Abstract
In this section of the research, I use POSA to facilitate more thorough comments upon how we perceive and understand three-dimensional abstract art. In my exposition of POSA, I take some time to delve into previous uses of the approach to reveal partial ordering of variables. I present examples of POSA’s previous theoretical deployment and I demonstrate its application using applied art-related examples. Specifically, in the chapter, I consider a partially ordered model of the eight characteristics from Crowther’s ontology. I assert the usefulness of POSA by referencing how I have used this approach in earlier research that incorporated Crowther’s characteristics within a mapping sentence which I analysed using POSA to facilitate greater understanding of two-dimensional abstract art. I conclude the chapter by considering how the mapping sentence models I have developed for two and three-dimensional art, and partial order analysis of facets from the mapping sentences for two and three-dimensional art, relate to each other.
Partial Ordering and Partial Order Scalogram Analysis (POSA) are data analysis procedures and I will use the terms interchangeably. Analyses that attempt to reveal partial ordering do so to display the variables within a research project that can be demonstrated to posses ordering in their associated responses.
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Notes
- 1.
The establishment of important evaluation criteria is often initially a theoretical question but the validity of criteria and sub-features of these is ultimately an empirical question.
- 2.
A monotonic relationship is a relationship where an increasing value for one characteristic is associated with increases in other characteristics and an increase in overall liking of the three-dimensional artworks.
- 3.
For the sake of this illustration I propose that we ignore the experiential and psychological differences between viewing two-dimensional representations of three-dimensional works of art and viewing the three-dimensional artworks themselves. The two experiences are obviously extremely different in their perceptual make-up but above I am illustrating the concept of size rather than investigating the perceptual process and experience involved in the apprehension of size between different dimensional formats of display.
- 4.
For instance, as abstract three-dimensional artwork A is larger than artwork B this may be written as A > B. In POSA the specification of a research project’s content is written: P being a data set of N elements (e.g., three-dimensional artworks, aspects of three-dimensional artworks) (p i …p n ) upon which a set of k different variables (v 1 …v k ) are observed. On this understanding, each element p i may be typified by a profile of attributes or dependent variables (v 1 i …v k i ) and where an attribute is a fully ordered category (as stated above). Each of these elements may be fully ordered in terms of their own variable’s content. However, rankings of elements are potentially different and thus a partial ordering may exist between elements. To enlarge upon Guttman’s understanding of POSA, and applying this to abstract three-dimensional artworks, any two three-dimensional artworks may be considered identical (p i = p j ) if and only if (iff) their respective profiles are identical (v m i …v m j for each and all m = 1…k); p i is greater than p j (p i > p j ) iff v m i ≥ v m j for every m = 1…k. What this means is that for any three-dimensional artwork p i is ranked before p j in terms of some aspect of abstraction, and p j is ranked before p i in terms of some other aspect of abstraction. On the latter understanding the three-dimensional artworks would be considered incomparable.
- 5.
As I have stated above for a comparison between pairs of three-dimensional artworks to be made the relationship ≤ must exist as a binary relation among the objects.
- 6.
It should be noted here that the use of the letters A,B,C, is indicative of the three abstract paintings by the three specified artists. These letters are used in axioms 1 to 3 for simplicity. However, in an actual partial ordering the paintings themselves are not rated; what are rated are characteristics of each of the paintings.
- 7.
I also make the obviously point that in the theoretical definition of reflexivity I am ordering example artworks A. B. C. However, in my example of the ratings of three abstract three-dimensional artworks given in Fig. 3.5, the ordering of paintings is B, A, C. The differences between these orderings is, irrelevant to either examples or analyses.
- 8.
Often the assignment of a qualitative variable to being either nominal or ordinal is determined by the reason for the research investigation and why items are being ranked.
- 9.
