L.E.J. Brouwer (1951)

Source: *Brouwer's Cambridge Lectures on Intuitionism* (1951) publ. Cambridge University Press, 1981. Most of first lecture plus the appendix of fragments reproduced here.

The gradual transformation of the mechanism of mathematical thought
is a consequence of the modifications which, in the course of
history, have come about in the prevailing philosophical ideas,
firstly concerning the origin of mathematical certainty, secondly
concerning the delimitation of the object of mathematical science.
In this respect we can remark that in spite of the continual
trend from object to subject of the place ascribed by philosophers
to time and space in the subject-object medium, the belief in
the existence of immutable properties of time and space, properties
independent of experience and of language, remained well-nigh
intact far into the nineteenth century. To obtain exact knowledge
of these properties, called mathematics, the following means were
usually tried: some very familiar regularities of outer or inner
experience of time and space were postulated to be invariable,
either exactly, or at any rate with any attainable degree of approximation.
They were called axioms and put into language. Thereupon systems
of more complicated properties were developed from the linguistic
substratum of the axioms by means of reasoning guided by experience,
but linguistically following and using the principles of classical
logic. We will call the standpoint governing this mode of thinking
and working the *observational *standpoint, and the long
period characterised by this standpoint the observational period.
It considered logic as autonomous, and mathematics as (if not
existentially, yet functionally) dependent on logic.

For space the observational standpoint became untenable when,
in the course of the nineteenth and the beginning of the twentieth
centuries, at the hand of a series of discoveries with which the
names of Lobatchefsky, Bolyai, Riemann, Cayley, Klein, Hilbert,
Einstein, Levi-Cività and Hahn are associated, mathematics
was gradually transformed **into **a mere science of numbers;
and when besides observational space a great number of other spaces,
sometimes exclusively originating from logical speculations, with
properties distinct from the traditional, but no less beautiful,
had found their arithmetical realisation. Consequently the science
of classical (Euclidean, three-dimensional) space had to continue
its existence as a chapter without priority, on the one hand of
the aforesaid (exact) science of numbers, on the other hand (as
applied mathematics) of (naturally approximative) descriptive
natural science.

In this process of extending the domain of geometry, an important
part had been played by the *logico-linguistic method,* which
operated on words by means of logical rules, sometimes without
any guidance from experience and sometimes even starting from
axioms framed independently of experience. Encouraged by this
the *Old Formalist School *(Dedekind, Cantor, Peano, Hilbert,
Russell, Zermelo, Couturat), for the purpose of a rigorous treatment
of mathematics *and logic *(though not for the purpose of
furnishing objects of investigation to these sciences), finally
rejected any elements extraneous to language, thus divesting logic
and mathematics of their essential difference in character, as
well as of their autonomy. However, the hope originally fostered
by this school that mathematical science erected according to
these principles would be crowned one day with a proof of its
non-contradictority was never fulfilled, and nowadays, after the
logical investigations performed in the last few decades, we may
assume that this hope has been relinquished universally.

Of a totally different orientation was the *Pre-intuitionist
School, *mainly led by Poincaré, Borel and Lebesgue.
These thinkers seem to have maintained a modified observational
standpoint for the introduction of natural numbers, for the principle
of complete induction, and for all mathematical entities springing
from this source without the intervention of axioms of existence,
hence for what might be called the 'separable' parts of arithmetic
and of algebra. For these, even for such theorems as were deduced
by means of classical logic, they postulated an existence and
exactness independent of language and logic and regarded its non-contradictority
as certain, even without logical proof. For the continuum, however,
they seem not to have sought an origin strictly extraneous to
language and logic. On some occasions they seem to have contented
themselves with an ever-unfinished and ever-denumerable species
of 'real numbers' generated by an ever-unfinished and ever-denumerable
species of laws defining convergent infinite sequences of rational
numbers. However, such an ever-unfinished and ever-denumerable
species of 'real numbers' is incapable of fulfilling the mathematical
function of the continuum for the simple reason that it cannot
have a positive measure. On other occasions they seem to have
introduced the continuum by having recourse to some logical axiom
of existence, such as the 'axiom of ordinal connectedness', or
the 'axiom of completeness', without either sensory or epistemological
evidence. In both cases in their further development of mathematics
they continued to apply classical logic, including the *principium
tertii exclusi, *without reserve and independently of experience.
This was done regardless of the fact that the noncontradictority
of systems thus constructed had become doubtful by the discovery
of the well-known logico-mathematical antonomies.

