[Keysh]

advances made during the early middle ages, because it needs to ignore those advances in order to make its claim. The simple fact that there were many advances in the fields of engineering, metallurgy, architecture, transportation, mathematics and others

I would specifically question the argument that were "significant advances" in mathematics in the area of the former Western Empire in the early medieval period. There were advances in the Islamic lands, but Western Europe was mostly stuck with Boethius's 6th Century summaries. The Arabic/Islamic advances (and the rediscovery of much of ancient Greek mathematics) did not really spread to Western Europe until the 12th Century, when you start getting translations into Latin of mathematicians like Euclid and al-Khwarizmi.