Reduction and Elimination in Philosophy and the Sciences

objects. But they also leave unanswered a variety of
foundational questions concerning what it means to regard
algorithms in this manner. For note that if algorithms are
indeed treated as intensional entities within computer
science, then we might fear that will be forced to posit a
novel class of non-extensional (and perforce nonmathematical)
abstract objects in order to account for the
truth conditions of statements of types II) and III). 4 This
concern serves to illustrate the importance of establishing
a positive answer to (A).
Some hope that such an answer may be given
comes from reflecting on the origins of computer science
within computability theory. The origins of the latter subject
can be traced to the call to provide a mathematical
definition of the class C of effectively computable functions
as it arose within the Hilbert programme. It this context,
there was general agreement that a function f: N k → N is
effectively computable just in case there exists an
algorithm for computing its values. But in order to show
that a given function is not effectively computable required
that C be given a precise definition.
The developments leading to the consensus that C
should be identified with the class of partial recursive
functions — i.e. to the claim now known as Church's
Thesis [CT] — are sufficiently familiar that they need not
be repeated here. What is less well recognized is that the
original arguments for CT did not proceed by first giving a
mathematical definition of a class A which could plausibly
be taken to consist of objects corresponding to algorithms
and then defining
(C) C =df {f : N k → N : f(x1, ..., xn) = A(x1, ..., xn) & A ∈ A}
Rather, Church, Turing, Gödel, and Post all proceeded by
defining a class of formal models M (which I will refer to
somewhat inaccurately as machines) which formalize
different notions of what it means to be a finitary
procedure. For instance, for Gödel, M consisted of the
class of general recursive definitions. Such definitions can
be taken to formalize a variety of ways in which functions
can be introduced so that their values can be explicitly
computed (e.g. by course of values recursion). But if we
define FM to be the class of functions computable by
members of M, for each choice of M, the question remains
as to whether the corresponding class FM exhausts all
effectively computable functions.
In order to demonstrate that we ought to accept C =
FM thus requires an additional argument that for any
informally characterized algorithm A, there exists a
machine M ∈ M such that A and M determine the same
function. This appears to be an extensional claim about
the relationship between two classes of functions. But note
that any argument in its favor must apparently proceed by
the following intensional route: i) given any algorithm A,
there is an M ∈ M such that each step in the informally
characterized operation of A can be correlated with one or
more steps in the operation of M; ii) hence the function
induced by the complete operation of M coincides with that
induced by that of A. Incipient arguments to this effect may
be found in the original papers of Church, Turing, and Post
from 1936. Better fleshed out versions appear in the
4 The gravity of this concern will ultimately depend upon how tightly the practice
of computer science pins down the identity conditions which must be
imposed on algorithms. It follows from the example of the previous note that
extensionally equivalent algorithms cannot be identified when they differ with
respect to a definite computational property such as asymptotic running-time
complexity, But at the same time, there do not appear to be cases in which
statements of algorithmic non-identity -- i.e. of the form A1 ≠ A2 -- are accepted
in computer science when no such property serves to distinguish A1 and
A2.
Algorithms and Ontology — Walter Dean
writings of more recent commentators such as (Rogers
1967), (Gandy 1980), and (Sieg & Brynes 2000).
Inasmuch as any sound argument for CT must
proceed in the manner just suggested, one might
reasonably conclude that at least certain choices for M will
include a formal representation of every algorithm. 5 And on
this basis, one might conclude that it is allowable to take
A = M in (C). But the members of M will generally be finite
combinatorial objects, and thus mathematical objects par
excellence. Thus one also might conclude that not only
should (A) be answered in the positive, but such an
answer is already implicit in our acceptance of CT.
The fact that such a conclusion is not warranted
follows by reflecting further on some basic results which
have emerged from algorithmic analysis. As noted above,
for instance, algorithms are individuated at least as finely
as their big-O running-times. But for certain choices of M,
we can find examples of algorithms A computing certain
functions f to which we assign running-time O(tA(|x|)) but
for which it may be shown that there is no M ∈ M
computing f with running-time O(tM(|x|)) ≤ O(tA(|x|)). 6 If we
take the property of having running-time O(tA(|x|)) to be a
property of A itself, then results like this suggest that we
cannot take algorithms to be identical to members of any
specific class M. For if we were to do so, there would be
no guarantee that there is a member of M which faithfully
represents A’s computational properties.
This situation highlights the kind of conceptual and
technical problems which arise when we attempt to settle
(A) directly by identifying algorithms with machines. For on
the one hand, the argument for CT sketched above
promises to show that for every informally presented
algorithm A, there will exist a machine M A ∈ M which
mimics its step-by-step operation. But on the other hand,
the question of determining when the existence of a
particular form of step-by-step correlation is sufficient to
allow us to conclude that M A is identical to A appears to
require that we have a prior characterization of the
properties of A itself. This is to say that before we can be
in a position to assess whether a given argument for (A) of
this form might be successful, we must first agree on how
our computational practices fix the properties of individual
algorithms.
At this point, a number of analogies between (A)
and various reductive proposals in the philosophy of
mathematics can be drawn. For note that if we agree that
algorithms are regarded as intensional objects in computer
science, (A) amounts to the claim that reference to such
entities can be eliminated in favor of extensional
mathematical ones. The desire to demonstrate such a
claim can thus be compared to the traditional nominalist
desire to show how reference to mathematical entities can
be eliminated in favor of reference to concrete ones.
This observation suggests that other strategies are
available in attempting to demonstrate (A) than simply
attempting to identify a mathematical object to correlate
with each individual algorithm — i.e. what (Burgess &
5 This point is put by (Rogers 1967) (p. 19) as follows: “[T]here is a sense in
which each of the standard formal characterizations appears to include all
possible algorithms ... For given a formal characterization..., there is a uniform
effective way to ‘translate’ any set of instructions (i.e. algorithm) of that characterization
into a set of instructions of one of the standard formal characterizations.”
6 A number of examples of lower-bound results of this nature are known for
the single-tape, single-head Turing machine model T which is most often
referenced in Rogers-style translation arguments. For instance, while there is
a trivial O(n) algorithm for determining whether a binary string is a palindrome,
no machine T ∈ T can solve this problem in time faster than O(n2) (cf. Hopcort
and Ulman 1979).
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