What is a Multiplication Table?
1) Introduction
The Sumerians made a cultural revolution with their introduction of the
positional arithmetic. What was in older times a heap of symbols which
needed active participation of the memory, is now a string of tokens in a
children’s game with reading off values that are on fixed positions
relative to each other. Position means: more, as one proceeds left from
right. The idea of a step is to be used as one progresses from one position
to the next. This kind of step is different to the step one makes among the
symbols that can be placed. Within the same place, the possible symbols are
gradated, too: but this is a different gradation to that which a step in
the gradation along the positions yields. We have a two-dimensional
counting with the positional arithmetic.
The positional algorithm is very near in its requirement, on the
capabilities of abstraction of a human child, to that of a multiplication
table. The table is learnt, in Europe, at the age of 7, mostly in the 2nd
class elementary. As grown-ups, we use it almost incessantly in our
everyday life, as a background etalon of a general sense of how many or how
much something is or can get, etc. The table gives our assumptions a
skeleton, a framework, a map of correct distances, a landscape of graded
extents.
2) The scientific relevance of a multiplication table
Our forefathers have imagined the perfect human to be able to fight, sing,
dance, ride, tell verses, hunt and calculate well, next to being
rhetorically elegant. They would not have accepted a master of the sword to
compete with a maestro of the mandoline, a maitre des dances, etc. “No
specialists!” is an implicit rule of the idle class, be they Brahmin or
other kinds of elite.
Mastering well the arts of the multiplication table is then not a
philosophical advantage but one of the craftsmen, the actualisers, the
realisers; not that of the thinkers. A technical tool, which professionals
use, which gives an advantage to those professionals who use them well,
would not be considered then a major discovery of a scientific nature.
Well, they have developed a better mousetrap, after all their efforts: but
this can hardly be a scientific progress. If there is anything commendable
as furthering the progress of science, human goals and eternal well-being,
then it would have to be something to do with the idea or principle of
catching mice, not an improvement on a well-worn, much-in-use basic
principle.
Inasmuch the tautomat is a large multiplication table, it has, strictly
spoken, no novelty value. It has always been so and it will always be so,
that in reordering a set from being ordered on *alpha *into an order that
is ordered on *beta*, the following elements will be joined into one and
the same corpus: {…}. The technical working principles have been brought
to the notice of the public by registering them in the OEIS.
The basic idea can’t be hijacked and proudly privatised, neither: *a+b=c*
is known since time immemorial, and the ability of ranking and ordering
sensually accessible units culturally predates even *{a,b,c,+,=}*, the
symbols abstracted from sensually accessible units. There is nothing
basically new caused by the imposition of an improved multiplication table
as the tool of state of the art of counting, in the sense of the philosophy
of knowledge.
3) The sociological relevance of the multiplication table
It is a basic human understanding that anyone may utter grammatically
correct, actually true sentences. We encourage little children to observe
clearly and to express themselves while they correctly apply the rules of
the language. They are rewarded if they can truthfully name some relations
between objects in their surroundings. By naming things by their names,
children can get into an unwelcome role by disrespecting taboos. A
multiplication table talks even less than a well-groomed child, in fact, it
does not talk at all, so it will not run the risk of being an unwelcome
intruder. The voice of reason talks anyway in a whisper.
The contents of the tautomat’s tables cannot cause uproar or adversity. One
has to come clear about the facts: and the facts are presented for anyone
to see, understand and put to use, but until one is taught to use them
routinely, one has to seek them out and actively want to get familiar with
them.
As long as there was no multiplication table in use, many priests,
clairvoyants, druids, shamans and gurus were employed to divine the
inexplicable, how from few there appears a many, and exactly that many
within the very many. The sacred algorithms were not yet desecrated by the
use of the general population. There is not much interest among the savants
to see their specific advantage – their unique sales proposal – drifting
away, being diluted by the use of simple tools. End of the monopoly to
calculate large numbers correctly.
4) Thought patterns simplified by the multiplication table
That region of imagination which was before the invention of the
multiplication table, monopolised by the savants, was not accessible in a
rational fashion for the general public. The multiplication table has
unveiled an order in a part-world of thinking, heretofore believed to be
inhabited by demons and being un-understandable; it created a structure
within the confusing multitudes that were always there, but had been
believed to be chaotic.
Wittgenstein knew, why he advised against colourful speculation about that
what is not the case. He was content to have been the midwife of a portion
of the world, that portion which is logical, where all sentences are true
and logical rules apply. Not having had computers at his disposal, it would
have been folly of him to consider what mechanical philosophers could say
about what is not the case, compared to, and contrasted to, that what is
the case, because no human philosopher can discuss the relations among more
than a very few objects, and the relevant values appear in their full
discordance in the orders of magnitude of EE+035, EE+094, EE+232.
Using the tools invented by the generations of our grandparents and
parents, it is possible to throw some light into other parts of the cave,
generate shadows for our own entertainment and education, and also to have
some reasonable, meaningful, even informative conversation about that what
is certainly not the case.
We have created a mirroring surface to that what is the case. It is a tool,
nothing more and nothing less. It is now possible to discuss whether
alternative *a* will be the case or rather alternative *b*, resp. one of
the alternatives *{b,c,d,e,f,etc.}*. If the terms “divisor”, “disjunctor”,
“contrastor” would not be in use yet, they could be used as names for some
of the algorithms; maybe “conflictor” would be acceptable to the users. The
tables of the tautomat do not show multiples, but they show ranks, places,
conflicts, successions, communalities, possibilities for compromises, and
ways of transmodalism and of discontinuation. The substance of the
conflictor tables, of which a tautomat consists, is not different to the
substance of multiplication tables. Their impact on social stratifications
could also turn out to be comparable.
We are now able to speak about the “otherwise”. Previously, the “.false.”
logical statement has had no 67 shades of being false. With the help of the
newly introduced tables, the otherwise is not only the negation of that
what is the case, but has many gradations, and may change its rhetorical
position with that what is the case. The *advocatus diaboli* has been
invited to the discussion about the basic truths. There are, after all,
rather many more alternatives to do wrong than to act virtuously. The
dialectic dialog between the opposing forces, represented by Mr. Bond and
Dr. No, or other versions of yin and yang, can now be numerically accessed,
thanks to the utilisation of a collection of databases. In an
epistemological sense, tautologies do not contribute anything to culture,
insight or progress, regardless whether they are simple or elaborate, very
practical, practical or useless; just like a multiplication table, which
can be useful or irrelevant, in dependence of the task.
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