1. Introduction

There is a recent trend among population geneticists to call the continent-level human genetic clusters inferred from structure-like algorithms ‘populations’.Footnote ¹ For example, Bamshad et al. (Reference Bamshad, Wooding, Watkins, Ostler, Batzer and Jorde2003, 584–85) call Africans, East Asians, Europeans, and Indians “continental populations” when conducting population assignment with Alu insertions and microsatellites. McEvoy et al. (Reference McEvoy, Lind, Wang, Moyzis, Visscher, van Holst Pellekaan and Wilton2010, 297) call Africans, East Asians, Eurasians, Native Americans, and Oceanians “geographic populations” when using frappe and structure to detect human population structure. Feldman and Lewontin (Reference Feldman, Lewontin, Koenig, Lee and Richardson2008, 90) call these same continental groups “geographical populations” when summarizing two landmark studies on human population structure. Furthermore, Zhivotovsky, Rosenberg, and Feldman (Reference Zhivotovsky, Rosenberg and Feldman2003, 1183) call both East Asians and Eurasians “a metapopulation” when doing human genetic clustering research. However, this linguistic pattern raises two philosophically interesting questions.

First, what kind of population are population geneticists referring to when they call continent-level human genetic clusters ‘populations’?Footnote ² Second, is the kind of population being referred to a biological population? The second question requires some clarification. Philosophers who study populations in biology agree that there are many ways that biologists legitimately divide living things into populations.Footnote ³ However, there is a consensus among these experts that the kind of population that is minimally appropriate for evolutionary biology is the biological population—where biological populations are groups that evolve (descend with modification).Footnote ⁴ Perhaps Millstein (Reference Millstein2009, 268) puts it best when she says, “Populations are the entities that evolve, prior to any evolution of species.” Sometimes the members of biological populations interact in evolutionarily significant ways, but sometimes they don’t. For instance, Gannett (Reference Gannett2003, 997) has shown that population geneticists sometimes identify populations as “breeding units” and sometimes as “genealogical units.” Hence, the second question can be restated as, “Is the kind of population being referred to one that requires populations to be groups that evolve?”

The two questions above are philosophically interesting for multiple reasons. Not only is it interesting to figure out whether continental groups like Native Americans cohere in the right way to form biological populations, but also the work done here will inform two ongoing debates in philosophy of biology. One is about whether there is a single, best conception of a biological population for evolutionary biology.Footnote ⁵ The other is about whether any of the human continental groups that population geneticists call ‘populations’ can be legitimately understood as races in any biological or ordinary sense.Footnote ⁶

My answer to the first question is that the kind of population being referred to in this context is the K population. This answer constitutes K population theory. While a K population cannot be understood without jargon from fuzzy set theory and perdurantist metaphysics, it’s roughly a biological population whose members are united by common genomic ancestry and whose members can have graded membership. My answer to the second question is that the K population is indeed a biological population. Note that, in this article, I do not answer whether continent-level human genetic clusters are biological populations. That would be an enormous project that would require defending the sampling scheme, study design, and other methodological assumptions that go into identifying continent-level human genetic clusters. Rather, what I do is provide the metaphysical framework for legitimately calling continent-level human genetic clusters ‘populations’.

I begin by presenting jargon from fuzzy set theory and perdurantist metaphysics that serves as background assumptions for K population theory. Second, I define the K population. Third, I defend the K population as the kind of population being referred to when population geneticists call continent-level human genetic clusters ‘populations’. Fourth, I defend the K population as a kind of biological population. Fifth, I address potential objections to K population theory. I end with concluding remarks.

2. Background Jargon

The mathematical and metaphysical jargon that I will use to define the K population is the following. Suppose that crisp sets are the objects called ‘sets’ in Zermelo–Fraenkel set theory. Let an object space X be a crisp set of objects. Suppose that a membership function μ is a function such that $X^{\underrightarrow{\mu }}\left[0,1\right]$. Then, a fuzzy set à is a pair (X_Ã, μ_Ã). Unlike crisp sets, ‘∈’ has no meaning for fuzzy sets. The analogous relation is belonging. Suppose that μ_Ã(x) is x’s grade of membership (or strength) in Ã. Then, xbelongs to à just in case x ∈ X_à and μ_Ã(x) > 0.

