How do logicism, intuitionism, and formalism differ in their approaches to mathematical foundations?

During the early 20th century, logicism, intuitionism, and formalism emerged as competing schools of thought aimed at securing a rigorous foundation of mathematics following the discovery of contradictions in naive set theory.12

• Logicism: This view contends that mathematics is essentially an extension of logic. Logicists, such as Gottlob Frege and Bertrand Russell, argued that all mathematical truths can be deduced from pure logical principles without requiring specifically mathematical concepts.13
• Intuitionism: Introduced by L.E.J. Brouwer, this approach views mathematics as a constructive activity. Intuitionists reject nonconstructive arguments and the classical law of the excluded middle, particularly when dealing with infinite sets, arguing that a mathematical statement is only true if it can be proven through construction.14
• Formalism: Championed by David Hilbert, formalism focuses on the syntax and formal language of mathematics. It treats mathematical systems as formal games governed by rules, aiming to prove the consistency of these systems through metamathematical analysis.52

While these schools offered different philosophical justifications, they collectively shaped modern mathematical inquiry by highlighting the necessity of formal rigor.12