FOM: Neo-Fregean reverse mathematics

[James Robert Brown]

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 that view is out of date which used to say “define your terms before 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.







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James Robert Brown
Department of Philosophy
University of Toronto
Toronto    M5S 1A1
Canada
Phones: office (416) 978-1727,  home (519) 439-2889
Email:  jrbrown at chass.utoronto.ca
Home page: http://www.chass.utoronto.ca/~jrbrown/index.htm
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