Concept

Formalism (philosophy of mathematics)

Overview

In 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 (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."

According to formalism, mathematical statements are not "about" numbers, sets, triangles, or any other mathematical objects in the way that physical statements are about material objects. Instead, they are purely syntactic expressions—formal strings of symbols manipulated according to explicit rules without inherent meaning. These symbolic expressions only acquire interpretation (or semantics) when we choose to assign it, similar to how chess pieces follow movement rules without representing real-world entities. This view stands in stark contrast to mathematical realism, which holds that mathematical objects genuinely exist in some abstract realm.

Formalism emerged as a response to foundational crises in mathematics during the late nineteenth and early twentieth centuries, particularly concerns about paradoxes in set theory and questions about the consistency of mathematical systems. It represents one of the three major philosophical approaches to mathematics developed during this period, alongside logicism and intuitionism, though formalism encompasses a broader spectrum of positions than these more narrowly defined views. Among formalists, the German mathematician David Hilbert was the most influential advocate, developing what became known as Hilbert's program to establish the consistency of mathematics through purely formal methods.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. Yet his ideas were speculative in spirit; whereas the system of Wolf, who carried the method out to the height of pedantry, was even in subject-matter a metaphysic of the understanding. The abuses which these methods with their formalism once led to in philosophy and science have in modern times been followed by the abuses of what is called 'Construction.' Kant brought into vogue the phrase that mathematics…

   Georg Wilhelm Friedrich Hegel, The Logic of Hegel

   States what Formalism is; names Formalism several times.

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2. The most remarkable result of modern methods in mathematics is the importance of symbolic logic and of rigid formalism. Mathematicians, under the influence of Weierstrass, have shown in modern times a care for accuracy, and an aversion to slipshod reasoning, such as had not been known among them previously since the time of the Greeks. The great inventions of the seventeenth century--Analytical Geometry and the…

   Bertrand Russell, Mysticism and Logic and Other Essays

   The author is tied to Formalism in the knowledge graph; names Formalism twice.

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3. The logical basis of the theory alone belongs to philosophy, and alone will occupy us to-night. The way the problem of continuity enters into philosophy is, broadly speaking, the following: Space and time are treated by mathematicians as consisting of points and instants, but they also have a property, easier to feel than to define, which is called continuity, and is thought by many philosophers to be destroyed when…

   Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy

   States what Formalism is; the author is tied to Formalism in the knowledge graph.

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Philosophers who discussed Formalism (philosophy of mathematics)

• Willard Van Orman Quine
• Imre Lakatos
• Gottlob Frege
• Rudolf Carnap
• Gerrit Mannoury

Related ideas

• Interpretation (logic)
• Integer
• Kurt Gödel
• Mathematics
• Intuitionism
• Ontology
• Paradox
• Rule of inference
• Set theory
• Semantics
• Arithmetic
• Completeness (logic)
• Logicism
• Formal system
• Bertrand Russell
• Logic
• Philosophy of mathematics
• Gottlob Frege
• Chess
• Syntax (logic)
• Rudolf Carnap
• Gödel's incompleteness theorems
• Hilbert's program
• Consistency

Place in the tradition

Part of Analytic philosophy.

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