Brouwer Develops Intuitionist Foundations of Mathematics

L. E. J. Brouwer pioneered the intuitionist reformulation of the logical foundations of mathematics.

Summary of Event

The first decade of the twentieth century was a crisis period in many areas of mathematics, particularly in geometry, arithmetic, set theory, and formal logic as well as their interrelations. These crises arose predominantly from a number of paradoxes in Georg Cantor’s theory of sets, from the attempted formalization of logic and arithmetic by Richard Dedekind and Giuseppe Peano, and from methodological disputes about the validity of different types of mathematical proof. From the mid-1880’s, these contradictions indicated to many mathematicians that unforeseen defects were arising from various attempts to reformulate classical geometry, arithmetic, and logic in ways noncontradictory with respect to linguistic uses and ontological commitments about mathematical terms and referents. As a response to these crises, three main schools of mathematical philosophy arose: logicism, formalism, and intuitionism. Mathematics;intuitionist foundations
[kw]Brouwer Develops Intuitionist Foundations of Mathematics (1904-1907)
[kw]Intuitionist Foundations of Mathematics, Brouwer Develops (1904-1907)
[kw]Mathematics, Brouwer Develops Intuitionist Foundations of (1904-1907)
Mathematics;intuitionist foundations
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[c]Mathematics;1904-1907: Brouwer Develops Intuitionist Foundations of Mathematics[00950]
Brouwer, L. E. J.

Philosophical discussion of arithmetic and geometry in Immanuel Kant’s Kant, Immanuel
Kritik der reinen Vernunft (1781; Critique of Pure Reason, 1838) Critique of Pure Reason (Kant) has been cited often as the earliest progenitor of mathematical intuitionism. Kant explicated his theory of the innate forms of spatial and temporal perception, respectively, as the intuitional basis of geometry and arithmetic. Kant argues that the way we first learn that 7 + 5 = 12 is precisely by a process like counting progressively from 7 to 12 via successive additions of 1, operating or constructed with a particular example of objects first given in mental intuition. In an algebraic sense, the set of numbers generated by the above, or related, procedures is united with the natural counting numbers insofar as it has an initial element 0 and a successor relation 1. The algebraist, according to Kant, gets arithmetic results by manipulating representational symbols according to certain rules of construction, which, Kant claims, cannot be obtained without these prior intuitions. Kant posited this construction with intuitions as the chief source of clarity and evidence in the fundamentals of mathematics.

In the mid-nineteenth century, German mathematician Leopold Kronecker Kronecker, Leopold published what many historians and philosophers of mathematics consider to be preintuitionist queries about the “existential proofs” of anti-Kantian mathematician-philosopher Bernhard Bolzano Bolzano, Bernhard in the theory of differential equations. In particular, Kronecker doubted the utility and validity of proofs that assume without demonstrating the existence of solutions to classes of differential equations, insofar as formalists like Bolzano sought to establish the existence of mathematically permissible solutions by indirect deductive arguments, without either explicitly constructing or providing a finite procedure for finding such a solution. Between 1880 and 1906, a more extensive constructivist school of French mathematicians, including Émile Borel, Henri-Léon Lebesgue, and, notably, Henri Poincaré, Poincaré, Henri likewise stressed the need of basic intuitions to define the sequence of integers and the general content on which all mathematics depends.

Many of these issues came to a focus with the publication of David Hilbert’s Hilbert, David
Grundlagen der Geometrie (1899; The Foundations of Geometry, 1902) Foundations of Geometry, The (Hilbert) on one hand and Edmund Husserl’s Husserl, Edmund
Logische Untersuchungen (1900-1901; Logical Investigations, 1970) Logical Investigations (Husserl) on the other, together with the subsequent acrimonious sessions on “logic and philosophy of science” at the Second International Congress of Philosophy in Paris in 1904. With this background, Gerrit Mannoury’s Mannoury, Gerrit views on the relations of ordinary language, psychology, and Kantian philosophy laid the foundations of mathematics. As a graduate student in 1904, L. E. J. Brouwer chose the foundations of mathematics as his dissertation topic. Brouwer closely reexamined the ongoing (1901 to 1906) debates among Poincaré, Louis Couturat, Couturat, Louis and Bertrand Russell Russell, Bertrand on the relations of priority and independence between (symbolic) logic and (axiomatic) mathematics, and on the criticality of logical deduction versus psychological intuition in mathematical creation and proof. In his first semitechnical publication on the origins of mathematical thinking and its “objects,” Brouwer sought a way to keep the methodological autonomy, purity, and rigor of Hilbert, together with the links to philosophical and everyday thinking of Husserl, while avoiding the logical and linguistic puzzles of Russell.

