Reduction and Elimination in Philosophy and the Sciences

Rosen 1997) call objectual reduction. Rather, we might
start out by treating our computational practice as
constituting a term-introducing “procedural” theory Tp.
Such a theory would contain not only standard mathematical
terms and quantifiers, but also terms (A1, A2, ...)
naming algorithms and quantifiers (∀X1, ∀X2, ...) ranging
over such entities.
The question as to whether Tp commits us to the
existence of a non-mathematical class of entities
corresponding to the range of the procedural quantifiers
can accordingly be formalized by asking whether it is
possible to interpret Tp over a purely mathematical theory
Tm ⊆ Tp. In particular, we can ask whether Tp is
conservative over Tm for purely mathematical statements
and also whether it is possible to formulate Tm in a manner
such that it is able to derive appropriate interpretations of
results of types II) and III).
Demonstrating the former fact is likely to be
straightforward as it requires only that Tm is able to prove
the correctness of various algorithms (in the sense of note
2) relative to some means of representing them which
need not reflect their intensional properties such as
running-time. However, constructing an interpretation
which is also capable of accounting for results in
algorithmic analysis will most likely require that we attend
to the details of how we make reference to individual
algorithms in practice. Reflection on this topic suggests
that: 1) the only linguistic means we have of referring to
individual algorithms is via expressions of the form “the
algorithm implemented by machine m” 7 ; 2) we generally
take it to be possible to refer to the same algorithm by
referring to different machines. As a consequence, Tp is
likely to contain many statements of the form imp(m1) =
imp(m2), and imp(m1) ≠ imp(m2) (where imp(⋅)) is intended
to formalize “the algorithm implemented by m”).
This latter observation illustrates why it is unlikely
that the interpretation of an algorithmic name like
MERGESORT can be taken to be identical to any particular
machine. 8 If Tp is to reflect the grammatical structure of
statements like those in II) and III), this suggests that we
7 The other option is to treat algorithms as corresponding to the denotations of
programs -- i.e. linguistic descriptions of procedures given over a formal programming
language. However, reference to algorithms via this route arguably
collapses into reference via machines as each program will be interpretable as
a machine via an appropriate form of operational semantics.
8 For if we take A = M for a fixed M there will generally be no way of defining
imp so that these identity and non-identity statements are satisfied. More
generally, such a proposal will entail that the computational properties of A are
identical to those of M. But this will generally be unacceptable since machines
possess a variety of “artifactual” properties which we generally do not attribute
to algorithms -- e.g. have a fixed number of states, having exact (as opposed
to asymptotic) running-time, etc.
60
Algorithms and Ontology — Walter Dean
must take the values of imp(⋅) to be equivalences classes
of machines under a definition of computational
equivalence ≈ defined over a suitable class of machines
M. 9 Such a definition would ideally serve to analyze the
meaning of statements of the form
(M) machines m1 and m2 implement the same algorithm
in a manner which additionally satisfied all statements of
algorithmic identity and non-identity contained in Tp. On
this proposal, the sustainability of (A) will rest on the
availability of such a definition of equivalence. If such a
definition could be given, we would have shown how it was
possible to contextually reduce procedural discourse to
mathematical discourse (again in the sense of Burgess &
Rosen). The ontological status of algorithms could
accordingly be taken to be that of (neo)-Fregean abstracts
over M relative to ≈.
Literature
Burgess, John & Gideon Rosen 1997 A subject with no object,
Oxford: Clarendon Press.
Gandy, Robin 1980 “Church’s thesis and principles for mechanisms”
in Jon Barwise,
H. J. Keisler, and K. Kunen (eds.) The Kleene Symposium, Amsterdam:
North-
Holland, 123–148.
Gurevich, Yuri 1999 “The sequential ASM thesis.” Bulletin of the
EATCS, 67, 93-125.
Hopcroft, John & Jeffrey Ullman 1979 Introduction to Automata
Theory, Languages, and Computation Boston: Addison-Wesley.
Knuth, Donald 1973 The art of computer programming, volumes I-I
I I. Boston: Addison Wesley.
Moschovakis, Yiannis 1998 “On the founding of a theory of algorithms”
H. G. Dales & G. Oliveri, (eds.), Truth in mathematics, 71–
104. Oxford: Clarendon Press.
Rogers, Hartley 1967 Theory of Recursive Functions and Effective
Computability.
9 This fact is recognized by Moschovakis (who identifies algorithms as equivalence
classes of computational models known as recursors) but it is ultimately
denied by Gurevich. Even for Moschovakis, however, the question remains
whether his chosen notion of equivalence either 1) serves to analyze the
meaning of statements of the form (M) and 2) induces identity questions on
algorithms which are consistent with those reflected by Tp.