An important point to note is that all of the objects that belong to a fuzzy set need not have a partial membership in that set. In fact, it could be the case that all of the objects that belong to a fuzzy set have a strength of 1. In such a case, we say that the fuzzy set’s membership function is a characteristic function, which is a μ such that $X^{\underrightarrow{\mu }}\left\{0,1\right\}$. But here are some more useful terms. Suppose that the support of à is the crisp set of all objects that belong to Ã.Footnote ⁷ Furthermore, à is empty iff nothing belongs to it.Footnote ⁸ Finally, the cardinality of à is equal to ∑_{x∈X Ã}μ_Ã(x).Footnote ⁹

Now suppose that we have a crisp set of fuzzy sets A. Then, a fuzzy set $\tilde{B}$ is the union of all members of A just in case (a) |A| ≥ 2, (b) the object space of $\tilde{B}$ is identical to the object space of each member of A, and (c) each ${\mu }_{\tilde{B}}\left(x\right)$ is identical to the maximum value among all strengths of x in each member of A.Footnote ¹⁰ Next, suppose that we have a finite, crisp set A. Then, a crisp set B is a fuzzy K partition of A just in case (i) each member of B is a fuzzy set, (ii) each member of B is nonempty, (iii) |B| = K, (iv) 2 ≤ |B| < |A|, (v) the support of the union of all members of B equals A, and (vi) for any object x that belongs to any member of B, the sum of each grade of membership that x has in each member of B is 1. Furthermore, let a member of a fuzzy K partition be a part.Footnote ¹¹ Thus, an important point to remember is that a fuzzy K partition is itself a crisp set, but it’s called ‘fuzzy’ because its parts are all fuzzy sets.

Now suppose that an object persists iff it exists at different times, an object perdures iff it persists and it is only partially present at any time it exists, an object that perdures is a perduring object, and a temporal part is any part of a perduring object that constitutes its partial presence when it exists.Footnote ¹² Now I am ready to define the K population.

3. The K Population

Let L be a sexually reproducing species that forms a lineage or a lineage of such species, but not both.Footnote ¹³ For instance, L could be the species Pan troglodytes (the common chimp) or the Pan genus (the common chimp and the bonobo).Footnote ¹⁴ Let L_t be a crisp set of organisms that constitutes all of the members of L at a time t. Now suppose that any member of L_t has a genome that consists of a sequence of distinct loci, with one or more alleles at each locus.Footnote ¹⁵ For instance, if L_t consists of diploid organisms, then each member of L_t will usually have two alleles at each locus. Now I will articulate how L can be subdivided into K populations. Let’s start with the notion of a genomic ancestry partition.

Suppose that a crisp genomic ancestry partition of L_t is a fuzzy K partition of L_t such that the object space for each part in the partition is L_t, each part in the partition is the temporal part of an isolated breeding group, each L_t member’s degree of membership in each part is equal to the proportion of its genome inherited from an organism in the breeding group that the part represents, and each part’s membership function is a characteristic function. Also, it will be useful to call each organism that belongs to a part in a crisp genomic ancestry partition an unmixed organism.

Now, suppose that L_u and L_v are distinct crisp sets that constitute all of the members of L at distinct times u and v, respectively, and that u occurs before v. Also, suppose that u is the last time before v when L has a crisp genomic ancestry partition into K parts due to hybridization after u. Thus, hybridization could have begun at v or at some time between u and v. Let C be a crisp genomic ancestry partition of L_u into K parts. Then, a fuzzy genomic ancestry partition of L_v into K parts, call it F, is a fuzzy K partition of L_v such that the object space for each part in F is L_v, there is a one-to-one correspondence between C and F, each L_v member’s degree of membership in each part is equal to the proportion of its genome inherited from an organism that belongs to the part in C that corresponds to the part in F under consideration or else an unmixed organism descended from such an organism, and not every part’s membership function is a characteristic function. Also, it will be useful to call any organism that belongs to a part in a fuzzy genomic ancestry partition and has a grade of membership in that part in (0,1) a mixed organism.

