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) x² -1141y² = 1
We might ask, does equation (5) have any solution in positive integers? We see from (5) that
x = sqrt(1141y² + 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 10¹²). We go overboard and check all y up to 1 trillion trillion (10²⁴). 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(1141y² + 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.