FOM: real numbers and the real world

Robert Black Robert.Black at
Tue Nov 14 04:32:04 EST 2000

Vladimir Sazonov:

>> (Contrast here the
>> nominalism of Hartry Field, for example, for whom although mathematical
>> objects are fictions, mathematical theories can be applied to the real
>> world in virtue of isomorphisms between the fictional realm and the real
>> realm.)
>He really says that? Is it serious? Say, abstract real numbers
>or set-theoretic cardinals can be isomorphically embedded in
>the real world? I am asking even independently on the assumptions
>on (in)finity of the universe.

Sincy Hartry, I think, follows this list, I'm a bit wary about presenting
his views. Also I'm being a bit loose because an isomorphism would be an
abstract object. But with that caveat, Field does think that the reals (not
set theoretical cardinals, which to my knowledge have no applications in
physics) can be isomorphically embedded in the physical world. That's
straightforward, since he's a realist about physical space (or spacetime)
and its points and regions. The classical example then would be synthetic
Euclidean geometry (with primitives like 'point', 'line', 'incident',
'congruent', regarded as physical predicates) and analytic geometry, whose
objects are abstract (triples of real numbers etc). It is, after all, well
known that you can move from a system of axioms for a synthetic geometry to
its coordinatization and back again, and that algebraic properties of the
coordinatization reflect features of the synthetic axiomatization and
vice-versa. That's the sense in which Eudoxus' theory of proportions is
implicitly a theory of the reals.


Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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