Selasa, 27 Desember 2016
Structuralism And Mathematical Thinking
Mathematical thinking is only one aspect of thinking
in general. Therefore, question concerning mathematical thinking raise
questions concerning all thinking. Althought mathematical thought is in many
ways very special, it admits of straightforward that throws like on all
thinking and suggests a powerful general approach to studying the latter. The
basis of our analysis of mathematical thinking involves the abstract concept of
structures. Futhermore, structures play the same essential role in all thinking
as they do in mathematics, whether or not the thinker is aware of their
presence.
A general point of view firmly based on structure is
developed in some detail in recent book
by the author entitled structuralism and structure (rickart, 1995). A main
objective of the book is to build for the reader a solid notion of abstract
structure as well as an understanding of how structures are involved in the
study of any serious field of information. Because the book is strongly
influenced (at least indirectly) by mathematics, much of this material is
obviously relevant to the material presented throughout this chapter and is
reffered to as needed.
The general assumption behind all of the following is
that information is coded and stored in the brain in the farm of structures,
and that the brain is especially “designed” for recording and processing these
structures. This is true despite the fact that an individual is seldom aware of
what is going on. Althought our ultimate objectives is to explain why this
point of view on thinking is both valid an useful, it is desirable for obvious
reasons first to take careful look at structures for their own sake. This
provides basic concepts and the necessary language for dealing with a serious
structural approach to any subjects. ( in this connection, a reader might also find
useful the discussion of structures and examples that appear in various
sections of rickart, 1995, especialy in the introductory chapters 1 throught 3)
Structures are present everywhere, either explicitly
or implicitly, and provide an approach by which and situation or mass of data
might be made inteligible. In fact, the general discipline that is known as
“structuralism” and is devoted to this covering and studying the structural
content of a field is called “the art of the intelligible” by caws (1988) in
his book entitled structuralism. In other words, the structural content
represents the contained information that is ultimately recorded is one’s
understanding. What all of this means becomes clearer as we proceed.
The subject of structure, for which we are about to
offer a formal introduction, is rather different from most subjects. This is
because virtually everyone already has a fairly clear notion of what a
structure is without the benefit of a definition. In other word they are able
to decide offhand which a many familiar objects in their surroundings may
legitimately be called a structure. Up to a point this may be an advantage, but
view people understand what it is that these familiar objects actually have in
common. In other words, they lack the general notion of structures, something
exceedingly helpful in a serious structural approach to any subject of
substance.
The next section contain some of the basic material
required for dealing sistematically with structures. The perceptive reader will
note the similarity of this situation with a common mathematical case in which
one can be familiar with certain mathematical concepts at, say, notational
level, without being able to think of them in the same way that a mathematician
might. The difference here is between concreteness an abstractions . in other
words, thus the concept exist in terms of the “ material world,” or is it independent of the latter? Note that an abstraction at least
for purposes of communication, usually requires a formal language to deal with
it.
STRUCTURE
DEFINITIONS AND PROPERTIES
A structure, by definition, consist of a set of
objects along with certain relations among those objects. The set of objects is
also said to have structure. A subset of the objects along with some of the
relations restricted to the subset is called a substructure of the given
structure. A substructure is obviously a structure in its own right, and
properties of a given structure are often expressed in terms of its
substructures.
Two
structures are said to be isomorphic if there exist a one-to-one corespondence
between their objects that preserves relations (in the sense that objects in
one structure are related if and only if their associates in the other
structure are also related). Isomorphic structures are said to have the same
structure, and what is common to them is an abstract structure.
We
regard abstract structures as having an independent existence. They are also
regarded as consisting of abstract objects and relations. Within an abstract
structure, an objects has only those properties that it gains by being a member
of the structure, including, for example. The property of being related to
certain other objects in the structure. A similar remark applies to mined by
the (possibly ordered) sets of objects that it relates.
A
concrete structure consist of concrete objects, and my be thought of as
isomorphic to the abstract structure that defines it is a structure. It may
also be regarded as a representation of, or as represented by, the associated
abstract structure. Observe that are quite irrelevant to their properties
derived from belonging to the given concrete structure.
