Contents Next: Epilogue Up:The Search for a Previous: The Finish Line

# Is This Really the End?

This is not the first time that a computer has played an important role in ``proving'' a theorem. A notable earlier example is the four-color theorem [3]. Yet, these are not proofs in the traditional mathematical sense. It is impossible for any human being to check through all the calculations. From personal experience, it is extremely easy to make programming mistakes. We have taken many precautions, including the use of two different programs to cross check selective sample cases and the checking of internal consistency when isomorphism testing is performed. Yet, I want to emphasize that this is only an experimental result and it desperately needs an independent verification, or better still, a theoretical explanation.

There is, moreover, the possibility of an undetected hardware failure. A common error of this type is the random changing of bits in a computer memory, which could mean the loss of a branch of the search tree. This is the worst kind of hardware error, because we might loose solutions without realizing it. The CRAY-1A is reported to have such errors at the rate of about one per one thousand hours of computing. At this rate, we expect to encounter two to three errors! We did discover one such error by chance. After a hardware problem, Patterson reran the 1,000 A2's just before the failure and the statistics have changed for the A2 processed just prior to the malfunction. How should one receive a ``proof'' that is almost guaranteed to contain several random errors? [an error occurred while processing this directive]

Unfortunately, this is an unavoidable nature of a computer-based proof --- it is never absolute. However, despite this reservation, we argued in [21] that the possibility of hardware errors leading us to a wrong conclusion is extremely small. Since each A2 is in a separate run and there are about half a million non-isomorphic A2's, the probability of one random hardware error affecting one specific A2 is about . Suppose we accept that the weight enumerator is correct. Then if an undiscovered plane of order 10 exists, it would contain 24,675 weight 19 codewords. The 19 points in each such codeword give rise to an A2. Since we have searched through all non-isomorphic A2's, we must have encountered these special A2's. If all these 24,675 special A2's are isomorphic, then only one out of about half a million non-isomorphic A2's can be extended to the undiscovered plane. Even under this assumption, the probability of this special A2 being affected by two or three undetected hardware errors is less than . Is it likely that all 24,675 A2's arising from an undiscovered plane are isomorphic? Since the plane is known to have a trivial collineation group [1,15,36], it is more likely that there are two or more non-isomorphic A2's amongst the 24,675 cases. In this situation, the probability of hardware errors affecting all of them is infinitesimal. The same argument can be used even if we do not assume the correctness of previous computer-based results. Basically, the argument depends on the observation that if a plane of order 10 exists, then it can be constructed from many different starting points. Random hardware failures are unlikely to eliminate all of them. In other words, the fact that no one has yet constructed one is a very strong indication that it does not exist.

Contents Next: Epilogue Up:The Search for a Previous: The Finish Line

[an error occurred while processing this directive]