Formal science is a branch of study that focuses on disciplines involving abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory, and theoretical linguistics.
Logic, the study of valid reasoning and inference, is also a branch of philosophy. It is the abstract study of propositions, statements, or assertively used sentences and of deductive arguments, focusing on the structures or logical forms they embody. It deals with the formal structure of statements and propositions, ensuring valid inferences from premises to conclusions.
The formal science of mathematics studies abstract structures described by formal systems, including logic, theoretical computer science, and statistics. These disciplines focus on developing formal systems and the rules governing them.
Statistics is the science concerned with developing and studying methods for collecting, analyzing, interpreting, and presenting empirical data. It encompasses both descriptive statistics, which summarize data, and inferential statistics, which draw conclusions from data using probability.
Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation, including topics like the theory of computation, formal language theory, the lambda calculus, and type theory. It explores the structure, processing, transmission, and reproduction of information through abstraction and formal modeling.
The formal science of artificial intelligence involves developing and studying methods and software that enable machines to perceive their environment, learn, and take actions to achieve defined goals. AI encompasses technologies like machine learning, deep learning, and natural language processing.
Information theory is the mathematical study of quantifying, storing, and communicating information, originally established by Harry Nyquist, Ralph Hartley, and Claude Shannon in the 1920s and 1940s. It resides at the intersection of probability theory, statistics, computer science, and electrical engineering. Applications range from data compression to error correction, impacting technologies like mobile phones, the Internet, and even art creation.
Game theory is the branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent, considering each other's possible strategies in formulating their own strategy.
Systems theory is a transdisciplinary study that examines cohesive groups of interrelated, interdependent components--whether natural or artificial--by considering their causal boundaries, context, structure, function, and relations with other systems. It emphasizes that a system is more than the sum of its parts, often exhibiting emergent behavior or synergy. Changes in one component can impact other parts or the entire system, and understanding these dynamics helps model and optimize various fields.
Also known as the theory of choice, decision theory is a branch of applied probability theory and analytic philosophy that investigates how decisions are made by assigning probabilities to various factors and numerical consequences to outcomes.
The formal science of theoretical linguistics aims to understand the universal and fundamental nature of language while developing a theoretical framework for describing it.
These fields provide essential tools for characterizing abstract structures and aiding natural and social sciences in understanding the physical world and making references.
The formal sciences rely on deductive reasoning and axiomatic systems, while natural and social sciences use empirical methods based on observation and experimentation. For example, mathematical laws are certain and indisputable, whereas, in the other sciences, theories can be debated and modified based on new evidence. The formal sciences contain analytic statements that are devoid of empirical content, unlike synthetic statements found in other sciences.
Due to the non-empirical nature of the formal sciences, they are constructed by delineating a set of axioms and definitions. Other statements, known as theorems, are logically derived from these foundational elements. Consequently, theories within formal sciences are considered to consist exclusively of analytic statements, devoid of synthetic ones.
Formal sciences provide the essential tools for understanding the world, bridging the gap between theory and reality.
The focus of this section of our web guide and its subsections are the formal sciences, as a whole or individually.
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Recommended Resources
Introduction to Theoretical Computer Science
Created by Boaz Barak, this textbook is intended for an introductory undergraduate course on theoretical computer science. Individual chapters can be viewed on the website, and interested viewers may also download the entire volume in a single PDF file. Prior versions of the book can be found on a repository release page, while the most updated version can be accessed through the index page. The work is licensed under a Creative Commons International License.
https://introtcs.org/
Created by Peter Smith, author of "Beginning Mathematical Logic," "Exploring Chaos," "An Introduction to Formal Logic," "An Introduction to Gödel's Theorems," "Godel Without (Too Many) Tears," and "Category Theory I," the site highlights and offers access to much of the content of his books, many of them available in full via a PDF file. Also included are links to a selection of freely available online resources for those interested in category theory. A guide is available.
https://www.logicmatters.net/
Founded in 2002, Statistics was one of the first institutions to embrace online teaching and learning, becoming the first online education to be approved by the American Council on Education in 2007, and a learning partner for several higher education institutions in 2017, offering courses recognized for college credit. Statistics provides a collaborative environment, peer networking, competitive compensation, and dedicated teaching assistants. Topics include data science and machine learning.
https://www.statistics.com/
Utilizing Moodle, an open-source learning platform or course management system, the site offers basic statistics courses for non-statisticians. The courses focus on linear models, experimental design, multivariate analysis, and the use of the statistical package R. The courses are available in different languages and taught by various presenters; as an interinstitutional development project, different partners create their own versions of the courses while maintaining a common structure.
https://www.statsforscience.com/
Endorsed by the ACM Special Interest Group on Algorithms and Computation Theory (ACM SIGACT), ToC is a free online journal that publishes research papers in theoretical computer science. Articles may be accessed by Volume (Year), beginning with the current volume and going back as far as 2005, or browse through other categories, such as special issues. Also available, ToC Library Graduate Surveys are peer-reviewed essays and monographs intended for graduate students and researchers.
http://theoryofcomputing.org/