With Kant, Brouwer believed the ultimate foundation of mathematics is the subjective awareness of time, divided into past/before and future/after. This, for Brouwer, gives rise to a basic intuition of “bare twoness,” not itself a number but an absolutely basic and simple mental intuition from which it is possible to construct all finite ordinal numbers. As Brouwer developed in his doctoral dissertation of 1907, he regarded the starting point for mathematics to be neither abstract logical rules nor functional mathematical axioms, but the individual’s originative form of understanding, without language or logical concepts, of the sequence of positive integers vis-à-vis repeated duplication of temporally sequential unit constructions with |, ||, |||, and so on (which might be indexed books on a shelf, sticks in a row, or any other heuristically convenient placeholder concept).

Some scholars consider Brouwer’s constructions with mental intuitions similar to the “thought experiments” of his contemporary the English philosopher Francis Herbert Bradley, Bradley, Francis Herbert whereas others see Brouwer’s as an implicit psychological theory of mentation. The paradoxes exist, Brouwer claimed, only because of logic and language and the fact that these symbolic and discursive concepts can never be made commensurate with temporal intuitions. In Brouwer’s intuitionist view, mathematics is apodictically certain and true precisely because it rests on this simple and direct mental awareness of self-evident temporal intuitions and constructs by the individual. Brouwer’s intuitionism sought to restrict mathematical knowledge to only that which can be actually and directly constructed in finite proofs: To know something mathematically is to have a specifically finite implementable proof based on temporal intuitions without existence assumptions. Moreover, Brouwer denied Hilbert and Russell’s claim of being able to circumscribe the full extent of any formal systems of mathematics, radically implying that the previously timeless, absolute, and impersonal realm of logic and mathematics in some ways exhibits the character of historical time, factual incompleteness, and personal mental activity.

Together with Friedrich Ludwig Gottlob Frege’s, Hugo Dingler’s, and Husserl’s philosophies of mathematics, Brouwer’s intuitionism takes the component statements of a mathematical theory to be really meaningful. Also, Brouwer did not agree with Poincaré’s neointuitionism or Hilbert’s formalism, which denied genuine intrinsic content or meaning to a given mathematical proposition apart from the complete set of axioms of which it is a part. Brouwer concurs with Russell’s logicism, insofar as intuitionism identifies a particular axiom (versus those dependent only on an entire system of axioms). In 1906, Russell’s paper “The Paradoxes of Logic” acknowledged that logicism cannot eliminate but can only regulate the use of mathematical intuition. Thus, for Brouwer, although propositions in and about mathematics can be ordered and ranked in an asymmetrical hierarchy according to their complexity of derivation, on explicitly philosophical grounds, Brouwer denied that the possibilities of mathematical construction, what he called an “open system,” can be confined a priori, within the bounds of any formal closed system of statements, conceptually anticipating some of the implications of Kurt Gödel’s later undecidability-incompleteness theorems. Brouwer likewise demonstrated at least nine pre- and metalevels with respect to operational mathematics and logic (in contrast to Hilbert’s subsequently published three-level system).

In his famous 1908 paper Over de Onbetrouwbaarheid der logische Principes
Over de Onbetrouwbaarheid der logische Principes (Brouwer) (on the untrustworthiness of logical principles), Brouwer rejected radically the traditional belief (held by Aristotle, Kant, Bolzano, Russell, Frege, Hilbert, Husserl, and many others) that classical logic has unquestionable validity. This rejection included denying the universally accepted laws of double negation (for example, that “the opposite of the opposite of a true statement is a false statement”), which Russell and Hilbert associated with the equation –(–1) = 1. Also rejected was the law of the excluded middle (that no options are possible but true or false statements), which latter concept Alfred North Whitehead and Russell identify with the law of noncontradiction in their Principia Mathematica (1910-1913). To illustrate the not infrequent inapplicability of the law of the excluded middle, Brouwer cited a number of constructionally incompletable sequences, such as algorithms to compute the final digit of pi or the second-to-last valid example of Pierre de Fermat’s conjecture, arguing that unless and until a proof is constructed to deduce their truth or falsity, the law of the excluded middle simply cannot be applied.


