Formal science

Main thing

Formal science studies abstract structures, using logical reasoning and formal systems, without relying on empirical evidence. It includes fields such as mathematics, logic, statistics, theoretical computer science, and linguistics.

It uses deductive reasoning to study these structures, working with formal systems that have clearly defined rules and axioms. Unlike natural and social sciences, formal science does not rely on empirical evidence or observations of the physical world. Instead, it uses logic and rigorous proofs to derive conclusions and expand knowledge within its abstract domains.

Formal science is essential because it provides the foundation for many other scientific fields and has numerous practical applications. Mathematics, for example, is used extensively in physics, engineering, and economics. Logic is fundamental to philosophy, computer science, and artificial intelligence. Statistics is crucial for analyzing data in social sciences, biology, and medicine. These interdisciplinary connections highlight the importance of formal science in advancing knowledge across various domains.

Real-life example: Mathematicians use formal systems to prove theorems, working with abstract concepts like numbers, shapes, and equations.

Terms

• Formal system - a set of abstract symbols and rules for manipulating those symbols. Example: propositional logic, with symbols like "and", "or", "not", and rules for combining them into valid statements.

• Deductive reasoning - a logical process where conclusions are drawn from premises, based on formal rules of inference. It involves applying general rules to specific examples to arrive at a conclusion. Example: If all men are mortal, and Socrates is a man, then we can deduce that Socrates is mortal.

• Axiom - a statement that is assumed to be true within a formal system, serving as a starting point for deductions. Example: the axioms of Euclidean geometry, such as "a straight line can be drawn between any two points."

• Mathematics - the study of abstract concepts such as quantity, structure, space, and change. Example: algebra, calculus, and geometry.

• Logic - the study of valid reasoning and inference. Example: propositional logic and predicate logic.

• Statistics - the study of collecting, analyzing, interpreting, and presenting data. Example: probability theory and regression analysis.

• Theoretical computer science - the study of the theoretical foundations of computation and information processing. Example: algorithms, data structures, and complexity theory.

• Linguistics - the scientific study of language and its structure. Example: syntax, semantics, and pragmatics.

An analogy

Formal science is like playing a game with a strict rulebook. Just as the game's rules define what moves are allowed, formal systems have specific rules for manipulating symbols and making deductions. The game is played in the abstract realm of the mind, without needing physical pieces or a game board.

Real-life example: Chess has a formal set of rules that govern how each piece can move, and players use strategic reasoning to plan their moves within those constraints.

A main misconception

Many people think that formal science is disconnected from the real world, since it deals with abstract concepts. However, formal methods are essential for fields like computer science, where logical reasoning is needed to design software systems and prove their correctness.

Real-life example: A common misconception is that programming is just about writing code, but it heavily relies on formal methods to create reliable and efficient software.

The history

1. Ancient Greece (6th century BCE): Early Greek philosophers like Thales and Pythagoras laid the foundations for formal reasoning and mathematics.

2. Islamic Golden Age (8th-14th century): Scholars like Al-Khwarizmi and Avicenna made significant advances in algebra, logic, and other formal disciplines.

3. European Renaissance (14th-17th century): Mathematicians like Descartes and Fermat developed analytic geometry and number theory.

4. 19th-20th century: Formal logic and set theory were developed by figures like Boole, Frege, and Cantor, leading to modern formal systems.

"Mathematics is the queen of the sciences and number theory is the queen of mathematics." - Carl Friedrich Gauss, influential German mathematician known for his groundbreaking work in number theory, algebra, and geometry.

Three cases how to use it right now

1. When designing a complex software system, use formal methods to specify the system's requirements and behavior, ensuring that it will work correctly and meet the necessary specifications.

2. In an argument or debate, apply the principles of formal logic to construct valid arguments and identify logical fallacies in others' reasoning.

3. When solving a mathematical problem, break it down into a series of formal steps, using axioms and deductive reasoning to arrive at a solution that is guaranteed to be correct.

Interesting facts

• The concept of zero, which is fundamental to modern mathematics, was first developed in ancient India around the 5th century.

• The word "algorithm" comes from the name of the 9th-century Persian mathematician Al-Khwarizmi, who wrote influential works on algebra and arithmetic.

• In 1931, Kurt Gödel published his incompleteness theorems, showing that any consistent formal system containing arithmetic is incomplete, meaning there are true statements that cannot be proven within the system.

• The P versus NP problem, which asks whether every problem whose solution can be quickly verified can also be quickly solved, is one of the most important unsolved problems in theoretical computer science, with a $1 million prize offered for its resolution.

• The field of cryptography, which is essential for secure communication and transactions online, heavily relies on formal mathematical principles to design unbreakable codes and ciphers.