"Intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality."
The debate over the foundation of mathematics, with intuitionists emphasizing the role of constructive methods and formalists emphasizing the role of logic and set theory.
Foundations of Mathematics: A study of the fundamental ideas and principles that underlie the subject of mathematics.
Logic: A systematic study of ideas and principles of reasoning argument.
Intuitionism: A school of thought in philosophy of mathematics that holds that mathematical objects are created by the human mind rather than discovered in reality.
Formalism: A school of thought in philosophy of mathematics that holds that mathematical objects are defined by their properties and relations rather than specific constructions.
Set Theory: A branch of mathematics that deals with the study of sets, which are collections of objects.
Mathematical Proof: A demonstration that a particular proposition or theorem is true by logical argument or deduction.
The Foundations of Arithmetic: A book written by Gottlob Frege that outlines the foundations of mathematics.
The Axiomatic Method: A method of constructing a mathematical theory by starting with a small set of axioms and deriving all other propositions by using logical rules of inference.
The Brouwer-Hilbert Controversy: A dispute between two mathematicians, L.E.J. Brouwer and David Hilbert, regarding the foundations of mathematics.
Constructivism: A school of thought in philosophy of mathematics that holds that mathematical objects are created by a process of construction rather than discovery.
Intuitionistic Logic: A formal logical system that is based on intuitionistic principles of reasoning.
Philosophy of Language: A branch of philosophy that studies the nature of language and its relationship to the world.
Metaphysics: A branch of philosophy that studies the fundamental nature of reality.
Epistemology: A branch of philosophy that studies the nature of knowledge and belief.
Semiotics: The study of signs and symbols and their use and interpretation.
Structuralism: A school of thought in philosophy that emphasizes the importance of the structure of systems and the relationships between their parts.
Hermeneutics: The study of interpretation and understanding, especially of texts.
Linguistics: The scientific study of language and its structure.
Objectivity: The property of being independent of subjective opinion, personal bias, or individual influence.
Rationalism: The position that reason is the source of knowledge and that true knowledge is acquired through rational thought and reflection.
Brouwerian intuitionism: Based on Brouwer's philosophy of mathematics, asserts that mathematical objects are mental constructions and that intuition is the only reliable source of knowledge.
Russian intuitionism: Developed by Poincare's student, L.E.J. Brouwer, who believed that mathematical concepts are not objective, but constructed by the mind. He rejected the idea of using an infinite set to define mathematical objects.
Finitism: A branch of intuitionism that argues that only finite mathematical objects and methods are acceptable.
Constructive predicativism: This branch is quite similar to intuitionism, but with its focus on predicatively defined sets. It argues that mathematical objects must be explicitly constructed and that the use of the "law of excluded middle" and the "axiom of choice" is problematic.
Hilbert's formalism: Based on David Hilbert's principal, the philosophy of mathematics does not concern itself with abstract objects, but with derived rules of manipulation, such as algorithms and axioms, and their implications.
Deductive nominalism: This form of formalism asserts that only the formal relationships between symbols matter, rather than the objects or properties they represent.
Proof-theoretic conceptions of mathematics: Formalism used to improve the study of proof theory, including the role of the axiom of choice.
The syntax-semantics debate: This debate focused on the interrelation between syntax and semantics in mathematics, and it highlighted the importance of adequately explaining how formal axiomatic systems can represent fully mathematics.
"Intuitionism is opposed to preintuitionism, which suggests that mathematics is the discovery of fundamental principles existing in an objective reality."
"Yes, in intuitionism, mathematics is considered as the application of internally consistent methods used to realize more complex mental constructs."
"No, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied in intuitionism."
"Humans are considered to be solely responsible for the constructive mental activity that results in mathematics in the context of intuitionism."
"No, intuitionism rejects the notion of fundamental principles existing independently in an objective reality."
"Yes, the methods used in mathematics are expected to be internally consistent according to intuitionism."
"No, the existence of mathematical constructs in an objective reality is not a relevant concern in intuitionism."
"Intuitionism views mathematics as the result of constructive mental activity."
"Yes, intuitionism considers mathematics to be purely the result of human constructive mental activity."
"No, intuitionism does not consider logic and mathematics to reveal and apply deep properties of objective reality."
"Intuitionism suggests that logic and mathematics are not considered as analytic activities revealing deep properties of objective reality."
"Intuitionism acknowledges that the independent existence of mathematical constructs in an objective reality is not essential."
"No, intuitionism does not believe that deep properties of objective reality are revealed and applied through logic and mathematics."
"Yes, intuitionism focuses on the constructive mental activity of humans as the source of mathematics."
"No, intuitionism does not emphasize the discovery of fundamental principles existing outside human mental activity."
"The primary purpose of logic and mathematics in intuitionism is to realize more complex mental constructs using internally consistent methods."
"No, logic and mathematics are considered as application tools for constructing mental constructs in intuitionism."
"No, intuitionism does not concern itself with the objective reality of mathematical principles."
"Yes, intuitionism considers the realization of complex mental constructs through internally consistent methods to be significant."