There have been four responses (unless I've regretfully missed
some).
[1] Gordon Fisher.
Mr. Fisher seems to approve of the notion that it is mathematics
if a (some? all?) mathematician recognizes it as mathematics. Then he
identifies the presence of mathematics with the presence of numbers and
their algebraical-logical treatment. (There are other layers to his
response.)
Since I was asking for a rule of thumb, it would be out of
place to ask "which mathematician", etc. I am reminded somewhat of
the philosopher Donald Davidson's criterion for recognizing a phenomenon
as the manifestation of a language -- only upon the "successful"
translation of (interpretation of) the phenomena in our language.
Being more elaborate in either case is beset with great difficulties.
But I suspect that Gordon Fisher's rule of thumb is that
typically employed by historians of mathematics.
[2] Luigi Borzacchini Walter Felscher.
Both Mr. Borzacchini and Mr. Felscher address the problem of the
threshold of mathematics through the matter of what is distinguishing
about mathematical thought independently of whether or not it is
expressed. Mr.Borzacchini is explicit; Mr. Felscher, implicit, but
I'll take the risk of ascribing the former's view to the latter.
One crosses the threshold of mathematics when one has insight or
understanding conferring something like necessity (objective rather than
subjective certainty), and once and for allness, transcending
(empirical) induction. The moment the child moves on from by trial and
error looking for a greatest number to _seeing that there can't be a
greatest number_ (aside: I once heard ascribed to L. Carroll -- but
never could find -- the following fable about how infinity came into the
world -- there once was a greatest number, until that bad child,
Pandora's brother, erased it). Likewise I think that Mr. Felscher's
examples from commercial arithmetic also have this character; that the
clerks, etc., did not rely on their calculational tricks because of a
long experience of the books balancing (or whatever), but rather
because they could _clearly see that_ (with an insight transcending the
merely empirical) that the tricks of calculation did the right thing,
even if they could not have verbally explained why the tricks did the
right thing.
Let me give a contrived by perhaps not unworthy example of the
distinction I have in mind. Suppose I overhear two learned gentlemen
discussing withdrawing balls from a bag full of black balls and white
balls. One of the gentlemen mentions that he has discovered that it
suffices to take three balls from such a bag to be certain of having two
that are the same color. I think that this is an amazing, magical
fact, so I set out to check it. In fact, I spend a lifetime with
many, many sacks of such balls, and have verified it for tens of
thousands, and yet I worry whether some future bag may yield a
counterexample, though I am as confident as its correctness as I am of
anything. Distinguish this from a moment of thought in which I saw
that it is impossible for this "fact" to fail, e.g., I imaginative
construct a worse case scenario, I have drawn one black and one white
ball from the bag. But the next ball (the terms of the statement of the
fact) must be either black or white; in either case two of the three
balls will then be the same color. Mathematics!
[3] Julio Cabillon.
He brings up quite a number of matters. He seems to approve of
the thought that we have mathematics where there are mathematical forms
in nature or out (space, symmetries manifest in living organisms,
etc.). [See by the way Darwin's analysis in _Origin of Species_ of how
the honeybee in building its comb hits upon optimal spherepacking without
actually doing mathematics.]
Now, insofar as mathematical forms are exactly what inspires
mathematical thought (or so we might say), the importance of noticing
the ubiquity of mathematical forms is this: mathematical thought can be
inspired anywhere.
I think that Mr.Cabillon brings home to me that what concerns me
is the rule of thumb by which we discern the presence of mathematical
thought, as addressed in [2] above.
My questions go all too quickly to the philosophical -- what is
there about the mathematical forms present in nature (say) that makes
them knowable by means of mathematical thought (thought based on insight
into necessities, in the spirit of [2] above).
Dr. Cabillon's point about multicultural concerns:
In any case, we can explain cultural differences in
"mathematics" in terms of a number of parameters:
[a] what were the preferred or interesting mathematical forms
and what are the important questions about them?
[b] How explicit is one expected to be about justifying (as for
example, demonstrating) a mathematical fact? Correlatively: How
explicit could one be in that culture.
[c] (Suggested by a distinction made by Michael Atiyah in a
lecture): how _legalistic_ should one's presentation of mathematics be?
(And correlatively, how legalistic can it be in a considered culture?)
[d] What are the significant mathematical forms -- acknowledged
in the culture -- on which e.g., political decisions might be
acceptably made or which might be regarded as significantly representing
nature/the world, etc.?
But I think I would hold that the sort of compelling insight
pointed up in [2] above is a cultural invariant.
Finally: Here's a kind of rule of thumb I like (illustrated by an
instance): I don't think that ranking hardness of materials by, say,
numbers 1-10 is mathematical; it would become mathematical when the
ranking was such that, e.g., ratios among those numbers had nontrivial
significance for appreciating the materials. Although I might find
myself compelled to say that I accomplish the first ranking only by
virtue of the mathematical thought that e.g. 7 is necessarily greater
than 3 (given what I have in mind by the numerals and greater than), so
that I am here in the presence of mathematical thought, I find myself
wanting to push a little further requiring (if only mildly) less trivial
mathematical facts.
greetings from,
Babbling West(running)brook, Connecticut usa