Formal science

# Formal science

Discussion

Encyclopedia
The formal sciences are the branches of knowledge that are concerned with formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

s, such as logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

, information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

, systems theory
Systems theory
Systems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can be applied to all types of systems at all nesting levels in all fields of research...

, decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, and some aspects of linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

.

Unlike other sciences, the formal sciences are not concerned with the validity of theories
Scientific theory
A scientific theory comprises a collection of concepts, including abstractions of observable phenomena expressed as quantifiable properties, together with rules that express relationships between observations of such concepts...

based on observation
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...

s in the real world
Reality
In philosophy, reality is the state of things as they actually exist, rather than as they may appear or might be imagined. In a wider definition, reality includes everything that is and has been, whether or not it is observable or comprehensible...

, but instead with the properties of formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

s based on definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

s and rule
Rule of inference
In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...

s. Methods of the formal sciences are, however, applied in constructing and testing scientific models dealing with observable reality.

## History

Formal sciences began before the formulation of scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

, with the most ancient mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

texts dating back to 1800 BC (Babylonian mathematics
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

), 1600 BC (Egyptian mathematics
Egyptian mathematics
Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...

) and 1000 BC (Indian mathematics
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

). From then on different cultures such as the Indian, Greek
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

and Islamic mathematicians
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...

made major contributions to mathematics, while the Chinese
Chinese mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....

and Japanese
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...

independently developed their own mathematical tradition.

Besides mathematics, logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

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
Indian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...

from the 6th century BC, China
Logic in China
In the history of logic, logic in China plays a particularly interesting role due to its length and relative isolation from the strong current of development of the study of logic in Europe and the Islamic world, though it may have some influence from Indian logic due to the spread of...

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
Logic in Islamic philosophy
Logic played an important role in Islamic philosophy .Islamic Logic or mantiq is similar science to what is called Traditional Logic in Western Sciences.- External links :*Routledge Encyclopedia of Philosophy: , Routledge, 1998...

. The Indian tradition also continued into the early modern period
Early modern period
In history, the early modern period of modern history follows the late Middle Ages. Although the chronological limits of the period are open to debate, the timeframe spans the period after the late portion of the Middle Ages through the beginning of the Age of Revolutions...

. The native Chinese tradition did not survive beyond antiquity
Ancient history
Ancient history is the study of the written past from the beginning of recorded human history to the Early Middle Ages. The span of recorded history is roughly 5,000 years, with Cuneiform script, the oldest discovered form of coherent writing, from the protoliterate period around the 30th century BC...

, though Indian logic was later adopted in medieval
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...

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
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

and Blaise Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

(1654), and Christiaan Huygens (1657) started the earliest study of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. In the early 1800s, Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...

and Laplace developed the mathematical theory of statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, 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-twentieth century, mathematics was broadened and enriched by the rise of new mathematical sciences and engineering disciplines such as operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

and systems engineering
Systems engineering
Systems engineering is an interdisciplinary field of engineering that focuses on how complex engineering projects should be designed and managed over the life cycle of the project. Issues such as logistics, the coordination of different teams, and automatic control of machinery become more...

. These sciences benefited from basic research in electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

and then by the development of electrical computing
Electrical computer
Electrical computer may refer to one of the following:* electrical analog computer.* electrical digital computer....

, which also stimulated information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

, numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

(scientific computing), and theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

. Theoretical computer science also benefits from the discipline of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, which included the theory of computation
Theory of computation
In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm...

.

## Differences from other forms of science

As opposed to empirical sciences (natural, social), the formal sciences do not always involve empirical procedures. They also do not always presuppose knowledge of contingent fact, 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 worlds – whereas, statements based on empirical theories, such as, say, General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

or Evolutionary Biology, do not hold in all possible worlds, and may even turn out not to hold in this world. 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.

• Rationalism
Rationalism
In epistemology and in its modern sense, rationalism is "any view appealing to reason as a source of knowledge or justification" . In more technical terms, it is a method or a theory "in which the criterion of the truth is not sensory but intellectual and deductive"...

• Abstract structure
Abstract structure
An abstract structure in mathematics is a formal object that is defined by a set of laws, properties, and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects...

• Abstraction in mathematics
Abstraction (mathematics)
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...

• Abstraction in computer science
Abstraction (computer science)
In computer science, abstraction is the process by which data and programs are defined with a representation similar to its pictorial meaning as rooted in the more complex realm of human life and language with their higher need of summarization and categorization , while hiding away the...

• Formal grammar
Formal grammar
A formal grammar is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax...

• Formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...

• Formal method
• Formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

• Mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

• Mario Bunge
Mario Bunge
Mario Augusto Bunge is an Argentine philosopher and physicist mainly active in Canada.-Biography:Bunge began his studies at the National University of La Plata, graduating with a Ph.D. in physico-mathematical sciences in 1952. He was professor of theoretical physics and philosophy,...

(1985). Philosophy of Science and Technology. Springer.
• Mario Bunge (1998). Philosophy of Science. Rev. ed. of: Scientific research. Berlin, New York: Springer-Verlag, 1967.
• C. West Churchman
C. West Churchman
Charles West Churchman was an American philosopher and systems scientist, who was Professor at the School of Business Administration and Professor of Peace and Conflict Studies at the University of California, Berkeley...

(1940). Elements of Logic and Formal Science, J.B. Lippincott Co., New York.
• James Franklin
James Franklin (philosopher)
James Franklin is an Australian philosopher, mathematician and historian of ideas. He was educated at St. Joseph's College, Hunters Hill, New South Wales. His undergraduate work was at the University of Sydney , where he attended St John's College and he was influenced by philosophers David Stove...

(1994). The formal sciences discover the philosophers' stone. In: Studies in History and Philosophy of Science. Vol. 25, No. 4, pp. 513–533, 1994
• Stephen Leacock (1906). Elements of Political Science. Houghton, Mifflin Co, 417 pp.
• Bernt P. Stigum (1990). Toward a Formal Science of Economics. MIT Press
• Marcus Tomalin (2006), Linguistics and the Formal Sciences. Cambridge University Press
• William L. Twining (1997). Law in Context: Enlarging a Discipline. 365 pp.