Abstract
In this chapter, we investigate by different types of citation analysis the structure and dynamics of Late Analytic Philosophy in order to shed light on the processes of fragmentation and specialization. We try to answer questions such as: When did these processes begin? What is their pace? How did they carve the overall structure of the field? The key notion of documental space is introduced to guide the analyses: the documental space is defined as the universe of documents that are cited by the articles published in the five analytic philosophy journals that form our bibliographic representation of Late Analytic Philosophy. The first set of analyses investigates the structure of Late Analytic Philosophy using a co-citation map. The next set of analyses focuses instead on the dynamics of the field. Patterns in the citation trends of the most cited documents are examined, a data-driven periodization of the documental space is introduced, and, lastly, longitudinal co-citation maps are analyzed. The chapter concludes with a theoretical reflection on the meaning of citation counts and co-citation clusters.
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Notes
- 1.
The interactive map is available at https://homepage.univie.ac.at/maximilian.noichl/full/zoom_final/index.html.
- 2.
The query can be launched directly using the link https://www.webofscience.com/wos/woscc/summary/c73662cc-5500-443b-bcd0-4853d20ded07-1b5c1b52/date-ascending/1 Note that the number of displayed results depends on the subscription of the user. Users with restricted access may obtain fewer results.
- 3.
The median is defined as the “middle” value that divides the greater and lesser halves of a distribution. An important property of the median is that it is not skewed by a small proportion of extremely large or small values. Therefore, it provides a better representation of the “typical” value of a skewed distribution.
- 4.
The average institution, by contrast, produces 11.2 articles with a standard deviation of 23.6!
- 5.
In February 2022, the only journal with at least one editorial board member who is not affiliated with an American, British, or Australian university is Mind.
- 6.
In the previous section, for instance, we cited Kreuzman (2001) several times, but in the bibliography at the end of the chapter, it appears only once.
- 7.
Formally, the first indicator C for the year y is equal to the sum of the number of cited references c of the N citing articles a published in year y:
$$\begin{aligned} C_y = \sum _{a=1}^{N_y} c_a \end{aligned}$$(4.1)The second (the average \(\bar{c}\) of year y) is the total divided by the number of citing articles N published in year y:
$$\begin{aligned} \bar{c}_y = \frac{1}{N_y} C_y = \frac{1}{N_y} \sum _{a=1}^{N_y} c_a \end{aligned}$$(4.2)Lastly, the standard deviation \(\sigma \) for year y is defined as
$$\begin{aligned} \sigma _y = \sqrt{\frac{\sum _{a=1}^{N_y} (c_a - \bar{c}_y)^2 }{N_y}} \end{aligned}$$(4.3).
- 8.
https://github.com/python/cpython/blob/main/Lib/difib.py. The script is available in the Supplementary Materials (Appendix A).
- 9.
As noted above, the computation of the pairwise similarity between strings requires a lot of computational resources. It was not possible to calculate a similarity matrix of \(72{,}385 \times 72{,}385\) strings. Hence, the calculation was restricted within the author oeuvre. More precisely, only authors with 10 citations or more were considered in the first step and authors with 5 citations or more in the second step. Furthermore, cited references pointing to articles were not considered as they were already cleaned with CRExplorer.
- 10.
A further limitation of the dataset, which however cannot be structurally addressed, concerns the quality of the WoS capturing of the cited references’ text strings. Based on our experience, we can say that the bibliographic records of philosophical publications in WoS sometimes contain errors in the cited references: some records lack cited references even if the original publications feature them, or cited references are linked to wrong documents because of indexing errors. The records in the Journal of Philosophy are particularly susceptible to these capturing errors because of the peculiar citation style of this journal, where cited references are not collected into a bibliography at the end of the articles but appear only in footnotes.
- 11.
We will return to the meaning of citation counts in the final section of the chapter (see Sect. 4.5.1).
- 12.
The Lorenz curve is shown in the Supplementary Materials (see Appendix A). Consider that the Gini coefficient may be overestimated because the list of cited authors is less clean than that of cited references. Remember that the matching algorithm was run only on authors with 10 citations or more in the corpus which, given the skewness of the distribution, represent only a fraction of all the cited authors. The ranking, by contrast, is reliable at least for the higher ranks.
