L.E.J. Brouwer > Quotes

“Most philosophers and moralists believe as little in what they write as the manufacturers of baby foods and meat extracts believe in their own products; neither do they act with more good faith than those who lead spiritualist séance; and very few poets have themselves experienced the happiness they describe.”
― L.E.J. Brouwer, Vita, arte e mistica

“In Philosophy structures and systems are useless (one wants to be struck by direct insight). Systems have value only when applied in the struggle with an enemy; philosophy should not be applied. Philosophy cannot work mathematically.”
― L.E.J. Brouwer

“Mathematics is created by a free action independent of experience; it develops from a single aprioristic basic intuition [Ur-intuition], which may be called invariance in change as well as unity in multitude.”
― L.E.J. Brouwer

“Alas, the spheres of truth are less transparent than those of illusion.”
― L.E.J. Brouwer

“Let the motivation behind mathematics be the craving for the good, not passion or brains.”
― L.E.J. Brouwer

“Nothing in art or science that is true has value (i.e. commercial value).”
― L.E.J. Brouwer

“In mathematics, just as in the arts, it is dangerous to depart from the ‘Schaffe Künstler, rede nicht’, since also here the basic principles cannot be expressed, but can only be read between the lines.”
― L.E.J. Brouwer

“A special case [of mathematical attention] is the construction of objects in thought, that is, of persistent, permanent things (simple or compound) of the perceptional world, so that at the same time the perceptional world becomes stabilized. [These] phases of mathematical attention are in no way merely passive attitudes; on the contrary, they are the acts of the will.”
― L.E.J. Brouwer

“Mathematical Attention as an act of the will serves the instinct for self-preservation of individual man; it comes into being in two phases; time awareness and causal attention. The first phase is nothing but the fundamental intellectual phenomenon of the falling apart of a moment of life into two qualitatively different things of which one is experienced as giving away to the other and yet is retained by an act of memory. At the same time this split moment of life is separated from the Ego and moved into a world of its own, the world of perception. Temporal twoity, born from this time awareness, or the two-membered sequence of time phenomena, can itself again be taken as one of the elements of a new twoity, so creating temporal threeity, and so on. In this way, by means of the self-unfolding of the fundamental phenomenon of the intellect, a time sequence of phenomena is created of arbitrary multiplicity.”
― L.E.J. Brouwer

“It [logical reasoning] serves only lawyers and demagogues, not to instruct other people but to deceive them, and that is because the vulgar herd unconsciously reasons: the language with its logical figures is there, so it will be useful and so they meekly let themselves be deceived; just as I heard several people defend their habit of gin drinking with the words: ‘What else is gin for?’ Whoever has the illusion to improve the world, may just as well agitate against the language of logical reasoning as against alcohol.”
― L.E.J. Brouwer, The Selected Correspondence of L.E.J. Brouwer

“Mathematics is nothing more, nothing less, than the exact part of our thinking.”
― L.E.J. Brouwer

“The theorem: ‘if a triangle is isosceles it is an acute triangle’ is expressed as a logical theorem: the predicate ‘isosceles’ in the case of triangles is considered to imply the predicate ‘acute’, i.e. one imagines all the triangles of a given plane represented by the points of an R6 and one then sees that the domain of R6 representing isosceles triangles is contained in the domain representing all acute triangles. This is in fact true, and logical formulation and logical language can therefore safely be applied.
But the thoughts of the mathematician, who because of the poverty of his language formulated this theorem as a logical theorem, proceed in a way quite different from the above interpretation. He imagines that he is going to construct an isosceles triangle, and then finds that either at the end of the construction all angles appear to be acute or that on the postulation of a right or obtuse angle the construction cannot be executed. In other words, he thinks the construction mathematically, not in its logical interpretation.”
― L.E.J. Brouwer

“On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the "principle of excluded middle", that is, the principle that for every system every property is either correct [richtig] or impossible, and in particular the principle of the reciprocity of the complementary species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property.”
― L.E.J. Brouwer

“Truth is only in reality i.e., in the present and past experiences of consciousness ...But expected experiences, and experiences attributed to others are true only as anticipations and hypotheses; in their contents there is no truth ...There are no non-experienced truths.”
― L.E.J. Brouwer

“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.”
― L.E.J. Brouwer, Brouwer's Cambridge Lectures on Intuitionism

