Formal science is a branch of science studying formal language disciplines concerned with formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, 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 symbolic systems. The formal sciences aid the natural and social sciences by providing information about the structures used to describe the physical world, and what inferences may be made about them.
The modern usage of the term formal sciences, in English-language literature, occurs at least as early as 1860,[non-primary source needed] in a posthumous publication of lectures on philosophy by Sir William Hamilton wherein logic and mathematics are listed as formal sciences. Going even further back to 1819, a German-language textbook on logic was published by Wilhelm Esser, elucidating the significance of the designation formal science (Formalwissenschaft) as applied to logic; an English-language translation of it is provided in William Hamilton's lecture:
Logic thus obtains, in common parlance, the appellation of a formal science, not indeed in the sense as if Logic had only a form and not an object, but simply because the form of human thought is the object of Logic; so that the title formal science is properly only an abbreviated expression.
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 Greek, Arab and Persian 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 6th century BC, China in the 5th century BC, and Greece between the 4th century BC and the 1st century BC. 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.
Therefore, straightly speaking, formal science is not an empirical science. It is a formal logical system with its content targeted at components of experiential reality, such as information and thoughts. As Francis Bacon pointed out in the 17th century, experimental verification of the propositions must be carried out rigorously and cannot take logic itself as the way to draw conclusions in nature. Formal science is a method that is helpful to empirical science but cannot replace empirical science.
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 – 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. For this reason, in Rudolf Carnap's logical-positivist conception of the epistemology of science, theories belonging to formal sciences are understood to contain no synthetic statements, being that instead all their statements are analytic.