Foundations of Set Theory (Volume 67) Quotes

“The Brouwerian believed that this conception was wholly wrong from the beginning. They accused it of misunderstanding the nature of mathematics and of unjustifiedly transferring to the realm of infinity methods of reasoning that are valid only in the realm of the finite. By regaining the right perspective, mathematics could be constructed on a basis whose intuitive soundness could not be doubted. The antinomies were only the symptoms of a disease by which mathematics was infected. Once this disease was cured, one need worry no longer about the symptoms. All Russellians thought that our naiveness consisted in taking for granted that every grammatically correct indicative sentence expresses something which either is or is not the case, and some — among them Russell himself — believed, in addition, that through some carelessness a certain type of viciously circular concept formation had been allowed to enter logico-mathematical thinking. By restricting the language — and proscribing the dangerous types of concept formation— the known antinomies could be made to disappear. Their faith in the consistency of the resulting, somewhat mutilated, systems was less strong than that of the Brouwerians, since certain intuitively not too well founded devices had to be used in order to restore at least part of the lost strength and maneuverability. Zermelians, finally, thought that our blunder consisted in naively assuming that to every condition there must correspond a certain entity, namely the set of all those objects that satisfy this condition. By suitable restriction of the axiom of comprehension, in which this assumption is formulated, they tried to construct systems which were free of the known antinomies yet strong enough to allow for the reconstruction of a sufficient part of classical mathematics.”
― Abraham Adolf Fraenkel, Foundations of Set Theory (Volume 67)

“The place of language in Brouwer's conception is that of a device for the transmission of will of the individuals that make up society. With respect to mathematics Brouwer considers language, including logic, as a phenomenon accompanying the wordless mathematical construction processes of the individual. As a consequence Brouwer maintains that logic does not precede mathematics but, on the contrary, is preceded by mathematics.”
― Abraham Adolf Fraenkel, Foundations of Set Theory (Volume 67)

“The mind of an individual experiences sensations. The individual identifies certain sensations and starts to recognize iterative sequences of sensations with the property that if one of these sensations occurs the others are expected to occur also, in a specific order. Such sequences are called causal sequences. The individual will try to use his knowledge of causal sequences to obtain certain desired sensations by producing a sensation that precedes the desired sensation in a previously experienced, causal sequence. This shift from end to means is called “cunning act” by Brouwer. Certain complexes of sensations are independent of the order in time, and their dependence on the individual is small or nil. These complexes are called things, e.g. external objects, human beings. The whole of things is called the external world of the individual. The relation of the individual with other individuals (which are again sensation complexes, i.e. things) is described by identification of causal sequences, observed by the individual, of itself and of other individuals. This identification justifies the term “acts of other individuals”. It is observed by the individual that causal acts (i.e. cunning acts based on knowledge of causal sequences) of itself and other individuals are highly dependent. Hence the need for cooperative causal acts arises. This is where scientific thinking, as an economical way to deal with large groups of these causal acts, is introduced. Scientific thinking as such is based on mathematics. The genesis of mathematics takes place at the creation of two-ities. Brouwer construes the two-ity from a move of time, which is a concept defined with respect to the individual. Namely: a move of time takes place when one sensation gives way to another. Both sensations are retained in their proper order and constitute a two-ity. The individual abstracts all quality of this two-ity and uses it as the basic ingredient for iterative processes. These iterative procedures can create predeterminately or more or less freely infinite proceeding sequences of mathematical entities previously produced.”
― Abraham Adolf Fraenkel, Foundations of Set Theory (Volume 67)