Philosophy of mathematics

Reality is the state of everything that exists, not how they might be imagined. Different cultures and academic disciplines conceptualize it in various ways.

thumb|The Sierpiński triangle contains infinitely many (scaled-down) copies of itself. Infinity is something which is boundless, limitless, endless. It is denoted by , called the infinity symbol.

subfield of mathematics

branch of philosophy that studies the assumptions, foundations, and implications of mathematics

study of the basic mathematical concepts

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 activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possibl

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, laws, or entities. The word was coined by Johann Eduard Erdmann as Psychologismus, being translated into English as psychologism.

interpretation of probability as a measure of the degree of belief of an individual assessing the uncertainty of a particular situation

notion that some mathematicians may derive aesthetic pleasure from mathematics

In philosophy of mathematics, logicism is a school of thought comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

view that statements of logic can be considered to be statements about the consequences of certain string manipulation rules

two propositions or events that cannot both be true

mathematical concept – an extension of the idea of infinity

thumb|Squaring the circle: the areas of this square and circle are both equal to [[pi|. It was proved in 1882 that this figure cannot be constructed in a finite number of steps with an idealized straightedge and compass. Nevertheless, "proofs" of such constructions were still published even 50 years later.]]

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 theoretical computer scientist Christos Papadimitriou. Character design and artwork are by Alecos Papadatos and color is by Annie Di Donna. The book was originally written in English, and was translated into Greek by author Apostolos Doxiadis for the release in Greece, which preceded the UK and U.S. releases.

mathematical proof at least partially generated by computer

reasoning about degrees of belief using Bayesian inference

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 quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions.

logical principle

Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy', or the epistemological view that reality is fundamentally mathematical. The term has been applied to a number of philosophers, including Pythagoras and René Descartes although the term was not used by themselves.

relationship

concept of infinite entities as given, actual and completed objects rather than non-terminating processes

In the philosophy of mathematics, ultrafinitism, ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism are various philosophies of mathematics with aspects of finitism and intuitionism. Common to these philosophies is their objection to the totality of number theoretic functions like exponentiation over natural numbers.

viewpoint in the philosophy of mathematics

emerging field of applied ethics

Finiteness, finitude, or being finite, is the state of being limited or having an end, and is a counter to the concept of infinity. Humans are considered to be in this state because of their limited life span, uniformly ending in death. Each natural number is considered to be in this state, because counting up to that number stops when the number is reached. The concept appears across disciplines, from mathematics and linguistics to philosophy, where it is used to describe quantities, structures, and conditions. In mathematics, a set or number is finite if it is limited in size, while in lingu

form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging

the theorem that, if a machine-learning algorithm does well on some problems, then it pays for that on all other problems