I think the question about the "threshold of mathematics" is something that
periodically everybody in the field endeavors to answer.
JGC replays to Robert Tragesser:
"In my opinion, your "when" seems first to beg the question of "what is"".
It is right. Then JGC writes:
"For scholars and laymen alike it is not
philosophy but _active play_ in chess
itself that alone can answer the basic
question: What is Chess? --"
Yes, but what about the ancient history of chess, the links between chess
and other board-games, the ancient relationship between chess, divining and
computation? Is it sufficient 'active play' to answer these questions?
Anyway RT looked for a "rule of thumb", and I think possible to say
something about his question searching in a field which had not to be
completely stranger to History of Mathematics: children cognitive psychology
as developed by Piaget and his followers.
Reading Piaget's books about the development of mathematical concepts, I
found some 'qualitative jumps' which could be the ontogenetic version of the
philogenetic breakthroughs we could consider as characterizations of the
above threshold.
Maybe the most interesting is the passage from the 'empirical' verification
of mathematical properties to their recognition as 'absolutely certain'. For
example the property that the total number of two sets of items does not
change if we move one item from the first to the second set, is first tried
dubiously by the child some times, but suddenly becomes something 'absolutely
certain', that must never more verified. Another example is the sudden
'light' by which a child understands that there is no limit to the sequence
of integers, after having painfully extended his range of numbers for
months.
These 'emergences' are, according to Piaget, neither empirical nor accepted
through external learning, but seem connected to the reflecting abstraction
from the 'action' and to the development of a general symbolic skill.
So that they are linked to the external world and to the socio-psychological
performances, without reducing mathematics simply to an empirical discovery
or to a personal invention.
The analysis of this "absolute certainty" shows that it is something prior
to the idea of 'proof', which seems to be its late formalization, and linked
to the idea of "symbolic representation" which appears something deeply
different from the earlier "iconic representation".
Could the emergence of "absolute certainty" and of "symbolic representation"
be the suitable "rule of thumb" for the "threshold of mathematics"?
Obviously to verify these features in archaeological findings is a problem I
do not try to answer.
Yours sincerely
Luigi Borzacchini