Perspex machine IX: transreal analysis

We introduce transreal analysis as a generalisation of real analysis. We find that the generalisation of the real exponential and logarithmic functions is well defined for all transreal numbers. Hence, we derive well defined values of all transreal powers of all non-negative transreal numbers. In particular, we find a well defined value for zero to the power of zero. We also note that the computation of products via the transreal logarithm is identical to the transreal product, as expected. We then generalise all of the common, real, trigonometric functions to transreal functions and show that transreal (sin x)/x is well defined everywhere. This raises the possibility that transreal analysis is total, in other words, that every function and every limit is everywhere well defined. If so, transreal analysis should be an adequate mathematical basis for analysing the perspex machine - a theoretical, super-Turing machine that operates on a total geometry. We go on to dispel all of the standard counter "proofs" that purport to show that division by zero is impossible. This is done simply by carrying the proof through in transreal arithmetic or transreal analysis. We find that either the supposed counter proof has no content or else that it supports the contention that division by zero is possible. The supposed counter proofs rely on extending the standard systems in arbitrary and inconsistent ways and then showing, tautologously, that the chosen extensions are not consistent. This shows only that the chosen extensions are inconsistent and does not bear on the question of whether division by zero is logically possible. By contrast, transreal arithmetic is total and consistent so it defeats any possible "straw man" argument. Finally, we show how to arrange that a function has finite or else unmeasurable (nullity) values, but no infinite values. This arithmetical arrangement might prove useful in mathematical physics because it outlaws naked singularities in all equations.

Perspex machine VIII: axioms of transreal arithmetic

Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. It allows the specification of the Universal Perspex Machine which unifies geometry with the Turing Machine. Here we axiomatise the algebraic structure of transreal arithmetic so that it provides a total arithmetic on any appropriate set of numbers. This opens up the possibility of specifying a version of floating-point arithmetic that does not have any arithmetical exceptions and in which every number is a first-class citizen. We find that literal numbers in the axioms are distinct. In other words, the axiomatisation does not require special axioms to force non-triviality. It follows that transreal arithmetic must be defined on a set of numbers that contains{-8,-1,0,1,8,&pphi;} as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

Linking human and machine brains

A look is taken here at how the use of implant technology is rapidly diminishing the effects of certain neural illnesses and distinctly increasing the range of abilities of those affected. An indication is given of a number of problem areas in which such technology has already had a profound effect, a key element being the need for a clear interface linking the human brain directly with a computer. In order to assess the possible opportunities, both human and animal studies are reported on. The main thrust of the paper is however a discussion of neural implant experimentation linking the human nervous system bi-directionally with the internet. With this in place neural signals were transmitted to various technological devices to directly control them, in some cases via the internet, and feedback to the brain was obtained from such as the fingertips of a robot hand, ultrasonic (extra) sensory input and neural signals directly from another human's nervous system. Consideration is given to the prospects for neural implant technology in the future, both in the short term as a therapeutic device and in the long term as a form of enhancement, including the realistic potential for thought communication potentially opening up commercial opportunities. Clearly though, an individual whose brain is part human - part machine can have abilities that far surpass those with a human brain alone. Will such an individual exhibit different moral and ethical values to those of a human.? If so, what effects might this have on society?