With
these definition and conventions, it is possible to state precisely what it is
that familiar objects recognized as structures have in common: Each one is a
concrete structure according to the aforementioned definition of structure.
It is worth noting at this point that term “system”
often appears in common usage as a synonim for “structure.” We prefer, however,
to use the term to mean any collection of interrelated objects along with all
of the “potential” structures that might be identified within it. This turns
out to be a useful concept taht includes many mathematical examples, of which
we mention only one—“a number system.” It is instructive with respect to both structures
and systems to spell out some of the details of this example. Recall that a
number system, which consists of the numbers themselves as objects, also admits
operations of addition and multiplication. The operations determine two “
ternary ” relations defined as follows : the numbers in an ordered triple (
x,y,z) are defined to be addition related provide x+y = z, or multiplication
related provided xy = z. This identifies two structures in the number system. A
complete description would, of course, require precisely stated axioms giving
definition along with individual and joint structure properties.
Although
the general structure definitions enable us to concentrate on basic structure
concepts, they do not bring out explicitly the fact that structures may also be
dynamic in character ( a subject discussed briefly in chapter 3 of Rickart,
1995). As might be expected, the analysis of almost any dynamic structure is
likely to present special problems. One example of such a structure is a
typical machine in operation, and it may be represented as an “ ordinary”
structure existing in four dimensional “
space time.” Another kind of dynamic structure is the classical model of an
atom. The latter admits a very different , but especially effective,
representation of a dynamic structure in which the orbit of a moving electron
around the nucleus of the atom is suggested by simply drawing the orbit.
The general structure definitions outlined here may be
a bit misleading because they represent structures as more or less isolated
entities. In actual practice, nothing could be farther from the truth. For
example, most structures that we encounter are embedded (as substructures) in
the larger structures that may or may not be relevant to whatever interest that
we might have in the initial structure.
At the very least, our attention to a structure automatically embeds a
representation of it in our ever-present system of mental structures.
Furthermore, our perception of a structure is not a passive experience, but is
shaped by previous experience that will determine much of what we see in the
presented structure.
One further comment on our approach is that the
emphasis on abstract structures is abviously a partial commitment to an
idealistic point of view. Although most mathematicians do not appear to be very
strongly committed to any particular philosophical position, most of them seem
to think and talk of mathematical structures as being abstract. In fact, it is
a bit awkward to do otherwise.
A
STRUCTURAL ANALYSIS OF THINKING
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.
SPECIAL
ASPECTS OF MATHEMATICAL THINKING
Although mathematical thinking is just one form of
ordinay thinking, it does exhibit special features that set is apart from most
other thinking. One thing that stands out here is the fact that its subject
matter is unambiguously abstract, consisting of pure mathematical structures.
Whether the content is ordinary arithmetic elementary algebra, or a very
advanced topic, it is necessarily abstract when correctly understood. Anyone
whose understanding falls short of this is to that degree handicapped in the
use of mathematics.
Another special feature of mathematics is its language. As far as ordinary thinking
is concerned, the language used when needed is ordinary language that already
exist, whereas in the cas of mathematics, the language is usually very special
and needs to be learned along with the subject matter. It is ordinarily quite
formal and also very close to the underlying mathematical content; so close, in
fact, that the formalism is sometimes naively confused with the mathematics
itself. As in ordinary thinking, mathematical thinking may involve only the
content or it may also involve the formal language. Involvement of only the
language, which amounts to either ignoring content or indentifying it with the
language structure, is usually not good in either case. There are exceptions, however, tht are
mentioned in the next section. The naïve confusion of content and formalism is,
in most cases, a low-level wakness that stands in the way of a correct
understanding of mathematics. This is a special problem in the case of
elementary topics such as elementary algebra. Restriction to content is not so
common and usually involves more experienced mathematicians.
Some of important special differences between
mathematical thinking and ordinary thinking can be brought out by examining how
children learn the most elementary material. A good place to start is with the
case of language. It is abvious that children have an instinctive drive to learn
the language, as attested by the spontaneous way they become involved in the
learning.
