[FOM] 486:Naturalness Issues
david.roberts at adelaide.edu.au
Thu Mar 15 19:59:54 EDT 2012
On 15 March 2012 01:20, Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> FEEDBACK FROM FOM SUBSCRIBERS IS REQUESTED
> *UNIFORM TRANSFORMATIONS*
> We say that T:Q* into Q* is a UNIFORM TRANSFORMATION (with respect to ~) if
> and only if for all x in Q*, (x,Tx) ~ (Tx,TTx).
> Note that Z+up:Q* into Q* is a uniform transformation. We can show that all
> uniform transformations are very much like Z+up.
I like this abstraction very much. Z+up, while perhaps nice in hindsight,
and certainly very concrete, always looked to me to be a tailor-made function
in order to arrive at the theorems. I have two questions
1) Are we stuck with the definition of ~ as given, or can that be altered
2) What is the definition of (-,-)? I guessed it was concatenation of strings,
but someone pointed out to me that this would mean x order equivalent
to Tx, which is not true for Z+up. Can you please clarify? (consider, for
example, Z+up(3/2,1) = (3/2,2))
I have asked at MathOverflow  if people can come up with examples
of uniform transformations, and if they can see a way to arrive at
Z+up from some natural combinatorial problem.
I can think of some nice ways to arrive at the condition (x,Tx) ~ (Tx,TTx)
in a very natural way (given some relation ~), but it depends on the definition
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