Thanks Dima. To my mind, the thread has shown that some
questions have to be settled first. I am trying to gather
the relevant comments, ignoring completely the genuine
real field (which is one motivation but not the purpose
of the ticket). This is a personal interpretation of what
happened. Feel free to correct me.
One proposal consisted to discard the "field" terminology
(J. Cremona, S. Lelievre)
"Real Floating-Point Numbers with x bits of precision"
or
"Real Floats with x bits of precision"
These proposals describe somehow accurately the set but
completely discard the importance of the underlying algebraic
structure. Here was proposed "pseudo-field" and "quasi-field"
(M. Jung). I think that both of these names are bad because
"pseudo" and "quasi" are used in many mathematical concepts
but here would refer to non standard terminology. I proposed
"numerical field" which is also an invented concept
but has the advantage to fit well with the "is_exact()"
method already present on some parents.
So question number 1:
1. Should we drop any reference to the algebraic structure
for numerical approximations?
(J. Cremona, S. Lelievre)
2. Should we have a common adjective for all approximations?
Which one "pseudo", "quasi", "numerical", "nonexact", ...?
Should we differentiate various kind of approximations (eg
floating-point vs balls/intervals, various flavors of p-adics)?
Parallel to this question is the "categorical" version
1'. Should RealField simply be a member of the Sets() category?
2'. Should we develop some categorical machinery to differentiate
exact fields from approximate fields? Which one?
Best
Vincent
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