It is also clear that one cannot separate the learning
of new material and the learning of the language in terms of which the material
is described. in the words, the learning process mixes the two. furthermore, if
anyone bothers to pay attention, they will be impressed by the way virtually
every adult who comes in contact with the child automatically assumes the role
of a teacher. they will adjust their voice and repeat phrases to make it easier
for the child to learn the correct use of the language.
A closer observation will suggest that children are
also equipped to learn numbers, along with their structure, just as they can
learn a language. Learning about a number system is, of course, very different
on the surface from learning about a language system. In the first place, few
are able to recognize the procces and the result does not compare in importance
with language. The development of an understanding of a number system could be
encouraged along with learning a language, but few parents, skilled or not in
elementary mathemarics, think to encourage and help the child grow in this way.
The difference is in spontaneity, which dominates the learning of ordinary language
but is ignored in learning numbers. The result is an individual that will have
so struggle later with the problem of elementary mathematics on an entirely
different level. We mention the problem here because it is importand and also
demands a very different treatment later. This is crucial in both the teaching
of arithmetic and elementary algebra.
I digress briefly at this point to cite a rather
different example, a personal one that involved my youngest son when he was an
early preschooler. One day I showed him an ordinary fork and asked him first
how many prongs the fork had and then how many “spaces” it had. The answer here
was 4 and 3, bot of which were very easy, so I asked how many spaces there were
when the fork had various other (reasonable) numbers of prongs. These answers
were also very easy,although I am sure that he had never seen a fork with more
than 4 prongs. The point is that he could visualize without any effort
something he had never seen and also understand immediately that it would haveto
prossess a specific structural feature. This is kind of intuitive understanding
that I regard as fundamentally significant and closely related to other
mathematical understanding. Incidentally, I did not understand the event then
as well as I do now, or I would have pursued it further.
We turn now to the question of how we learn and come
to understand elementary mathematics at the level of elementary arithmetic and
algebra. Although the general problems here are superficially much the same as
the problems associated with learning elementary material of any kind, the
mathematics is different in the details.
Virtually all students in elementary school have long
since passed the stage of being able spontaneously to learn about the structure
of the ordinary number system. Therefor, these thing need to be addressed in a
different way. Ideally, the number system should be introduced and its
structure some how communicated to the student so that the basic idea
predominates apart from a word description of it. At the elementary level, it
is not practicial to do this in a formal way. The massage can ordinarily only
be implicit in the way arithmetic is taught, so much of the picture has to be
communicated in the attitude and expressions of a teacher who understands what
is going on. This is a very delicate matter and is often flawed. Therefor, it
is not unusal for many students to wind up with either an empty or a seriously
defective notion of the arithmetic number system. It becomes very difficult to
help students when such handicaps become estabilished, even if a teacher
understands the nature of the problem-something that may to be common. Students
who avoid this problem often are able to do so because they retain some of
their instinctive ability to make sense out of numerical material.
This bring us to the stage of trying to teach students
about elementary algebra. The ideal situations is to deal with students who
have derived from their experience with arithmetic a correct intuitive picture
of the ordinary number system. Unfortunately, such students are probably
somewhat rare. Too often, success with students is a hit –or-miss matter with
many who never are able to understand the true nature of a number system apart
from a list of rules. It should be admitted here that a similar situation can
exist in the learning of anything new of substance. The problem isespecially
difficult in mathematics, however, because students may have a very wrong, but
very solidly estabilished, picture of the elementary number system because of
their bad experiences with arithmetic.
Even with students who have a correct notion of the
elementary number system, there remains a problem with the learning of algebra.
Although the algebra system may be motivated by the earlier understanding of
the elementary number system, the fact remains that the algebra system is more
inclusive. It includes in addition, for example, the much larger system formed
by all polynomials. The number system obviously has special properties that are
not held by the algebra system. This is not unusual, of course, because a
structure need not address all of the properties of a concrete structure that
represents it. In any case, an appropriate representation of the algebra system
will not be isomorphic with a full representation of the number system. The
transition between the number system and the algebra system is obviously not a
minor step in understanding.
