Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers.

Overview

The term ''

Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...

'' is used because such a view is seen to parallel

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

Theory of Forms The Theory of Forms or Theory of Ideas, also known as Platonic idealism or Platonic realism, is a philosophical theory credited to the Classical Greek philosopher Plato. A major concept in metaphysics, the theory suggests that the physical w ...

and a "World of Ideas" (Greek: ''eidos'' (εἶδος)) described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular ''

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...

'' of ancient Greece, who believed that the world was, quite literally, generated by

number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

s. A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the

Ultimate Ensemble In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. According to the hypothesis, the universe ''is'' a ...

, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.

Views

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things

Edmund Husserl Edmund Gustav Albrecht Husserl (; 8 April 1859 – 27 April 1938) was an Austrian-German philosopher and mathematician who established the school of Phenomenology (philosophy), phenomenology. In his early work, he elaborated critiques of histori ...

said about mathematics, and supports

Immanuel Kant Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German Philosophy, philosopher and one of the central Age of Enlightenment, Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works ...

's idea that mathematics is

synthetic Synthetic may refer to: Science * Synthetic biology * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic elements, chemical elements that are not naturally found on Earth and therefore have to be created in ...

a priori ('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, Justification (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...

''.)

Philip J. Davis Philip J. Davis (January 2, 1923 – March 14, 2018) was an American academic Applied mathematics, applied mathematician. Biography Davis was born in Lawrence, Massachusetts. He was known for his work in numerical analysis and approximation theor ...

and

Reuben Hersh Reuben Hersh (December 9, 1927 – January 3, 2020) was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. Although he was generally known as Reuben Hersh, late in life h ...

have suggested in their 1999 book ''

The Mathematical Experience ''The Mathematical Experience'' (1981) is a book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective. The book discusses the psychology of mathematicians, and gives ...

'' that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.

Full-blooded Platonism

Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the

excluded middle In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and th ...

, and the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from a single consistent set of axioms.

Set-theoretic realism

Set-theoretic realism (also set-theoretic Platonism) a position defended by

Penelope Maddy Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Distinguished Professor Emerita of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the ...

, is the view that

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

is about a single universe of sets. This position (which is also known as naturalized Platonism because it is a

naturalized Naturalization (or naturalisation) is the legal act or process by which a non-national of a country acquires the nationality of that country after birth. The definition of naturalization by the International Organization for Migration of the ...

version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of

Paul Benacerraf Paul Joseph Salomon Benacerraf (; 26 March 1930 – 13 January 2025) was a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement ...

's epistemological problem.

Platonized naturalism

A similar view, termed Platonized naturalism, was later defended by the

Stanford–Edmonton School 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 mathematics is reducible to logic, or some or all ...

: according to this view, a more traditional kind of Platonism is consistent with naturalism; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of

abstract object In philosophy and the arts, a fundamental distinction exists between abstract and concrete entities. While there is no universally accepted definition, common examples illustrate the difference: numbers, sets, and ideas are typically classif ...

References