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L E J Brouwer
Times obituary
Dr. G. T. Kneebone writes:
The eminent Dutch mathematician L. E. J. Brouwer, who died on December 2 at the age of 85, was one of the men who decisively influenced the 20th-century evolution of mathematical thought. His main work was done during the period 1907–1927, in the two fields of topology and foundations of mathematics. Although topology has now become one of the most fundamental branches of mathematics, it was at that time a very new discipline, only just beginning to take shape, and the methods which Brouwer introduced carried it a long way forward. His historical importance was thus very great; and among his specific discoveries which are now classical is his well-known fixed point theorem.
Brouwer is most famous, however, for his contribution to the philosophy of mathematics and his attempt to build a new mathematics on an intuitionist foundation in order to meet his own searching erudition of heretofore unquestioned assumptions. Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic. He, in fact, rejected the universally accepted logic of deductive reasoning, which had been codified initially by Aristotle, handed down with little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism. This logic, as he saw, had been formulated at a time when deductive reasoning was applied exclusively to finite totalities But post-Newtonian mathematics deals largely with infinite totalities, and the carrying over of Aristotelian principles to these is not legitimate. Thus a new logic was called for to match the changed situation. Infinite totalities have only potential existence and are not already there to be surveyed; and intuitionist logic is accordingly "constructive" in character, since we cannot make assertions that go beyond that finite part of the potentially infinite which we have ourselves actually made. This is the motivation of the celebrated intuitionist rejection of the Law of Excluded Middle.
Brouwer's projected reconstruction of the whole edifice of mathematics remained a dream, but his ideal of constructivism is now woven into our whole fabric of mathematical thought, and it has inspired, as it still continues to inspire, a wide variety of inquiries in the constructivist spirit which have led to major advances in mathematical knowledge. Brouwer's name is indeed worthily linked with those of Russell and Hilbert, his two great contemporaries in the study of the foundations of mathematics, and his intuitionism is now an integral part of mathematical philosophy.
Dr. G. T. Kneebone writes:
The eminent Dutch mathematician L. E. J. Brouwer, who died on December 2 at the age of 85, was one of the men who decisively influenced the 20th-century evolution of mathematical thought. His main work was done during the period 1907–1927, in the two fields of topology and foundations of mathematics. Although topology has now become one of the most fundamental branches of mathematics, it was at that time a very new discipline, only just beginning to take shape, and the methods which Brouwer introduced carried it a long way forward. His historical importance was thus very great; and among his specific discoveries which are now classical is his well-known fixed point theorem.
Brouwer is most famous, however, for his contribution to the philosophy of mathematics and his attempt to build a new mathematics on an intuitionist foundation in order to meet his own searching erudition of heretofore unquestioned assumptions. Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic. He, in fact, rejected the universally accepted logic of deductive reasoning, which had been codified initially by Aristotle, handed down with little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism. This logic, as he saw, had been formulated at a time when deductive reasoning was applied exclusively to finite totalities But post-Newtonian mathematics deals largely with infinite totalities, and the carrying over of Aristotelian principles to these is not legitimate. Thus a new logic was called for to match the changed situation. Infinite totalities have only potential existence and are not already there to be surveyed; and intuitionist logic is accordingly "constructive" in character, since we cannot make assertions that go beyond that finite part of the potentially infinite which we have ourselves actually made. This is the motivation of the celebrated intuitionist rejection of the Law of Excluded Middle.
Brouwer's projected reconstruction of the whole edifice of mathematics remained a dream, but his ideal of constructivism is now woven into our whole fabric of mathematical thought, and it has inspired, as it still continues to inspire, a wide variety of inquiries in the constructivist spirit which have led to major advances in mathematical knowledge. Brouwer's name is indeed worthily linked with those of Russell and Hilbert, his two great contemporaries in the study of the foundations of mathematics, and his intuitionism is now an integral part of mathematical philosophy.
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