It is time now to discuss in more
detail the structure basis for thinking in general. As we have already noted,
information is obviously recorded is some manner of other in the mind-brain
sytem. Although “nerve cell” details are relatively sparse as to how such
material is actually recorded, it is reasonable to assume that the information
is somehow coded into a structure which is in turn represented by a nerve
structure. The mind-brain construct is often referred to as a mental structure.
The term can mean either the associated concrete nerve structure or the
abstract structure that represents it. Observe that the second point of view focuses
more attention on the structure concept. For this reason, we usually think of a
mental structure as an abstract entity.
As was already mentioned, our
assumption is that the thinking process is one of the special function of the
general structure processing activity of the human brain, and the material of
thought consists of the information recorded as tructures in one’s mind-brain
system. This material consist of ideas and concepts; cinther structures
extracted from current sense data, or retrieved from previously recorded
information.
Despite the difficulties of
determining just how structures are generally recorded of coded in the brain, a
very important and better known intermediate device involves the use of
language in any form. In this case, a structure is represented by language
structure that involves time. In other words, the structure is presented
piecewise and strung out in time. This form of representation is of special
importance because is provides a mechanism for communicating the content of a
structure from one person to another (deSaussure, 1966; see Rickart 1995,
chapter 5). From our point of view, use of language is only one among many
special ways that structures may be manipulated. In other words, thinking is a
remarkable tool for dealing with mental structures but is generally independent
of language, although the possibility of communication provided by language is
of an unquestioned special importance.
Thinking, with or without
language, invlovesideas or concepts directly in terms of their basic intrinsic
relationship. They are somehow represented so that they can evolve and
interact, largely independently of the outside world. The system may, of
course, use material out of memory banks and even occasional inputs from the
outside. In this context, language becomes just another method of manipulating
structures. Although thinking is normally independent of language, it may shift
into the communication mode, if needed.
Language can play a very
important positive role in thinking, but it can also play a seriously negative
role. For example, it is possible for language to function quite independently
of underlying ideas for which it was ostensibly created. It is, of course, by
virtue of its structure that language is able to represent a system of ideas.
At the same time, the language structure may exist independently of idea
structures that is usally organizes. When thins separation occurs, the
underlying substance idas may be lost and we are left with the empty language it
self. Everyday examples of this are found, for example, in the use of clichés.
We see it in mathematics when the formalism is used without the underlying
mathematical ideas. In other words, the structure of the formalism is adopted
for its own sake. This comes up again later, when we examine, for example, some
of the problems associated with the teaching of elementary algebra.
Despite our insistence that
thinking is not dependent on language, there are some who maintain that all
thinking is actually self-communication, and thus is dependent on language in
some form or another. Although this point is obviously dependent on ones
definition of thinking, the language restriction would exclude many instances
of mental activity that, in my opinion, should be classified as genuine
thinking. Included, for example, are certain mental experiences of any creative
mathematician. I offer specific examples later.
Thinking is generally regarded as
taking place in the conscious part of the mind. However, again because of some
rather vivid mathematical experiences, plus certain other ordinary phenomena,
it appears that there does not exist a sharp distinotion between the conscious
and un conscious thinking. The process can evidently take place in either the
conscious or unconscious mind and even shift back and forth between the two.
What is missing when the unconscious is involved is the monitoring or censoring
of the whole process by the conscious. The unconscious setting, despite a lack
of discipline, may allow a freedom of mental activity that can be far more
creative than when restricted by the conscious. Illustrative examples are
presented later.
At this point it is important to
make a distinotion between the general unconscious, which we have in mind here
when we refer to the unconscious, and the Freudian unconsciousness. There is
not a sharp line dividing the general unconscious and the conscious mind.
Although there are regions in the general unconscious that are not easy to
success, much of it is very close to the conscious so that passage between the
two is not difficult. This is not the case with the Freudian unconscious, which
is difficult to access and is normally widely separated from the conscious. Any
influence that it has on mathematical thinking is probably limited to familiar
Freudian phenomena, which are not easily related to the highly rational mental
phenomena characteristically associated with mathematical thought.
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