- 13.
VOSviewer was developed by Ludo Waltman and Nees Jan van Eck and is freely available at www.vosviewer.com. We used the version 1.6.17.
- 14.
Changing the threshold in the interval \([40-60]\) citations does not change the overall structure of the map. A lower threshold, by contrast, makes it difficult to distinguish the signal from the noise.
- 15.
We will return on the interpretation of the clusters in Sect. 4.5.2.
- 16.
The Philosophical Investigations are almost external to the cluster. Wittgenstein’s work is in fact placed in the area of Philosophy of Mind, suggesting that the clustering algorithm and the placing algorithm of VOSviewer partly conflict about the correct classification of Wittgenstein. This is expected given the variety of topics addressed by this work.
- 17.
The citation density CD is defined as the ratio between the number of citations of the cluster and the number of cited references in the cluster. The normalized citation density \(\overline{CD}\) of cluster c is defined as the ratio between the citation density of the cluster and the citation density of the entire map \(CD_A\):
$$\begin{aligned} \overline{CD_c} = \frac{CD_c}{CD_A} \end{aligned}$$(4.4)A normalized citation density higher than 1 means that the cluster is cited over the average, and a citation density lower than 1 means that the cluster is cited under the average.
- 18.
Technically, this is due to the fact that the algorithm belongs to the family of hard clustering algorithms. Other classes of algorithms are able to produce fuzzy clustering in which an item can belong to more than one cluster at the same time (Hennig et al., 2016).
- 19.
From this point of view, also the 2-dimensional nature of the map causes a loss in the information originally contained in the dataset. The 2D representation is obtained by VOSviewer by a (variant of a) statistical technique called Multi-Dimensional Scaling (MDS) starting from the co-citation matrix that contains the co-citation frequencies of each pair of cited references (van Eck et al. 2010). The input of MDS are the relative pair-wise distances between data points. From these data, the goal of MDS is to find for each data point a set of coordinates in a low-dimension space (usually a 2D space, i.e., a plane) such that the between-point distances are preserved as well as possible (Borg and Groenen, 2010). The task of MDS is conceptually analogous to the process of reconstructing on a 2D map the coordinates of a group of cities based on the table reporting their pairwise distance. MDS, however, can produce also artifacts, i.e., structures or patterns that are visible on the map but that are not present in the original data. They are generated by the process of dimensionality reduction and can be easily understood by the following example. Imagine that we have four points in a three-dimensional space, each one located at the same distance from the others, like the vertices of a three-sided pyramid, all sides of equal length (see Borg & Groenen 2010, Chap. 13.3). When we try to place the four points in a two-dimensional plane, we can respect the equal distance only for three points out of four. The fourth point will lie almost at the center of a bi-dimensional triangle (as if we were looking at the pyramid from above) so that its distance from the other points will always be shorter than the distances between the three points themselves. Without knowing the original three-dimensional structure and by looking only at the two-dimensional map, we would wrongly conclude that the fourth point is closer to the other three. The wrong conclusion arises from the fact that the two-dimensional map necessarily distorts the three-dimensional structure because it suppresses the third dimension, which however carries essential information (the equal distance between the fourth point and the other three points). By losing such information, it introduces an artifact. Interestingly, MDS can generate also the opposite artifact: points that are placed far away in the map can be however connected by “tunnels” in hidden dimensions (Leydesdorff and Rafols, 2009).
- 20.
In rare cases, old publications are re-discovered by the community and start a new life by receiving fresh citations. In bibliometrics, they are called “sleeping beauties” and are a very rare phenomenon in the scientific literature (Van Raan, 2004).
- 21.
In rare cases, books receive citations in the form of drafts before their publication. These cases were however ignored.
- 22.
Formally, the relationships between the three indicators \(\overline{C'}\), \(\overline{C''}\), and \(\overline{C'''}\) for the reference r in year y are the following:
$$\begin{aligned} \overline{C'_r} = \frac{C_r}{C_y} \le \overline{C''_r} = \frac{C_r}{R_y} \le \overline{C'''_r} = \frac{C_r}{D_y} \end{aligned}$$(4.5)where \(C_y\) is the total number of citations in year y, \(R_y\) is the total number of distinct cited references in year y, and \(D_y\) is the number of citing documents in year y. The ordering of the indicators depends on the different sizes of the denominator: in most common cases, \(C_y \ge R_y \ge D_y\). The trend of these three indicators in time is different, with D showing the slightest growth rate (+35%) and C the highest (+385%). By contrast, note that the numerator is always the raw number of citations \(C_r\).
