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At least three arguments establish the existence and reality of biological race:

Argument (1) from Michael Hardimon’s (2017) book “Rethinking Race: The Case for Deflationary Realism” (The Argument for the Existence of Minimalist Races, see Chapters 2, 3, and 4):

The conditions of minimalist racehood are as follows:

(C1) … a group, is distinguished from other groups of human beings by patterns of visible physical features

(C2) [the] members are linked be a common ancestry peculiar to members of that group, and

(C3) [they] originate from a distinctive geographic location (Hardimon, 2017: 31).

This is the argument to prove the existence of minimalist races:

P1) There are differences in patterns of visible physical features which correspond to geographic ancestry
P2) These patterns are exhibited between real groups, existing groups (i.e., individuals who share common ancestry)
P3) These real, existing groups that exhibit these physical patterns by geographic ancestry satisfy the conditions of minimalist race
C) Therefore race exists and is a biological reality

Argument (2) from Michael Hardimon’s (2017) book “Rethinking Race: The Case for Deflationary Realism” (The Argument for the Existence of Populationist Races, see Chapters 5 and 6):

P1) The five populations demarcated by Rosenberg et al (2002) are populationist races; K = 5 demarcates populationist races.
P2) Populationist race=minimalist race.
P3) If populationist race=minimalist race, then everything from showing that minimalist races are a biological reality carries over to populationist races.
P4) Populationist races capture differences in genetic variation between continents and this genetic variation is responsible for the distinctive patterns of visible physical features which correspond to geographic ancestry who belong to biological lines of descent which were initiated by geographically isolated founding populations.
C) Therefore, since populationist races=minmalist races, and visible physical features which correspond to geographic ancestry are genetically transmitted by populations who belong to biological lines of descent, initiated by reproductively isolated founding populations, then populationist races exist and are biologically real.

Argument (3) from Quayshawn Spencer’s (2014) paper “A Radical Solution to the Race Problem” (The argument for the Existence of Blumenbachian Populations):

P1) The term “race” in America refers to biologically real entities; when speaking of race in America, Americans defer to the US Census Bureau who defers to the Office of Management and Budget (OMB).
P2) The OMB refers to race as “sets of” categories, while considering “races” to have 5 members, which correspond to the five major geographic regions.
P3) Rosenberg et al show that, at K = 5, meaningful, though small (~4.3 percent) genetic variation exists between continental-populations
C) Since Americans defer to the US Census Bureau who defers to the OMB, and the OMB refers to race as “sets of” categories which then correspond to five clusters found by Rosenberg et al’s (2002) analysis, race (what Spencer, 2014 terms “Blumenbachian populations”) must exist, though “race” is both socially constructed and biologically real.

Put another way, Spencer’s (2014) argument could also be:

P1) The US meaning of “race” is a referent, which refers to the discourse used by the US Census Bureau; the US Census Bureau refers to the discourse used by the Office of Management and Budget (OMB).
P2) The referent of “race”, in US ontology, refers to a set of human population groups, not a biological kind (sets of human population groups as denoted by the OMB), which refer to “Africans”, “Caucasians”, “East Asians”, “Native Americans”, and Pacific Islanders/Oceanians.
P3) The US meaning of race is both biologically real and socially constructed; Americans refer to real, existing groups when they talk about race.
C) If the US meaning of race is a referent which refers to the discourse used by the US Census Bureau and they refer to the OMB who discuss “sets of” population groups, then when Americans talk about race they talk about Blumenbachian partitions, since race is both biologically real and socially constructed.

The claim “Race exists” is now established. Note how Argument (1) establishes the claim that “races” are real, existing groups who are phenotypically distinct populations with differing geographic ancestry. Note how Argument (2) establishes the claim that populationist race = minimalist race and that “races” are a group of populations that exhibit a distinctive pattern of genetically transmitted phenotypic characters which then correspond to that group’s geographic ancestry who belong to a biological line of descent which was initiated by a geographically separated and reproductively isolated founding population. (This definition of “race” a subdivision of Homo sapiens is the best I’ve come across so far.) Finally, note how Argument (3) establishes the claim that race, in the American sense, is both biologically real and socially constructed. All three arguments are sound and logically valid.

Now, which groups fall into which of the five racial categories?

Caucasians

Caucasians denote a wide-range of groups; Europeans, MENA (Middle Eastern/North African) peoples, Indians are a very diverse group, racially speaking, with “Caucasoids”, “Mongoloids” and “Australoids” (Australoids would mean Pacific Islander/Oceanian) (see Kashyap et al, 2006 for an overview of ethnic, linguistic and geographic affiliations of Indians in the study). Ashkenazi Jews are taken to be a specific race in today’s modern racial ontology, however, Ashkenazi Jews do not exhibit a distinctive pattern of genetically transmitted phenotypic characters which then correspond to their geographic ancestry; they do represent a “geographically isolated and reproductively isolated founding population”, but the fact that they do not exhibit a distinctive pattern of genetically transmitted phenotypic characters means they are not a race, according to Arguments (1) and (2). Ashkenazi Jews are Caucasian, and not their own race. Of course, skin color does not denote race, it is only one marker to use to infer which groups are races.

Africans

Africans comprise all of Sub-Saharan Africa. Africa has the most genetic diversity in the human species (see Campbell and Tishkoff, 2010). Africans, in general, have long, slim bodies with a broad nose, dark skin, kinky hair (lip size is different based on the ethny in question). There are over 3,000 different ethnic groups in Africa, who all comprise the same race. Now, since Africans have the most genetic diversity this does not necessarily mean that they are so phenotypically distinct that there are tens, hundreds, thousands of races on the continent. One only needs to refer back to Arguments (1) and (2) to see that brash claims that “all Xs are Ys” don’t make any sense—especially with the arguments laid out above.

