Without mathematics (understood broadly) we are doomed to make vague statements and to arrive at wrong conclusions. How can we test theoretical predictions with data, if we are not even sure that the “prediction” in fact follows from the theory’s premises?
Many, probably most of historians have a strong dislike of mathematics. Additionally, they may feel threatened by intruders from natural sciences on their turf. Some argue that history had its share of great thinkers, and there is no need for mathematical approaches.
The core activity of science is the empirical testing of theories. As a scientific field matures and addresses theories of increasing complexity, it needs to become progressively more “mathematized”. Mathematics provides a formalized language for precise description, and rigor ensuring that conclusions indeed follow from premises.
Mathematics is not just about quantities (it includes such fields as mathematical logic, abstract algebra, and topology). However, if we are interested in understanding the dynamics of such historical processes as population change, territorial expansion/contraction, and the spread of religions, we must get involved with numbers and rates. Furthermore, a “naked” human mind, unaided by mathematical formalism and computers, is a poor tool for predicting dynamical processes characterized by nonlinear feedbacks, or grasping such complex behaviors as mathematical chaos.
I have nothing but deep respect for the giants of historical thought from Polybius and Ibn Khaldun to Fernand Braudel and William McNeill. But I argue that it is not enough. In addition to admirable research already performed by historians, we need a systematic effort addressed at translating verbal theories into mathematical models, putting together large collections of historical data, and testing model predictions on this empirical material. Contrasting predictions of rival theories with data will allow us to reject some theories in favor of others. This is one of the best measures of scientific progress, but rarely happens in history.
One of the few examples of successfully applying this approach is the work of the Nobel laureate Robert Fogel and colleagues on the economic feasibility of slavery in the Antebellum United States. We need more such studies, and not just in the field of economic history.
At present history ranks rather low in prestige among sciences (many would even deny that it is a science). This is reflected, among other things, in a woefully inadequate financial support for historical research. However, what is a better strategy for correcting this deplorable situation? Fighting a rearguard action against the introduction of new methods and approaches from the natural sciences? Or accepting the change and integrating these methods with what historians do? In the twentieth century biology made a transition from a descriptive to explanatory science. Now biology has large numbers of both splitters and lumpers; some study the private life of warblers and others investigate mathematical models. I am confident that history can make a similar transition, and only will be enriched in the end.