# FOM: Neo-Fregean reverse mathematics

James Robert Brown jrbrown at chass.utoronto.ca
Sat Mar 24 14:27:31 EST 2001

```In his post on contextual definitions, Alasdair Urquhart make two key points:

1. So-called Hume's Principle is being treated as a contextual definition;
but this is wrong, since it implies a theorem of infinity
2. Contextual definitions in general are nonsense; the only legitimate
definition is a stipulation.

As to the first, I don't see the objection.  Hilbert took his axioms in the
Foundations of Geometry to be contextual definitions of the terms
involved.  He put it as follows: "...each axiom contributes something to
the definition, and therefore each new axiom alters the concept.  “Point”
is always something different in Euclidean, non-Euclidean, Archimedean, and
no-Archimedean geometry respectively." (Hilbert in Frege, On Foundations of
Geometry and Formal Theories of Arithmetic, 1971, p 13)  However, this set
of axioms has consequences galore.  So it can be no objection that Hume's
principle (if it is indeed a contextual definition) has consequences (such
as a theorem of infinity).  There may be problems with saying Hume's
principle is a contextual definition, but it is not a further objection
that it implies various theorems.

There is, however, (and this supports Urquhart's second point) a powerful
objection to contextual definitions.  It's a point made by Frege.  If
"line" is, for example, defined contextually by the parallel postulate,
then "line" does not mean the same thing in the the negation of that
postulate.  And this will undermine Hilbert's attempt to show the
independence of that postulate.

The whole issue is exceedingly tricky, since there are plausible ground for
saying that key terms of many theories (especially scientific terms) are
indeed defined by their use in the theory.  This doctrine in turn leads to
the infamous incommensurability arguments.  After all, if "mass" is defined
in use, then Kuhn and Feyerabend are right: "mass" does not mean the same
thing in Newtonian and in relativistic physics.

It's rare for working scientists to take an interest in this issue, but
some FOMers might find the following passage notable.  It's a powerful
statement of the rightness of contextual definitions and it comes from
Gravitation, one of the most influential recent physics texts.

"... that view is out of date which used to say “define your terms before
you proceed”.  All the laws and theories of physics ... have this deep and
subtle character, that they both define the concepts they use ... and make
statements about these concepts.  Contrariwise, the absence of some body of
theory, law, and principle deprives one of the means properly to define or
even use concepts.  Any forward step in human knowledge is truly creative
in this sense: that theory, concept, law, and method of measurement —
forever inseparable — are born into the world in union." (Misner, Thorne,
and Wheeler 1973, 71)

Jim B.

*********************************************************
James Robert Brown
Department of Philosophy
University of Toronto
Toronto    M5S 1A1