Derived changes in genes associated with primary microcephaly (MCPH) have been suggested to be “currently sweeping to fixation” i.e., increasing in frequency in most populations, with the likely outcome that the derived allele will completely displace the ancestral allele over time. Possible causes for this sweep include effects on human reasoning and language. Here we test the hypothesis that these derived alleles are associated with current variation in spoken or written language and related traits. The association of derived alleles of the ASPM, MCPH1, CDK5RAP2 and BRCA1 genes was tested against well-validated measures of dyslexia, specific language impairment, working memory, IQ, and head-size in a family-based association study of over 1776 subjects from 789 families of twins. No evidence for association was found for any gene to any trait. The results strongly did not support the hypothesis that derived alleles in MCPH-related genes are related to the evolution of human language or cognition. Results were compatible with the alternate hypothesis, suggesting that adaptations in these genes associated with a dramatic increase in brain size have long since reached fixation and are now maintained by stabilizing selection.