I do not deny but nature, in the constant production of particular beings, makes them not always new and various, but very much alike and of kin one to another: but I think it nevertheless true, that the boundaries of the species, whereby men sort them, are made by men.
—John Locke (1689), An Essay Concerning Human Understanding
Abstract
Philosophy of biology widely acknowledges that taxonomic practice conditions the delimitation of species; however, this impact has remained predominantly theoretical, lacking precise formal quantification. This article presents a mathematical framework utilizing morphospaces to model this influence across three hierarchical levels: the theoretical morphospace (\(\:\Omega\:\)), encompassing the combinatorially possible; the ontological morphospace (\(\:{{\Omega\:}}_{O}\)), a subset of the theoretical morphospace constrained by biological viability; and the epistemic morphospace (\(\:{{\Omega\:}}_{E}\)), a quotient space of the ontological morphospace defined by taxonomic rules. By treating diagnostic criteria as logical constraints, it becomes possible to quantify the impact of taxonomic decisions on both the number of recognized species and the information cost within the system. Furthermore, the necessary conditions for treating systematics as a formal problem are established. Finally, the critical importance of adopting objective and computable diagnostic criteria to ensure greater rigor in biodiversity classification is highlighted.
Data and Software Availability
The “Hierarchical Morphospace Simulator”, an interactive browser-based simulation environment, can be executed online via GitHub Pages (https://albertogonzalezcasarrubios.github.io/hierarchical-morphospace-simulator/ Its complete versioned code is archived on Zenodo (https://doi.org/10.5281/zenodo.20693984 This tool allows users to visually explore the hierarchical structures of the theoretical (\(\:\Omega\:\)), ontological (\(\:{{\Omega\:}}_{\text{O}}\)), and epistemic (\(\:{{\Omega\:}}_{E}\)) morphospaces while calculating and monitoring the real-time behavior of both the Lumping Factor (\(\:L\)) and the Epistemic Gap (\(\:{I}_{E}\)). The application is open-source and fully available online.
Notes
It is crucial to distinguish the ontological morphospace (\(\Upomega_{o}\)) from the classical concept of the realized morphospace. Whilst the latter is limited to forms produced by historical contingency (the factual), \(\Upomega_{o}\) represents the space of biological viability (the possible under natural constraints). Consequently, \(\Upomega_{o}\) contains all combinations of characters that are structurally and functionally coherent, regardless of whether they exist in the current or fossil record.
Strictly speaking, since the epistemic constraints \(\mathcal C_{\epsilon}\) are formulated over the same basis set of morphological characteristics (Requirement 3), the resulting equivalence relation (\(\sim_{E}\)) can be formally evaluated over the entire theoretical morphospace (\(\Upomega\)). However, because alpha-taxonomy operates on realized or biologically viable organisms, \(\Upomega_{E}\) is defined specifically as the quotient space of the ontological morphospace (\(\Upomega_{o}\)). In formal terms, the epistemic relation inducing \(\Upomega_{E}\) constitutes the restriction to \(\Upomega_{o}\) of a more general equivalence relation defined over \(\Upomega\), thereby avoiding the formalization of taxonomic categories composed of biologically impossible morphotypes.
The abbreviation NP stands for nondeterministic polynomial time. In computational complexity theory, an NP problem is a decision problem (requiring a “yes” or “no” answer) where any proposed solution can be verified efficiently by a computer, even though finding that solution from scratch may be exceptionally difficult. The class #P represents the counting counterpart to NP; instead of asking whether a valid configuration exists, it determines how many valid configurations exist; therefore, it constitutes a counting problem rather than a decision problem. A problem is designated as #P-complete if it belongs to the most computationally demanding problems within #P, meaning that no efficient, generalized algorithm is known to exist for its resolution.
In this regard, this interpretation closely parallels the species concept proposed by Andersson (1990), who stated that species may be visualized as clusters of individuals (in this context, morphotypes) within a multidimensional space where each dimension represents a character axis. Consequently, speciation constitutes a compartmentalization of character hyperspace, whilst species descriptions serve to circumscribe, in an oblique way, such compartments.
The preference for Hartley entropy over Shannon entropy is based on epistemic considerations. In an a priori analysis of the space of taxonomic possibilities, observed frequencies (probabilities) for each theoretical morphotype are unavailable. Furthermore, it is not desirable to bias the measure by the current ecological abundance of species. Hartley entropy represents the maximum information capacity of the system, based on the assumption that, within the theoretical plane, any biologically valid form constitutes an equiprobable state for the purposes of classification.
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Acknowledgments
AGC was supported by a predoctoral contract from the Community of Madrid (PIPF-2023).
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González-Casarrubios, A. How Much Diversity Is Lost in Classification? Morphospaces as a Metric of Taxonomic Impact. Biol Theory (2026). https://doi.org/10.1007/s13752-026-00549-4
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DOI: https://doi.org/10.1007/s13752-026-00549-4
Facts Only
* John Locke noted that nature makes particular beings alike and of kin, but men set species boundaries.
* A mathematical framework uses morphospaces to model taxonomic influence across three levels: theoretical ($\Omega$), ontological ($\OmegaO$), and epistemic ($\OmegaE$).
* Diagnostic criteria are treated as logical constraints to quantify the impact of taxonomic decisions on species numbers and information cost.
* The work establishes necessary conditions for treating systematics as a formal problem.
* The ontological morphospace ($\OmegaO$) represents biological viability, including all functionally coherent combinations regardless of current existence.
* The epistemic morphospace ($\OmegaE$) is defined as the quotient space of the ontological morphospace ($\OmegaO$).
* Hartley entropy is preferred over Shannon entropy because observed frequencies for theoretical morphotypes are unavailable, and classifcation should assume equiprobability among biologically valid forms.
* Species may be visualized as clusters of morphotypes in a multidimensional space, where speciation is the compartmentalization of character hyperspace.
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This text appears to be a piece of original, highly specialized academic writing synthesizing philosophy, biology, and computational mathematics into a formal framework regarding taxonomic impact.