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What is understanding?



How Can We Identify a True Statement and Will that Help Us Understand
Understanding?

The funny thing about truth is that we all do in fact have a pretty
decent intuitive notion of what it is. We all know what is meant by "It
is raining" and can look out the window and see for ourselves if that is
the case. But, to define a true statement to the satisfaction of a
twentieth-century logician ain’t easy.

Existence

Previously, we asserted that we consider Existence to be divided into (1)
the Universe, U, with three space-like dimensions and one time-like
dimension (and, perhaps, a few extra (compact) dimensions to account for
the fundamental forces), (2) Mind, M, which may or may not intersect the
universe to an undetermined extent, (3) the Ideals, I, which contains
among other (incorporeal) things the complete and perfect Euclidean
geometry replete with all of its theorems and proofs and with nothing
missing or corrupted, (4) the Relations, R, (the relations among all
things everywhere and, perhaps, for all time) and (5) Everything Else,
EE, about which I have nothing to say except that it could be anything
or nothing. Also, I have no idea whether all of the past and future of
the Universe exists or not. This was illustrated in Fig. 1-1 and
discussed briefly in Chapter 1 of *On the Preservation of Species* [1a].

Events

Let us explore further the components of the statement, "Truth is the
congruence of statements with events." What is an event? In the
universe of space and time, an event is a point set. For example, a
baseball is thrown from the pitcher’s mound to home plate at the
Astrodome. We are free to take the event to be the space and time
occupied by the ball from the time the pitcher "comes set" until the catcher feels the ball strike his mitt. We can take ancillary activities to
be part of the event if we wish. We can define the event to be all of
the space within the convex hull of the Astrodome over the corresponding
period of time. This is a larger point set in four-dimensional Minkowski
space.

Clearly, such an event generates an infinite number of relations that
exist in R, our symbol for the Relations, a subset of Existence. The
Relations, unlike the Ideals, continue to be created in time. The
current distance from the end of my nose to the source of the Nile is a
relation. So are the similarities and differences between my philosophy
of ethics and that of the Stoics.

[Note in proof (7-9-97). Regarding "the congruence of statements with
events", events are composed of phenomena, which are presented to our
senses as surrogates for the noumena or "things in themselves" of which
we have no knowledge. In the following discussion, we accept the
phenomena as actual events. Truth, then, applies to phenomena not to noumena!]

We would like to define events outside the Universe or in the intersection or union of the Universe with Mind, the Ideals, the Relations, and Everything Else. Therefore, we allow that events in the mind occupy space,
although perhaps not measurable space, and they certainly occupy time.
(It will be unnecessary to justify, in this exposition, the notion that
thoughts, for example, occupy some type of space in the mind.) Events
in the realm of the Ideals are incorporeal objects with an existence completely invariant with respect to time. Thus, we use ‘event’ in a
generalized sense in this space. But, the difference between a point set in some sort of generalized (topological?) space and the Cartesian
product of such a space with the time line is of little interest to the
logician. Therefore, Euclidean geometry can be taken to be an event just as the number one can be taken to be an event or the perfect curve ball
to a left-handed batter can be taken to be an event even though it is
only an Ideal. (Please note that the ideal curve ball (a pitch in the
game of baseball) does not occupy time even though its "real"
counterparts do. As stated above, its actual counterparts generate an
infinite number of relations in R.) We can discuss it, say true and
false things about it, and, in fact, make discoveries about it. We may refer to events outside the part of existence known to ourselves, but we
cannot say much about them. Probably, though, they are not described in
a book such as the Bible or in the cosmology of any religion. These
descriptions are myths and, in some cases, are beautiful myths that do no harm unless they are taken to be true!

Statements

What a Statement Is from the Existential Viewpoint

Statements, including generalized statements (discussed elsewhere) and
compound-complex statements, e.g., Newtonian mechanics, map mental
images of events, i.e., point sets in space-time, as in special relativity, into minds capable of imagining similar events in the reader's own
conception of the proper setting for the events, normally the Universe,
in any case, the World or some part of it, e.g., a subset of the Ideals,
wherein lies differential geometry or what some overly strict person
might call a lie, but what I call an idealization, namely, an
incompressible Newtonian viscous fluid. (Obviously, no such fluid
exists, but it approximates the behavior of many real fluids well enough
for practical purposes. In addition, it is a good candidate for the
study of physics.)

