Aspects of Numeracy in Von Neuman Computing Systems: Part One
- Aspects of Numeracy in Von Neuman Computing Systems
Metaphors and the things themselves.
Natural language users, human speakers and listeners and readers/writers, by the time they reach late teenage typically are well endowed cognitively to utilize knowledge from both their spatial world (the world that they see) and from the numeric world. Numeracy can be taken to mean a number of things and so here I will leave it undefined so as to promote the widest audience to the ideas of this article.
Numeracy is a different cognitive thing than is natural language. While numeracy deals with concepts, being a means of representing numerical concepts with symbols and processes (processes which may be unnamed, but appropriately activated)), natural language is a symbolic form of external conveyance of symbolized concepts. The 'mechanism' we use to transform concepts such as experiencing a green colour (so called qualia) into natural language is (apparently) the self same 'mechanism' we use to transform numeric concepts into natural language. It would seem that numeric concepts, mathematical concepts, are different in some non-trivial way from concepts like colour, acoustic pitch and such. It is because of our cognitive understanding of things like magnitude, difference and sameness, and count, that we can apply these concepts as descriptions to other concepts. Example: 6 as a number value, foot as a measure of linear distance and we can say "Joe is 6 feet tall." In other words, the tallness attribute of Joe is 6 and the unit for that measure is foot (6 foots). Back in the days of the pyramids people used pebbles (calculi) and counting-sticks, where each pebble or stick stood for or represented one unit of count.
Joe had uh uh goats in his flock. The uh uh was the collection of sticks or pebbles he had in his hand. Natural language allows us to transfer those pebbles or counting sticks into organized noises (speech), and we can therefore replace those objects with sound.
Well so what. Computers were invented to make calculations, like where will the cannonball land shot out of that cannon over there? That means simple iteration and perhaps just integer arithmetic. (lots of people would want to use decimal or floating point.) The number values involved, really just a way of processing bit strings, are nothing more than bit strings to the computer. Numeric values per se are not in the computer's ken ,which is very restricted. The ken of a computer really amounts to is bit string a identical to bit string b? That's it, that is the total intrinsic logic capability of the typical Von Neuman computing system (VNC). There is no such thing as apprehension of 'magnitude' in a VNC. There is no such thing as apprehension of 'experiencing a green colour' (so called qualia). So. Putting an aspect of Numeracy into a VNC is not trivial or obvious.
In this part one of this topic discussion I'll say that the first step towards putting any aspect of Numeracy into a VNC means being able to relate 'mathematical' symbols, such as the numeral characters (0 through 9), the operator signs, like + - multiply / (divide) sigma integral derivative quadrature ad naseum, to 'mathematical' concepts. In the next part, part two, we'll see how MathML3 and OWL ontologies permit a complex mapping to be defined which relates 'mathematical' symbols (via MathML3) with 'mathematical' concepts (via the terms in math ontologies). We'll look at NASA JPL SWEET2 ontologies. We will also look at XML Link and Pointer; and see why it is necessary to use XTM (XML Topic Maps) instead, to represent the complex of links/associations. In part three we will examine how 'learning machines' (programs) are used to map multiple instances of 'mathematical' symbols into one or more 'mathematical' concepts, such as quadrature. Use of this capability to perform "mathematical 'spell-checking' " is to be described there. Also we will look at the process of "by inspection"; a phrase my first year maths instructor insisted on using just before he lost most of the class in a sea of math symbols covering acreage of blackboard.