.
In his primer on Intuitionism,
Heyting [5] frequently relies on the occurrence or
non-occurrence of the sequence 0123456789 in the decimal
expansion of to highlight issues of classical versus
intuitionistic (or constructivist) mathematics.
At the time that Brouwer developed his theory (1908) and even at the
time that [5] was written, it seemed well-nigh impossible
that the first occurrence of any 10 digit sequence in could
ever be determined.
The confluence of faster computers and better algorithms, both for
and more importantly for arithmetic (fast Fourier transform
based, combined with Karatsuba, multiplication (see [3])) have rendered their intuition false.
Thus, in June and July 1997, Yasumasa Kanada and Daisuke Takahashi at
the University of Tokyo
completed two computations of on a massively parallel Hitachi
machine with 
processors.
The key algorithms used are as in the recent survey in this Journal
[1], with the addition of significant
numerical/arithmetical enhancements and subtle flow management.
During their computational tour-de-force, Kanada and Takahashi
discovered the first occurrence of 0123456789 in
beginning at the 17,387,594,880th digit (the `0') after the decimal point.
It is worth noting that years become hours on such a parallel machine
and also that the effort of multiplying two seventeen billion digit integers
together without FFT based methods is also to be measured in years.
The underlying method reduces to roughly 300 such multiplications
([3],[2]).
Add the likelihood of machines crashing during a many-year sequential computation. Hence without fast arithmetic and parallel computers, Brouwer and Heyting
might indefinitely have remained safe in using this particular and
somewhat natural example.
We may emphasize how out of reach the question
appeared even 35 years ago with the following anecdote.
Sometime after Shanks and Wrench computed 100,650 places of (in 1961 in
9 hours on an IBM 7090), Philip Davis asked Dan Shanks to fill in the
blank in the sentence ``mankind will never determine the
of .'' Shanks, apparently almost immediately
replied ``the billionth.''