Hello! It’s rather shocking that you’re here given the infinitesimal cyberspace odds so let me start by offering a very hearty 🎉 Welcome to my Blog 🎉.
Let me follow that by trying to clinch your attention with an exceedingly well-crafted purpose statement: the purpose of this blog to explore specific intersections of mathematics, ontology, and dialectics. Not intrigued? Kthxbye 😢. Yes intrigued? Yay well-crafted 😍.
Ok enough with the emojis. What specific intersections, you might ask? Well, the purpose statement has an obvious philosophical flavor in addition to the explicit “mathematics” bit, so I can start by identifying the most common intersection of math and philosophy and continue by telling you that that’s not the one I care about. The most common intersection of math and philosophy is logic, in particular higher-order logic. I don’t care about it, at least not in this blog. I’m not interested in determining the truth-values of chains of propositions quantified over variables. (And in one swoop I draw my curtain on Analytic Philosophy.)
So what mathematical constructs am I interested in instead? Briefly, I’m interested in spaces and trees, rather than higher-order logic. A little less briefly, I’m interested in two specific types of spaces and two specific types of trees, and the ways in which all of these four interact. The two types of spaces are, well, actual space—that three-dimensional thing we all float through that is often discussed with time—and parameter spaces—those n-dimensional things most commonly encountered as scatterplots and histograms. The two types of trees are taxonomies and partonomies. The taxonomy par excellence is the Linnaean tree of life (kingdom, phylum, … genus, species). Partonomies are less familiar but are represented by, for instance, the hierarchical breakdown of jurisdictions in a country, the folder/file structure of the contents of a hard drive, and the syntax trees of natural language utterances.
I am interested in these four things because they construct a framework for parsing everything in the Universe because everything exists in space, is built from smaller things or their interactions, and can be compared to similar things. Rather than pick one domain and analyze the specific classes of things therein as most academics do, I’m interested studying the patterns of composition and typology in general and how they are expressed by spaces and trees. I’m also interested in these four things because they interface seamlessly with the most advanced machine learning and AI techniques, and because any philosophy that does not zealously embrace these moving forward must be discarded IMHO.
I can give some examples of the interactions among physical space, parameter spaces, taxonomies, and partonomies:
Physical space and parameter spaces:
1) In almost all languages, words and idioms describing physical proximity double as words and idioms of abstract similarity, where similarity merely corresponds to proximity in a parameter space. The spatial metaphor for similarity and difference is so entrenched it might represent a deep fact about cognitive statistical systems in general and not just the specifically human kind.
2) The right-left political spectrum exists in the parameter space of political positions. When legislators arrange themselves in their halls, they almost never seat themselves according to their geographic provenance but instead according to their place on the political spectrum. The spatial metaphor of similarity thus ceases to be merely a metaphor in these situations, because humans have arranged themselves in space by similarity.
Parameter spaces and taxonomies:
1) The diversity of life is extremely well characterized by the Linnaean taxonomy, which is a tree. The taxonomy flounders among sexually reproducing entities at sufficiently small amounts of variation because genetic recombination enacts a cross product among all the features. At this level, the best characterization ceases to be a tree and is instead a parameter space whose dimensions are the ways in which the members of the population vary. Given gradualistic evolution, the spaces for every (sexually reproducing) population connect together into a tree. However, because these spaces do not have the same dimensions, there is no “Space of Life” (pace Dawkins), instead something that is an odd combination of a tree and parameter spaces, which I call a metatope.
Physical space, parameter spaces, and taxonomies:
1) According to the wave model of language change, linguistic innovations start from a center and spread out in all directions. The two-dimensional surface of the Earth imposes a geometry on such spread, so that it is not possible for all 16 combinations (24) resulting from the cartesian product of 4 (or more) innovated/uninnovated pairs to exist. If innovations are constrained to the boundaries of previous innovations, then not only is the discrete space of cartesian possibilities not filled, but the structure of linguistic diversity begins to look like a tree. Additionally, dimensional analysis of linguistic variants in certain societies would discover not two dimensions, but perhaps three, such as in the case of social stratification producing low- to high-class variants.
All four together:
1) The Universe can be described by locating all of the point-particles in space with their velocities, or by locating the one point in the Universe’s Hilbert space corresponding to its quantum state (which, to bastardize, takes the locations and velocities of point-particles as dimensions). On one extreme there is one 3-dimensional space containing as many points as there are particles, on the other extreme there is one n-dimensional space containing 1 point (where n is proportional to the number of particles, again bastardizing a bit). Between these two extremes, there can be cases with many m-dimensional parameter spaces each containing as many points as there are subsystems describable by those parameters, where m varies by subsystem class. These parameter spaces of different dimensions can be stitched together vertically by using the contents of some as dimensions of others (i.e. where composition is involved) with the lowest subsystems having as dimensions the relative positions and velocities of the point particles, or horizontally by adding or removing dimensions as needed to move from subsystem class to class (producing a graph with nodes for parameter spaces and vertices for the addition or removal of a dimension). This produces a fuzzy mathematical object combining point-filled spaces and graphs (a dendrotope in the case of biology, as trees are a subtype of graph) that depends on the identities of the elementary point-particles and what they have actually composed, which bridges the points-in-3D and point-in-Hilbert-space descriptions of the Universe in a very specific way.
You can also checkout a former direction for this blog as a vehicle for a book or two in the same vein, which I have since suspended.