Formal system

A formal system is a special way of organizing ideas and rules to figure things out. Think of it like a set of instructions that help us decide what is true based on a few basic ideas called axioms. From these axioms, we can use certain rules to prove new ideas called theorems.

In 1921, a mathematician named David Hilbert thought that formal systems could be the best way to build all of mathematics. He wanted to use them as the strongest base for knowledge. But later, in 1931, another mathematician named Kurt Gödel discovered something important. He showed that any formal system strong enough to handle simple math could not prove that it itself was complete. This meant that Hilbert's big plan had a limit.

Sometimes, the word formalism is used to mean the same thing as a formal system. But it can also talk about special ways of writing things, like the bra–ket notation made by Paul Dirac to make math easier to read. Formal systems help us understand deep ideas in math and logic by giving us clear rules to follow.

Concepts

A formal system is a way to build ideas and rules for math and logic. It has three main parts:

• Formal language: This is a special kind of language made of symbols and rules. It helps us write clear, exact sentences.
• Deductive system: This is a set of rules and starting points (called axioms) that help us prove new ideas (called theorems). These rules make sure each step follows logically from the last.
• Inductive system: In some cases, we can start with a simple idea and make it more general.

These systems help us understand how math and logic work by using clear rules and symbols.

History

Main articles: Formalism (philosophy of mathematics) and Formal logical systems

Early logic systems include Indian logic from Pāṇini, Aristotle’s logic, the logic of Stoicism, and Chinese logic from Gongsun Long (around 325–250 BCE). Later contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic grew in the 19th century in Europe.

David Hilbert started a movement called Hilbert’s program to solve problems in the basics of mathematics. This idea was later changed by Gödel's incompleteness theorems. The QED manifesto tried to write down all known mathematics but has not finished yet.