ssrn-id897085 - Paradox, Natural Mathematics, Relativity and

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Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas by John Ryskamp philneo2001@yahoo.com This is the story of an error. We identify the error, describe it, and tell how it came to be. New historical research shows that twentieth-century thought was expressed in terms of the “natural” mathematics developed at the turn of the century. This was a new incarnation of the ancient constructivist orientation, and was developed in order to cope with the supposed “paradoxes” generated by Cantorian set theory. Economics, physics, biology—apparently no area of inquiry has escaped being made part of the “natural” mathematics project. This mathematics asserts that mathematical formulations—indeed, all arguments—are inherently anomalous; the evidence of this is that they generate paradoxes. Therefore, the idea that mathematics is an aspect of human perception, must be made a part of mathematical formulations even though it deprives any “natural” mathematical formulation of logical content. This is done by an arbitrary intervention in the argument. Our theme is, the precise location of the constructivist intervention. The polemical nature of “natural” mathematics—its frank and unapologetic embrace of bad faith—is nowhere more clearly stated than in this recent formulation of constructivism: “Constructivism is a point of view (or an attitude) concerning the methods and objects which is normative: not only does it interpret existing mathematics according to certain principles, but it also rejects methods and results not conforming to such principles as unfounded or speculative (the rejection is not always absolute, but sometimes only a matter of degree: a decided preference for constructive concepts and methods). In this sense the various forms of constructivism are all ‘ideological’ in character….Characteristic for the constructivist trend is the insistence that mathematical objects are to be constructed (mental constructions) or computed; thus theorems asserting the existence of certain objects should by their proofs give us the means of constructing objects whose existence is being asserted.” 1 We shall further explore the “means” of this bizarre— although historically explicable—idea, keeping in mind of course that arguments formulated in “natural” mathematics can never be trusted. Their advocates do not want to convince—they want power. Their normative point of view sanctions any intellectual crime. It is a huge task to identify this point of view in the disciplines and remove every example of it. The role of “natural” mathematics has gone unremarked for the very reason it was influential in the first place. Whether the researcher was the physicist Albert Einstein, the economist Piero 1 A. S. Troelstra, “A History of Constructivism in the 20th Century,” University of Amsterdam, ITLI Prepublication Series ML-91-05 (1991), 1 (http://staff.science.uva.nl~anne/hhhist.pdf). 1
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ssrn-id897085 - Paradox, Natural Mathematics, Relativity and

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