In point of fact, pre-intuitionism seems to have maintained on the one hand the essential difference in character between logic and mathematics, and on the other hand the autonomy of logic, and of a part of mathematics. The rest of mathematics became dependent on these two.

Meanwhile, under the pressure of well-founded criticism exerted
upon old formalism, Hilbert founded the *New Formalist* *School,
*which postulated existence and exactness independent of language
not for proper mathematics but for meta-mathematics, which is
the scientific consideration of the symbols occurring in perfected
mathematical language, and of the rules of manipulation of these
symbols. On this basis new formalism, in contrast to old formalism,
*in confesso *made primordial practical use of the intuition
of natural numbers and of complete induction. It is true that
only for a small part of mathematics (much smaller than in pre-intuitionism)
was autonomy postulated in this way. New formalism was not deterred
from its procedure by the objection that between the perfection
of mathematical language and the perfection of mathematics itself
no clear connection could be seen.

So the situation left by formalism and pre-intuitionism can be summarised as follows: for the elementary theory of natural numbers, the principle of complete induction and more or less considerable parts of arithmetic and of algebra, exact existence, absolute reliability and non-contradictority were universally acknowledged, independently of language and without proof. As for the continuum, the question of its languageless existence was neglected, its establishment as a set of real numbers with positive measure was attempted by logical means and no proof of its non-contradictory existence appeared. For the whole of mathematics the four principles of classical logic were accepted as means of deducing exact truths.

In this situation intuitionism intervened with two acts, of which the first seems to lead to destructive and sterilising consequences, but then the second yields ample possibilities for new developments.

*Completely separating mathematics
from mathematical language and hence from the phenomena of
language described by theoretical logic, recognising that intuitionistic
mathematics is an essentially languageless activity of the
mind having its origin in the perception of a move of time. This
perception of a move of time may be described as the falling
apart of a life moment into two distinct things, one of
which gives way to the other, but is retained by memory.
If the twoity thus born is divested of all quality, it
passes into the empty form of the common substratum of
all twoities. And it is this common substratum, this empty
form, which is the basic intuition of mathematics.*

Inner experience reveals how, by unlimited unfolding of the basic
intuition, much of 'separable' mathematics can be rebuilt in a
suitably modified form. In the edifice of mathematical thought
thus erected, language plays no part other than that of an efficient,
but never infallible or exact, technique for memorising mathematical
constructions, and for communicating them to others, so that mathematical
language by itself can never create new mathematical systems.
But because of the highly logical character of this mathematical
language the following question naturally presents itself. *Suppose
that, in mathematical language, trying to deal with an
intuitionist mathematical operation, the figure of an application
of one of the principles of classical logic is, for once,
blindly formulated. Does this figure of language then accompany
an actual languageless mathematical procedure in the actual
mathematical system concerned?*

A careful examination reveals that, briefly expressed, the answer
is *in the affirmative, as far as the principles of contradiction
and syllogism are concerned,' if one allows for the inevitable*
*inadequacy of language as a mode of description and communication.
But with regard to the principle of the excluded third,* *except
in special cases, the answer is in the negative, so that this*
*principle cannot in general serve as an instrument for discovering
new mathematical truths.*

Indeed, if each application of the *principium tertii exclusi
in* mathematics accompanied some actual mathematical procedure,
this would mean that each mathematical assertion (i.e. an assignment
of a property to a mathematical entity) could be *judged, *that
is to say could either be proved or be reduced to absurdity.

Now every construction of a bounded finite nature in a finite mathematical system can only be attempted in a finite number of ways, and each attempt proves to be successful or abortive in a finite number of steps. We conclude that every assertion of possibility of a construction of a bounded finite nature in a finite mathematical system can be judged, so that in these circumstances applications of the Principle of the Excluded Third are legitimate.

But now let us pass to infinite systems and ask for instance if
there exists a natural number *n* such that in the decimal
expansion of *pi* the nth, (*n*+1)th, ..., (*n*+8)th
and (*n*+9)th digits form a sequence 0123456789. This question,
relating as it does to a so far not judgeable assertion, can be
answered neither affirmatively nor negatively. But then, from
the intuitionist point of view, because outside human thought
there *are* no mathematical truths, the assertion that in
the decimal expansion of *pi* a sequence 0123456789 either
does or does not occur is devoid of sense.

