- "Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics..."
The study of the logical and philosophical basis of mathematics, including the axioms on which mathematics rests and the nature of mathematical truth.
The nature of mathematical objects: This covers debates about whether mathematical objects such as numbers and sets actually exist and, if so, what their nature is.
The role of logic in mathematics: This covers topics such as the different types of logic used in mathematics, formal systems, and the relationship between logic and truth.
The axiomatic method: This covers the use of axioms to define mathematical systems and the role of proof in mathematics.
Mathematical ontology: This covers the study of what exists in mathematics, such as the nature of mathematical objects and the relationships between them.
The philosophy of mathematics and its history: This covers the history of mathematics and its relationship to philosophy, including debates about the nature of mathematical knowledge and the role of mathematics in society and culture.
Mathematics and its relation to physics: This covers the relationship between mathematics and physics, including the use of mathematical models to describe physical phenomena and the role of mathematics in shaping the development of physics.
The philosophy of probability and statistics: This covers philosophical issues related to probability and statistics, including the interpretation of probability, the role of statistics in scientific inquiry, and the ethical implications of statistical analysis.
Mathematical pluralism: This covers the idea that there are multiple valid ways to do mathematics and questions about the relationship between different mathematical systems.
Mathematical intuitionism: This covers the philosophy of mathematics that emphasizes the role of intuition and constructive methods in mathematical reasoning.
Mathematical realism and anti-realism: This covers debates about whether mathematical objects actually exist or whether they are simply abstract entities invented by mathematicians.
Mathematical explanation: This covers the philosophical questions surrounding the nature of mathematical explanation and how it differs from other types of explanation.
Metaphysics of mathematics: This covers philosophical questions about the nature of mathematical reality and its relationship to other aspects of reality, such as space and time.
Mathematical concepts: This covers the philosophical questions surrounding the nature of mathematical concepts and their relationship to other types of concepts, such as empirical and metaphysical concepts.
Mathematics and the philosophy of language: This covers philosophical questions about the nature of mathematical language, the relationship between meaning and reference, and the role of language in mathematical communication.
The relationship between mathematics and computer science: This covers philosophical questions about the use of computers in mathematics and the relationship between mathematical and computational systems.
Logicism: This type of foundation posits that mathematics is essentially a branch of logic, and that mathematical concepts can be reduced to logical concepts. The goal of logicism is to provide a logical foundation for all of mathematics.
Formalism: Formalists believe that mathematics is simply a manipulation of symbols, and that the meaning of these symbols is irrelevant. The goal of formalism is to provide a purely syntactic foundation for mathematics, without any reference to meaning or objects.
Intuitionism: Intuitionists believe that mathematical objects and concepts are constructed through human intuition, and that they do not exist independently of our minds. The goal of intuitionism is to provide a foundation for mathematics that is based on our intuition and experience.
Platonism: Platonists believe that mathematical objects and concepts exist independently of human minds and are discovered, rather than invented. The goal of platonism is to provide a foundation for mathematics that is based on the existence of these objects.
Constructivism: Constructivists believe that mathematical objects only exist if they can be constructed by a finite process. The goal of constructivism is to provide a foundation for mathematics that is based on the ability of humans to construct mathematical objects.
- "...the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics."
- "...the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague."
- "Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts..."
- "...the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts..."
- "The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges."
- "The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic."
- "Mathematics plays a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially Physics)."
- "Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole."
- "The systematic search for the foundations of mathematics started at the end of the 19th century..."
- "...formed a new mathematical discipline called mathematical logic..."
- "It went through a series of crises with paradoxical results..."
- "...until the discoveries stabilized during the 20th century..."
- "...a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.)..."
- "...whose detailed properties and possible variants are still an active research field."
- "Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences."
- "...which may help to connect it with the rest of human knowledge."
- "The development, emergence, and clarification of the foundations can come late in the history of a field..."
- "...might not be viewed by everyone as its most interesting part."
- "which later had strong links to theoretical computer science."