For practical purposes, to this point I have
restricted attention to ordinary arithmetic and elementary algebra. As far as
the level of mathematics is concerned, however, geometry should also be
included, but it is actually rather different from others. For example, the
subject matter of elementary geometry consists of familiar figures, such as
straight lines, triangles, and circles plus, perhaps, some three-dimensional
objects. These are acquired very early and painlessly. They also have a
different relationship to the language. In fact, up to a point, the associated
language is close to ordinary language.
This is probably one reason many students report that
elementary geometry was easy, whereas elementary algebra was impossibly
difficult . in other words, elementary geometry is learned much like anything
else is learned, so can be dealt with in a relatively routins manner.
Ultimately, of course, it is necessary in geometry to face questions similar to
the ones that ultimately underlie any serious mathematical field.
FORM VERSUS
CONTENT
The problem of teaching elementary algebra is similar
to the problem of teaching (or learning) any new piece of mathematics. There
are two questions that must be addressed. The first involves the degree of
understanding of the background subject matter leading up to immediate subject
of interest, and the second involves the formalism associated with the subject.
The latter item is very special in mathematics, making it different from
learning other material. It refers very explicitly to the structural content of
the subject.
Notice that there is an ambiguity here because a
mathematician may react to the material in two different ways. There are times
when the content is of vital importance, which means that the emphasis is
almost wholly on the mathematical nature of the system described by the formal
language. At other times, the user’s attention may be absorbed by a related
mathematical system and the given operations are reduced to little more than
pure formalism. What is important, however, is the fact that the
mathematician’s attention can be shifted immediately to the content behind the
given formalism when needed. I also mention in passing that there are a few
formalists who equate mathematics to its formalism. This, however, is a
sophisticated philosophical position having nothing in common with the views of
elementary students.
One of the problems in the routine teaching of
elementary algebra is that it sometimes reduces to nothing more than teaching
the formalism. This can be a very serious problem when a student’s
understanding of the underlying arithmetic structure is inadequate, and
therefore the only recourse is to fall back on the formalism. It is not
uncommon for many tests to be constructed to measure mastery of the algebra
formalization rather than the mathematical content behind it. For this reason,
it is not uncommon to find student who have a pretty good record as far as
grades are concerned, so are fairly adept at the formalization of elementary
algebra, but do not understand elementary algebra well enough to deal with its
application to calculus. Needless to say, such student are almost beyond
salvaging. The farther they have gone in this direction, the more difficult it
is to supply the missing understanding so necessary for many applications of
elementary algebra.
It is only fair to point out that most courses in elementary
algebra do not address directly the problem of understanding. Nevertheless,
some of the students automatically fill in a correct understanding.
This is a late manifestation of an ability that small
children tend to show automatically if they are allowed or encouraged to do so.
It is also an example of a drive to complete, in one way or other, an
“unfinished” structure. Such student repair, at least partially, some of
deficiences in their previous training. The tendency of many teachers, however,
to avoid the problem of understanding tends to blunt the self-correction thay
may save a student. The teacher’s objective in algebra should include a
constant awareness of the desirability of helping the student to supply or
develop the basic understanding of an algebra system.
The student who master the formal algebra operation
without being able to associated with them the concept of an algebra system has
somehow replaced the desired algebra concept with the relatively superficial
structure of the associated formal language. We call this form without content
(or structure). It is a special case of substituting the structure of the
language for the structure of the material that the language is supposed to
described. This can actually occur at a level much higher than algebra, as
shown by the following personal example.
A number of years ago I was invited to give an
hour-long talk before a meeting of the American mathematical society. This was
an important assignment, so I devoted considerable effort to the preparation of
my talk. In fact, I prepare it much to well! As a result, when I gave the talk
I found myself standing before the audience and listening to, rather than
thingking about, my lecture. In other words it reduced to a clear case of form without
content. The content of the lecture was reduced to the language rather than to
ideas, and instead of enjoying a flow of ideas, I experienced only a flow of
words. Needless to say, this was a very
unusual and unpleasant experienced for me and I can only hope that the audience
did not notice what was happening. Incidentally, this is probably why
mathematicians seldom give lectures by “reading” a manuscript to the audience.
