Immanuel Kant, in his Critique of Pure Reason, posited that mathematics is a priori and synthetic, meaning it is known independently of experience and provides new knowledge.

Logical Positivism, also known as Logical Empiricism, is a philosophical school that emphasizes the importance of logic and mathematics in understanding the world and verifying knowledge.

Henri Poincaré, a French mathematician and philosopher, argued that mathematics is a human invention and not a discovery of objective truths, emphasizing the role of human creativity in mathematical thought.

Gödel's Incompleteness Theorem, formulated by Kurt Gödel, states that any sufficiently complex formal system is either incomplete or inconsistent, challenging the idea of a complete and consistent foundation for mathematics.

Bertrand Russell, in his work Principia Mathematica, argued that mathematics is a language that describes the structure of reality, emphasizing the logical foundations of mathematics and its relationship to the world.

Mathematical Empiricism is a philosophical theory that emphasizes the importance of mathematical models and simulations in understanding scientific phenomena, arguing that mathematics is an empirical tool for exploring the world.

Paul Feyerabend, in his book Against Method, argued that mathematics is a social construction and is shaped by cultural and historical factors, challenging the idea of a universal and objective foundation for mathematics.

Mathematical Aesthetics is a philosophical theory that emphasizes the importance of mathematical beauty and elegance in understanding the nature of reality, arguing that mathematical beauty is a guide to truth.

Nelson Goodman, in his book Ways of Worldmaking, argued that mathematics is a form of art and is not subject to the same standards of truth and objectivity as science, emphasizing the creative and subjective aspects of mathematical thought.

Mathematical Structuralism is a philosophical theory that emphasizes the importance of mathematical structures and patterns in understanding the world, arguing that mathematics is a study of abstract structures that are independent of any particular interpretation.

John Stuart Mill, in his System of Logic, argued that mathematics is a tool for describing and predicting the behavior of the physical world, emphasizing the instrumental value of mathematics in scientific inquiry.

Mathematical Formalism is a philosophical theory that emphasizes the importance of mathematical logic in understanding the foundations of mathematics, arguing that mathematics is a formal system based on axioms and rules of inference.

L. E. J. Brouwer, the founder of intuitionism, argued that mathematics is a form of intuition and is not derived from experience, emphasizing the role of the human mind in constructing mathematical concepts.

Mathematical Naturalism is a philosophical theory that emphasizes the importance of mathematical naturalism in understanding the relationship between mathematics and the natural world, arguing that mathematics is a natural phenomenon that arises from the structure of the universe.

Galileo Galilei, in his book The Assayer, argued that mathematics is a universal language that can be used to describe the structure of the universe, emphasizing the importance of mathematics in understanding the laws of nature.