Reduction and Elimination in Philosophy and the Sciences

The Four-Color Theorem, Testimony and the A Priori
Kai-Yee Wong, Hong Kong, China
1. Computer proof as experiment
The computer played an essential role in providing a key
lemma, which requires certain combinatorial checks too
long to do by hand, in the proof given by Kenneth Appel
and Wolfgang Hanken of the Four-Color Theorem (4CT).
The proof of the 4CT has generated a flurry of philosophical
discussions about its significance. Some of them focused
on the arguments put forth by Thomas Tymoczko
1979. According to Tymoczko, the argument for the 4CT is
like ‘an argument in theoretical physics where a long argument
can suggest a key experiment which is carried out
and used to complete the argument’ (Tymoczko 1979: 78)
because there is an unavoidable reliance on computers to
produce the proof of the theorem. Since belief in the reliability
of computers ultimately rests on empirical considerations,
the proof establishes the 4CT on grounds that
are in part empirical. So the 4CT, Tymoczko concludes, is
a substantial piece of knowledge which can be known only
a posteriori. Almost three decades on, the issues raised by
Tymoczko’s paper are still of great interest to many. (For a
recent discussion see, see Arkoudas and Bringsjord 2007.
See also Brown 1999, Burge 1998, Coady 1992: ch. 14.)
In this paper I shall examine the central thread in Tymoczko’s
reasoning and Michael Detlefsen and Mark Luker’s
(1980) contention that it in fact leads, rightly, to the much
more drastic conclusion that mathematical proof is typically
empirical.
Tymoczko argues that the appeal to computers, in
the case of the 4CT, involves a claim of this kind:
(1) If computer C running program P produces result
R, then mathematical statement M is true.
The truth of (1) turns on the reliability of C and P, which in
turn involves the claim that
(2) C does what it is supposed to do, namely, to
correctly execute P.
Evidently (2) may turn out true or false, depending on a
complex set of empirical factors and thus its truth is ‘ultimately
a matter for engineering and physics to assess’
(Tymoczko 1979: 74). Mathematical knowledge obtained
by computer proof is therefore grounded on a ‘wellconceived
computer experiment’. So, Tymoczko concludes,
accepting computer-assisted proof means giving
up the traditional notion of mathematical proof as something
surveyable, non-empirical and a priori.
2. Proof, archives, and testimony
Expressing this argument in terms of testimony can help
us articulate a problem for Tymoczko. What he argues for
is that since testimony is empirical in character and the
proof of the 4CT is based partly on the testimony of a
computer, accepting the 4CT as a theorem requires
changing the concept of proof.
Indeed, it seems arguable that testimony has a
significant role in the transmission of mathematical
knowledge and this in turn may have interesting bearing
on the notion of proof. Mathematics investigation, of
course, builds on past results. Now suppose Kodel, a
mathematician, has produced with great care a proof with
gaps filled by a set of established, ‘archived’ results. It is
implausible to think that Kodel’s proof therefore fails to be
a proof in the proper sense, otherwise much of
mathematics as we know it would go down the drain
because mathematical investigation has always involved
appeals to, so to speak, testimony of archived results.
Reliance on such testimony is part and parcel of a
mathematician’s work. Thus, in a crucial respect, Kodel’s
proof, or for that matter, much of traditional mathematical
investigation is relevantly similar to the proof of the 4CT in
their reliance on testimony. If we adopt Tymoczko’s view,
we must say that the use of testimony injected into Kodel’s
proof empirical ingredients and rendered the theorem
proved a piece of a posteriori knowledge, just as the 4CT
is. Notice whether or not Kodel can survey the archived
proofs has no bearing on this point. To see this, we can
imagine that another mathematician, Podel, produced a
proof with gaps filled by a vast set of archived results.
Surveying the enormously complex and numerous proofs
for these results is such a mammoth task that no
mathematician can finish. (If this sounds far-fetched,
consider the now accepted proof of the classification of all
simple finite groups. The proof, carried out over many
years by a large number of mathematicians, spans across
about 15,000 pages on journals. This example is from
Brown (1999), p. 158). The fact that these proofs have
been surveyed in the past will not change the situation. For
without a first-hand survey of each proof, Podel must rely
on empirical evidence that the archives are a reliable
source of mathematical results. It is arbitrary to claim that
Kodel’s theorem does not rely on any empirical evidence
because his proof is not as extensive as Podel’s in its
appeal to the archives.
If so, one should find puzzling if not inconsistent
Tymoczko’s claim that the appeal to computers in the 4CT
forces a change of the traditional concept of proof. For if
this claim is to be true, it must be the case that
mathematical proof had not come to include empirical
elements before the proof of the 4CT. But by dint of
reasoning similar to Tymoczko’s own, as shown by the
case of Kodel and Podel, it should be held, albeit in my
view implausibly, that traditional mathematics is mostly
empirical.
Yet one may think that such a conclusion is not
implausible. As a matter of fact, some have argued that
much of traditional mathematics is partially empirical.
Detlefsen and Luker (1980) argue that if one follows the
logic of Tymoczko’s reasoning where it leads, one is forced
to see empirical ingredients in proofs that are generally
held to be paradigms of a priori mathematical arguments,
such as the following one attributed to the young Gauss.
Gauss proved that the sum of the first one hundred
positive numbers was 5,050 in the following way: write
down the following pairs: , , …, ,
observe that the numbers of each pair together make 101
and that there are 50 pairs, and then conclude that the
sum of the first one hundred numbers is 5,050. According
to Detlefsen and Luker, the episode of calculation by which
we determinate that 101 multiplied by 50 is 5,050 is
needed in the reasoning from the ‘observation’ to the
‘conclusion’. Among the assumptions, they add, required
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