FOM: Sazonov's "What is the standard model for PA ?" and Brian Rotman postmodern semiotics of mathematics

[Patrick Peccatte]

fom digest:
Position: Software engineer and philosopher
Institution: Software private company and Paris 7 University
Research interest: History and philosophy of sciences and mathematics
(specially quasi-empiricism, experimental mathematics, ontology)
More information: http://peccatte.rever.fr
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Try to compare the ideas expressed by Vladimir Sazonov many times on fom
list and the Brian Rotman postmodern semiotics of mathematics.
Following, I quote from the Rotman's book "The Ghost in Turing's
Machine. Taking God Out of Mathematics and Putting the Body Back In",
Stanford University Press, 1993:

"The starting point for any examination of arithmetic and number is the
activity of counting, that is, the production of signs through the
process of repeatedly concatenating instances of some arbitrary but
fixed signifier. Thus the primary object of arithmetic study must always
be a sequence of iterates such as
0, 00, 000, 0000, 00000, 000000, 0000000, ...
or
1, 11, 111, 1111, 11111, 111111, 1111111, ...
or, more formally, through the introduction of the successor function S,
the sequence
0, S(0), S(S(0)), S(S(S(0))), S(S(S(S(0)))), ...
where the sign "..." is to be interpreted as the instruction to the
Agent to repeat iterating, insofar as this is possible within the
constraints imposed by the condition of realizability. We shall think of
the progression of iterates as beeing bounded by or as tending to a
limit denoted by $, and shall write i < $ for any iterate i. No
assumptions will be made about $ other than the fact it exists as the
upper bound to the iterates. Of course, $ is not itself an iterate.
Using the ideogram introduced earlier we shall write our primary object
of study as
0, 1, 2, 3, ..., $
and call this sequence the progression of realizable iterates, or simply
the $-sequence."
op. cit. p 161

"... the production of integers via repeated addition of "1" becomes
increasingly difficult, so that one reaches a region where their further
production through such repetition becomes impossible".
op. cit. p 57

"Indeed, as the contemporary manifestation of logistica, computer
science signals the instatement of the slave - the one-who-counts - onto
the mathematical scene. An essential part of this instatement is the
recognition that any act of counting/calculating, whether by fingers,
abacus beads, written marks, or electronic pulses, requires energy,
space, and time for its realization. As such, the emergence of computer
science represents a large and growing de facto challenge to the subject
matter and internal organization of traditional, infinistically
conceived mathematics fostered by arithmetica."
op. cit. p 149

"... the dream of endless counting is a dream..."
op. cit. p 158

Regards
Patrick Peccatte