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Specieslike clusters based on identical ancestor points

Samuel Allen Alexander

TL;DR

The paper develops a formal, genealogical framework for defining species in an infinite biosphere by introducing the identical ancestor point axiom (IAP) and the convexity axiom, which together define specieslike clusters as connected sets that resist permanent, intra-cluster splits. It then establishes a genericity-based program to guarantee the existence of maximal specimen-sets under additional constraints, notably the common ancestor property (CA) and the reflection property (REF). The main contributions are (i) the formalization of IAP and CONV into specieslike clusters, (ii) a generator-based objective perspective that leads to an almost-disjoint overlap structure among infinite clusters, (iii) a genericity framework proving the existence of maximal IAP∩CONV∩CA∩REF-sets containing any given organism, and (iv) a discussion of objections and ring-species considerations to reinforce the plausibility and robustness of the proposed axioms. Collectively, the work provides an abstract, mathematically grounded path toward an objective classification of species based solely on genealogical relations, with potential relevance for philosophical debates on species concepts and for theoretical explorations in phylogenetics.

Abstract

We introduce several axioms which may or may not hold for any given subgraph of the directed graph of all organisms (past, present and future) where edges represent biological parenthood, with the simplifying background assumption that life does not go extinct. We argue these axioms are plausible for species: if one were to define species based purely on genealogical relationships, it would be reasonable to define them in such a way as to satisfy these axioms. The main axiom we introduce, which we call the identical ancestor point axiom, states that for any organism in any species, either the species contains at most finitely many descendants of that organism, or else the species contains at most finitely many non-descendants of that organism. We show that this (together with a convexity axiom) reduces the subjectivity of species, in a technical sense. We call connected sets satisfying these two axioms "specieslike clusters." We consider the question of identifying a set of biologically plausible constraints that would guarantee every organism inhabits a maximal specieslike cluster subject to those constraints. We provide one such set consisting of two constraints and show that no proper subset thereof suffices.

Specieslike clusters based on identical ancestor points

TL;DR

The paper develops a formal, genealogical framework for defining species in an infinite biosphere by introducing the identical ancestor point axiom (IAP) and the convexity axiom, which together define specieslike clusters as connected sets that resist permanent, intra-cluster splits. It then establishes a genericity-based program to guarantee the existence of maximal specimen-sets under additional constraints, notably the common ancestor property (CA) and the reflection property (REF). The main contributions are (i) the formalization of IAP and CONV into specieslike clusters, (ii) a generator-based objective perspective that leads to an almost-disjoint overlap structure among infinite clusters, (iii) a genericity framework proving the existence of maximal IAP∩CONV∩CA∩REF-sets containing any given organism, and (iv) a discussion of objections and ring-species considerations to reinforce the plausibility and robustness of the proposed axioms. Collectively, the work provides an abstract, mathematically grounded path toward an objective classification of species based solely on genealogical relations, with potential relevance for philosophical debates on species concepts and for theoretical explorations in phylogenetics.

Abstract

We introduce several axioms which may or may not hold for any given subgraph of the directed graph of all organisms (past, present and future) where edges represent biological parenthood, with the simplifying background assumption that life does not go extinct. We argue these axioms are plausible for species: if one were to define species based purely on genealogical relationships, it would be reasonable to define them in such a way as to satisfy these axioms. The main axiom we introduce, which we call the identical ancestor point axiom, states that for any organism in any species, either the species contains at most finitely many descendants of that organism, or else the species contains at most finitely many non-descendants of that organism. We show that this (together with a convexity axiom) reduces the subjectivity of species, in a technical sense. We call connected sets satisfying these two axioms "specieslike clusters." We consider the question of identifying a set of biologically plausible constraints that would guarantee every organism inhabits a maximal specieslike cluster subject to those constraints. We provide one such set consisting of two constraints and show that no proper subset thereof suffices.
Paper Structure (15 sections, 10 theorems, 2 equations, 7 figures)

This paper contains 15 sections, 10 theorems, 2 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Infinite Biospheres
  3. Two Species Axioms
  4. The identical ancestor point axiom
  5. The convexity axiom
  6. Specieslike clusters
  7. Objective Species
  8. Generic Properties of Organism-Sets
  9. Positive genericity results
  10. Negative genericity results
  11. An Answer to Informal Question \ref{['mainqstn']}
  12. Replies to Anticipated Objections against the Identical Ancestor Point Axiom
  13. Intuitive arguments
  14. The ring species argument
  15. Summary and Conclusion

Key Result

Lemma 1

(König's Lemma) If $G$ is an infinite biosphere and $v\in G$ has infinitely many descendants, then there is an infinite directed path in $G$ starting at $v$.

Figures (7)

  • Figure 1: Different types of permanent splits. Left: A two-way permanent split. Right: A one-way permanent split.
  • Figure 2: An example of an infinitary biosphere in which a vertex fails to be contained in any maximal specieslike cluster.
  • Figure 3: Negative genericity examples. (a) The example in Case 1 of Proposition \ref{['firstnegativeprop']}. (b) The example in Case 2 of Proposition \ref{['firstnegativeprop']}. (c) The example in Case 3 of Proposition \ref{['firstnegativeprop']}. (d) The example in Proposition \ref{['secondnegativeprop']}.
  • Figure 4: Three $\mathrm{IAP}\cap\mathrm{CONV}\cap\mathrm{CA}\cap\mathrm{REF}$-maximal sets, all equivalent (in the sense of Definition \ref{['simequivalencerelndefn']}), in the simple infinite biosphere of Example \ref{['atomicspeciesexample']} part 1 with $k=3$. The fact that some vertices have $>2$ parents is of course biologically unrealistic, but the example was chosen for its mathematical elegance.
  • Figure 5: Left: an infinite biosphere consisting of a single two-way permanent split, and the two corresponding $\mathrm{IAP}\cap\mathrm{CONV}\cap\mathrm{CA}\cap\mathrm{REF}$-maximal sets. Right: an infinite biosphere consisting of a single one-way permanent split, and the two corresponding $\mathrm{IAP}\cap\mathrm{CONV}\cap\mathrm{CA}\cap\mathrm{REF}$-maximal sets.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Remark 3
  • Definition 5
  • Lemma 5
  • ...and 25 more