## 1. The Person




Brouwer studied at the (municipal) University of Amsterdam where his
most important teachers were Diederik Korteweg and Gerrit Mannoury.



Korteweg (1848-1941) was professor of mathematics, mechanics and
astronomy at the University of Amsterdam from 1881-1918; the
last five years as *extraordinarius*, so as to make place for
Brouwer. Around 1900, Korteweg was the most important and influential
mathematician in the Netherlands, and he contributed much to the
internationalisation of Dutch mathematics. A mathematical physicist
trained by van der Waals, he had a particular interest in mechanics
and thermodynamics. Among his many results, the best-known is probably
the Korteweg-de Vries equation, describing the behaviour of waves in a
shallow channel. Korteweg also had a strong historical interest, and
was chief editor of vols. XI-XV of Christiaan Huygens' collected
works (1908-1925). Part of the correspondence with his PhD
student Brouwer concerned the unification of physical theories (see
van Stigt, 1990, pp. 490-498).




Mannoury (1867-1956) was a philosopher, mathematician,
psychoanalyst, accountant, and political activist (Kirkels, 2019). Largely an
autodidact, it was he who introduced in the Netherlands both topology
(in a series of papers of 1897-1900) and Peano's symbolic logic
(in a lecture in 1903). In 1903 he was
appointed *privaatdocent* (unpaid lecturer) in the logical
foundations of mathematics at the University of Amsterdam, where
Brouwer was among his first and most enthusiastic students. His
lectures have been published, in shortened and revised form, as
Mannoury, 1909. But it was Mannoury's earlier, topological papers that
influenced the young Brouwer decisively: the "undertone"
of that work, Brouwer said in a speech in 1946, had changed his view
of mathematics from a collection of "truths fascinating by their
immovability, but horrifying by their lifelessness, like stones from
barren mountains of disconsolate infinity" to a concern with
what is "built ... out of the structural elements of our
thinking" (Brouwer, 1946B, pp. 192-193). Brouwer was also
very interested in Mannoury's reflections on the uses and misuses of
language in mathematics and philosophy. In 1917 Mannoury became
professor *extraordinarius*, in 1918 *ordinarius* at the
University of Amsterdam, succeeding Korteweg at the chair of analytic
and descriptive geometry, mechanics and philosophy of mathematics,
until 1937. A large part of his career was devoted to furthering the
development of *significs*, an analysis of communicative acts
starting from the ideas laid down by Victoria, Lady Welby in her paper
"Sense, Meaning and Interpretation" (Welby, 1896). Brouwer
was also involved in that project.



Brouwer's principal (intuitionistic) students were Maurits Belinfante,
Arend Heyting, and Johan de Iongh; Heyting, in turn, was the teacher
of Anne Troelstra and Dirk van Dalen, and de Iongh of Wim
Veldman. Brouwer's classes were also attended by Max Euwe, later to
become world chess champion, who published a game-theoretical paper on
chess from the intuitionistic point of view (Euwe, 1929), and who
would much later deliver Brouwer's funeral speech. Among Brouwer's
assistants were Heyting, Hans Freudenthal, Karl Menger, and Witold
Hurewicz, the latter two of whom were not intuitionistically
inclined. The most influential supporter of Brouwer's intuitionism
outside the Netherlands at the time was, for a number of years,
Hermann Weyl (whose adaptation of intuitionistic analysis was not
wholly coherent); also Husserl's student Oskar Becker should be
mentioned.




Brouwer seems to have been an independent and brilliant man of high
moral standards, but with an exaggerated sense of justice, making him
at times pugnacious. As a consequence, in his life he energetically
fought many battles. To Mannoury's daughter, Brouwer once said:
"Indeed, your father is one of the few people with whom I have
never had a quarrel. But he brought out the good in people, and I the
bad" (Schmitz, 1990, p. 383).




From 1914 to 1928, Brouwer was member of the editiorial board of the
*Mathematische Annalen*, and he was the founding editor of
*Compositio Mathematica*, which first appeared in 1934.




He was a member of, among others, the Royal Dutch Academy of
Sciences, the Royal Society in London, the Preussische Akademie
der Wissenschaften in Berlin, and the Akademie der Wissenschaften in
Gottingen.




Brouwer received honorary doctorates from the universities of Oslo
(1929) and Cambridge (1954), and was made Knight in the Order of the
Dutch Lion in 1932.


