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WHEN IS IT RATIONAL TO BELIEVE A MATHEMATICAL STATEMENT 193
proved that the existence of a finite projective plane of order 10 can be reduced to checking
codewords of weight 12, 15, and 16. In other words, if there is a finite projective plane of order
10, it must have a corresponding incidence matrix, which in turn contains a codeword of
weight 12, 15, or 16. Given such a codeword, we can consider its behavior and generate a
number of submatrices of the n=10 incidence matrix. At least one such submatrix can be
expanded to the full matrix. So if we try to find the incidence matrix itself, we can start by
considering the submatrices generated by the possible codewords and check whether each
submatrix can be expanded to the full matrix. If we find a way to do so, we have just solved
the problem: we have successfully constructed a finite projective plane of order 10. If we
cannot complete any of the submatrices, the full incidence matrix does not exist, and neither
does the projective plane.
As soon as this had been worked out, a group of mathematicians proved that w 15=0, i.e., there
are no codewords of weight 15, after around three hours of computer time. 14 Tackling the
remaining cases proved to be more difficult. By 1974, Carter made significant headway with
the n=16 case in his dissertation 15 , thereby opening the field for the team that eventually was
to solve the problem: the mathematicians Clement Lam, Larry Thiel and Stanley Swiercz.
This was the situation when we entered the picture. While we knew w 15=0, the search
for weight 16 codewords was about three-quarters done and the search for weight 12
codewords was presumed to be too difficult. (Lam 1991: 312)
Things moved along further when the three started to tackle the problem through
intensive use of computers. The details of the story, leading up to the finished run of
calculations, are of no specific importance to us here, interested readers might wish to
consult Lam 1991, where the whole history is given in much more detail. Suffice it to say
that during the 1980s, more and more cases were successfully calculated, until on
November 11th, 1988, the CRAY supercomputer that was running most of the calculations
by that point was finished, no full incidence matrix had been found and the finite
projective plane of order ten was pronounced dead.
This would be where our story ends, if not for what happened next. So far, the search for the
order ten plane is, like the Four Color Theorem, merely another computer-assisted proof. But
here is where things get interesting.
One week after the CRAY finished its calculations, Nick Patterson (the deputy director of the
Communication Research Division at the Institute for Defense Analyses, where the CRAY
supercomputer was running) called Lam. Patterson, who had “looked after the day-to-day
running of the program for over two years”, and whom Lam calls “the unsung hero in the
successful completion of our work” (Lam 1991:314), had checked the computation logs and
found cause for concern. He reported that there had been an error code four in one of the
partial computations. 16 Code four errors meant that this part of the computation was so large
that the software the CRAY was running could not handle the computation. What to do? Lam,
Thiel and Swiercz split the problem up into 200 subcases and used many different machines,
among them the CRAY, to solve them part by part. On November 29th, after between 2000
and 3000 hours of computing time on the CRAY supercomputer alone, the problem was
pronounced solved again. The authors were flooded with attention from Science, the New
York Times, the Scientific American, and, of course, interested mathematicians.
After the burst of publicity, we finally managed to read the magnetic tape containing
the statistics. To our horror, we found another A2 with an error number four. However,
14
The resultant publication is MacWilliams, Sloane & Thompson 1973.
15
See Carter 1974.
16
These partial computations, called A2's, are submatrices of the suspected incidence matrix that have
been constructed in a specific way. For more details see Lam, Thiel & Swiercz 1989: 1118.