Mind, Body, World- Foundations of Cognitive Science, 2013a

in general, but because these errors are often systematic, and their systematicity
reflects the underlying algorithm.
Because we rely upon their accuracy, we would hope that error evidence would
be difficult to collect for most calculating devices. However, error evidence should
be easily available for calculators that might be of particular interest to us: humans
doing mental arithmetic. We might find, for instance, that overtaxed human calculators
make mistakes by forgetting to carry values from one column of numbers
to the next. This would provide evidence that mental arithmetic involved representing
numbers in columnar form, and performing operations column by column
(Newell & Simon, 1972). Very different kinds of errors would be expected if a different
approach was taken to perform mental arithmetic, such as imagining and
manipulating a mental abacus (Hatano, Miyake, & Binks, 1977).
In summary, discovering and describing what algorithm is being used to calculate
an input-output mapping involves the systematic examination of behaviour.
That is, one makes and interprets measurements that provide relative complexity
evidence, intermediate state evidence, and error evidence. Furthermore, the algorithm
that will be inferred from such measurements is in essence a sequence of
actions or behaviours that will produce a desired result.
The discovery and description of an algorithm thus involves empirical methods
and vocabularies, rather than the formal ones used to account for input-output regularities.
Just as it would seem likely that input-output mappings would be the topic
of interest for formal researchers such as cyberneticists, logicians, or mathematicians,
algorithmic accounts would be the topic of interest for empirical researchers
such as experimental psychologists.
The fact that computational accounts and algorithmic accounts are presented in
different vocabularies suggests that they describe very different properties of a device.
From our discussion of black boxes, it should be clear that a computational account
does not provide algorithmic details: knowing what input-output mapping is being
computed is quite different from knowing how it is being computed. In a similar vein,
algorithmic accounts are silent with respect to the computation being carried out.
For instance, in Understanding Cognitive Science, Dawson (1998) provides an
example machine table for a Turing machine that adds pairs of integers. Dawson
also provides examples of questions to this device (e.g., strings of blanks, 0s, and
1s) as well as the answers that it generates. Readers of Understanding Cognitive
Science can pretend to be the machine head by following the instructions of the
machine table, using pencil and paper to manipulate a simulated ticker tape. In
this fashion they can easily convert the initial question into the final answer—they
fully understand the algorithm. However, they are unable to say what the algorithm
accomplishes until they read further in the book.
Multiple Levels of Investigation 45