For clarity, the hybridization model I am presupposing is the following:

Model 1 is one of three biological possibilities for hybridization between breeding groups. The second and third are below:

The fact that I am presupposing model 1 as the hybridization model for K population theory will be relevant to how K population theory avoids certain objections. In any case, let a genomic ancestry partition be a crisp genomic ancestry partition or a fuzzy genomic ancestry partition. Now, I will define K population parts, which are the temporal parts of K populations.

Suppose that we call a part in a genomic ancestry partition a K population part (or KP part for short). Thus, while KP parts must be indexed to at least one time, they need not exist at exactly one time. Rather, KP parts can exist through time. Now I’m going to make some distinctions among KP parts in order to define K populations as, roughly, sequences of KP parts through time.

Suppose that à and $\tilde{B}$ are KP parts. Then, $\tilde{B}$ is an offspring of à and à is a parent of $\tilde{B}$ iff $\tilde{B}$ originated from a change in Ã’s object space or membership function. For instance, suppose that at time t ₁ we have a fuzzy set of Hadza people that counts as a KP part of humans, call it ‘${\tilde{H}}_1$’. Then, at a later time t ₂ a human dies but not a Hadza person. Also suppose that the latter event produces a new KP part of humans, call it ‘${\tilde{H}}_2$’, that has the same support and membership function as ${\tilde{H}}_1$. Then, it is safe to say that ${\tilde{H}}_2$ originated from changing the object space of ${\tilde{H}}_1$, and thus ${\tilde{H}}_2$ is an offspring of ${\tilde{H}}_1$ and ${\tilde{H}}_1$ is a parent of ${\tilde{H}}_2$.

Now, suppose that two or more KP parts merge iff their offspring is their union, two or more KP parts fuse iff they merge and they’re approximately equal in cardinality, and two or more KP parts join iff they merge but do not fuse. Furthermore, let a parent with much lower cardinality in a joining be a minor parent. Then, a KP beginning is any KP part that originated from its parent(s) through hybridization, budding, splitting, or fusion. In contrast, a KP end is any KP part that has no offspring after multiple generation times for its species, has offspring that arose from splitting, has offspring that arose from fusion, or is a minor parent and has offspring that arose from joining. We are finally in a position to define a K population.

Thus, K populations are, roughly, perduring objects whose temporal parts are fuzzy sets of conspecific organisms that cohere via common genomic ancestry. Here are some final terms that will be useful for talking about K populations. First, an organism is a member of a K population at a time t iff it belongs to that K population’s temporal part at t. Second, an organism’s grade of membership in a K population at a time t is equal to its grade of membership in that K population’s temporal part at t. Now that I have introduced what a K population is, we can turn to answering whether population geneticists are actually referring to K populations when they call continent-level human genetic clusters ‘populations’.

4. The Evidence for K Population Theory

There is plenty of evidence that population geneticists are referring to the K population when they call continent-level human genetic clusters ‘populations’. But first, here’s some clarification. The linguistic context in which continental population talk is used in population genetics is population structure analysis that utilizes structure-like clustering algorithms. Structure-like clustering algorithms (structure, frappe, admixture, etc.) attempt to infer population structure in a group of organisms using an algorithm that searches for the fuzzy K partition that maximizes allele frequency differences among parts using genotype data from the organisms. Sometimes what’s investigated is the optimal partition into K parts, and other times the task is to find the optimal partition at each K level. Suppose that we call any population structure analysis that uses a structure-like clustering algorithm ‘structure-like PSA’. Then, two examples of structure-like PSA are Gonder et al.’s (Reference Gonder, Locatelli, Ghobrial, Mitchell, Kujawski, Lankester, Stewart and Tishkoff2011) use of structure to identify the population structure of chimpanzees and Li et al.’s (Reference Li2008) use of frappe to identify the population structure of humans.

