John von Neumann laid foundations for:
- Computers (‘von Neumann’ and connectionist architectures), self-reproducing machines (the Game of Life), and artificial intelligence,
- the quantum – classical interface,
- the quantum measurement problem,
- axiomatic quantum field theory, and
- game theory, utility theory, and economics.
These fields are all characterized by intractable problems.
We ask whether these problems have a different cause within each field, or whether these problems have some common cause within the mathematical foundations underlying each field. This is a possibility as the common underlying mathematical foundations of each field were provided by von Neumann.
According to practitioners within each field:
- Artificial Intelligence (AI)
- Status: no longer researching “GOFAI”, or “Good Old Fashioned AI”.
- Maths: Optimized tree search over all pathways taking account of all possible correlations and unable to reproduce "chunking" behaviours of human cognition.
- Status: Neural network research is “balkanized” and not researching the big questions like “qualia” or “consciousness”.
- Maths: Optimized search over a single landscape which takes account of all possible correlations.
- Quantum Field Theory (QFT)
- Status: Plagued by infinities and requiring renormalization which subtracts yet further infinities to correct the initial infinities.
- Maths: Renormalization averages over many variables to obtain one variable to reduce the dimensionality of the space. (The only legitimate way to reduce spatial dimensionality is via correlated variables, whenever and are correlated.)
- Axiomatic Quantum Field Theory
- Status: Has never achieved a formally defined axiomatic basis.
- Maths: As noted by Dirac, the evolution operator lies in a space of too many dimensions to have Hilbert space coordinates, rendering the Schroedinger picture unworkable. (Some sort of spatial constraint, likely to do with correlation has been violated.)
- The Quantum – Classical Divide
- Status: Current methods to model the “cut” are ad hoc.
- Maths: Information is discarded when quantum variables are forced to be independent and uncorrelated via to approximate classical variables. This method discards correlation information.
- Quantum Measurement Theory
- Status: Imposes non-unitary wave function collapse onto the quantum probability state.
- Maths: Decoherence models a quantum system interacting with a reservoir and then discards any information in the reservoir by setting . This method discards correlation information.
- Game Theory (and Expected Utility Theory and Economics)
- Status: Plagued by paradoxes, puzzles and conundrums, and people who outright fail to act as predicted.
- Maths: A strategic tree search over all pathways taking account of all possible correlations.
Our hypothes is that there is a single underlying problem common to all these areas within their common mathematical foundations as provided by von Neumann. We examine this prospect first in game theory. And then consider models in strong artifician intelligence.