East Asians

East Asians denote a minimalist and populationist race (Arguments (1) and (2)) and Blumenbachian partition (Argument (3)). East Asians denote, obviously, those that derive from East Asia (Chinese, Japanese, Koreans, Vietnamese). These peoples are relatively short, on average, have a distinct yellow-ish tint to their skin (which is why they are sometimes called “yellow”), epicanthic folds and shorter limbs (more likely to have the endomorphic phenotype).

Native Americans

Native Americans are derived from a Siberian population that crossed the Bering Land Bridge about 14kya. They then spread throughout the Americas, becoming the “Natives” we know today. They are what used to be termed “red” people, due to their skin color. Native Americans are derived from Siberians, who share affinities with East Asians. (This will be discussed in further depth below.) They have black hair, and dark-ish skin. Populations that lived in the Americans pre-1492 expansion are part of the Native American racial grouping.

Pacific Islanders/Oceanians

The last racial grouping are Pacific Islanders. Spencer (2014: 1032) writes that we can define Oceanians (Pacific Islanders):

as the most inclusive human population born from East Asians in Oceania (Sahul and the Pacific Islands) and from the original human inhabitants of Oceania. Since Sahul was a single landmass composed of present-day Australia, New Guinea, and Tasmania 50,000–60,000 years ago, when humans first inhabited it, and since we know that the original human inhabitants of Oceania interbred to create modern Oceanians, and since temporal parts of populations are genealogically connected, it should be the case that most Oceanians have genealogical connections to the original peoples of some Pacific island. The only Oceanians who will not will be individuals who became Oceanian from interbreeding alone and Oceanians descended from indigenous peoples of Sahul but not indigenous peoples of a Pacific island (e.g., Aboriginal Australians). The final source of evidence comes from counterfactual cases. [Pacific Islanders and Australian Aborigines share a deep ancestry, see McEvoy et al, 2010.]

A group is in race X, if and only if they share a pattern of visible physical features and common geographic ancestry. If they do not share a pattern of visible physical features which correspond to common geographic ancestry then they do not constitute a race. Keep this in mind for the next two sections.

Are Oceanians black?

One claim that gets tossed around a lot (by black nationalists) is the claim that Oceanians are black due to their skin color, certain phenotypic traits. But this could just as easily be explained by convergent evolution, not that they are, necessarily, the same racial grouping. If this were true, then Australian Aborigines would be black, by proxy, since Australian Aborigines and Oceanian are the same race. The claim, though, holds no water. Just because two groups “look similar” (which I do not see), it does not follow that they are the same race, since other conditions need to be met in order to establish the claim that two separate groups belong to the same race.

Are Native Americans Mongoloid?

Lastly is the claim that Native Americans do not denote an actual racial grouping, they are either Mongoloid or a sub-race of Mongoloids.

Many authors throughout history have presumed that Native Americans were Mongoloid. Franz Boas, for example, said that the Maya Indians were Mongoloid, and that, American populations had features the most similar to Mongoloids, so they are thusly Mongoloid. Wikipedia has a great overview of the history of the “Mongoloid” terminology, with examples from authors throughout history. But that is irrelevant. Native Americans genetically transmit heritable phenotypic characters which correspond with their geographic ancestry and are genetically and geographically isolated population groups.

Although the claim that “Native Americans are Mongoloid” has been echoed for hundreds of years, a simple argument can be erected to take care of the claim:

P1) If Native Americans were East Asian/Mongoloid, then they would look East Asian/Mongoloid.
P2) Native Americans don’t look East Asian/Mongoloid, they have a distinct phenotype which corresponds to their geographic ancestry (See Hardimon’s minimalist/populationist race concepts).
C) Therefore, Native Americans are not East Asian/Mongoloid.

Establishing the claim that Native Americans are not East Asian/Mongoloid is simple. Some authors may make the claim that since they look similar (whatever that means, they don’t look similar to me), that they, therefore according to Arguments (1) and (2) they are a separate race and not a sub-race of East Asians/Mongoloids; Argument (3) further establishes the claim that they are a separate race on the basis that they form a distinct cluster in clustering analyses (Rosenberg et al, 2002) and since, Americans defer to the US Census Beureau and the US Census Beureau defers to the OMB who discusses sets of populations, then when Americans talk about race they talk about Native Americans as separate from East Asians/Mongoloids, since, according to Arguments (1) and (2) they have a distinct phenotype.

Generally, they have distinct skin colors (of course, skin color does not equal race, but it is a big tell), they have similar black, straight hair. But they are, in my opinion, just too phenotypically distinct to call them the same race as Mongoloids/East Asians. For the claim “Native Americans and Mongoloids/East Asians” to be true, they would need to satisfy P1 in Argument (1) and P4 in Argument (2). Native Americans do not satisfy P1 in Argument (1) nor do they satisfy P4 in Argument (2). Therefore, Native Americans are not Mongoloid/East Asian.

Conclusion

The claim “Race exists and is a biological reality” is clearly established by three sound, valid arguments—two from Hardimon (2017; chapters 2-6) and one from Spencer (2014). These arguments show, using the latest of genetic clustering studies, that races, as classicly defined, do indeed, exist and that our old views of race hundreds of years ago were, largely, correct. These arguments establish the existence of the old folk-racial categories. Races have distinct phenotypes which are genetically transmitted and are correlated with geographic ancestry. Some may make certain claims that “Oceanians are black” or “Native Americans are Mongoloid”, but these claims do not hold. These two groups in question are phenotypically distinct, and they come from unique geographic locations, therefore they are not a part of the races that some purport them to be.