Mathematical Truth and Knowledge

Mathematical truth is generally considered to be objective and independent of human minds, discovered through logical reasoning and proof, not revealed through divine inspiration.

Platonism is the philosophical theory that states that mathematical objects exist independently of the human mind.

Mathematical objects are generally considered to be abstract and not accessible to our senses, not created by the human mind, and subject to change.

Formalism is the philosophical theory that states that mathematical truth is discovered through logical reasoning and proof.

Formalism is generally criticized for reducing mathematics to a game of symbols, ignoring the role of intuition in mathematical discovery, and not being able to account for the creativity and beauty of mathematics, not for being too abstract and divorced from the real world.

Constructivism is the philosophical theory that states that mathematical knowledge is constructed by the human mind.

Constructivism is generally criticized for making mathematics subjective and dependent on the individual's beliefs, not being able to account for the objectivity and universality of mathematical truth, and being too narrow and not allowing for the possibility of mathematical discovery, not for being too abstract and divorced from the real world.

Mathematical knowledge is generally considered to be a priori and independent of experience, synthetic and extending our knowledge of the world, not a posteriori and derived from experience.

Kantianism is the philosophical theory that states that mathematical knowledge is synthetic and extends our knowledge of the world.

Kantianism is generally criticized for making mathematics too subjective and dependent on the individual's mind, not being able to account for the objectivity and universality of mathematical truth, and being too abstract and divorced from the real world, not for being too narrow and not allowing for the possibility of mathematical discovery.

Mathematics is generally considered to be a tool for describing and understanding the real world, a language for expressing abstract ideas, and a game played with symbols, not a self-contained system with no connection to the real world.

Formalism is the philosophical theory that states that mathematics is a self-contained system with no connection to the real world.

Formalism is generally criticized for making mathematics too abstract and divorced from the real world, ignoring the role of intuition in mathematical discovery, and not being able to account for the creativity and beauty of mathematics, not for being too narrow and not allowing for the possibility of mathematical discovery.

Mathematical proof is generally considered to be a logical argument that demonstrates the truth of a mathematical statement, a formal demonstration that a mathematical statement is true, and a convincing argument that a mathematical statement is true, not a process of discovering new mathematical knowledge.

Intuitionism is the philosophical theory that states that mathematical proof is a process of discovering new mathematical knowledge.