Algorithms and continuous processes

[from Peter Cariani, 960313, 1100]

Excerpts of Cariani, 3/12/96:

Algorithms, as Bill P. points
out, are external, symbolic constructs that operate on the basis of rules
even the process of folding up a protein involves complex
"analog" dynamics that are not readily (and perhaps cannot be)
described in terms of rules operating on discrete symbols.
there can be "computational" or "algorithmic" processes
in nature, once it is made clear what the "operational states"
of the system in question are
many neural informational processes that
involve all-or-nothing or discrete response alternatives can also
be seen in this way given the right observables.

Martin Taylor replied:

I hope you don't mean your intervention in the way it seems to read.

You seem to suggest that a process that produces as its result the
continuous sum of two continuous variables is _not_ an algorithmic process.
You don't mean that, do you?

I do mean that, but one needs to think hard about what it means to
have an "effective procedure" for computing the value of the
continuous sum of two continuous variables. Colloquially, we talk
and act as if the discrete numerical approximations that we use when
we go to "compute" these variables are the same as the continuous
variables themselves, but they are not. The quantity pi is only
exactly defined in terms of circumferences and diameters of circles,
not in terms of ratios between integers. We do not have an effective
procedure (one that yields an exact result in a finite number of
steps) for computing the value of pi, although we do have such
procedures for approximating that value. There is a qualitative
difference here between the two situations, and it is a ultimately
the result of the incommensurability of notational systems
based on circles and those based on integers.

What kind of effective procedure could be proposed for finding the
sum of two continuous variables that would permit different
observers to 1) ascertain the exact values of the two variables,
2) ascertain their sum, and 3) compare their results so as to
determine whether there is replicability across observers. This
is not a matter of the conceptual representation of the process
(a + b = c), but its operational implementation.

If you can't express it in a finite, discrete notation, then
the entity in question is, operationally speaking, not uniquely
distinguishable and exhaustively defined (i.e. it is ill-defined).
[I should say that I also have great problems with the
Dedekind cut and the usual means by which the real numbers
are constructed. A continuum is not an infinite series of
individuated entities, it is lack of individuation.
This is not to say that I in any way doubt the reliability
or consistency of the (finite,
discrete) mathematical operations that we use every day,
only their interpretation. There has yet to
be a Bohr or a Bridgman in the foundations of mathematics
who would keep mathematicians honest by observing what they
do with their hands, not what they say they are doing.]
Operationally, since one does not have access to the exact value
of the continuous variables,the situations
where one is postulating continuity and where one is carrying out the
formal procedures for manipulating the variables are very, very different.

This is the reason that geometrical proofs by physical construction using
continuous-valued analog devices (compass, straightedge, pen)
do not have the same status as those based on discrete symbolic
arguments (logic, algebra). In order, as observers,
to get access to continuous valued
quantities (e.g. distances) in order to compare them, one must make a
measurement (which discretizes the variable) that can contain error
(the deviation of the observed value from the "real" value).

In my opinion, terms such as "computation" and "algorithm" have been
used much too promiscuously, to encompass any process which is vaguely
"informational" and/or reasonably orderly (I admit that I do it myself
when discussing "neural computations" or "auditory computations", but
in these contexts one is speaking very loosely.) The current, looser
usage is different from the earlier meanings of the terms that
related to concrete "effective procedures" for carrying out
a calculation to reliably reach a unique result.

Let's be more concrete and say you have a device that has an
analog sensor A that produces a continuous voltage a and another
one B that produces b, and you have an element C that (you think)
sums them together to produce voltage c.
One (postulates that one) can describe the device by the equation
a + b = c, where a, b, and c are quantities that can (in one's mind) take on
a continuum of values. One can get approximate values for the
voltages by measuring them and converting them into numerical quantities,
and one can specify a numerical "algorithm" for approximating the behavior of
the device. I think one can say then that the device's behavior can be
approximated using a numerical algorithm (in the same way that a
differential equation is approximated by a numerical procedure), but not
that the device itself is performing an algorithm. If the device has
discrete and distinguishable states and has a deterministic state-transition
structure, such as an electronic digital computer,
then the operation of the device can be exactly described in terms of discrete
operational states and state-transition rules, and one can say, strictly
speaking, that the device is implementing an algorithm.

I think we are bogging down in the semantics of "algorithm", so I'll end
soon. It may be another semantics impasse, and I'm probably the only person
on earth who cares about these distinctions.

But, conceptually, if "algorithm" can be used to describe continuous processes,
then what is definitely <not> an algorithm?

Peter Cariani