cluster representing the self-concept, which increases the connection of the manipulated concept to the self-concept. Source effectiveness comes not from the source’s credibility but rather from its position in the space and its relationship to the self-concept. The model can also account for the boomerang effect—that is, attitude change in the opposite direction from what was intended. The model predicts that this effect can occur when the sum of the message vectors results in a vector whose endpoint is farther from the self-point than from the manipulated concept. In addition, the model provides a useful way to demonstrate how different attitudes and beliefs can result in different types of resistance to change, which is measurable and subject to disconfirmation. This type of modeling is supported by a growing number of refereed articles across several disciplines (see Woelfel 2020), including psychometrics, sociology, communication, neuroscience, computer science, and computational linguistics (Evans 2012, 2013). Like any scientific theory, multidimensional scaling is subject to revision and replication. One important variable that has been left out of this discussion, for example, is time, which requires precise longitudinal data beyond the scope of this experiment. We note that several longitudinal studies exist that use the Woelfel and Fink (1980) model. One of the most prominent follows Korean immigrants to Hawaii over fifteen years (Kincaid et al. 1983); other studies examining longitudinal data include Evans (2012), Newton et al. (1986), and Gillham and Woelfel (1977). Galileo theory is a model that is neither true nor false, but it is more or less accurate for modeling the processes under study. It provides a computational basis for describing and influencing cognitive and cultural processes that are not available in the type of classic model that proposes that the immaterial mind is motivated by hope of reward and fear of punishment and is equipped with free will. Further, multidimensional models are consistent with emerging research in computer science, neuroscience, and other disciplines, and they fit measured data as well or better than attitude change models such as the theory of planned behavior (Ajzen 1991).