2.1. Hierarchy in Mathematical Number Theory, in Geometry, in Topology

Practical implementing, exploiting and benefiting from hierarchical clustering can be very effective and efficient (Murtagh Reference Murtagh2017). Number systems can be very worthwhile to consider. The real number system is commonly used and a natural tree or hierarchy representation of any decimal, also termed 10-ary (or 10-adic), number is a regular 10-way tree. Apart from having any other base system for number representation, there is often consideration of p-adic number systems, where p is a prime number. A p-adic number has a natural tree or hierarchy representation as a regular p-way tree. Consider just how computers are based on binary numbers, also termed bits, binary digits. In earlier times, there were proponents for computers to be based on ternary, i.e. 3-adic numbers (Stakhov Reference Stakhov2009).

In topology in mathematical terms, a hierarchy, also termed a tree structure, is a representation of an ultrametric topology. Such a structuring of the data as a tree is defined by considering distance between the points defining the nodes of this tree. Given any two points, i, j, symmetry and positive definiteness properties are: d(i,j)≥0, d(i,j)=0 implies i=j; d(i,j)=d(j,i). The further inequality that holds is the strong triangular inequality, also termed the ultrametric inequality: d(i,k)≤max{d(i,j),d(j,k)} for distance d and for a triplet of points, i, j, k. This is informally expressed as ‘stronger’ relative to the fundamental property of what constitutes a metric or distance: d(i,k)≤d(i,j)+d(j,k). Now, the natural geometric ordering of metric valuations is: a real line. But when we have partial order – alternatively expressed, hierarchical order – then this is the ultrametric case: at issue is a hierarchy or rooted tree.

Informally expressed, in terms of genealogy – a genealogical tree – the ultrametric distance is the least common ancestor distance.

In regard to terminology, a Euclidean metric endowed space can be termed an Archimedean space, while an ultrametric or tree topology, that is, an ultrametric endowed space, is termed a non-Archimedean space.

The latter, triangular inequality that defines a metric, can be most informally described as follows: consider travelling from location i to location k. If there is deviation via location j, then the travel distance will be longer, or at the very best, equal to what it is without this deviation. That is just a very informal description. Now an ultrametric is a metric but with a stronger requirement in the strong triangular, or ultrametric, inequality. For this to always hold, then all triplets of points comprise an isosceles triangle, i.e. with two equal length sides, with a small base. Additionally, an equilateral triangle also satisfies the strong triangular or ultrametric inequality. An illustration is given in Figure 1 (right) of such an approximation. Informally, in the graph displayed here, we may claim that we are seeking the closest points to the upper right-hand-side point, and we are prepared to approximate, with quite a good approximation, the isosceles triangle with approximately equal longer sides. From the point of view of the two black points in the base of this triangle, the upper right-hand-side point is an exception or anomaly relative to their (i.e. these black points) proximity. In Figure 1, left, there is the explanation, based on the display, of the ultrametric inequality.

Figure 1.

Illustration of how a hierarchical display can depict what is exceptional or anomalous. (Left) Hierarchy, or rooted tree, depiction. (Right) Illustration of how hierarchy can be a graphical representation of exceptionality.

In Figure 1, the strong triangular inequality defines an ultrametric: every triplet of points satisfies the relationship: d(x,z)≤max{d(x,y), d(y,z)} for distance d (cf. by reading off the hierarchy, how this is verified for all x,y,z: d(x,z)=3.6; d(x,y)=3.6; d(y,z)=1.1). In addition, the symmetry and positive definiteness conditions hold for any pair of points. Once we either set up, or have as an approximation here, a triangle of points that form an isosceles triangle with small base, then there is a hierarchical depiction.

2.2. Metric Geometry and Ultrametric Topology, Hierarchy and Symmetry

An ultrametric representation is quite appropriate for genealogical relations. The ultrametric, i.e. hierarchy-derived or tree-derived distance, can be seen in Figure 1(left). This can also be expressed as the closest common ancestor distance. So this refers to genealogical or genetic, and sometimes even creative processes starting with the root, and then growing the outputs and outcomes in a tree or hierarchy structure. Of course, a tree itself is relevant here. Phylogeny and evolution are very relevant for a tree, hierarchical, hence ultrametric representation or expression. This is relevant also for, e.g., a thesaurus, or all that is related to the brain cerebellum.

Our presentation in this article has focused on the ultrametric or tree topology, and on how number systems can be associated too, with such topologies. For practical data analytics, the computational construction of a hierarchical clustering from data is usually carried out as follows. The analysis starts with the set of terminal nodes. This is followed by carrying out, stepwise, an agglomeration into a new cluster that is associated with a non-terminal node in the rooted tree. Mathematically, a hierarchy is a rooted tree, i.e. it has the root node. In practical applications it may be a case of constructing the hierarchy, termed a dendrogram, and then cutting it horizontally in order to yield a partition of the object set at issue, i.e. the set of terminal nodes in the dendrogram.

In Figure 2, the data on which this was based were the frequencies of occurrence of the eight nouns in 24 successive, non-overlapping segments of Aristotle’s Categories. The eight terms comprise the terminal (or leaf) nodes. In the hierarchy these are singleton clusters. The non-terminal nodes, i.e. clusters, are numbered in sequence (the numbers beside the relevant nodes). If one cuts horizontally through the dendrogram or tree or hierarchy, one gets a partition of the set of eight terms. The clusters associated with the non-terminal nodes are shown with ellipses here.

Figure 2.

Hierarchical clustering of eight terms.