## 2. Chronology



**1881** February 27, born in Overschie (since 1941
part of Rotterdam), The Netherlands.



**1897** Enters the University of Amsterdam to study
mathematics and physics.



**1904** Obtains *doctorandus* title (MSc degree)
in mathematics; first publication (on rotations in four dimensional
space); marries Lize de Holl (born 1870). They would have no children,
but Lize had a daughter from an earlier marriage. They move to
Blaricum, near Amsterdam, where they would live for the rest of their
lives, although they also had houses in other places.



**1904-1907** Doctoral studies. His mathematical
and philosophical notebooks towards his dissertation show strong
influence from notably Cantor, Hilbert, Russell, Poincare, and
Couturat. He is aware of the Frege-Hilbert debate, and in that
matter sides with Frege. Some remarks betray a form of anti-semitism
that, unfortunately, often appeared in language at the time, as
codification of a common prejudice; an occasional remark goes further
and contrasts Jewish people unfavourably to Germanic ones. No such
utterances are known from other periods in his life; compare also the
entries for 1934-1935 and 1940-1945 below. There is no meaningful
connection between this fleeting anti-semitism and the philosophical
views on mathematics, language, and society he set out in his
dissertation and developed over the rest of his life.




**1907** Obtains *doctor* title with
dissertation *On the Foundations of Mathematics*, under
supervision of Korteweg at the University of Amsterdam. (The original
Dutch title of the dissertation was 'Over de Grondslagen der
Wiskunde'. The present entry will for the most part refer to works
cited by using the titles of English translations, with the
bibliography providing information about the original publications.)
It marks the beginning of his intuitionistic reconstruction of
mathematics. Later that year, Brouwer's wife graduates and becomes a
pharmacist. All his life, Brouwer did the bookkeeping for her and
filled out the tax forms, and sometimes he assisted behind the
counter.



**1908** First publication of his argument against the
general validity of the Principle of the Excluded Middle in "The
unreliability of the logical principles". First participation in
an international conference, the Fourth International Conference of
Mathematicians in Rome.



**1909-1913** In a very productive four years,
Brouwer founds modern topology, as a chapter of classical mathematics.
Highlights: invariance of dimension, fixed point theorem, mapping
degree, definition of dimension. As van Dalen (2005, p. 450) observes,
Brouwer's definition of dimension seems to be the first reformulation
of an inductive definition as a game. A pause in his intuitionistic
program.



**1909** Becomes *privaatdocent* (unpaid lecturer)
at the University of Amsterdam. Inaugural lecture "The Nature of
Geometry".



**1909** Meets Hilbert in the Dutch seaside resort of
Scheveningen. Brouwer much admires Hilbert and describes their meeting
in a letter to a friend as "a beautiful new ray of light through
my life" (Brouwer & Adama van Scheltema, 1984, p. 100). Twenty
years later, Brouwer's relation with Hilbert would turn sour.



**1911** First appearance of the names
"formalism" and "intuitionism" in Brouwer's
writings, in a review of Mannoury's *Methodological
and philosophical remarks on elementary mathematics* (1909).



**1912** Elected member of the Royal Academy of
Sciences (during World War II "Dutch Academy of Sciences",
afterward "Royal Dutch Academy of Sciences").



**1912** Appointed full professor
*extraordinarius* in the field of set theory, function
theory, and axiomatics. His philosophical inaugural lecture
"Intuitionisme en Formalisme" is translated into English as
"Intuitionism and Formalism" and thus becomes, in 1913, the
first publication on intuitionism in that language.



**1913** Appointed full professor *ordinarius*,
succeeding Korteweg, who had generously offered to vacate his chair for
the purpose.



**1914** Invited to join the editorial board of the
*Mathematische Annalen*; accepts the honour.



**1918** Brouwer begins the systematic intuitionistic
reconstruction of mathematics with his paper "Founding
Set Theory Independently of the Principle of the Excluded Middle. Part
One, General Set Theory".



**1919** Receives offers for professorships in
Gottingen and in Berlin; declines both.



**1919** "Intuitionistic Set Theory" is the first piece of
intuitionistic mathematics in a widely read international journal, the
*Jahresbericht der Deutschen Mathematiker Vereinigung*.