At the same time I am mystified by the fact that distinguished representatives
of nonscientific fields often deliver a lecture by reading a prepared
manuscript.
CREATIVE
MATHEMATICAL EXPERIENCES
It is exceedingly difficult for a mathematician to
convey to any nonmathematician some idea of the nature of a creative experience
in mathematics. Much of the problem is due to the fact that some of the richest
experience with mathematicial thinking involves nontrivial mathematics that
cannot be adequately described without use of highly technical mathematical
language. Despite of the problem, a very distuingished mathematician, Henri
Poincare, published an essay, Mathematicial Creation (Poincare, 1913, pp,
383-394).
The essay contains an account of his personal
experience in discovering some definitely nontrivial mathematics. A fairly
detailed discussion of Poincare’s essay is given in Rickart (1995, section 44),
where the emphasis tends to be more in the way that mathematics fits into a
general structuralism approach. All of my references to Poincare are to the
essay.
Needless to say, the Poincare essay does not provide a
technical analysis of the mathematics on which his remark are based, but rather
is devoted to a discussion of the way his mind dealt with the ideas. Although
there is a mystery as to what actually happened with the mathematics, one can
appreciate to a degree how the ideas were manipulated. An interesting fact is
that a significant part of the experience took place in Poincare unconscious.
Although all approaches to this subject have much in common, it is important to
understand that the Poincare account describes only one of many possible
variations on experience of this kind.it is also important to understand that
what occurs is not uniquely determined by the fact that it concerns
mathematics. Experiences of this kind can occur in any subject and at any level
of understanding. What is unique about mathematics is the relatives case with
which events can be isolated and studied.
Poincare emphasizes that the creative experience was
proceed by a lot of hard mathematical work on the problem, which did not yield
a solution. This was followed by a relaxation or preoccupation with something
totally unrelated to the unsolved problem. Although Poincare conscious mind
thus ceased to deal with the problem, his unconscious (“subliminal self”) continused
to work on the problem and eventually came up with a solution. He thinks of the
solution as a “good combination” of known mathematical entities and suggested
that the result has an aesthetic value that brings it into consciousness. He
also attributed the aesthethic value to a character of beauty and elegance.
Such entities “are those whose elements are harmoniously disposed so that the
mind without effort can embrace their totally without realizing the
details.” These are the most useful and
beautiful because they lead to a mathematical law. It is their beauty and
usefulness that distuighishes therm from the ordinary combinations and brings
them to consciousness before the many others.
Poincare emphasized the importance of the preliminary
conscious work that provides the unconscious with an enormous supply of
combinations, most of which are useless. He figured that “the future elements
of our combinations are something like the hooked atoms of Epicurus.” Normally
motionless and “hooked to the wall.” Under certain nonrandom circumstances
(perhaps resulting from the preliminary conscious work), certain atoms are
detached from the wall and move about “like the molecules of gas in the
kinematic theory of gases,” so new combinations are produced by their mutual
impacts. The point is that the ultimately selected combinations tend to be good
combinations.
It is possible to give a more structural description
of what may occur in the Poincare picture. The following account, based on the
brain structure, is a sketch of a more detailed discussion given in Rickart
(1995). It views the brain structure as a massive electrical network, despite
the fact that it is considerable more complex than that. In this picture, allof
our mental structures appear as electrical networkseach of which is a
substructure of all inclusive brain structure.
A brain network is inactive until it can become an element in the general
structure processing. We may there fore think of the “active” structures as
being highlighted and therefore candidates for general processing. The process
relevant to mathematical creation is an extension process. This means the
creation of larger structure that includes the given one as a substructure. The
extension may be either a new structure or a structural joining of the given
structure to another previously existing structure. Either case may be regarded as growth of the given or known
structure and the result normally a new mathematical structure. The question
now is, “How can the Poincare experience be described in terms of such a
structure extension?”
