Formalism is a mathematical philosophy that emphasizes the rigor of mathematical proofs over their interpretation. Formalists believe that mathematics is a purely formal system that can be described by a set of axioms and rules, and that the meaning of mathematical statements is irrelevant to their truth.
An example of formalism in mathematics is the use of axiomatic systems. An axiomatic system is a set of axioms, which are statements that are assumed to be true, and rules of inference, which are rules that allow new statements to be derived from the axioms. Axiomatic systems can be used to define mathematical structures, such as groups, rings, and fields and theorems can be derived from the axioms using the rules of inference. This way, the truth of mathematical statements can be established without having to interpret their meaning.
In Philosophy:
In philosophy, formalism is the view that the meaning of a statement is entirely determined by its logical form. This means that the meaning of a statement is independent of its context, its speaker, and its audience.
An example of formalism in philosophy is the work of the logician Gottlob Frege. Frege argued that the meaning of a sentence is determined by the way its parts are arranged, and not by the way it is used in language. For example, the sentence "The cat is on the mat" has the same meaning regardless of who says it, to whom it is said, or in what context it is said.
In art and literature:
Formalism is a school of thought in art and literature that emphasizes the importance of form over content. Formalists believe that the primary goal of art and literature is to create a beautiful or aesthetically pleasing object, and that the meaning of a work of art or literature is irrelevant to its value.
An example of formalism in art is the work of the painter Wassily Kandinsky. Kandinsky believed that the goal of art was to create a harmonious composition of colors and shapes, and that the meaning of a painting was irrelevant to its artistic value.