The mathematics presented here is both interesting and formal. However, the interesting mathematics is not formal, and the formal mathematics is not interesting.

Please note that the mathematical essays are written in PDF. If you have a modern browser it will probably open the essays automatically. Failing that, save the PDF files to disk and open them with Acrobat Reader. If the PDF files do not open properly then down load a free, up to date copy of Acrobat Reader from the web. Install it and use the newly installed version of Acrobat Reader to open the files.

Please note that the source files are ASCII text with Unix convention newline = carriage return + linefeed. Your browser will probably do something sensible with them, but, if not, save the files to disk and read them with a text editor.

All of the source files are in the language Pop11. Pop11 is part of the Poplog suite of AI languages.

If you want to browse a directory of mathematics news, books, and web resources then visit the Mathematics Directory.

If you wish to discuss any aspect of the Book of Paragon with the author, then email me.

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Perspex Machine

This conference paper is a very interesting introduction to the perspex machine. The theoretical perspex machine can do more than any other computer and a practical perspex machine might run faster than any of today's computers. An unexpected spin-off is an experiment to measure fluctuations in time-flow. If the result of the experiment is positive it should be possible to transmit signals into the future and the past. If the result is negative it rules out some conceivable models of the internal structure of time. That's the best sort of experiment there is - however it turns out, it tells us something!

Perspex Machine (Presentation)

These are the presentation slides for the above conference paper.

How to Build a Time Machine? (Presentation)

These seminar slides describe a simpler experiment to detect time-flow oscillations. The seminar took place on Tuesday the 4th of March 2003 in Room 107 in the Department of Physics at the University of Reading, England.

The proposals presented in the seminar were criticised on the grounds that quantum physics is entirely deterministic; the apparently stochastic elements arise only through ignorance of the position and motion of particles imposed by Heisenberg’s uncertainty principle.

If true, this completely undercuts everything in the seminar and the proposed experiments are pointless. [However, this interpretation of quantum physics was challenged after the seminar on the grounds that all "hidden variables" theories of physics fail. Therefore, so far as is known, quantum physics is irreducibly, statistically random - as required for the proposed experiments to be meaningful.]

The proposals were criticised on the grounds that physics, for example, beta-decay, is temporally asymmetrical.

If this is generally true then everything in the seminar is undercut, but the temporal asymmetry of the given example of decay and fusion has no bearing on the proposed experiments.

The proposals were criticised on the grounds that during an oscillation in time a particle could be in the same place at the same time, but moving both forward and backward in time. This cannot be described in spacetime, so mathematical analysis will fail. Physics cannot handle time being inherent in particles, it must exist as an objective reality viewed from reference frames.

The first part of this criticism is unfounded. During oscillating time the universe is reversible, so there is no difficulty with a single particle being in the same place at the same time, moving both forwards and backwards in time, during arbitrarily many cycles of an oscillation. All that happens is that the history of the universe winds back and evolves many times.

The second part of the criticism is misplaced. It is true that oscillating time cannot be described in geometrical spacetime. It is also true that this will prevent mathematical analysis of the kind used in physics. But it does not prevent simulation and, specifically, it does not prevent simulation by a Perspex machine. Computer Scientists can as easily simulate a universe with oscillating time as with spacetime.

The third part of the criticism is a hostage to fortune. If a Computer Scientist simulates particles in oscillating time and spacetime emerges from the simulation, then General Relativity can be unified with particle models, such as quantum theory. Such emergence of physical phenomena has already been observed in particle models of fluid dynamics. There is a big literature on particle modelling in Computer Science.

It was suggested that if quantitative predictions could be made, these could be tested against existing data in similar experiments, thereby testing the proposed effect without the cost and effort of constructing any experimental devices.

Fair enough. I'll learn some more physics and do some simulations with the Perspex machine. It will take a while, though, because I have other uses for the Perspex machine.

Reading 1

Working notes on the experiment Reading 1 to demonstrate oscillations in time flow.

Exact Numerical Computation of the Rational General Linear Transformations

This conference paper is an interesting introduction to transrational arithmetic - arithmetic where division by zero is meaningful. The paper also introduces transrational trigonometry that can deal with the infinities that arise in the tangent and similar functions. It is mildly interesting to note that in transrational trigonometry cos2(∞) + sin2(∞) = 1 and tan(∞) = sin(∞) / cos(∞).

Exact (Presentation)

These are the presentation slides for the above conference paper.

Exact (Source Code)

The following source code implements the transrational arithmetic and trigonometry described in the above conference paper. It also provides an implementation of transrational tan2 and arctan2.

Transrational arithmetic

Transrational trigonometry

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