**1920** Start of the "Grundlagenstreit"
(Foundational Debate) with Brouwer's lecture at the
"Naturforscherversammlung" in Bad Nauheim, published in
1921, "Does Every Real Number Have a
Decimal Expansion?"; the debate was amplified by Weyl's defence of intuitionism
in 1921, "On the New Foundational Crisis of
Mathematics"; Hilbert responds in 1922,
"The New
Grounding of Mathematics".



**1922** Co-founds, with Gerrit Mannoury, the man of letters 
Frederik van Eeden, and others, the "Signific Circle", 
aiming at spiritual and
political progress through language reform. The Circle ends its meetings in
1926, but Mannoury continues its work.



**1926** Lecture in Gottingen; as a result of a
group dinner at Emmy Noether's house, Hilbert and Brouwer are (for a
brief period) on good terms again.



**1927** Lecture series in Berlin; Hans Reichenbach and
Andre Weil are in the audience, as is Brouwer's later assistant
Hans Freudenthal. The newspaper *Berliner Tageblatt* proposes a
public debate between Brouwer and Hilbert, to be held in its pages,
but for some reason this is not realised. Neither does Brouwer
complete the book he is invited to write by the German publisher
Walter de Gruyter. The lectures and an incomplete book are published
posthumously (Brouwer, 1992).



**1928** March 10 and 14: two lectures in Vienna.
Godel is in the audience, as is Wittgenstein. It is said that the
first lecture made Wittgenstein return to philosophy. Brouwer spends a
day with Wittgenstein.



**1928** April: conversations with Husserl, who is in
Amsterdam to lecture.



**1928** Conflict over the Bologna conference. The German
mathematicians are, for the first time since the ending of World War
I, admitted to an international conference again, but not quite as
equals. Brouwer insists that this is not fair, and that the conference
should therefore be boycotted. Hilbert, who does not share this view,
is much chagrined by Brouwer's action and attends the conference as
the leader of the German delegation, the largest present.



**1928-1929** "Mathematische Annalenstreit",
the conflict in the editorial board of *Mathematische
Annalen*. Hilbert, thinking he is about to die, feels a need to
make sure that after his death Brouwer will not become too
influential, and expels him from the board in an unlawful way.
(Hilbert's motivation as described here is documented in letters from
people close to him: Caratheodory to Einstein, October 20,
1928; Blumenthal to the publisher and editors of the *Mathematische
Annalen*, November 16, 1928; Born to Einstein, November 20,
1928. Copies of these letters are in the Brouwer Papers 
at the Noord-Hollands Archief in Haarlem. 
Relevant quotations from these can be found in van Dalen,
2005, p. 604 and p. 613.) Einstein, also member of the board,
refuses to support Hilbert's action and does not want to have anything
to do with the whole affair; most other board members do not want to
irritate Hilbert by opposing him. Brouwer vehemently protests. In the
end, the whole board is dissolved and immediately reassembled without
Brouwer, in a strongly reduced size (in particular, Einstein and
Caratheodory decline). The conflict leaves Brouwer mentally
broken and isolated, and puts an end to a very creative decade in his
work. Now that the two main contestants are no longer able to carry it
on, the "Grundlagenstreit" is over.



**1928-1930** Conflict with Karl Menger over the
priority for the first correct definition of the notion of
dimension.



**1929** August: theft of Brouwer's briefcase on the
tram in Brussels, and with it of his mathematical notebook. When
neither the police nor a private detective hired for the purpose is
able to find it again, he despairs of ever being able to reconstruct
its contents. Brouwer later said that this loss was instrumental in the
shift of his main interest from mathematics to philosophy.



**1929** Begins preparations for the foundation of a
new mathematical journal.



**1934** Appearance of the first issue of Brouwer's own
international journal, entitled *Compositio Mathematica*.



**1934** Lecture series in Geneva, on invitation of
Ferdinand Gonseth. Plans for a course on intuitionistic mathematics
in Gottingen come very far, but are, in the end, not realised.



**1934-1935** Ludwig Bieberbach, a German member of
the editorial board of
*Compositio Mathematica* who in 1933
had joined the National Socialist movement by entering the SA,
asks Brouwer to remove the Jewish members from that board. 
Brouwer refuses; Bieberbach resigns.




**1935-1941** Member of the municipal council of
Blaricum for the local Neutral Party (in 1939 he wins the elections by
receiving 310 of the 1601 votes).



