Formal sciences are language disciplines concerned with formal systems, such as logic, mathematics, statistics, theoretical computer science, information theory, game theory, systems theory, decision theory, and theoretical linguistics. Whereas the natural sciences and social sciences seek to characterize physical systems and social systems respectively using empirical methods, the formal sciences are language tools concerned with characterizing abstract structures described by sign systems. The formal sciences aid the natural sciences by providing information about the structures the latter use to describe the world, and what inferences may be made about them.
Formal sciences began before the formulation of the scientific method, with the most ancient mathematical texts dating back to 1800 BC (Babylonian mathematics), 1600 BC (Egyptian mathematics) and 1000 BC (Indian mathematics). From then on different cultures such as the Indian, Greek and Islamic mathematicians made major contributions to mathematics, while the Chinese and Japanese, independently of more distant cultures, developed their own mathematical tradition.
Besides mathematics, logic is another example of one of oldest subjects in the field of the formal sciences. As an explicit analysis of the methods of reasoning, logic received sustained development originally in three places: India from the , China in the , and Greece between the and the . The formally sophisticated treatment of modern logic descends from the Greek tradition, being informed from the transmission of Aristotelian logic, which was then further developed by Islamic logicians. The Indian tradition also continued into the early modern period. The native Chinese tradition did not survive beyond antiquity, though Indian logic was later adopted in medieval China.
As a number of other disciplines of formal science rely heavily on mathematics, they did not exist until mathematics had developed into a relatively advanced level. Pierre de Fermat and Blaise Pascal (1654), and Christiaan Huygens (1657) started the earliest study of probability theory. In the early 1800s, Gauss and Laplace developed the mathematical theory of statistics, which also explained the use of statistics in insurance and governmental accounting. Mathematical statistics was recognized as a mathematical discipline in the early 20th century.
In the mid-20th century, mathematics was broadened and enriched by the rise of new mathematical sciences and engineering disciplines such as operations research and systems engineering. These sciences benefited from basic research in electrical engineering and then by the development of electrical computing, which also stimulated information theory, numerical analysis (scientific computing), and theoretical computer science. Theoretical computer science also benefits from the discipline of mathematical logic, which included the theory of computation.
One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.
As opposed to empirical sciences (natural and social), the formal sciences do not involve empirical procedures. They also do not presuppose knowledge of contingent facts, or describe the real world. In this sense, formal sciences are both logically and methodologically a priori, for their content and validity are independent of any empirical procedures.
Although formal sciences are conceptual systems, lacking empirical content, this does not mean that they have no relation to the real world. But this relation is such that their formal statements hold in all possible conceivable worlds (see valid formula) - whereas, statements based on empirical theories, such as, say, general relativity or evolutionary biology, do not hold in all possible worlds, and may eventually turn out not to hold in this world as well. That is why formal sciences are applicable in all domains and useful in all empirical sciences.
Because of their non-empirical nature, formal sciences are construed by outlining a set of axioms and definitions from which other statements (theorems) are deduced. In other words, theories in formal sciences contain no synthetic statements; all their statements are analytic.