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Formal science is a branch of
science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

science
studying
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
disciplines concerned with
formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essen ...
s, such as
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
,
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
,
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

theoretical computer science
,
artificial intelligence Artificial intelligence (AI) is intelligence Intelligence has been defined in many ways: the capacity for logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, ...

artificial intelligence
,
information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the painful divisions between ...
,
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
,
systems theory Systems theory is the interdisciplinary study of system A system is a group of interacting Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in th ...
,
decision theory Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent's choices. Decision theory can be broken into two branches: normative Normative generally means relating to an evaluative standard. Normativi ...
, and
theoretical linguistics Theoretical linguistics is a term in linguistics which, like the related term general linguistics, can be understood in different ways. Both can be taken as a reference to theory of language, or the branch of linguistics which inquires into the Phi ...
. Whereas the
natural science Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

natural science
s and
social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

social science
s seek to characterize
physical systems
physical systems
and
social system In , social system is the patterned network of relationships constituting a coherent whole that exist between individuals, groups, and institutions. It is the formal of role and status that can form in a small, stable group. An individual may be ...
s, respectively, using empirical methods, the formal sciences are language
tools A tool is an object that can extend an individual's ability to modify features of the surrounding environment. Although many animals use tool use by animals, simple tools, only human beings, whose use of stone tools dates back Paleolithic, hund ...

tools
concerned with characterizing abstract structures described by symbolic systems. The formal sciences aid the
natural science Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

natural science
,
social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

social science
and
actuarial science Actuarial science is the discipline that applies mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country ...
all through providing information about the structures used to describe the physical and the contemporary world, and what inferences may be made about them.


Etymology

The modern usage of the term ''formal sciences'', in English-language literature, occurs at least as early as 1860, 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:


History

Formal sciences began before the formulation of the
scientific method The scientific method is an empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence ...

scientific method
, with the most ancient
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
texts dating back to 1800 BC (
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') denotes the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Bab ...
), 1600 BC (
Egyptian mathematics Ancient Egyptian mathematics is the mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
) and 1000 BC (
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 (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
). From then on different cultures such as the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
, Arab and Persian made major contributions to mathematics, while the
Chinese Chinese can refer to: * Something related to China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's most populous country, with a populat ...
and
Japanese Japanese may refer to: * Something from or related to Japan , image_flag = Flag of Japan.svg , alt_flag = Centered deep red circle on a white rectangle , image_coat = Imperial Seal of J ...
, independently of more distant cultures, developed their own mathematical tradition. Besides mathematics,
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

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 India, officially the Republic of India (Hindi Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, ...
from the ,
China China (), officially the People's Republic of China (PRC; ), is a country in East Asia East Asia is the eastern region of Asia Asia () is Earth's largest and most populous continent, located primarily in the Eastern Hemisphere ...
in the , and
Greece Greece ( el, Ελλάδα, Elláda, ), officially the Hellenic Republic, is a country located in Southeastern Europe Southeast Europe or Southeastern Europe () is a geographical subregion A subregion is a part of a larger region In geogr ...
between the and the . The formally sophisticated treatment of modern logic descends from the Greek tradition, being informed from the transmission of
Aristotelian logic In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, langu ...
, which was then further developed by Islamic logicians. The Indian tradition also continued into the
early modern period The early modern period of modern history Human history, or world history, is the narrative of Human, humanity's past. It is understood through archaeology, anthropology, genetics, and linguistics, and since the History of writing, adve ...
. The native Chinese tradition did not survive beyond
antiquity Antiquity or Antiquities may refer to Historical objects or periods Artifacts * Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages In the history of Europe, the Middle Ages ...

antiquity
, though Indian logic was later adopted in
medieval In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
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 (; between 31 October and 6 December 1607 – 12 January 1665) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (fren ...

Pierre de Fermat
and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic Church, Catholic theologian. He was a child prodigy who was educated by his father, a tax collector i ...

Blaise Pascal
(1654), and
Christiaan Huygens Christiaan Huygens ( , also , ; la, Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is regarded as one of the greatest scientists of all time and a major fig ...

Christiaan Huygens
(1657) started the earliest study of
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
. In the early 1800s,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
and
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ) is a classical language belonging to the I ...

Laplace
developed the mathematical theory of
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

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 science The mathematical sciences are a group of areas of study that includes, in addition to mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s and engineering disciplines such as operations research and
systems engineering Systems engineering is an field of and that focuses on how to design, integrate, and manage s over their s. At its core, systems engineering utilizes principles to organize this body of knowledge. The individual outcome of such efforts, an e ...
. These sciences benefited from basic research in
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics The field of electronics is a branch of physics and electrical enginee ...

electrical engineering
and then by the development of
electrical computing
electrical computing
, which also stimulated
information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the painful divisions between ...
,
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
(
scientific computing Computational science, also known as scientific computing or scientific computation (SC), is a field that uses advanced computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes th ...
), and
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

theoretical computer science
. Theoretical computer science also benefits from the discipline of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, which included the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
.


Branches

Branches of formal science include
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
,
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
,
information science Information science (also known as information studies) is an academic field which is primarily concerned with analysis, collection, classification Classification is a process related to categorization Categorization is the human ability a ...
and
systems science Systems Science, also referred to as Systems Research, or, simply, Systems, is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic discipline An academic discipline or ac ...
.


Differences from other sciences

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 ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
'', 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 Francis Bacon, 1st Viscount St Alban, (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General for England and Wales, Attorney General and as Lord Chancellor of K ...

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 General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
or
evolutionary biology Evolutionary biology is the subfield of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interacti ...
, 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 An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
and
definitions A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical me ...

definitions
from which other statements (
theorems In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
) are deduced. For this reason, in
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle The Vienna Circle (german: Wiener Krei ...
's logical-positivist conception of the
epistemology of science Philosophy of science is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...
, theories belonging to formal sciences are understood to contain no synthetic statements, being that instead all their statements are analytic.


See also

*
Philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

Philosophy
*
Science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

Science
*
Rationalism In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, ...
* Abstract structure * Abstraction in mathematics * Abstraction in computer science *
Formalism (philosophy of mathematics) In the philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient ...
*
Formal grammar In formal language theory In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λ ...
*
Formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
* Formal method *
Formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essen ...
* Form and content *
Mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environm ...
*
Mathematics Subject Classification The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH ...
*
Semiotics Semiotics (also called semiotic studies) is the study of sign processes (semiosis Semiosis (, ), or sign process, is any form of activity Activity may refer to: * Action (philosophy), in general * Human activity: human behavior, in sociology ...

Semiotics
*
Theory of forms#REDIRECT Theory of forms {{Redirect category shell, 1= {{R from move {{Redirect from other capitalisation {{Redirect unprintworthy ...


References


Further reading

*
Mario Bunge Mario Augusto Bunge (; ; Florida Oeste, September 21, 1919 – Montreal, February 24, 2020) was an Argentina, Argentine-Canadian philosophy, philosopher and physicist. His philosophical writings combined scientific realism, Systemics, systemism, ...
(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 (1940). ''Elements of Logic and Formal Science'', J.B. Lippincott Co., New York. * James Franklin (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.


External links

*{{Commonscat-inline, Formal sciences
Interdisciplinary conferences — ''Foundations of the Formal Sciences''
Branches of science