We may think of a given structure as a potential for
growth through the existence of sensitive connector points on the surface of
the structure. These are points with the potential of entering into an electrical
nerve connection with material outside the given structure, possibly with
another structure. Mathematical creativity consist in replacing a given
familiar mathematical structure with a larger one that is a previously unknown
good mathematical structure. Just as for Poincare, the extensions are anplanned
aside from things learned in the preliminary work on the problem, and good extension are the beautiful ones, which
are forced into consciousness by their beauty. Needless to say, just what
constitute the beauty aside from usefulness, is rather difficult to detail. In
any case, to the experienced mathematician the beauty is often quite apparent,
although perhaps difficult to explain.
It is worthwhile to describe another far more ordinary
example of a creative experienced that has something in common with the earlier
discussion. This is another personal experience and would probably not rate
mentioning in another context. (it is also discussed, along with another
example, in Rickart 1995)
The example involves the proofs of a paper that I had
written to be published in a standard reaserch journal. Everything was quite
routine because the corrections were only trivial ones, so normal practice
would consist of returning the sheets in the mail. For some reason unknown at
the time, I put the sheets aside and postponed returning them to the journal.
In the meantime, I noticed that I was automatically reviewing the proof of one
of the lemmas in the paper. The occurred at odd times even when the paper was
not in my thoughts.
Cause the lemma is an important one and I am generally
happy with it. In fact, given the evidence that's analogous to the mechanical
melody of reviewing a piece of favorite music. The only difference here is that
I remember the evidence at the time that strange or uncomfortable and there are
certain about everything compulsiveness. Under the growing anxiety, finally I
force myself to sit down and review the proof of lemma. The problem, of course,
is that the proof of lemma contained errors. Fortunately, the mistake was not
bad, so it's rather easy to fix, allow me to return the evidence without
further delay.
Note that the experience is one of the unconscious and
the conscious not only demonstrated that without a hitch as the creator, but
also a fine critic as well. At the same time, we can ask why conscious not to
give answers along with a warning of the error. Although it is not entirely
clear why so, may not be aware of the exploration or development can handle
structures, but the Joes have no facilities to make references about the
structure. Therefore, it can be "exhibit a" defect structure but cannot
explain it. In fact, the exhibition structure defects may be nothing more than
a direct result ran to defect while exploring the structure. Poincare also
pointed out the fact that the subconscious seems to never present one with the
details of the solution. Apparently conscious is always required in filling in
the details of the confirmation, or in the provision of any communication.
Readers will find additional discussion of
mathematical creativity by Hadamard (1954).
One of the problems with the oft-quoted account of
creativity is that they demonstrate that creativity is the experience provided
for great minds. This is not the case, because a similar experience similar to
that of ordinary people when they experience something com-monplace as face
recognition. The problem is that these experiences happen so often and so
relaxed that they are not recognized as true ¬ crea tive. A simple analysis of
the many experiences of this kind would reveal their creative quality.
A
DESCRIPTION OF THE QUALIFICATIONS
I included here some comments on our previous use of
nerve structures of the brain. readers may be aware of the fact that my
attraction to nerve structures are not tight and functions only as a convenient
way to visualize what is going on. To determine how the structure is actually
represented in the structure of the nerve is actually a very difficult problem,
so it may not be surprising that not much is known about this issue. In spite
of the difficulties, however, the use of the symbol representation of a neural
structure comfortably until a certain point. At the same time, a certain
mathematical object indicates that the structure of the nerve is not what
ultimately needed.
Note, for example, the concept of a triangle. Note
that this is some kinda thing ¬ right and we hope the concept will be recorded
as a more or less sharp images a triangle. It is difficult to understand,
however, how a net figure of the triangle can be represented in a chaotic mass
of nerve fibers. Everything becomes more difficult when it realized that the
concept of a triangle should cover a wide range of triangle; including, for
example, a triangle is equilateral, right elbow, acute or obtuse an gled ¬, and
so on. Perhaps the brain structures are being considered at the wrong level.
