Paradox, Natural Mathematics, Relativity and TwentiethCentury
Ideas
by
John Ryskamp
[email protected]
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 twentiethcentury 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 ML9105 (1991), 1 (http://staff.science.uva.nl~anne/hhhist.pdf).
1
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 Winter '11
 BARNARD
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