FOM: Reply to Kanovei on counting
kanovei at wminf2.math.uni-wuppertal.de
Sat Oct 10 07:57:05 EDT 1998
There is a big mess in NT's arguments on arithmetic.
Indeed there are two theses:
the notion of the number of elements in a well-defined
(finite) collection C of physical objects is physically
(exactly, by QM reasons) not sound, or, practically,
it is not sound for very big collections.
We cannot say that at the moment C has some number of
elements because we do not have full information about C.
For instance take as C the "set" of all cardinals in the
Roman church. We know, in principle, there are say 9 of them
(whom we could know by names). But since they are, generally elderly
people one of them could decease (god forbids !) a minute ago,
so for safety reasons is is better not to bet on 9 in this case
and rather write something like
(N) #xF(x) = n* iff there are exactly n F's
as the ultimate knowledge on the subject.
My further comment.
Of course, there is a difference between
(a) the number of cardinals at 2-53 PM ET October 10, 1998
(b) the information available to NT or yours truly relevant to (a),
see e.g. Kant on this difference.
But we are educated people
looking in the next millennium (x fingers) so we would
claim that a possible miserability of (b) does not
question THE VERY EXISTENCE of (a).
After this (true or false) has been established, we may go ahead
and see that
gathering of 9 cardinals and 3 pagans results in one and
the same amount of people which does not depend whom we start
counting with, cardinals or pagans.
This number we will call 12.
This observation has led to the commutativity of the number
addition. We can go ahead with the rest of PA axioms.
Now (the key point !) the number 12 above is REALLY 12, not something
like 12.0001 or 12 with probability 0.999 and 13 with probability 0.001.
And this striking physical fact surprisingly does not depend on what
a Frege or whoever may have to say about it or on which next theory
a Professor Sazonov can suggest where there is no numbers bigger than 11
or some unspecified n.
This phenomenon 1) has been most likely known since ancient
Egipt, 2) I call STABILITY OF COUNTING (perhaps there has been
a better name), and 3) was EXACTLY the reason for someone
(Kant ? sorry, have forgotten) to declare the arithmetic to
be divine in opposite to the man-created rest of mathematics.
Now we return to K's thesis at the beginning, which poses a
problem to the whole setup.
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