Godel’s first incompleteness theorem Summarised from various sources Godel’s first incompleteness theorem is the starting point for Penrose’s approach to an explanation of the nature of mathematical understanding. In the later development of Penrose’s ideas by Stuart Hameroff mathematical understanding has been enlarged to mean consciousness, presumably on the argument that we need to consciously feel or appreciate the truth of a mathematical statement that is not provable by the immediately Read more […]

# Archive for the ‘Artificial Intelligence, AI / Computing / Robotics’ Category

## Godel & Turing

Posted byGodel & Turing Chaitin Godel showed that the connection between proof and truth was shaky. In mathematics and in other formal systems statements can be true but unprovable. Some mathematical propositions might be undecidable, and this demolishes the idea of a closed consistent body of rules, and replaces it with incompleteness. Chaitin discusses randomness. Something is random if there is no pattern or abbrevaited description. Then there is no algorithm shorter than the thing itself. Read more […]

## Consciousness in mathematical cognition

Posted byThe Essential role of consciousness in mathematical cognition Robert Hadley, Simon Fraser University Journal of Consciousness Studies, 17, No. 1-2, 2010, pp. 27-46 http://ingentaconnect.com/journals/browse/imp/jcs Hadley puts forward alternative possibilities to Penrose’s argument from the Godel theorem, in order to reach a Penrose-type conclusion about brains and computers. He argues that a system that lacked consciousness would be incapable of certain concepts and certain proofs. Hadley Read more […]

## Singularity & Chalmers

Posted byThe Singularity: Commentary on David Chalmers Susan Greenfield Journal of Consciousness Studies, 19, No. 1-2, 2012, pp. 112-118 http://ingentaconnect.com/journals/browse/imp/jcs Greenfield is sceptical of the excitement surrounding the singularity, the point at which machine intelligence will exceed human intelligence. She argues that computational processing can only deal with specified problems that are capable of a single clear solution, but are less useful with more complex social, Read more […]

## Machine consciousness

Posted byCan machines be murdered? Tate, M.A. et al This chapter in ‘Consciousness and the Universe’ emphasises the lack of depth of thinking that can be seen as a hall mark of the functionalist approach to consciousness theory. The first sentence makes the assumption that particular actions or more especially the combining of particular actions will produce consciousness. Characteristically in this sort of writing, no argument is presented, and conclusions are merely asserted. Computers/robots are Read more […]

## Computable universe

Posted byForeword to A Computable Universe: Understanding computation and exploring nature as computation Roger Penrose World Scientific 2012 Penrose’s argues that Gödel’s incompleteness theorem provides a strong case for human understanding being non-computable. This relates to our ability to demonstrate the truth of certain mathematical propositions. These are described as π1 sentences. These sentences assert that a particular computation, such as Lagrange’s theorem which asserts that every natural Read more […]

## Singularity is logically brittle

Posted byBelief in the singularity is logically brittle Selmer Bringsjord, Rennsselaer Institute Journal of Consciousness Studies, vol. 19, nos. 7-8, (2012) The singularity here refers to the forecast that it will be possible to build computers that are more intelligent than humans, and that these super-intelligent machines will be able to design still more intelligent computers. This is widely forecast to happen within a few decades. Against this, Bringsjord takes the view that the idea of a Read more […]

## Quantum information systems

Posted byRules for a complex quantum world Michael Nielsen University of Queensland Scientific American – November 2002 Nielsen’s article discusses the development of quantum information systems. He begins by pointing out that information has a physical basis, which applies to both classical and quantum types of information. With classical information the basic unit is the ‘bit’, which is either ‘0’ or ‘1’. With quantum information the basic unit is a ‘qubit’. The special Read more […]

## NP hard problems

Posted byNP Hard Problems Paul Parsons New Scientist – 24th January 2004 The idea of quantum computing was originated in the 1980’s by the physicist, Richard Feynman, best known for the theory of quantum electrodynamics (QED). He realised that no conventional or classical computer could simulate the complexities of the universe. Some equations for relationships in the macroscopic or classical world and for some mathematical problems have to be solved by equations that also describe aspects of quantum Read more […]

## Neurons and logic gates

Posted byNeurons and logic gates from the webmaster The idea that the brain is a computer has been popular since the invention of the latter, and this remains a mainstream consensus. The feeling of many others that there is an essential difference between computers and humans has tended to be pushed aside as an uneducated delusion. However, although much brain processing can be accepted as being analagous to computing, it may be that there is a fairly easy to detect structural difference between Read more […]