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| (] in Probability Theory: The Logic of Science | (] in "Probability Theory: The Logic of Science") | ||
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Revision as of 00:39, 14 November 2005
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In this article, "Cantor's Theory" refers to the naive set theory introduced by Georg Cantor in the latter part of the nineteenth century. The "anti-Cantorians" are people who claim that Cantor created a fantasy world. The "Cantorians" defend Cantor's Theory. This terminology seldom appears in the mathematical literature, but it has become almost standard in Usenet discussions of Cantor's Theory.
The anti-Cantorians claim that while abstraction is the essence of mathematics, it is possible to go too far: what Cantor did was to take an argument (the diagonal argument), which is perfectly valid in concrete mathematics, and recklessly apply it to the abstractions of the infinite, ultimately producing something which has no potential to help us understand, from a scientific perspective, the world in which we live.
The Cantorians (which includes almost all modern pure mathematicians) claim that since Cantor's Theory can be formalized in an (apparently) logically consistent way (e.g. ZFC), there is no room for debate about Cantor's Theory.
Early attitudes
Right from the start, Cantor's Theory was controversial.
"I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there" (Kronecker)
"Later generations will regard set theory as a disease from which one has recovered" (Poincare 1908, see endnote)
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.
"Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already" (Poincare quoted from Kline 1982)
Cantor's Theory is built on the premise that the infinite is something that has an actual existence (i.e. exists as a completed totality). Many mathematicians, along with Leopold Kronecker argued that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.
Cantor's ideas ultimately were accepted, strongly supported by David Hilbert, amongst others. Even constructivists and the intuitionists, who developed their schools of mathematics as a rebellion against Cantor's infinitary ideas, generally no longer argue that mathematicians should abandon Cantor's Theory. It would appear that Hilbert's prediction has proven accurate:
"No one will drive us from the paradise which Cantor created for us" (Hilbert, 1926)
Recent attacks
Contemporary anti-Cantorians today are typically lay mathematicians (that is, not pure mathematicians), who professionally apply mathematics. Younger researchers in the field of artificial intelligence seem to be especially attracted to the anti-Cantorian point of view.
"...those of us who work in probability theory or any other area of applied mathematics have a right to demand that this disease , ... be quarantined and kept out of our field" (E. T. Jaynes in "Probability Theory: The Logic of Science")
The anti-Cantorians are attracted to the view that the computer may be thought of as a microscope which helps us peer into a world of computation, and that Mathematics is a science which studies the phenomena observed in that world of computation. The only place where infinity enters into this picture is through the potential infinity of the tape in a Turing machine, and the potential of a Turing machine to fail to halt.
The anti-Cantorians propose that a reality criterion should be added to mathematics: we must take steps to guarantee that formal conclusions reached in the world of abstractions can be translated back into assertions about the concrete world (and now that we have a microscope for mathematics (i.e. the computer), it makes sense to think of the world of computation as real and concrete); infinite sets and power sets of infinite sets (and hence, real numbers etc.) exist only as useful fictions (abstractions) which help us reason about the concrete reality underlying mathematics; axioms and the rules of inference for abstractions should guarantee that any statement about the infinite should have implications for approximations to the infinite. Statements which have no implications observable in the world of computation, are fictions.
It's not clear that anyone has produced a collection of axioms and rules of inference that satisfy these criteria, and are powerful enough to do all potentially useful mathematics. The constructivists have made progress towards that goal.
Connections between recent views and past views
The basic anti-Cantorian argument really hasn't changed much since Cantor's time, except for the use of the computer (in the abstract) as a conceptual aid for reasoning about the foundations of mathematics. Poincare's proposed cure for the disease of Cantor's Theory, and the basic ideas behind constructivism, capture the basic idea behind the more modern anti-Cantorian arguments:
"The important thing is never to introduce entities not completely definable in a finite number of words" (Poincare 1908)
"...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory ..." (Weyl, 1946)
"If God has mathematics of his own that needs to be done, let him do it himself." (Errett Bishop, in the Introduction to Foundations of Constructive Analysis)
Mathematicians on axiomatic set theory
While there is almost no debate over the validity of Cantor's Theory in the contemporary mainstream mathematical literature, nevertheless, many mathematicians informally admit that modern set theory (i.e. the formalization of Cantor's Theory) goes far beyond what could reasonably be called reality. Consider:
"Set theory is based on polite lies, things we agree on even though we know they're not true. In some ways, the foundations of mathematics has an air of unreality." (William P. Thurston)
" have followed a gleam that has led them out of this world... The fact that mathematics is valuable because it contributes to the understanding and mastery of of nature has been lost sight of... the work of the idealist who ignores reality will not survive." (Kline, 1982)
Discussion
For the anti-Cantorians, the bottom line is that Cantor's Theory is a mythology (a story about a world unrelated to our experience), and that disqualifies it as a foundational theory of mathematics. If mathematicians are to accept Gauss' dictum that in mathematics, there is no true controversy, then the source of the controversy must not be accepted as part of mathematics.
"Maybe the next century will, under the increasing influence of the computer, bring a greater appreciation of the reality of (constructive) mathematics, evoked by, but lying deeper than, the virtual reality - beautiful and seductive though it may be - of the platonist/formalist" (Douglas S. Bridges in `Reality and virtual reality in Mathematics', Bull. European Assoc. for Theoretical Computer Science)
Endnote
The quote "Later generations will regard set theory as a disease from which one has recovered" is from Kline, and is apparently his translation of a quote from Poincare's speech "The future of mathematics" given in 1908. There has been considerable dispute about what Poincare actually intended to imply. Another translation reads "I think, that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case." So Poincare's proposed cure for the disease, "never to introduce any conception which may not be completely defined by a finite number of words" would undermine Cantor's seminal idea underlying set theory; he was calling set theory a disease.
References
- David Hilbert, 1926. "Über das Unendliche". Mathematische Annalen, 95: 161—90. Translated as "On the infinite" in van Heijenoort, From Frege to Gödel: A source book in mathematical logic, 1879-1931, Harvard University Press.
- Morris Kline, 1982. Mathematics: The Loss of Certainty. Oxford, ISBN 0195030850.
- Henri Poincare, 1908. "The Future of Mathematics". Address to the Fourth International Congress of Mathematicians . Published in Revue generale des Sciences pures et appliquees 23.
- Hermann Weyl, 1946. "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell". American Mathematical Monthly 53, pages 2—13.