Following this, a very different type of example, is suggestive.
Consider the image of the character (mathematics) on a
sheet of paper. This structure can be very sharp and may contain a lot of
information. In addition, the paper brings the structure drawn is also a
complex structure which is the object of molecules of various types. Although
the paper played an important role in the representation of the figure of the
mathematics, which struc-forthcoming, at least at the molecular level, it seems
really irrevalant. Is it possible that the underlying neural structures of the
brain is not relevant to the representation of the structure of that interest
us? If there is a rate at which the brain can be seen as a structure that can
serve a role analogous to the gross structure is served by a piece of paper in
case the figure drawn? Needless to say, I could not answer this question and do
not know where the structural level may be more relevant.
MATHEMATICAL
ABILITY
It is clear that individuals vary greatly in their
ability to understand the thir mathe-matics ¬ abilities vary with age. Some
children may have DIF. .. ¬ ficulty from scratch, while others are good at
mathematics up to the point where suddenly became impossible. Difficulties with
math suddenly can occur at almost every stage. Finally, there are some very
talented with math and seemingly limitless in their ability to develop into the
subject. What is the difference here? Are they genetic or they are the result
of a particular combination of experience? If there are some who, for some
reason, just can't think beyond a certain point mathematically? The difference
of ability in a variety of subject matter is hard to analyze, but it might be
easier to deal with math than with other areas because the more obvious involvement
of the structure in the second.
We've seen some difficulties arise with children who
are dealing with issues of learning math. Some of these go back to the first
experience with numbers that occur at the same level as language learning. It
is clear that the difficulty on this level will increase the difficulty of
developing a proper idea of the number system in the process of learning about
numbers in basic arithmetic. We have also seen that the problem with the
regular number system automatically makes the document a dif ficulties ¬ when
it comes to learning about the algebraic system in the course of learning basic
algebra. Unfortunately, these problems are piling up and the problem of
correcting them is becoming increasingly difficult. Note that in all these
cases the problem is not built in but the result of gaps or errors in teaching
or learning. In other words, trouble could have been avoided by proper handling
of the issue, so it's not something that an individual was stuck with the
because of hereditary factors.
It is clear that anyone who has had a number of
misconceptions about the algebraic system repaired or will be better able to
deal with more advanced math. At the same time, it is important to point out
that there can be lasting disability remains from the experience. The mind is
not like a computer: when something corrected it not cleared, but only
"marked" and then followed by a correction. items marked are always
there and probably not usually arise in consciousness, but it is still there
and inadvertently can force its way into attention. Therefore, although the cor
rection ¬, mistakes are never really deleted and can slow down the use of
someone from the concepts involved. I believe that most of mathematics can
dredge the mathematical experience to support this statement, but perhaps it
would be better not to cite daily experience, now I do.
A few years ago we planted a peach tree in our
backyard. This is the first season surprised us by producing a few peaches that
unusually large. My first unthinking reaction to this event be amazed that such
a small tree that produced the big Peach. I've been known for. a long time, of
course, that the size of peaches had nothing to do with the size of the tree,
but sometimes before I learned that I actually had to be associated with the
size of the tree fruit size. It's part of the information is wrong, except it
can still be keenly monitored, forced its way into my consciousness. Items may
vary, of course, in their tendency to be called back. It is not surprising that
a piece of misinformation math can appear in the same way. An event in
consciousness may be rare, but it is easy to see that it still may often slow
down the thought of someone about a topic of interest.
Although we have emphasized the basic mathematical
items in this chapter, the same phenomenon can occur with more advanced
material. There fore, ¬ piece corrected information mathematics at any level
may end his career in the subject. In other words, a type of mathematical ideas
are so important to understand many things that will surprise if restrictions
on learning mathematics-is limited to the default quality. The exception to
this may be alleged in order to very rare, very talented mathematicians.
Admittedly, their ability may be based on something that even absent in many
highly successful mathematician.