Furthermore, in circumstances where a small number of items (artworks) are considered, it is possible to relax the requirements of spatial ordering. In this research, such a relaxation is due to the three-dimensional artworks all being discretely identifiable art works on the one hand whilst potentially possessing similarity in their ranked assessment. Consequent upon this, three-dimensional artworks may be treated as different three-dimensional artworks but equivalent with identical data matrix rows. Moreover, when equivalent three-dimensional artwork profiles exist removing equivalent items and leaving a single three-dimensional artwork to represent this profile simplifies the data matrix. Later it is possible when writing up the research project to reintroduce equivalent profiles to facilitate understanding of the complete partial ordering analysis (Patil and Taillie 2004).
- 10.
The resulting diagram is a Hasse diagram, also called a partial order set diagram, order diagram or line diagram.
- 11.
However, flexibility exists in the drawing of Hasse diagrams. Bruggemann and Patil (2011) have noted that isomorphic Hasse diagrams result where the same partial order relationship is differently drawn and represented in different Hasse diagrams.
- 12.
However, Crowther allows for possible increases in evaluative intricacy through the sub-division of facet elements. Crowther also notes that dimensional combinations potentially exist and that such combinations could play such an important a role in structuring the contextual space of abstract art that the combinations should be considered to exist as separate dimensions in their own right.
- 13.
The other relationship being transivity, which I consider later in this chapter.
- 14.
Guttman’s usage of these symbols, and consequently our use of these symbols, is specific and slightly different to its more usual meaning as a logic symbol in which instance it is taken to mean implies.
- 15.
On the other hand, transitive relationships exist between entities of the same kind.
- 16.
As I have stated earlier, mapping sentences are research tools that have the ability to be used to embody and display the interrelationships between complex research designs that have linked components.
- 17.
Rather than using a single stemmed arrow.
- 18.
The actual artist and three-dimensional artworks that I have chosen in this example are unimportant and my example could be applied to any set of art objects.
- 19.
As well as constructing the Spiral Jetty earthwork, Smithson also filmed its construction and the film exists as documentation of the process.
- 20.
The following section draws heavily upon: Shye and Amar (1985).
- 21.
I earlier claimed that it seems unlikely that the complex psychological processes involved in perceiving two-dimensional art were all perfectly ordered (Hackett 2016).
- 22.
Within the context of this essay, the scalogram may best be thought of in tandem with the notion of Guttman scaling (or cumulative scaling). In Guttman scaling the objective is to develop a uni-dimensional continuous measure for the concept under scrutiny. Under the rubric of a Guttman scale, items are developed and included such that if a person agrees with a statement or item they will have agreed with all earlier items or statements in the Guttman scale. If we are able to construct a reliable Guttman scale then the total score a person receives on the scale will have the following characteristic: knowledge of the score will enable researchers to tell flawlessly in advance a subjects item responses (on a perfect Guttman scale, if a person scores 7 out of ten then it will be the first 7 items with which the respondent agrees rather than any other combination of item responses). In applied settings it is usual for a measure constructed as a Guttman scale to approximate the perfect notion of this scale. Under these conditions the extent to which a set of test items approach being a perfect Guttman scale is investigated using scalogram analysis. A scalogram is defined as “A rectangular matrix in which the columns represent items and the rows subjects. The matrix indicates, for each subject, the category to which it belongs in each item.” Zvulun (1978), p. 238.
- 23.
This research is not a direct testing of Krauss’ models or claims but instead I have chosen the same artworks as she did to permit not only the development of my own conceptual model but also to allow comparative statements to be made between my results and hers.
- 24.
Including other forms of sculpture is beyond the remit of Crowther’s ontology in this current research but is part of my future research.
- 25.
It is an obvious point that rating of images of three-dimensional artwork is very different to rating the works themselves. I faced the same issue to a lesser degree in the research I conducted using a similar assessment procedure with two-dimensional works of art (Hackett 2016). However, in this research I felt the evaluations were meaningful and only slightly impacted upon the dimensions of Crowther’s ontology. I will further discuss this issue later in this book.
- 26.