“Will hypothetical human beings with an unlimited memory, who use words only as invariant signs for definite elements and for definite relations between elements of pure mathematical systems which they have constructed, have room in their verbal reasonings for the logical principles for tacking together mathematical affirmations? Or what comes to the same: Will human beings with an unlimited memory, while surveying the strings of their affirmations in a language which they use for an abbreviated registration of their constructions, come across the linguistic images of the logical principles in all their mathematical transformations. A conscientious rational reflection leads to the result that this may be expected for the principles of identity, of contradiction and of syllogism, but for the principium tertii exclusi only in so far as it is restricted to affirmations about part of a definite, finite mathematical system, given once and for all whilst a more extensive use of the principle would not occur, because in general its application to purely mathematical affirmations would produce word complexes devoid of mathematical sense . . . . It follows that the language of daily intercourse between people with a limited memory, being necessarily imperfect, limited and of insecure effect, even if it is organized with the utmost practically attainable refinement and precision, will only be suitable for its task of mnemotechnic, economy of thought and understanding in mathematical research and mathematical intercommunication, if any application of the principium tertii exclusi which is not restricted to a well defined system is avoided.”
― L.E.J. Brouwer

“True mathematics never lacks significance, because it is never without a social cause. Man came to think mathematically because only by applying this method of thinking was he able to prevail in the struggle for life which became more and more complicated and difficult.”
― L.E.J. Brouwer

“The primordial phenomenon is simply the intuition of time in which repetition of "thing in time and again thing" is possible, but in which (and this is a phenomenon outside mathematics) a sensation can fall apart in component qualities, so that a single moment can be lived through as a sequence of qualitatively different things. One can, however, restrict oneself to the mere sensation of these sequences as such, independent of the emotional content, i.e. independent of the various degrees to which objects perceived in the world outside are to be feared or desired. (The attention is reduced to an intellectual observation.)”
― L.E.J. Brouwer

“A complete empirical corroboration of the inferences drawn [about the “world of perception”] is usually materially excluded a priori and there cannot be any question of even a partial corroboration in the case of (juridical and other) inferences about the past.”
― L.E.J. Brouwer

“[T]he wording of a mathematical theorem has no sense unless it indicates the construction either of an actual mathematical entity or of an incompatibility (e.g., the identity of the empty two-ity with an empty unity) out of some constructional condition imposed on a hypothetical mathematical system.”
― L.E.J. Brouwer

“But the continuum of two and more dimensions can only be conceived as a continuous cardinality, if an unknown point can be approximated in a denumerable sequence (in all coordinates together, for if I want to handle one coordinate first, that would never terminate, and the other coordinates never got their turn). But that approach is only possible if the decimal sequence ω is ordered as a certain ordinal number, so that all coordinates are treated in turn. Hence the sequence of coordinates is a part of that denumerable number, so it is also denumerable. And the denumerably unfinished number of dimensions is a part of it.”
― L.E.J. Brouwer

“In the deductions of mathematical logic one lacks all guiding stimulus if one does not keep in mind the meaning. It has always to be viewed as abstracted from something living, and that is only successful for something mathematical.”
― L.E.J. Brouwer

“Hilbert’s pseudo-geometries are (in contrast to the non-Euclidian) of little importance, because they have been built within a rather ‘farfetched’ building [i.e., construction] (while the non-Euclidian in the ordinary Cartesian space).”
― L.E.J. Brouwer

“The way in which Hilbert escapes the Russell paradox, completely without his logic, amounts to that he only speaks of a class of earlier constructed objects.”
― L.E.J. Brouwer

“[T]his basal intuition of mathematics, in which the connected and the separate, the continuous and the discrete are united, gives rise immediately to the intuition of the linear continuum, i.e., of the "between," which is not exhaustible by the interposition of new units and which therefore can never be thought of as a mere collection of units.”
― L.E.J. Brouwer

“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.”
― L.E.J. Brouwer, Brouwer's Cambridge Lectures on Intuitionism

“It is just as foolish to view a tree merely as a weight of planks, as it is one-sided to view mathematics as an axiomatic system.”
― L.E.J. Brouwer

“Nature simply has not yet fully determined all objects.”
― L.E.J. Brouwer

“Science places whatever is perceived, outside the self, in a world of perception independent of the self; the bond with the self, its only source and guide, is lost. It then constructs a mathematical-logical substratum which is completely alien to life, an illusion, one which acts in life as a Tower of Babel with its confusion of tongues.”
― L.E.J. Brouwer

“In mathematics no truths could be recognized which had not been experienced, and that for a mathematical assertion a the two cases formerly exclusively admitted were replaced by the following four:
1. a has been proved to be true;
2. a has been proved to be false, i.e. absurd;
3. a has neither been proved to be true nor to be absurd, but an algorithm is known leading to a decision either that a is true or that a is absurd;
4. a has neither been proved to be true nor to be absurd, nor do we know an algorithm leading to the statement either that a is true or that a is absurd.”
― L.E.J. Brouwer