If the continent-level human genetic clusters inferred in structure-like PSA are K populations, then we can explain how population geneticists model, sample, talk, and otherwise behave in structure-like PSA. With respect to modeling, the representations in the models used in structure-like algorithms are what we would expect if the populations being inferred are K populations.

For example, one important model assumption of structure-like algorithms is that populations leave behind a genetic trail. This is because structure-like algorithms represent populations as sequences of allele frequencies (Pritchard et al. Reference Pritchard, Stephens and Donnelly2000, 947; Tang et al. Reference Tang, Peng, Wang and Risch2005, 290). However, K population theory predicts this model assumption nicely because the fact that K populations leave behind a genetic trail in the form of a sequence of allele frequencies can be derived from what a KP part is.

Second, K population theory explains why population geneticists sample in the way that they do. For example, the widespread practice in fuzzy PSA of sampling “ancestry informative markers” for cluster analysis suggests that what is being sought are genealogical groups, which K populations are by definition (Tishkoff and Kidd Reference Tishkoff and Kidd2004, S23, S25). In fact, one population geneticist, Marcus Feldman (Reference Feldman, Markus and Moya2010, 157), coined the term ‘ancestry group’ to talk about continent-level human genetic clusters.

Third, K population theory predicts how population geneticists talk in fuzzy PSA. For example, population geneticists have no problem saying that continent-level human genetic clusters originated thousands of years ago, which would make little sense unless the clusters were perduring objects, like K populations.Footnote ¹⁶ Finally, K population theory explains many other behaviors of population geneticists in structure-like PSA that would otherwise be peculiar. For one, K population theory explains why population geneticists check the accuracy of structure-like algorithms in the way that they do. For example, Shringarpure and Xing (Reference Shringarpure and Xing2014, 904) tested the accuracy of admixture’s population membership grade assignments by judging how well it predicted “the proportion of YRI ancestry” in individuals of a fabricated admixed population.Footnote ¹⁷

Nevertheless, one objection is that K population theory is pointless because the actual population concept being used here is inconsistent. For example, some population geneticists, such as Feldman and Lewontin (Reference Feldman, Lewontin, Koenig, Lee and Richardson2008, 81, 90), have no problem calling the continental clusters both ‘geographical populations’ and ‘ancestry groups’, while others, such as Weiss and Long (Reference Weiss and Long2009, 709), are fine calling the clusters ‘geographic populations’ but find the ‘ancestry group’ label misleading.

While it is true that there is no consensus among population geneticists about how to define ‘population’ when using it to talk about continent-level human genetic clusters in structure-like PSA, K population theory is not a project in conceptual analysis; it’s a theory about what ‘population’ refers to in a particular context. This project is compatible with there being inconsistent beliefs about what ‘population’ means as long as we adopt a referential semantics for ‘population’. Yet another concern is that even if K population theory is right, it still remains to be shown that the K population is a biological population. So to that I now turn.

5. Why K Populations Are Biological Populations

Remember that biological populations must be, at the very least, groups that evolve. Thus, if K populations are not, at least, groups that evolve, then the population geneticists who are acknowledging continental populations in humans are using the word ‘population’ in a way that is not helpful for evolutionary research. However, K populations are groups that evolve, and this fact can be derived from the definition of a K population.

Suppose that we have an arbitrary K population p and we want to know how it originated. By definition, the first temporal part of p is a KP beginning. Furthermore, KP beginnings, by definition, are offspring from one or more KP parts that are distinct from it. But for a KP part to be an offspring it must be a modified version of its parent, either in its object space or in its membership function. Thus, p will be a modified descendant the moment its first temporal part originates. Nevertheless, it’s an empirical question whether p originated from another K population since the KP part that p originated from could be a temporal part of a different kind of biological population.Footnote ¹⁸ In any case, it’s easy to see how K populations form evolutionary networks.