For the sake of simplicity, we are considering only written statements
that will be read by someone other than the author at a later time.
(Other types of statements can be handled similarly with little
difficulty.) We need to look at such statements from three distinct
viewpoints. We may call the first viewpoint the existential viewpoint.
At the outset, we must agree that the typographical marks "In the
beginning God created the heaven and the earth" on Page 7 of my copy of
the King James Bible do not by themselves constitute the statement in
quotes. The statement is at least the equivalence class of all such
typographical marks whether they be Times Roman or Helvetica, whether
they be 8-point or 10-point, and whether they be in one copy of the Bible or another. These equivalence classes are the statements from the
existential viewpoint.

What a Statement Is from the Functional Viewpoint

But, further, the truth of the statement may depend on whether or not the
statement be *lifted* from this equivalence class of typographical marks to the point in time and space when and where the first member of the class was placed on the printed page for the first time by its author. This
process of lifting is a very important part of what I mean by a
statement. I wish to give a trivial example. Suppose an author writes,
"It is March 23, 1934. I am in Detroit, a city in Michigan, and it is
snowing." This statement most assuredly does not mean that today is
March 23rd, 1934, and that the author is in Detroit and that it is
snowing. The statement is taken to be true or false at the time and
place when and where it was written and that is the time and place when
and where we take it to have meaning. If the statement be true, independently of time and place, the lifting process will do no harm. It will
simply remind the reader when and where the statement was written.

We have restricted this discussion to written statements that will be
read by someone other than the author at a later time. We may as well
specialize further as complete generality is easily recovered. Let us assume that we are considering statements from physics, which are, more or
less, immutable. Therefore, we may neglect considerations of time and
place, etc. that affect the truth of some statements. We shall assume
that the statements that we wish to understand, or wish to be
understood, are all of a type. We shall include compound statements
such as Newtonian mechanics, though. This theory has never placed a restriction on size or complexity. Frequently, the physicist doesn't study
statements about the Universe but rather statements about an
incompressible, Newtonian viscous fluid, say, which exists in the
Ideals. Whether the strict critic considers this concept a "lie" or not, it provides an excellent candidate to test the understanding of a
physics student of a useful theory - based upon an idealization of an
actual fluid.

The Purpose of a Statement

The purpose of a statement is to transfer a mapping of an event from one
mind to another. Clearly, if I wish to describe an event to you, I
cannot produce a motion picture of the event that will play inside your
head. Even if I could, you would have witnessed only a facsimile of the
event - not the event itself. Even the witnessing of an event
constitutes a mapping of one view of the event into one’s mind.
Normally, statements do much less. They transfer a corrupted and
incomplete facsimile of a viewpoint of the event from one mind into another mind or into the same mind at a later time. We shall refer to this
as a mapping of the event. The truth or falsehood of the statement
depends on how closely the event in the reader’s mind, say, projected
into the reader’s perception of reality, which we agree is the same reality of which the writer has a conception, corresponds to the actual event
in reality. What is close enough under some circumstances is not close
enough under others, therefore truth and falsehood depend on context. We
shall make this abundantly clear by examples - hopefully, in such a way that objections amount to mere quibbles. Let us describe the diagram
shown in Fig. 3-2, omitted in this posting.

Analysis of Statements Using a Model Inspired by Mathematical Categories
and Functors

This drawing is supposed to be taken as a mathematical representation of
what occurs when a true statement is understood and interpreted properly
by a reader. All of the processes represented by arrows labeled by
numbers in circles are assumed to have taken place correctly. Arrow 1
represents the process of perception in which an image of part of
reality is transferred to the mind of the writer. Although the process
begins with light striking the retina of the writer, most of the process
takes place in the mind of the perceiver. For example, a part of his
mind operates on the incoming signal and determines what part of it to
process further. Anyone who has taken peyote or one of a number of
similar drugs knows to what extent Mr. Huxley was correct in naming his book *The Doors of Perception* [23]. What we choose to see and how we
see it is determined to an amazing degree by us. Most of us could not
tolerate unfiltered perception for long. Every element in the image in
the perceiver’s mind corresponds to an element of reality, but not every
element of reality ends up mapped into the perceiver’s mind. Hence we
represent perception as a surjection, f(e(x(E))), a morphism whose domain is part of the writer's perception of reality and whose range is all of
the image of the phenomena in the writer's mind. The mathematical object
e(x(E)) is the perception, e, of the viewpoint, x, of the event, E, in
the mind of the writer, X. The surjection, f(e(x(E))) equals m(x(E),
which is X's mental image, m, of his viewpoint, x, of the event, E.
This all occurs in the Category P of minds and events.

[Note. Since we have mentioned peyote, the skeptical reader may argue
that the perceiver might be hallucinating and Arrow 1 is not a true
surjection. Let us exclude that possibility in our discussion and
finesse the question of whether or not hallucinations are "real". I
have a genuine bias when it comes to the notion that psychiatrists have
anything to treat. I question the notion of mental illness and I abhor
the medicalization of every deviation from "normal behavior".]

