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3/11/2004 8:37 AM
I have perhaps misstated Paul Werbos's position on the notion that anything natural can be modeled by mathematics.
http://www.ontologystream.com/beads/nationalDebate/fortyfour-two.htm
and ask that he make a careful clarification if in fact he does agree that there are natural processes that cannot be modeled by Hilbert mathematics.
(For the three "schools of mathematics" see:
http://www.ontologystream.com/beads/nationalDebate/fortyfour-three.htm
)
Paul Werbos has established a leading edge in artificial neural networks beginning in 1973 with his dissertation in which the very well known "back-propagation of error" artificial neural network algorithm was first published. Since his dissertation he was developed new work in the mathematics of quantum field theory and is well know to have created an advanced adaptive critic architecture based on an extension of the back propagation network as well as on a philosophical grounding of the critic architecture in psychology.
The problem that many in the text understanding community are trying to address is an extension of the Brower notion about the construction of a formal system from precise statements of specific intuitions. The notions of counting and the more subtle notion of “next too” might be illustrative examples of specific intuitions that are codified into modern Hilbert mathematics. In Hilbert mathematics we also have at least the additional notions of consistency and what is called the “axiom of choice”. The axiom of choice allows one to make a selection of one representative from a category.
Godel and others noted that the set of all sets couldn’t be consistently defined in Hilbert style mathematics without setting aside the axiom of choice in regard at least to the membership of the set of all sets itself. The details here are perhaps not so important, so only a brief indication of the problems of consistency and completeness are indicated.
There is controversy at this point. Many feel that the problem raised by Godel, and then extended by Sir Roger Penrose to apply to the problem of modeling cognition and thought, is not particularly relevant to foundational issues related to the characterization of human knowledge.
So we acknowledge the controversy. How might one address what appears to many of us to be the need to recognize the value of Hilbert – type mathematical constructions, while at the same time positively addressing the mistaken application of mathematical methods to information theory?
The example of the Hilbert engine, patented in 2003 by Primentia, is the most illustrative example where structure intuited as numbers with less then and greater then characteristics is used to create computer information systems having very high value when used in a flexible fashion. The use of various forms of latent semantic indexing (patents held by Telcordia Inc, and Recommind Inc – and others) provides other illustrative examples, as do the neural network categorizers and classifiers.
The other approach is to establish structure based on shallow or deep analysis of the co-occurrence of words, word meaning similarity and dissimilarity. This other approach does not use the same set of intuitions that are codified into the Hilbert mathematics type formalism.
The codification results into atomic signs that are in some way symbolized so that the symbols evoke an interpretation that is consistent with an experience of knowledge commonly held within the shared experiences of a community. To understand what is going on here, i.e. to do “science”, one needs a background in scholarly non-mathematical literatures (as well as in mathematics, logic, and semiotics literatures).
Bringing these scholarly
literatures together
is the purpose of the National Project.