The aforesaid property, suppositionally assigned to the number
*n*, is an example of a *fleeing property*, by which
we understand a property *f*, which satisfies the following
three requirements:

(i) for each natural number *n* it can be decided whether
or not *n* possesses the property *f*,

(ii) no way of calculating a natural number *n* possessing
*f* is known;

(iii) the assumption that at least one natural number possesses
*f* is not known to be an absurdity.

Obviously the fleeing nature of a property is not necessarily
permanent, for a natural number possessing *f* might at some
time be found, or the absurdity of the existence of such a natural
number might at some time be proved.

...

The belief in the universal validity of the principle of the excluded
third in mathematics is considered by the intuitionists as a phenomenon
of the history of civilization of the same kind as the former
belief in the rationality of *pi*, or in the rotation of
the firmament about the earth. The intuitionist tries to explain
the long duration of the reign of this dogma by two facts: firstly
that within an arbitrarily given domain of mathematical entities
the non-contradictority of the principle for a single assertion
is easily recognized; secondly that in studying an extensive group
of simple every-day phenomena of the exterior world, careful application
of the whole of classical logic was never found to lead to error.
[This means de facto that common objects and mechanisms subjected
to familiar manipulations behave as if the system of states they
can assume formed part of a finite discrete set, whose elements
are connected by a finite number of relations.]

The mathematical activity made possible by the first act of intuitionism seems at first sight, because mathematical creation by means of logical axioms is rejected, to be confined to 'separable' mathematics, mentioned above; while, because also the principle of the excluded third is rejected, it would seem that even within 'separable' mathematics the field of activity would have to be considerably curtailed. In particular, since the continuum appears to remain outside its scope, one might fear at this stage that in intuitionism there would be no place for analysis. But this fear would have assumed that infinite sequences generated by the intuitionistic unfolding of the basic intuition would have to be fundamental sequences, i.e. predeterminate infinite sequences proceeding, like classical ones, in such a way that from the beginning the nth term is fixed for each n. Such however is not the case; on the contrary, a much woder field of development, including analysis and often exceeding the frontiers of classical mathematics, is opened by the second act of intuitionism.

*Admitting two ways of creating new
mathematical entities: firstly in the shape of more or less freely
proceeding infinite sequences of mathematical entities previously
acquired* (so that, for decimal fractions having neither exact
values, not any guarantee of ever getting exact values admitted);
*secondly in the shape of mathematical species, i.e. properties
supposable for mathematical entities previously acquired, satisfying
the condition that if they hold for a certain mathematical entity,
they also hold for all mathematical entities which have been defined
to be 'equal' to it*, definitions of equality having to satisfy
the conditions of symmetry, relfexivity and transitivity. ...

Theorems holding in intuitionistic, but not in classical, mathematics
often originate from the circumstance that for mathematical entities
belonging to a certain species the inculcation of a certain property
imposes a special character on their way of development from the
basic intuition; and that from this compulsory special character
properties ensue which for classical mathematics are false. Striking
examples are the modern theorems that *the continuum does not
split*, and that *a full function of the unit continuum is
necessarily uniformly continuous*.

Introvert science, directed at beauty, does not carry risks for consequences.

The stock of mathematical entities is a real thing, for each person, and for humanity.

The inner experience (roughly sketched):

twoity;

twoity stored and preserved aseptically by memory;

twoity giving rise to the conception of invariable unity;

twoity and unity giving rise to the conception of unity plus unity;

threeity as twoity plus unity, and the sequence of natural numbers;

mathematical systems conceived in such a way that a unity is a
mathematical system and that two mathematical systems, stored
and aseptically preserved by memory, apart from each other, can
be added;

etc.

Classical logic presupposed that independently of human thought
there is a truth, part of which is expressible by means of sentences
called 'true assertions', mainly assigning certain properties
to certain objects or stating that objects possessing certain
properties exist or that certain phenomena behave according to
certain laws. Furthermore classical logic assumed the existence
of general linguistic rules allowing an automatic deduction of
new true assertions from old ones, so that starting from a limited
stock of 'evidently' true assertions, mainly founded on experience
and called axioms, an extensive supplement to existing human knowledge
would theoretically be accessible by means of linguistic operations
independently of experience. Finally, using the term 'false'
for the 'converse of true', classical logic assumed that in virtue
of the so-called 'principle of the excluded third' each assertion,
in particular each existence assertion and each assignment of
a property to an object or of a behaviour to a phenomenon, is
either true or false independently of human beings knowing about
this falsehood or truth, so that, for example, contradictorily
of falsehood would imply truth whilst an assertion *a* which
is true if the assertion *b* is either true or false would
be universally true. The principle holds if 'true' is replaced
by 'known and registered to be true', but then this classification
is variable, so that to the wording of the principle we should
add 'at a certain moment'.