**1940-1945** During the German Occupation of the
Netherlands in World War II, Brouwer assists the resistance by letting
them use storage space in the Mathematical Institute, of which he was
the director. He offers hiding places to Jews and other persecuted
persons in small houses on his premises (as testified by a neighbour's
daughter, herself involved in these operations, in van Dalen 2005,
vol. 2, p. 772-773). In 1943, he advises the students to sign
the declaration of loyalty demanded by the Germans; when some students
refuse and go into hiding, Brouwer and Heyting visit them to continue
their education. Part of Brouwer's explanation, after the war, for
the advice to sign is that signing would provide the students with the
relative peace needed to build up and carry out resistance
activities. He is met with skepticism (but is supported by Mannoury).
Because of this and some similar perhaps unfortunate attempts at
shrewdness during the occupation, after the liberation he is suspended
for a few months. Deeply offended, Brouwer considers emigration to
South Africa or the USA.



**1941 and 1944** Fires in Brouwer's house destroy
significant parts of his archives and library. 



**1942** Publishes three short notes again on
intuitionistic foundations, the first since 1933.



**1945-1950** Conflict over *Compositio
Mathematica*. The journal had not appeared during the war, and an
effort is made to bring it back to life. Difficulties in assembling a
new board of editors arise because of Brouwer's damaged reputation. In
the end, Brouwer's name remains on the title page, but in effect he is
removed from the board of the journal he had founded.



**1947-1951** Annual lecture series in Cambridge,
England. Brouwer plans to turn them into a book, but this does not
happen. However, he completes five of the planned six chapters, and
these are published poshumously (Brouwer, 1981).



**1948** Resumes his foundational program with a paper
that exploits the notion of the creating subject. Beginning of another
creative period.



**1949** Opposes a plan to have his collected papers
published, on the ground that he has no time to write annotations that
reflect his original as well as his present views on them, which he
considers would be the scientifically responsible thing to do.



**1951** Retires from the University of
Amsterdam. Cooling off of his relationship with Arend Heyting, his
successor at the post of director of the Mathematical Institute, as a
result of disagreement over the exact role the retired Brouwer could
still play there.



**1952** Lectures in London and in Cape Town.



**1953** Lectures in Helsinki, where he stays with Von
Wright. Lecture tour through the USA (among others MIT, Princeton,
University of Wisconsin-Madison, Berkeley, Chicago) and Canada
(Canadian Mathematical Congress in Kingston, Ontario). In Princeton,
he visits Godel.



**1955** Publishes his last new paper (based on his
lecture at the Boole conference in Dublin the year before).



**1959** Death of Mrs Brouwer, 89 years old. Brouwer
declines an offer for a 1-year position at the University of British
Columbia in Vancouver.



**1962** Brouwer is offered a position in Montana.



**1966** December 2: dies in Blaricum, The Netherlands,
85 years old, when he is hit by a car in front of his house.
His library and archives are dispersed.
Beginning in the 1970s, 
Walter van Stigt and Dirk van Dalen reconstitute the archives
as much as possible;
the Brouwer Papers are now kept at the Noord-Hollands Archief in Haarlem.


## 3. Brief Characterisation of Brouwer's Intuitionism




Based on his philosophy of mind, on which Kant and Schopenhauer were
the main influences, Brouwer characterised mathematics primarily as
the free activity of exact thinking, an activity which is founded on
the pure intuition of (inner) time. No independent realm of objects
and no language play a fundamental role. He thus strived to avoid the
Scylla of platonism (with its epistemological problems) and the
Charybdis of formalism (with its poverty of content). As, on Brouwer's
view, there is no determinant of mathematical truth outside the
activity of thinking, a proposition only becomes true when the subject
has experienced its truth (by having carried out an appropriate mental
construction); similarly, a proposition only becomes false when the
subject has experienced its falsehood (by realizing that an
appropriate mental construction is not possible). Hence Brouwer can
claim that "there are no non-experienced truths" (Brouwer,
1975, p.488).




Brouwer was prepared to follow his philosophy of mind to its ultimate
conclusions; whether the reconstructed mathematics was compatible or
incompatible with classical mathematics was a secondary question, and
never decisive. In thus granting philosophy priority over traditional
mathematics, he showed himself a revisionist. And indeed, whereas
intuitionistic arithmetic is a subsystem of classical arithmetic, in
analysis the situation is different: not all of classical analysis is
intuitionistically acceptable, but neither is all of intuitionistic
analysis classically acceptable. Brouwer accepted this consequence
wholeheartedly.