Following Brouwer’s initial publications between 1904 and 1911, both formalist and logicist camps reexamined their criticisms, consistency, and opportunities in the light of intuitionism. Initially, it was objected that intuitionist rejection of the pure axiomatic approach of Hilbert and the classical and symbolic logics of Aristotle and Peano and Russell left unaccounted for a large body of well-established pure and applied mathematics. In 1918, in response to attacks against intuitionism’s alleged impoverishment, Brouwer explicitly set out to reconstruct classical mathematics in accord with his own intuitionist tenets, seeking greater natural clarity in construction without too great an increase in mental labor, proof duration, and complexity. Brouwer’s own positive contributions included a new theory of rational numbers, an intuitionist theory of sets and topology, and a theory of finite and transfinite ordinal numbers. Arend Heyting, Brouwer’s principal student, later contributed to extending intuitionist reconstruction into the theory of functions as well as formal and some classes of symbolic logic.

Although working intuitionist analogues are still unavailable for many important areas of mathematics (for example, the implicit function theorem and the fundamental theorem of calculus), certain powerful and perplexing results can be proved in intuitionist theory that cannot be demonstrated in classical mathematics. Perhaps the most notable example of the latter is the intuitionist theorem that any real-valued function that is defined everywhere on a closed interval of the real number continuum is uniformly continuous on that interval. Since the 1950’s, a number of philosophical (metamathematical) objections have been raised to some intuitionist claims—notably, to the status of negative propositions referring to objects (such as a square circle) that do not exist and to occasional intersubjective conflicts between intuitionist mathematicians on whether a given intuitive proof is sufficiently self-evident.

Some intuitionist-influenced mathematicians—such as Hermann Weyl, George Polya, Ferdinand Gonseth, and Evert Willem Beth—have sought to reformulate Brouwer’s introspective notion of intuition from a purely subjective psychology of mental experience to a more general genetic, or heuristic, model for the historic or individual development of concrete empirical concepts facilitating understanding and using abstract mathematics. Many mathematicians and logicians do not consider Brouwer’s or subsequent intuitionist mathematics capable of ever being a self-sufficient and fully successful renovation of classical mathematics and logic. Nevertheless, in addition to intuitionism’s continuance as an identifiable school of thought with its own proceedings and journal, most acknowledge its many and lasting influences on mathematics, such as in the work of Gödel’s later recursive function theory and Paul Lorenzen’s efforts to develop Dingler’s notion of constructivist protologic for arithmetic, geometry, and other pure and applied mathematics. Mathematics;intuitionist foundations

Further Reading

  • Beth, Evert W., and Jean Piaget. Mathematical Epistemology and Psychology. Translated by W. Mays. Dordrecht, the Netherlands: D. Reidel, 1966. A complementary approach that preserves and updates Brouwer’s psychology of mental intuitions.
  • Brouwer, L. E. J. Collected Works. 2 vols. New York: Elsevier, 1975-1976. A critical complete edition of Brouwer’s collected and translated works. Volume 1 includes key sections of Brouwer’s 1907 thesis as well as several book and essay reviews on Kant, Frege, Russell, and others.
  • Dummett, Michael. Elements of Intuitionism. 2d ed. New York: Oxford University Press, 2000. This intermediate-level monograph is a philosophical reconstruction of basic concepts by Brouwer and Heyting by a longtime student of Russell’s and Frege’s logic. Offers an interpretation of Brouwer’s position.
  • Fischbein, Efraim. Intuition in Science and Mathematics. Dordrecht, the Netherlands: D. Reidel, 1987. Offers a thorough synopsis of the diversified possible meanings of intuition in mathematics.
  • Hesseling, Dennis E. Gnomes in the Fog: The Reception of Brouwer’s Intuitionism in the 1920s. Cambridge, Mass.: Birkhäuser Boston, 2004. Presents the foundational debate in mathematics that took place in the 1920’s, including both its contributions and its shortcomings.
  • Kleene, Stephen C., and Richard E. Vesley. The Foundations of Intuitionistic Mathematics. Amsterdam: North Holland, 1965. Requires some background in formal logic and proof theory. Spells out intuitionist efforts as well as the relations of Brouwer’s formal insights to Gödel’s undecidability theorem.
  • Van Heijenoort, Jean, comp. From Frege to Goedel. Cambridge, Mass.: Harvard University Press, 1967. Offers a carefully detailed outline of several stages in Brouwer’s early thought.

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