Gottlob Frege is widely regarded as the father of modern logic for his groundbreaking work on the foundations of mathematics and logic.

Russell's Paradox, also known as the Russell-Zermelo Paradox, exposed a fundamental contradiction in set theory and led to a crisis in the foundations of mathematics.

Ludwig Wittgenstein, in his later work, argued that mathematical statements are not true or false in the traditional sense, but rather are part of a language game with its own rules and conventions.

Intuitionism, championed by mathematicians like L.E.J. Brouwer and Hermann Weyl, rejected the idea that mathematics is solely based on formal logic and axioms, emphasizing the role of intuition and the creative process in mathematical discovery.

Gottlob Frege, in his influential work 'The Foundations of Arithmetic', argued that mathematical objects, such as numbers, have an objective existence independent of the human mind and are discovered through logical reasoning.

Gödel's Incompleteness Theorem, formulated by Kurt Gödel in 1931, demonstrated that any formal system capable of expressing basic arithmetic must either be incomplete or inconsistent.

Kurt Gödel, in his later writings, embraced mathematical Platonism, the idea that mathematical objects exist independently of the human mind and have an objective reality.

Structuralism, championed by mathematicians and philosophers like Paul Benacerraf and Michael Dummett, emphasizes the importance of mathematical structures and their relationships, rather than the specific objects or elements within those structures.

W.V. Quine, in his influential work 'Word and Object', argued that the truth or falsity of mathematical statements is determined by their coherence with the rest of our beliefs and knowledge, rather than by any inherent or objective properties.

Social Constructivism, championed by scholars like Thomas Tymoczko and Sal Restivo, emphasizes the role of social and cultural factors in shaping mathematical knowledge and practices, arguing that mathematics is not a purely objective and universal discipline.