The debate over whether mathematical entities are real objects that exist independently of us, or ideal constructs that we invent and manipulate.
Realism vs. nominalism: The debate between realism and nominalism is fundamental to the philosophy of mathematics. Realism argues that mathematical objects exist independently of the human mind, while nominalism claims that they exist only as useful human constructions.
Platonism: Platonism is the view that mathematical objects exist in a non-physical realm of reality as pure forms or ideas, independent of physical objects or human minds.
Nominalism: Nominalism argues that mathematical objects do not exist independently, but are simply names or labels we use to describe relationships between physical objects.
Mathematical logic: Mathematical logic is the study of formal systems of reasoning and deduction, and is a fundamental tool in modern mathematics.
Axioms and logical systems: Axioms are basic assumptions or principles that form the basis of a logical system. Different logical systems can have different sets of axioms, leading to different mathematical structures and results.
Set theory: Set theory is the study of collections of objects and the relationships between them, and provides a foundation for modern mathematics.
Frege's logicism: Frege's logicism is the view that all of mathematics can be reduced to logic and set theory, and provides a foundation for Platonism.
Intuitionism: Intuitionism is a branch of mathematics that rejects the existence of infinite sets and the law of excluded middle, and argues that mathematics is a human construct.
Formalism: Formalism is a view of mathematics that emphasizes the manipulation of symbols and formal rules, rather than the existence of mathematical objects.
Godel's incompleteness theorems: Godel's incompleteness theorems demonstrate that any consistent formal system of mathematics must contain true propositions that cannot be proven within the system itself.
Ontology: Ontology is the study of the nature of existence, and is relevant to Platonism and nominalism in terms of the ontological status of mathematical objects.
Epistemology: Epistemology is the study of knowledge, and is relevant to Platonism and nominalism in terms of how we can know about mathematical objects.
Metaphysics: Metaphysics is the study of the fundamental nature of reality, and is relevant to Platonism and nominalism in terms of the nature of mathematical objects and their relationship to reality.
Phenomenology: Phenomenology is the study of the structure of conscious experience, and is relevant to intuitionism in terms of the role of intuition in mathematical knowledge.
Philosophy of language: Philosophy of language is the study of the nature of language and its relationship to the world, and is relevant to nominalism in terms of the role of language in constructing mathematical objects.
Realism: This is the belief that mathematical objects exist independently of the human mind and that they have an objective existence.
Full-blown Platonism: This view goes beyond realism and asserts that mathematical objects are not just objectively real, but are also timeless and immutable entities.
Moderate Platonism: This view holds that mathematical objects have a real existence, but they are not entirely separate from the human mind. According to this view, human concepts and language play a role in shaping mathematical truths.
Semantic Platonism: This view contends that mathematical statements and concepts have an objective truth value, independent of our beliefs or understanding of them.
Structuralism: This is the view that mathematical objects are not things in themselves, but rather abstract structures that can be studied and understood through their relationships and properties.
Conceptualism: This view holds that mathematical objects do not exist independently of the human mind, but rather are constructs of our mental concepts and language.
Fictionalism: This view asserts that mathematical objects do not actually exist, but are useful fictions invented by humans to explain empirical phenomena.
Conventionalism: This view holds that mathematical truths are not objectively real, but rather are conventions agreed upon by humans in a particular cultural and social framework.
Empiricism: This is the view that mathematical knowledge is derived from experience and observation, rather than from abstract reason or independent existence of mathematical objects.
Instrumentalism: This view contends that mathematical objects do not have an independent existence, but are simply tools that humans use to solve problems and make predictions about the world.