# Category: Philosophy of mathematics

Bayesian epistemology
Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast
Ludwig Wittgenstein's philosophy of mathematics
Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. As with his philosophy of language, Wi
Multiverse (set theory)
In mathematical set theory, the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though so
Quasi-empirical method
Quasi-empirical methods are methods applied in science and mathematics to achieve epistemology similar to that of empiricism (thus quasi- + empirical) when experience cannot falsify the ideas involved
Ethics in mathematics
Ethics in mathematics is an emerging field of applied ethics, the inquiry into ethical aspects of the practice and applications of mathematics. It deals with the professional responsibilities of mathe
Centipede mathematics
Centipede mathematics is a term used, sometimes derogatorily, to describe the generalisation and study of mathematical objects satisfying progressively fewer and fewer restrictions. This type of study
Formalism (philosophy of mathematics)
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (al
Psychologism
Psychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, l
Logical harmony
Logical harmony, a name coined by Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system.
Bloch's principle
Bloch's Principle is a philosophical principle in mathematicsstated by André Bloch. Bloch states the principle in Latin as: Nihil est in infinito quod non prius fuerit in finito, and explains this as
Definitions of mathematics
Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All proposed definitions are controversial i
Platonic realism
Platonic realism is the philosophical position that universals or abstract objects exist objectively and outside of human minds. It is named after the Greek philosopher Plato who applied realism to su
Relationship between mathematics and physics
The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators. General
Weyl's tile argument
In philosophy, the Weyl's tile argument, introduced by Hermann Weyl in 1949, is an argument against the notion that physical space is "discrete", as if composed of a number of finite sized units or ti
Hume's principle
Hume's principle or HP says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs. HP can be stated formally in
Arithmetization of analysis
The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.
Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary. The term is also used to refer to the ontological status of things, indicat
Continuum (measurement)
Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variation us
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of
Foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosoph
Rational agent
A rational agent or rational being is a person or entity that always aims to perform optimal actions based on given premises and information. A rational agent can be anything that makes decisions, typ
Logical reasoning
Two kinds of logical reasoning are often distinguished in addition to formal deduction: induction and abduction. Given a precondition or premise, a conclusion or logical consequence and a rule or mate
Quine–Putnam indispensability argument
The Quine–Putnam indispensability argument, also known simply as the indispensability argument, is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such
Informal mathematics
Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies o
Controversy over Cantor's theory
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in s
Actual infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These m
Impredicativity
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quanti
Quasi-empiricism in mathematics
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and
Pseudomathematics
Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are soluti
Logicomix
Logicomix: An Epic Search for Truth is a graphic novel about the foundational quest in mathematics, written by Apostolos Doxiadis, author of Uncle Petros and Goldbach's Conjecture, and at the time Ber
Mathematical practice
Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the
Limitation of size
In the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox.
Aristotelian realist philosophy of mathematics
In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world (or in any
Bayesian probability
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation represent
Benacerraf's identification problem
In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entit
Infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity
Mathesis universalis
Mathesis universalis (from Greek: μάθησις, mathesis "science or learning", and Latin: universalis "universal") is a hypothetical universal science modelled on mathematics envisaged by Descartes and Le
Bayesian program synthesis
In programming languages and machine learning, Bayesian program synthesis (BPS) is a program synthesis technique where Bayesian probabilistic programs automatically construct new Bayesian probabilisti
Ultrafinitism
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitioni
Mathematical monism
No description available.
No free lunch theorem
In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and appears in the 1997 "No Free Lunch Theorems for Optimization". Wolpert had previously derived no
Certainty
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic ce
Philosophy of mathematics education
No description available.
Mathematical beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics
Object of the mind
An object of the mind is an object that exists in the imagination, but which, in the real world, can only be represented or modeled. Some such objects are abstractions, literary concepts, or fictional
Functional decomposition
In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recompose
Absolute Infinite
The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconcei
Paraconsistent mathematics
Paraconsistent mathematics, sometimes called inconsistent mathematics, represents an attempt to develop the classical infrastructure of mathematics (e.g. analysis) based on a foundation of paraconsist
Philosophy of logic
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at wo
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activ
Logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathem
Structuralism (philosophy of mathematics)
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their pla
Mathematical folklore
In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specia
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and f
Mutual exclusivity
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin tos