[FOM] 207:On foundations of special relativistic kinematics 2

[Jeffrey Ketland]

Harvey Friedman wrote:

>This is implemented as follows. A special sort is reserved for all
>mathematical entities, and special constant, relation, and function symbols
>are reserved for purely mathematical notions involving only mathematical
>entities. Other sorts are reserved for physical entities. All axioms are in
>an appropriate many sorted first order predicate calculus.

Suppose for simplicity it's 2-sorted:
  (i) variables X, X', X'', etc., for mathematical entities (reals, sets,
functions, etc.)
  (ii) variables e, e', e'', e''', etc., for non-mathematical ones (e.g.,
events)

(You'll need "mixed" relation symbols to express relations between reals and
events: e.g., "the co-ordinates of event e relative to co-ordinate frame K
are (X, X', X'', X''')".)

This raises a huge number of questions that philosophers of mathematics have
intensively studied for around 30 years, since Putnam's _Philosophy of
Logic_ (1971) and Field's _Science Without Numbers_ (1980).

Does it bother you that our scientific theory will contain quantification
over mathematical entities? Won't this contradict nominalism? (The view that
only physical things---events, etc.---exist).
Do you think that this quantification over mathematical entities is somehow
eliminable from physical theory?
Would the non-eliminability of quantification wrt mathematical variables
contradict nominalism?
Is it possible that richer and richer mathematical assumptions (e.g., set
existence assumptions) could yield new consequences in the purely physical
domain? (Just as richer set existence assumptions yield new arithmetic
consequences).

There's a large literature on this sort of question. The best single volume
summary is Burgess & Rosen 1997, _A Subject with No Object: Strategies
Towards Nominalist Interpretation of Mathematics_.

Just wondering what you think.

--- Jeff
~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
Faculty of Philosophy
Sidgwick Avenue
CAMBRIDGE CB3 9DA
U.K.
JJK32 at cam.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~