[FOM] 486:Naturalness Issues

[David Roberts]

Dear Harvey,

On 15 March 2012 01:20, Harvey Friedman <friedman at math.ohio-state.edu> wrote:

> FEEDBACK FROM FOM SUBSCRIBERS IS REQUESTED
>

Will do!

> *UNIFORM TRANSFORMATIONS*
(snip)
> 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
as well?

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 [1] 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
of (-,-).

Best regards,

David Roberts

[1] http://mathoverflow.net/questions/91238/