## 4. Brouwer's Development of Intuitionism



The following is a brief history of Brouwer's ideas in philosophy,
mathematics, and logic. There is also a more detailed separate article
on
 The Development of Intuitionistic Logic.




Brouwer's little book *Life, Art and Mysticism* of 1905, while
not developing his foundations of mathematics as such, is a key to
those foundations as developed in his dissertation on which he was
working at the same time and which was finished two years later. Among
a variety of other things, such as his views on society and women
in particular, the book contains his basic ideas on mind, language,
ontology and epistemology. 




These ideas are applied to mathematics in his dissertation *On the
Foundations of Mathematics*, defended in 1907; it is the general
philosophy and not the paradoxes that initiates the development of
intuitionism (once this had begun, solutions to the paradoxes
emerged). As did Kant, Brouwer founds mathematics on a pure intuition
of time (but Brouwer rejects pure intuition of space).




Brouwer holds that mathematics is an essentially languageless
activity, and that language can only give descriptions of mathematical
activity after the fact. This leads him to deny axiomatic approaches
any foundational role in mathematics. Also, he construes logic as the
study of patterns in linguistic renditions of mathematical activity,
and therefore logic is dependent on mathematics (as the study of
patterns) and not vice versa. It is these considerations that motivate
him to introduce the distinction between mathematics and
metamathematics (for which he used the term "second order
mathematics"), which he would explain to Hilbert in conversations
in 1909 (Brouwer, 1928A2, p. 375n.2).




With this view in place, Brouwer sets out to reconstruct Cantorian
set theory. When an attempt (in a draft of the dissertation) at making
constructive sense out of Cantor's second number class (the class of
all denumerably infinite ordinals) and higher classes of even greater
ordinals fails, he realises that this cannot be done and rejects the
higher number classes, leaving only all finite ordinals and an
unfinished or open-ended collection of denumerably infinite ordinals.
Thus, as a consequence of his philosophical views, he consciously puts
aside part of generally accepted mathematics. Soon he would do the same
with a principle of logic, the principle of the excluded middle (PEM),
but in the dissertation he still thinks of it as correct but useless,
interpreting \(p \vee \neg p\) as \(\neg p \rightarrow \neg p\).




In "The Unreliability of the Logical Principles" of 1908,
Brouwer formulates, in general terms, his criticism of PEM: although in
the simple form of \(p \vee \neg p\), the principle will never lead to a contradiction,
there are instances of it for which one has, constructively speaking,
no positive grounds. Brouwer names some. Because they
do not in the strict sense refute PEM, they are known as "weak
counterexamples". See the
supplement on Weak Counterexamples.




The innovation that gives intuitionism a much wider range than other
varieties of constructive mathematics (including the one in Brouwer's
dissertation) are the choice sequences. These are potentially infinite
sequences of numbers (or other mathematical objects) chosen one after
the other by the subject. Choice sequences made their
first appearance as intuitionistically acceptable objects in a book
review in 1914; the principle that makes them mathematically
tractable, the continuity principle, was formulated in Brouwer's
lectures notes of 1916. The main use of choice sequences is the
reconstruction of analysis; points on the continuum (real numbers) are
identified with choice sequences satisfying certain conditions. Choice
sequences are collected together using a device called 
"spread", which performs a function similar to that of the
Cantorian set in classical analysis, and initially, Brouwer even uses
the word "Menge" ("set") for it. Brouwer
develops a theory of spreads, and a theory of point sets based on it,
in the two-part paper "Founding Set Theory Independently of the
Principle of the Excluded Middle" (1918/1919).




The answer to the question in the title of Brouwer's paper
"Does Every Real Number Have a Decimal Expansion?" (1921A)
turns out to be no. Brouwer demonstrates that one can construct choice
sequences satisfying the Cauchy condition that in their exact
development depend on an as yet open problem. No decimal expansion can
be constructed until the open problem is solved; on Brouwer's strict
constructivist view, this means that no decimal expansion exists until
the open problem is solved. In this sense, one can construct real
numbers (i.e., converging choice sequences) that do not yet have a decimal
expansion.




