—“Lewontin and Gould were Marxist biologists who were so shameless about their ideology shaping their research that even left-leaning colleagues like Dawkins called them out.”— Matthew Genack

Lewontin: ‘greater variation within than across groups’
Gould: ‘mismeasure of man – cranium size is irrelevant’

What is “Lewontin’s Fallacy”?

By Justin Smith, PhD Genetics and Heredity, Stanford University (2016)

This is copied directly from Wikipedia but I think explains it well. Basically Lewontin’s argument was that because common genetic variation varies more between individuals than between races, race/ethnicitiy doesn’t really mean anything biologically, and that races/ethnicities aren’t real genetic categories.

Human Genetic Diversity: Lewontin’s Fallacy
“Lewontin’s argument
In the 1972 study “The Apportionment of Human Diversity”, Richard Lewontin performed a fixation index (FST) statistical analysis using 17 markers, including blood group proteins, from individuals across classically defined “races” (Caucasian, African, Mongoloid, South Asian Aborigines, Amerinds, Oceanians, and, Australian Aborigines). He found that the majority of the total genetic variation between humans (i.e., of the 0.1% of DNA that varies between individuals), 85.4%, is found within populations, 8.3% of the variation is found between populations within a “race”, and only 6.3% was found to account for the racial classification. Numerous later studies have confirmed his findings.[5] Based on this analysis, Lewontin concluded, “Since such racial classification is now seen to be of virtually no genetic or taxonomic significance either, no justification can be offered for its continuance.”
This argument has been cited as evidence that racial categories are biologically meaningless, and that behavioral differences between groups cannot have any genetic underpinnings.[6] One example is the “Statement on ‘Race'” published by the American Anthropological Association in 1998, which rejected the existence of races as unambiguous, clearly demarcated, biologically distinct groups.[7]

Edwards’ critique:
Edwards argued that while Lewontin’s statements on variability are correct when examining the frequency of different alleles (variants of a particular gene) at an individual locus (the location of a particular gene) between individuals, it is nonetheless possible to classify individuals into different racial groups with an accuracy that approaches 100 percent when one takes into account the frequency of the alleles at several loci at the same time. This happens because differences in the frequency of alleles at different loci are correlated across populations — the alleles that are more frequent in a population at two or more loci are correlated when we consider the two populations simultaneously. Or in other words, the frequency of the alleles tends to cluster differently for different populations.[8]
In Edwards’s words, “most of the information that distinguishes populations is hidden in the correlation structure of the data.” These relationships can be extracted using commonly used ordination and cluster analysis techniques. Edwards argued that, even if the probability of misclassifying an individual based on the frequency of alleles at a single locus is as high as 30 percent (as Lewontin reported in 1972), the misclassification probability becomes close to zero if enough loci are studied.[9]
Edwards’s paper stated that the underlying logic was discussed in the early years of the 20th century. Edwards wrote that he and Luigi Luca Cavalli-Sforza had presented a contrasting analysis to Lewontin’s, using very similar data, already at the 1963 International Congress of Genetics. Lewontin participated in the conference but did not refer to this in his later paper. Edwards argued that Lewontin used his analysis to attack human classification in science for social reasons.[9]”

There are also real traits that vary a lot my ethnicity. Another argument against the Lewontin’s argument has to with rare or functional variation. For example sickle cell anemia is much more prevalent in subsaharan african populations than in the rest of the human population, and cystic fibrosis is much more prevalent in european populations than in the rest of the world.