- 23.
In truth, the total number of available citations, which is a sum, is conditioned by articles with a disproportionately long bibliography (i.e., that cite a lot of cited references).
- 24.
Formally, time-series clustering is the process of unsupervised partitioning of a dataset \(D = {F_1, F_2, ..., F_n}\) made of n time-series data into a set of clusters \(C = {C_1, C_2, ..., C_k}\), in such a way that homogeneus time-series are grouped together based on some similarity measure. \(C_i\) is called a cluster, where \(D = \bigcup _{i = 1}^{k} C_i\) (the partitioning is complete, i.e., all time-series are attributed to at least one cluster) and \(C_i \cap C_j = \emptyset \) for \(i \ne j\) (the partitioning is exclusive, i.e., no time-series can belong to more than one cluster at the same time).
- 25.
To calculate the similarity between clusters, several methods are available. We used a standard complete-linkage method (also known as farthest neighbor clustering), in which the distance between two clusters equals the distance between the two farthest elements in the clusters. This method does not suffer from the “chaining phenomenon” that affects single-linkage method, in which the distance between two clusters equals the distance between the two closest elements. By experimenting, we found that it gives better results also than the average-linkage method.
- 26.
Its main drawback is that it cannot adjust the clusters after merging: after two clusters are merged based on their similarity, they can no longer be split. This may generate imprecise solutions.
- 27.
The cosine is the dot product of the vectors divided by the product of their lengths. Formally, given two vectors \(\textbf{a}\) and \(\textbf{b}\), their cosine similarity \(S_c\) is the cosine of the angle \(\theta \) between them:
$$\begin{aligned} S_c := \cos {\theta } = \frac{\textbf{a}\cdot \textbf{b}}{\Vert \textbf{a}\Vert \cdot \Vert \textbf{b}\Vert } \end{aligned}$$(4.6)where \(\Vert \cdot \Vert \) is the length of the vector. Since the values in our vectors are always positive integers, the coefficient ranges from 0 (no similarity) to 1 (maximum similarity).
- 28.
Note, by contrast, how different they would result if we decided to use as metric the distance between the tips of the vectors, that are placed far away from each other.
- 29.
The Cosine similarity is not the only measure available for assessing the similarity between vectors. In the earlier experiments in bibliometrics with author co-citation networks, for instance, the Pearson correlation coefficient R was frequently used McCain (1990). However, later studies have questioned the reliability of Pearson’s R as a similarity measure (Ahlgren et al., 2003; van Eck and Waltman, 2008). In addition to the Cosine, we tested on our data the Association Strength, the similarity metric used in VOSviewer to normalize co-citation frequencies. The final results are qualitatively very similar: the dataset is partitioned into three periods with slightly different limits than those found with the Cosine similarity.
- 30.
Note that the cells on the diagonal are blank to highlight the shade of the off-diagonal cells, but they should be the darkest as they are always equal to 1.
- 31.
The cells on the diagonal of the matrix correspond to the links that the nodes have with themselves. Since this information is redundant, these self-links were removed from the network.
- 32.
Note that the inclusion threshold was increased compared to the previous two maps to compensate for the citation inflation. A lower threshold introduces too much noise into the visualization.
- 33.
Research evaluation frameworks that reward citations as such have induced diffuse gaming strategies such as the strategic use of self-citations to boost indicators (Baccini et al., 2019).
- 34.
A clear example of the limits of the clustering algorithm is its inability to attribute a document to more than one cluster. There is no reason why a taxonomy of philosophy should avoid overlapping categories.
- 35.
A philosophical discussion of the ontology of the documental forces may take inspiration from debates on the nature of forces such as natural selection.
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Petrovich, E. (2024). Structure and Dynamics of Analytic Philosophy. In: A Quantitative Portrait of Analytic Philosophy . Quantitative Methods in the Humanities and Social Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-53200-9_4
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