6. Objections and Replies

There are a few objections that one could have with K population theory. First, in a recent article, Millstein (Reference Millstein2015, 9–10) has offered some healthy skepticism about interpreting continent-level human genetic clusters as biological populations. Millstein rightly points out that another possible interpretation is that continent-level human genetic clusters are not populations, but are just ancestry groups (in Feldman’s sense) of organisms descended from members of past populations.

While Millstein’s suggestion is a plausible interpretation of some inferred clusters in structure-like PSA, it’s not a plausible interpretation of continent-level human genetic clusters. This is because the hybridization model that Millstein is using, which is model 2, is not the model that best fits how population geneticists talk about continental clusters. In order for Millstein’s model to be correct, population geneticists would have to think that continent-level human genetic clusters represent only past nonrandom mating structure. However, population geneticists regularly talk about continent-level human genetic clusters as reflecting current nonrandom mating structure. Here is an example: “Each of the large population groups (the Sub-Saharan African farmers, Eurasia, and East Asia) can be considered as a metapopulation consisting of populations with some genetic exchange between them and with a common ancestry” (Zhivotovsky et al. Reference Zhivotovsky, Rosenberg and Feldman2003, 1183).Footnote ¹⁹ These authors do not sound like they’re just talking about past nonrandom mating structure. Of course, this way of talking is easily explained if we accept that population geneticists intend continental clusters to represent extant breeding groups. Furthermore, if that is true, then the appropriate hybridization model is model 1, not model 2, which is the model used in K population theory. For clarity, I am not saying that population geneticists are correct in using model 1 to model hybridization among continental groups of humans. All I am saying is that the hybridization model that they are in fact using when they call continent-level human genetic clusters ‘populations’ is model 1.

Second, both Millstein (Reference Millstein2009, 269) and Gildenhuys (Reference Gildenhuys2014, 434–37) have claimed that a respectable notion of biological population must be able to explain how populations are modeled in evolutionary dynamics. For instance, Millstein (Reference Millstein2009, 269) claims that migration modeling in population genetics (the source-sink model, the island model, etc.) requires that migration amounts to organisms wholly leaving one population and wholly joining another population. However, since organisms are born into K populations in virtue of their genomic ancestry, it’s hard to imagine how the members of K populations can migrate in the sense given above.

This is a good concern. The members of K populations are indeed born into them. And barring the extinction of one’s K population or one’s species, it’s impossible for a member of a K population to leave her population. However, the latter is a concern only if one presupposes “monism” about classifying individuals into biological populations, as Millstein (Reference Millstein2015, 6) does.Footnote ²⁰ However, if one does not presuppose monism, but rather accepts the possibility of “population pluralism”—which Stegenga (Reference Stegenga2016, 1) defines as the view that “there are many ways that a particular grouping of individuals can be related such that the grouping satisfies the conditions necessary for those individuals to evolve together”—then the fact that the members of K populations don’t migrate is not a flaw.Footnote ²¹ Rather, it’s just an example of how classifying organisms into K populations is not going to be useful in all research projects in population genetics.

In fact, Mishler and Brandon (Reference Mishler and Brandon1987, 401) would call K populations “historical entities” because they’re essentially genealogical groups (e.g., monophyletic groups). Furthermore, because they’re historical entities, they’re useful in history-oriented population-genetic research, such as research on “the history of human migrations” and “human evolutionary history” (Rosenberg et al. Reference Rosenberg, Pritchard, Weber, Cann, Kidd, Zhivotovsky and Feldman2002, 2381, 2384).Footnote ²²

7. Conclusion

In this article, I have attempted to show that there is a legitimate metaphysical basis for calling continent-level human genetic clusters ‘populations’. Namely, these groups could be biological populations in virtue of being K populations. I defended my position by first articulating a K population as, roughly, a biological population whose members are united by genomic ancestry and in which population membership is graded. However, the precise definition was given in definition 1. Next, I provided evidence that population geneticists are referring to the K population in population structure analyses that use structure-like clustering algorithms. After this, I defended K populations as authentic biological populations. Finally, I defended K population theory against two objections.