Some main geometrical properties of an ultrametric space are as follows (Murtagh Reference Murtagh2017; Lerman Reference Lerman1981). (1) In an ultrametric space, all triangles are either isosceles with small base, or equilateral. (2) Every point of a circle in an ultrametric space can be considered, from the foregoing ultrametricity properties, as a central point of the circle. (3) In an ultrametric topology, every ball is both open and closed, termed clopen. (4) An ultrametric space is 0-dimensional. (5) Informally, in an ultrametric space everything ’lives’ in a hierarchy expressed by a tree. Later there will be a reference to such properties of ultrametric spaces when Matte Blanco’s Bi-Logic is considered. Matte Blanco’s symmetry, i.e. unconscious reasoning, is most appropriately described in terms of an ultrametric topology (Lauro Grotto Reference Lauro Grotto2008).

In Murtagh (Reference Murtagh2009a) there is description of various symmetries that are integral to hierarchy: a hierarchy is an example of a wreath product group in mathematical group theory – this is just the fact that each and every node in the hierarchical tree structure can rotate or, in general, permute its child nodes. A major symmetry too, to be at issue in Matte Blanco’s, is the taking as similar and at least practically, and application-wise, the cluster contents. This will be related, later in this article, as a perspective on Freudian condensation and how Matte Blanco has this as the fundamental property of one’s unconscious reasoning.

2.3. Ultrametric Topology: in Physics, Time Series Analysis, Literature

Hierarchy, as a branching process, is a most appropriate means of expressing suboptimal and/or discrete energy states or levels. That includes quantum physics and lots of forms of genealogy, evolutionary processes, etc. A hierarchy, as seen, incorporates a sequence of partitions. There is a partial order on the clusters/nodes, and a total order on the partitions. Related to this can be the important role of p-adic representation for physics on very small scales (i.e. for quantum physics). Volovich (Reference Volovich2010) poses the general principle that the fundamental physical laws should be invariant under the change of the number field. This leads to the following ambitious statement: ‘If these ideas are true then number theory and the corresponding branches of algebraic geometry are … the ultimate and unified physical theory’ (Volovich Reference Volovich2010, 85).

Physics has been considered now, and the following, related to physics, is from Khrennikov (Reference Khrennikov2004, xiii):

Ultrametric topology was introduced by Marc Krasner in Reference Krasner1944 (referenced in Murtagh Reference Murtagh2012a). The ultrametric inequality was formulated by Hausdorff in Reference Hausdorff1934.

Above, it has been indicated that (i), according to Simon, ultrametric, i.e. hierarchical perspectives are crucial for the handling of complex systems; (ii) that ultrametric and closely linked p-adic perspectives are important for physics; in addition, (iii) later it will be shown how fundamental this all is for psychoanalysis and the human unconscious and conscious modes of being; (iv) and (the following is really important), how fundamental this is for its implications, and also for such aspects as the remarkable simplicity of all that is associated with high dimensional spatial representations for data and information.

As ambient dimensionality increases, distances become more and more ultrametric (Rammal etal. Reference Rammal, Toulouse and Virasoro1986; Murtagh Reference Murtagh2009b; Hall etal. Reference Hall, Marron and Neeman2005; Donoho and Tanner Reference Donoho and Tanner2005). That is to say, a hierarchical embedding becomes more and more immediate and direct as dimensionality increases. Hence there is inherent hierarchical structure in high dimensional data spaces. Points in high dimensional spaces become increasingly equidistant with an increase in dimensionality (Donoho and Tanner Reference Donoho and Tanner2005, 9453): ‘not only are the points [of a Gaussian cloud in very high dimensional space] on the convex hull, but all reasonable-sized subsets span faces of the convex hull. This is wildly different than the behavior that would be expected by traditional low-dimensional thinking’.

There are remarkable symmetries in very high dimensional spaces: very simple structures in very high dimensions are not necessarily trivial. Even very simple structures (hence with many symmetries) can be used to support fast and perhaps even constant time worst-case proximity search. In the machine learning framework, there are important implications ensuing from the simple high dimensional structures. Very high dimensional clustered data contain symmetries that in fact can be exploited to read off the clusters in a computationally efficient way. What we might want to look for in contexts of considerable symmetry are (what we might characterize as) the impurities or small irregularities that detract from the overall dominant picture.

In Murtagh (Reference Murtagh2005) there is the ultrametric embedding of time-varying signals, including biomedical, meteorological, financial and other data. At issue are the inherent hierarchical properties in the data. One finding is that eyegaze trace data, including saccades, which are the jumping around of human eye direction, can be very high in ultrametricity. Otherwise expressed, eyegaze trace data can be well expressed, or represented, as a hierarchy. A further outcome from EEG data is in regard to ultrametricity characterizing normal sleep behaviour and forms of epilepsy.

In Murtagh (Reference Murtagh2012b) there is a degree of ultrametric, hence inherently hierarchical, content in literature and in other forms of written narratives and text. Included are tales from the Brothers Grimm, Jane Austen novels, dream reports, air accident reports, and James Joyce’s Ulysses. Dream reports and Joyce’s Ulysses are the most ultrametric, i.e. the most inherently hierarchical in their semantic content.

Finally, it is the case that ultrametric spaces offer excellent computational benefits. Nearest neighbour search in ultrametric space can be carried out in O(1) or constant time (Murtagh Reference Murtagh2004, Reference Murtagh2017).

Quite clearly, one’s conscious mental processes are in the context of three-dimensional ambient space, and the time dimension, so that Euclidean-distance endowed space is most relevant. Then if the ambient space is very high dimensional, and perhaps even of infinite dimensionality, the ultrametric is fully appropriate. That is, such a very high dimensional space is inherently ultrametric, hierarchical. No wonder, therefore, that the human unconscious is to be considered as an infinite space.