Arrow 2 represents the imagination of the reader, Y. He reads a statement, for example, and he projects the image that appears in his mind into
(hence injection) existence as he perceives it. This all happens in the Category I, principally of imagination. We conceive of imagination, then,
as a mapping of an image in the mind into reality. Although the reality
where the mapping ends up is the reader’s perception of reality, it is a
perception of the same reality perceived by the writer. The injective
morphism is g(p(y(E))), where p(y(E)) is Y's understanding, p, of the
event, E, obtained by reading the statement. It maps this understanding
into i(y(E)), the way in which Y, the reader, imagines the content of the
statement about E in its appropriate setting, e.g., the Universe or the Ideals, for example. If we may generalize our usage slightly, the
statement is like a functor, F, mapping perception into imagination.

Since every element mapped by the reader into his conception of reality
originates from a unique element in the reader’s mental image of the
content of the statement, the inverse map of Arrow 2 is well-defined on
the relevant part of the reader’s conception of reality. The statement
(essentially, but not precisely) maps the author’s perception into the
inverse of the reader’s imagination (Arrow 3). (See equation below.)
It maps the image in the mind of the writer (a point set in his mental
space) into a point set in the mental space of the reader, one dimension
of which is time (Arrow 4). Also, the statement maps the event in
reality perceived by the writer into the event in reality imagined by
the reader (Arrow 5). Arrows 3, 4, and 5 comprise the statement from
the (pseudo-) functorial view. Arrow 4: F(m(x(E))) = p(y(E)). Arrow
3: F(f(e(x(E)))) = g^-1(i(y(E))). Arrow 5: F(e(x(E))) = i(y(E)).

The success of this process is determined by Arrow 6, P(i(y(E))) ---->
e(x(E)), which is not part of the functor. If the mapping P, for
pointer, points toward reality in the sense of Russell [18]; and, the
correspondence, element-by-element, is close enough (which depends on context), we say that the statement is true. If Arrow 6 points away from
reality, we say the statement is false if the writer is at fault. If the
reader reads badly, the arrow may point away from reality even though the
statement would have been true otherwise. The problem, then, is to
determine whose fault it has been when Arrow 6 points in the wrong
direction. That is why Arrow 6 cannot be said to be a proper part of
the statement. In many cases, particularly in modern advertising, the writer has deliberately set a trap for the reader to ensure that the arrow
will point in the wrong direction. However, in our case, Arrow 6, P, is (modulo the writer's competence) the indicator of understanding, provided
the statement be free of error in content, construction, and clarity.
Most teachers teach well, at least coherently, i.e., they have said what
needed to be said, using correct syntax, to facilitate understanding,
i.e., reasonable congruence between i(y(E)) and e(x(E)). I am delighted
- - except for a little cloud that casts its shadow over our mathematical
landscape.

Drawback of this Approach

The expert at category theory will note immediately that our categories
and our functor do not satisfy the conditions of that theory exactly.
See Hungerford [17]. The surjections and injections do not satisfy the associative condition because the events are distinct from the other
objects in the pseudo-categories. Our use of the term functor should be
taken to be an approximate analogy borrowed for our purposes for lack of
a better term. Nevertheless, there must be dozens of decent formulations
of models of communication among intelligent beings that employ functors
and category theory more or less properly. I shall present my latest efforts, such as they are, in the next section.

Second Approach

The Category W consists of various phenomena q, r, s, ... spawned by a distinguished event, which shall remain nameless. The surjective morphism,
t, maps deeper (and further removed from human perception) phenomena into
more observable phenomena, which some people take to be the objects that are explained by the deeper (and less apparent) phenomena, for example, q
could be the events associated with the collision of two neutrons under
extremely high impact energy, and r = t(q) could be a photograph of a
vapor trail in a cloud chamber, which recorded a small portion of what
occurred. The surjection is a partial explanation of q.

The functor F maps Category W into Category P, the objects of which are
various mental states of the mind of the writer, X, an eye-witness (or
very nearly) of the phenomena. The functor F maps q onto p(x(q)), X's
perception of the phenomenon q from his viewpoint, x: F(q) = p(x(q)).
Similarly, F(r) = p(x(r)). Finally, since u is an injection of p(x(r))
into p(x(q)), it is like an explanation of r in the manner according to
which we usually explain the more perceivable events with "myths" about the underlying barely knowable phenomena. F(t(q)) = u^-1(p(x(q))), which
is more like a description of how q occurred. Since u is an injection,
it is invertible on its range.