The plotting of items in a partial order scalogram analysis is always inaccurate to a certain extent. This inaccuracy is quantified in terms of a series of metrics. For the analysis presented in Figs. 4.6 the coefficients were as follows: proportion of profile pairs correctly represented = 0.8352; proportion of comparable pairs correctly represented = 0.8636; proportion of incomparable pairs correctly represented = 0.8085; score—distance weighted coefficient = 0.9927. When using the POSAC an assessment of how well the plotted items are located compared to where they exist experientially for respondents. This measure of goodness-of-fit is termed CORREP; an abbreviation of ‘correctly represented’. If a POSA perfectly represented items in its plots then the CORREP would be 1. However, this situation is unlikely in empirical social research. See Raveh and Landau (1993) for a description of the POSA/POSAC technique and for details of when a CORREP coefficent would be expected to be 1.0. In the present research the coefficients discovered imply that the plot is broadly accurate with an acceptable degree of inaccuracy.
- 27.
By items being placed in incorrect regions I mean that when straight lines were drawn to capture similar values for the analysis of number 8 (fantasy) this resulted in a significant number of values being located in partitioned regions that were dominated by other values.
- 28.
Abstractness as defined by the summation of all of the characteristics upon which I rated the artworks.
- 29.
I have included characteristic 8: fantasy, as legitimately partitioning captured a large proportion of image items.
- 30.
Whilst my analyses discovered five characteristics to be pertinent to my understanding of the twelve selected works of art, it may be that the assessments I made of reproductions of the chosen artwork lacked three-dimensional and other experiential characteristics and therefore that there is a clearly identifiable dimensional based difference between the experience of looking at abstract art “in the flesh’ and in reproductions (this is true whenever a person views an artwork in a catalogue, magazine, on a website, etc.,). This difference could easily be incorporated into a research design by including the format of presentation as a facet of the analysis of research findings.
- 31.
It should be kept in mind that all of these statements are relative to the other eleven three-dimensional works incorporated in the analysis. If other items were assessed with Andre’s Cuts then it is likely that the positioning of this item on all of the characteristics would be different to its present location.
- 32.
Here I have made the case for using respondent meaningful characteristics in survey design. The same holds true of any research approach that is attempting to facilitate understanding of respondents’ understanding and perceptions, such as in-depth interviews, focus groups, projective techniques, etc.
- 33.
By the possibilities of forming a generic understanding of abstract art assessments I mean an understanding that is equally pertinent to both two-dimensional and three-dimensional forms of abstract art.
References
Bruggemann R, Patil GP (2011) Ranking and prioritization for multi-indicator systems: introduction to partial order applications (Environmental and ecological statistics). New York, Springer
Crowther P (2007) Defining art, creating the canon: artistic value in an era of doubt. Oxford University Press, Oxford
Guttman L (1991) Louis Guttman in memoriam: Chapters from an unfinished textbook on facet theory. The Israel Academy of Science and Humanities and the Hebrew University of Jerusalem, Jerusalem, Israel
Hackett PMW (2016) Psychology and philosophy of abstract art: neuro-aesthetics, perception and comprehension. Palgrave, Basingstoke
Krauss R (1979) Sculpture in the expanded field. October 8:30–44
Patil G, Taillie C (2004) Multiple indicators, partially ordered sets, and linear extensions: multicriterion ranking and prioritization. Environ Ecol Stat 11:199–228
Raveh A, Landau S (1993) Partial order scalogram analysis with base coordinates (POSAC): its application to crime patterns in all the states in the United States. J Quant Criminol, 9(1):83–99
Shye S, Amar R (1985) Partial-order scalogram analysis by base coordinates and lattice mapping of the items by their scalogram roles. In: Canter D (ed) Facet theory: approaches to social research. Springer, New York, pp 277–298
Zvulun E (1978) Multidimensional scalogram analysis: the method and its application. In: Shye S (ed) Theory construction and data analysis in the behavioral sciences. San Francisco, Jossey Bass
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Hackett, P.M.W. (2017). A Partial Ordering Mereology for Non-traditional Three-Dimensional Art. In: The Perceptual Structure of Three-Dimensional Art. SpringerBriefs in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-48452-5_4
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