### Words, Images and Concepts

*5/15/99 Version*

The origin of language contains issues of image as well as sound. Cave
paintings showing animals are like spoken words. Both are ways to convey
something about nature to others. That says that language is fundamentally
a way to communicate. One thing to note is that written signs as well as
spoken sounds can comprise a language.

Linguistics traces kinship of spoken languages by comparing how *five* is
represented in sound. It is the most stable word, a fact probably originating
in its origin as sounds meaning *hand*, something that is true in many
languages. The image involving repetitions of numbers
Fives is part of a complex. That complex is a manipulation of multiple
concepts, just as a cave painting that shows arrows along with animals.
In the case Fives sound in the form of a poem
led to the image.

Neither the poem nor the image would
be the same if there were no five numeral, *i.e., 5 *. By creating
the independent concept, the stylized figure numeral 5, different from a hand,
there is indeed a new concept. One way to see that looks at the
likely origin of numeric symbols. Numbers shows the
first five by joining dots.

Some ways language throws words into the range of concepts is via
complex expressions: *multiplying* is just such a term. So are
*infinity*, *logarithm*, *calculus*, *combinatorics*
and many others. One of them is essentially the concept of counting by
numbers when there is no last item. Another was invented to make
simple adding operations able to carry out multiplications. While these
concepts are not explicitly used in the following, there are ways that
they come in to the issue of testing what one has learned. The first
item below shows the three-alternative thirteen-response system for
testing what one has learned that
uses a logarithmic weighting. The second is the Computer Science
Department report on its use. That report includes a description of
similar concepts used in two courses, and students' test scores showing
some had difficulty understanding the fundamental definitions. The last
item is a conference paper that is a shorter version of the report.
A way to find out whether learning of a subject has occurred can begin
with Learning Accomplished and the files there.

Computing began with numbers but has moved on. It includes many other
concepts, but having ability to deal with the mathematical words and
symbols is helpful to understanding them. Consider things about circles
as a simple instance of this concept, and one specific symbol, namely
.

The value of the symbol is that it enables speedy *writing* to
clearly communicate. Symbols are ways to economize. The
symbol
eliminates the need to create
an image to show a circle, and to portray visually the actual ratio of
its boundary length or circumference to the diameter. Instead, just
as in the case of 5, we use the Greek letter
as a generic quantity or concept.

Here is a quote from the book by Harold M. Stark,* An Introduction to Number Theory* Second printing, 1979,
MIT Press paperback edition, 1978, Copyright 1970. This is a thorough
exposition of an area of abstract mathematics that has relevance to
several important contemporary computer issues.

On pp. 7-8 Stark's book
attempts to explain the importance of the idea of universal truth
compared to experimental test in the realm of numbers. This is done with
respect to a certain equation. The remainder is directly taken from that
book.

We see that x = 1, y = 0 satisfies the Diophantine equation

(5) x^{2} -1141y^{2} = 1

We might ask, does equation (5) have any solution in positive integers? We see
from (5) that

x = sqrt(1141y^{2} + 1)

Thus the question is: Is 1141 y + 1 ever a perfect square? This may be checked
experimentally. It turns out that the answer is no for all positive y less than
1 million. In view of the previous example, perhaps we should experiment
further. The answer is still no for all y less than 1 trillion (1 million
million, or 10^{12}). We go overboard and check all y up to 1 trillion trillion
(10^{24}). Again the answer is, no. No one in his right mind would really
believe that there could be a positive y such that

x = sqrt(1141y^{2} + 1)

is an integer if there is no such y less than 1 trillion trillion. But there is.
In fact there are infinitely many of them, the smallest among them having
26 digits.