As long as mathematics was considered as the science of space
and time, it was a beloved field of activity of this classical
logic, not only in the days when space and time were believed
to exist independently of human experience, but still after they
had been taken for innate forms of conscious exterior human experience.
There continued to reign some conviction that a mathematical
assertion is either false or true, whether we know it or not,
and that after the extinction of humanity mathematical truths,
just as laws of nature, will survive. About half a century ago
this was expressed by the great French mathematician Charles Hermite
in the following words: '*Il existe, si je ne me trompe, tout
un monde qui est l'ensemble des vérités mathématiques,
dans lequel nous n'avons d'accés que par l'intelligence,
comme existe le monde des réalités physiques; l'un
et l'autre indépendant de nous, tous deux de création
divine et qui ne semblent distincts quà cause de la faiblesse
de notre esprit, par contre ne sont pour une pensée puissante
qu'une seule et même chose, et dont la synthèse se
rélève partiellement dans cette merveilleuse correspondence
entre les Mathématiques abstraites d'une part, I'Astronomie,
et toutes les branches de la Physique de I'autre*'.

Only after mathematics had been recognized as an autonomous interior
constructional activity which, although it can be applied to an
exterior world, neither in its origin nor in its methods depends
on an exterior world, firstly all axioms became illusory, and
secondly the criterion of truth or falsehood of a mathematical
assertion was confined to mathematical activity itself, without
appeal to logic or to hypothetical omniscient beings. An immediate
consequence was that for a mathematical assertion *a* the
two cases of truth and falsehood, formerly exclusively admitted,
were replaced by the following three:

(1) *a* has been proved to be true;

(2) *a* has been proved to be absurd;

(3) *a* has neither been proved to be true nor to be absurd,
nor do we know a finite algorithm leading to the statement either
that *a* is true or that *a* is absurd. [The case that
*a* has neither been proved to be true nor to be absurd,
but that we know a finite algorithm leading to the statement either
that *a* is true, or that *a* is absurd, obviously is
reducible to the first and second cases. This applies in particular
to assertions of possibility of a construction of bounded finite
character in a finite mathematical system, because such a construction
can be attempted only in a finite number of particular ways, and
each attempt proves successful or abortive in a finite number
of steps.]

In contrast to the perpetual character of cases (1) and (2), an assertion of type (3) may at some time pass into another case, not only because further thinking may generate an algorithm accomplishing this passage, but also because in modern or intuitionistic mathematics, as we shall see presently, a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property which it did not possess before.

See lecture above on *fleeing property*

One of the reasons [*incorrect, the extension is an immediate
consequence of the self-unfolding; so here only the utility of
the extension is explained.*] that led intuitionistic mathematics
to this extension was the failure of classical mathematics to
compose the continuum out of points without the help of logic.
For, of real numbers determined by predeterminate convergent
infinite sequences of rational numbers, only an ever-unfinished
denumerable species can actually be generated. This ever-unfinished
denumerable species being condemned never to exceed the measure
zero, classical mathematics, in order to compose a continuum of
positive measure out of points, has recourse to some logical process
starting from at least an axiom. A rather common method of this
kind is due to Hilbert who, starting from a set of properties
of order and calculation, including the Archimedean property,
holding for the arithmetic of the field of rational numbers, and
considering successive extensions of this field and arithmetic
to the extended fields and arithmetics conserving the foresaid
properties, including the preceding fields and arithmetics, postulates
the existence of an ultimate such extended field and arithmetic
incapable of further extension, i.e. he applies the so-called
axiom of completeness. From the intuitionistic point of view
the continuum created in this way has a merely linguistic, and
no mathematical, existence. It is only by means of the admission
of freely proceeding infinite sequences that intuitionistic mathematics
has succeeded to replace this linguistic continuum by a genuine
mathematical continuum of positive measure, and the linguistic
truths of classical analysis by genuine mathematical truths.

However, notwithstanding its rejection of classical logic as an instrument to discover mathematical truths, intuitionistic mathematics has its general introspective theory of mathematical assertions. This theory, which with some right may be called intuitionistic mathematical logic, we shall illustrate by the following remarks.