The objects in Category I are imaginings in the mind of the reader, Y,
prompted by reading the statement. The surjection, v, maps i(y(q)) onto
i(y(r)), i.e., v(i(y(q))) = i(y(r)). The functor G maps p(x(q)) onto i(y(q)): G(p(x(q))) = i(y(q)). Also, G(p(x(r))) = i(y(r)). Finally,
G(u^-1(p(x(q))))) = v(i(y(r))). The functor G is the statement.
Whether or not it is true and understood depends upon the resemblance of
i(y(q)), v, and i(y(r)) to q, t, and r, which can vary from the identity
(nearly) to the opposite.

Third Approach

If, as in Fig. 3-3, we turn the Fig. 3-2 (of *On the Preservation od
Species*) on its side, i.e., let all mental images, (in our minds' eyes)
of a particular external event, E, belong to the Category E. A(x(E)) is
the mental image of E in the mind of X, the writer, whilst B(y(E))) is
the mental image of E in the mind of Y, the reader. Place our cerebral
percepts (a(x(E)) for the writer, X) and concepts (b(y(E)) for the
reader, Y) corresponding to that event in another Category M. We can
employ category theory correctly; but, true statements will be
associative morphisms not functors. The functor F will map mental images of a particular external event E into individual minds by what we call
perception. Let us denote, by a surjective morphism f(B(y(E))) the
difficult business of transferring the reader’s mind’s eye from his own
head into the writer’s head in such a way that he is able to capture a
mental image of whatever the writer, X, an eye-witness, is able to see
of the event E. Let g(a(x(E))) be the simple associative injective
morphism that amounts to no more than a statement by X, the writer, providing Y, the reader, with an account of the event E.

Then, the functor F maps, in addition, the extremely awkward morphism
f(B(y(E))) into the extremely convenient morphism g(a(x(E))). The
written statement g maps the percept a(x(E)), which is the writer’s
perception, a, of his mental image, x, of the event E, into b, the
reader’s conception of the statement g.

The inverse of the functor F is the process, namely, imagination, that
the reader Y employs to visualize the image the writer X is trying to
convey. In this model of written communication, the imagination of the
reader is not part of the written statement, which seems fair enough, as
it is not the writer’s fault if the communication fails due to the
reader’s lack of imagination.

I hope it is understood that, although B is Y’s mental image in Y’s
mind’s eye, Y’s mind’s eye has been in X’s head; therefore, Y’s mental
image is A - exactly that which X would have captured in his mind’s eye
if his mind’s eye had not been replaced by Y’s mind’s eye by f.
Strange, isn’t it? It’s as though Y got inside X’s head without giving
up being Y, so his mental image is what X would have captured, but it’s
still Y’s and we call it B not A.

The associative morphisms that are mapped by the functor F into
associative true statements are almost the morphisms that change one
person into another as in the expression "If I were you, ...".
Obviously, writing is easier and a lot more fun than having oneself
changed into someone else - even if we are permitted to disregard the
impossibility of doing so. Then, if X’s perception be correct and X’s
statement to Y be true, the condition of Y’s mind is nearly the same as
it would have been if Y had become X (part or all of the relevant
content of Y’s mind had been mapped onto all of the relevant content of
X’s mind) and Y had perceived the event (thus informing the relevant
part of the content of Y’s mind, which would then be occupying all of the relevant part of X’s head). What we mean by "nearly the same" was
discussed in connection with Arrow 6, above.

The responsibilities of the teacher and the student are clearly separated
in this model too, and we need say no more about it.
Perhaps it is worth noting that the functor F in the third approach is
our first contravariant functor. I hope this does not have serious repercussions. Also, please notice that the inverse of imagination is
*memory*. Thus, F^-1(b(y(E))) = B(y(E)) is Y's memory. F(B(y(E))) is
Y's imagination.

Finally, we can simplify the second approach by simplifying the
categories to categories each containing one object (only), such that
the sole morphism is the identity. Suit yourself as to which you prefer
- if any.

Fourth Approach

Category W has one phenomenon, call it q. Category P has only X's
perception p(x(q)). Category I has Y's imagination of what the
statement G means, namely, i(y(q)). The functors F and G map F(q) =
p(x(q)) and G(p(x(q))) = i(y(q)), which last is the statement. It is
successful if i(y(q)) resembles q.

Epilogue

I do not wish to apologize for the mountain of machinery required to make
a concept like a true statement or "understanding" precise. However, I
am interested in corrections and amendments in the spirit of this
process.

Also, please consider the immense importance of memory in acquiring a
broad (comprehensive) understanding. We must be able to collect all of
our understandings of essentially everything we know. Eventually,
everything we know appears in our mind's eye as one spectacular, beautiful panorama - all of which is accessible instantaneously.

Literature References

On request.

Yours very truly / Tom Wayburn