Reduction and Elimination in Philosophy and the Sciences

for confidence in the result of any computation is the
following:
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(3) The computing agent correctly executes the program.
Since the validity of (3) rests on evidence of an empirical
sort, the reasoning embodied by Gauss’s proof must be
seen as utilizing empirical premises. Furthermore, they
argue, (3) is analogous (2); so, Gauss’s proof is straightly
analogous to the 4CT in its appeal to experience. They
then conclude that Tymoczko’s reasoning, which they endorse,
forces us to hold that traditional mathematics typically
utilizes empirical evidence. Let’s call the form of reasoning
behind Tymoczko’s and Detlefsen and Luker’s
views the Empirical-Reliability-of-X Principle (ERXP):
If X is something we must rely on in order to see the
correctness of a proof, then the empirical evidence
for the reliability of X is part of the proof, or, in
slightly different terms, the proof must be seem as
utilizing empirical premises.
On this principle, no knowledge acquired on the basis of a
proof obtained, partly or fully, through testimony of any
kind can be a priori because the empirical evidence regarding
the reliability of the source of testimony would
have to be part of the evidence for what is claimed to be
known. The problem with ERXP is that it commits us to a
highly restrictive account of a priori proof, and more seriously,
demonstrative reasoning in general. (Though someone
like Chisholm would say that such an account is the
correct one, see Chisholm 1989, 28-30). For in any inference
involving more than a few steps, we need to rely on
our memory and perception to store and retrieve premises.
But the reliability of our memory and perception, and ultimately
the proper functioning of our brain, is not amenable
to demonstration by reason alone. One would then have to
conclude that mathematical argument and demonstrative
reasoning typically utilize empirical premises and are thus
a posteriori. Why would Tymoczko and others take ERXP
seriously, despite its implausible consequences? They
seem to think that we cannot but hold that the appeal to
computers must force our warrant for the 4CT to be empirical
because there is no way for an agent to obtain in an
a priori way any knowledge through testimony. This, however,
shows that ERXP is ill-motivated because, as I shall
argue, with a proper explication of the a priori we can claim
that it is possible to obtain a priori knowledge through the
testimony of a rational source. Space does not allow me to
argue for the rational source of testimony in computer
proof (but see Tyler 1998 and Arkoudas & Bringsjord
2003). Instead I shall outline an account of the a priori
which shows that the empirical character of a testimony
need not render the knowledge obtained through it empirical.
3. Apriority and two roles for experience
The key idea here is a broad construal of the nonjustificatory
role that experience is held to play in knowledge
and cognition. The standard view distinguishes a
non-justificatory role from a justificatory role of experience.
We are familiar with the idea that sense experience is
needed to enable us to acquire the concept of, say, triangle,
before we can be said to know a priori that a triangle
has three sides. In general, sense experience needed for
the acquisition of a concept does not thereby become part
of any warrant one may have that would make a belief
expressed with the concept knowledge. I think the nonjustificatory
role of experience should not be limited to the
The Four-Color Theorem, Testimony and the A Priori — Kai-Yee Wong
acquisition of concepts. A non-justificatory role for experience
in this broader sense I shall call an ‘enabling role’.
Burge’s discussions of computer proof and content preservation
illustrate very well how one can have a very broad
construal of the enabling role of experience provided that
one is prepared to bring into question a whole lot of widely
held opinions about rationality, content preservation, and
testimony. (His ‘Acceptance Principle’ is a case in point,
see Burge 1993: 469, see also Burge 1998).
To further explain the notion of an enabling role of
experience, let us consider the appeal to memory. It is
often the case that memory constitutes a part of the
warrant of a remembered belief, as in the case in which I
remember vividly that I was convinced by someone’s
demonstration of a certain logical statement p (call this
episode of remembering M). Contrast this with the case
where I remember a certain logical statement q and utilize
it to fill a gap in a proof for a certain conclusion c (call this
episode M*). In the first case, as Steffen Borge puts it
(Borge 2003: 109), my cognitive attention is not merely
focused on p but also on the attitude of my earlier self
towards the proposition. Here M makes a substantive
contribution to the justification of a remembered belief. In
the second case, my memory serves not to supply
propositions about particular mental events. Rather its role
is to supply for the derivation a certain step, q, that is part
of the demonstration that entitles me to believe c. c is
underwritten by the demonstration consisting of, among
other steps, q. The memory that supplies q, M*, is not what
the demonstration is about, neither does it make any
substantive contribution to the warrant provided by the
demonstration. My belief that c is warranted because I
have proved it. My warrant needs no further justificational
forces to be supplied by M*. The role of M* is only
enabling, not justificatory. It serves to give me access to
that warrant, but not part of my entitlement to c. Typically,
in a complex demonstration (a proof or a deduction)
memory is called upon to play an enabling role. Contingent
propositions about what we happen to remember do not
thereby become part of the premises or make a
substantive contribution to the warrant provided by these
premises. The use of paper and pencil extends our means
of checking of longer deductions. (See Teller 1980 for a
related idea.) So I think precisely in the case of hand
calculation the same reasoning can be used to argue for
the mere enabling role of sense experience and perceptual
belief. In the context of computer proof we can see the
appeal to computers as a yet further extension of the
means of checking and executing a formally sound
algorithm which serves mainly to enable our access to the
a priori warrant for a mathematical result. One might say
that whereas the justification provided by a first-hand proof
is not dependent on experience essentially, the justification
provided by the proof of the 4CT is dependent on
experience essentially because it is too long for any
human to check. I can agree with this if ‘essentially’ is used
here just to mark out the complexity of the computer’s
proof. But the epistemic significance of ‘essential
dependence on experience’ in this sense can be doubted.
Given that it makes epistemic sense to see computers as a
way to extend our means of checking proofs, it is difficult to
see why the kind of dependence on experience involved in
a warrant provided by a computer proof should be
accorded special epistemic significance merely because of
the complexity involved. Since the relevant notion of
apriority pertains to the independence from experience in
its justificatory, not enabling, role, the above
considerations, if correct, allows us to hold that the appeal
to computers does not force the warrant for relying on the
testimony of a computer to be essentially dependent on