What is Positivism and why was it prominent in early 20th century and late 19th century historiographies?
Positivism is a philosophical theory that states that all knowledge must be derived from experience and observation rather than from other sources such as intuition, metaphysics, or theology. It is based on the idea that the only way to truly know something is to observe it with the senses and to use reason and logic to interpret the data. Positivism holds that any knowledge that is not based on empirical evidence is unreliable and thus should be discarded.

It was prominent in the late 19th and early 20th centuries because of the influential work of Émile Durkheim. Durkheim's work and methodologies had a major impact on the academic discipline of sociology, and his ideas and theories continue to be influential today. He established the notion of objective _sui generis_ “social facts” as a unique empirical object for the science of sociology to study, which allowed for an analysis of a given society to determine whether it was “healthy” or “pathological”. His work set the stage for modern sociology and continues to be influential in the field.
Interesting. Doesn't this type of philosophy have a blind spot insofar as all empirical evidence and experience must be somehow evaluated with knowledge that is known a priori? I mean, as an example, I know how to use math to verify a theorem used when designing bridges but how do I learn math in the first place? Does any Positivist philosophy address this topic?
Yes, this is a valid concern which is addressed by the philosophy of Positivism. The main idea behind the Positivist philosophy is that knowledge must be based on observed facts and testable evidence, rather than on conjecture or unverifiable beliefs. This allows for empirical data to be used in creating new theories about the world around us, allowing for more accurate predictions and solutions to problems. Additionally, Positivism also emphasizes the importance of applying scientific methods to all areas of study and life in general. Thus, it can provide an effective way to evaluate empirical evidence through its application of scientific methods.