In a lecture in 1923, again using choice sequences and open problems,
Brouwer devises a general technique, now known as "Brouwerian
counterexamples", to generate weak counterexamples to classical
principles: "On the Significance of the Principle of the
Excluded Middle in Mathematics" (Brouwer 1924N).




The basic theorems of intuitionistic analysis -- the bar theorem,
fan theorem, and continuity theorem -- are in "On the
Domains of Definition of Functions" of 1927. The first two are
structural theorems on spreads; the third (not to be confused with the
continuity principle for choice sequences) states that every total
function \([0,1] \rightarrow \mathbb{R}\) is continuous and even
uniformly continuous. The fan theorem is, in fact, a corollary of the
bar theorem; combined with the continuity principle, which is not
classically valid, it yields the continuity theorem, which is not
classically valid either. The bar and fan theorems on the other hand
are classically valid, although the classical and intuitionistic
proofs for them are not exchangeable. The classical proofs are
intuitionistically not acceptable because of the way they depend on
PEM; the intuitionistic proofs are classically not acceptable because
they depend on reflection on the structure of mental proofs. In this
reflection, Brouwer introduced the notion of the "fully
analysed" or "canonical" form of a proof, which
would be adopted much later by Martin-Lof and by Dummett. In a
footnote, Brouwer mentions that such proofs, which he identifies with
mental objects in the subject's mind, are often infinite.




"Intuitionist Reflections on Formalism" of 1928 identifies
and discusses four key differences between formalism and intuitionism,
all having to do either with the role of PEM or with the relation
between mathematics and language. Brouwer emphasises, as he had done
in his dissertation, that formalism presupposes contentual mathematics
at the metalevel. He also here presents his first strong
counterexample, a refutation of PEM in the form
\(\forall x\in \mathbb{R}(Px \vee \neg Px)\), by showing that it is false that every
real number is either rational or irrational. See the supplement on
 Strong Counterexamples.





Of the two lectures held in Vienna in 1928 -- "Mathematics,
Science and Language" (1929A) and "The Structure of the
Continuum" (1930A) -- the first is for the most part of a
philosophical nature while the second is more mathematical. In
"Mathematics, Science and Language", Brouwer states his
general views on the relations between the three subjects mentioned in
the title, following a genetic approach, and stressing the role of the
will. A longer version of this lecture was published as
"Will, Knowledge and Speech" (1933A2); it contains the
first explicit remarks about a notion that had been present from the
start, now known as that of the "ideal(ised) mathematician"
or "creating subject".




"Consciousness, Philosophy and Mathematics" (1949C) once
again goes through Brouwer's philosophy of mind and some of its
consequences for mathematics. Comparison with *Life, Art, and
Mysticism*, "Mathematics, Science and Language", and
"Volition, Knowledge, Language" reveals that Brouwer's
general philosophy over the years developed considerably, but mainly
in depth.




In 1949, Brouwer (1949A) publishes the first of a new class of
strong counterexamples, a class that differs from Brouwer's earlier
strong counterexample (1928, see above) in that the type of argument,
which now goes by the name of "creating subject argument",
involves essential reference to the temporal structure of the creating
subject's mathematical activity (Heyting, 1956, chs. III and VIII; van
Atten, 2003, chs.4 and 5; van Atten, 2018).




Brouwer's example shows that there is a case where the double negation
principle in the form of
\(\forall x\in \mathbb{R}(\neg \neg Px \rightarrow Px)\), leads to a contradiction ("The
Non-equivalence of the Constructive and the Negative Order Relation on
the Continuum", in Dutch). The first publication of a strong
counterexample in English had to wait till 1954, in "An Example
of Contradictority in Classical Theory of Functions"
(1954F). This polemical title should be understood as follows: if one
keeps to the letter of the classical theory but in its interpretation
substitutes intuitionistic notions for their classical counterparts,
one arrives at a contradiction. So it is not a counterexample in the
strict sense of the word, but rather a non-interpretability result. As
intuitionistic logic is, formally speaking, part of classical logic,
and intuitionistic arithmetic is part of classical arithmetic, the
existence of strong counterexamples must depend on an essentially
non-classical ingredient, and this is of course the choice
sequences.




The creating subject argument is, after the earlier introduction of
choice sequences and the proof of the bar theorem, a new step in the
exploitation of the subjective aspects of intuitionism. There is no
principled reason why it should be the last.