Brouwer's Cambridge Lectures on Intuitionism Quotes

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

“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

“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

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

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

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