[FOM] The Gold Standard

[Harvey Friedman]

On 2/19/06 10:43 AM, "Arnon Avron" <aa at tau.ac.il> wrote:

> On Sat, Feb 18, 2006 at 02:01:50AM -0500, Harvey Friedman wrote:
> 
>> ZFC + large large cardinals. Justification:
>> "inconsistencies should be easy
>> and not take long to find, like Kunen's for ZFC + j:V into V, and this
>> hasn't happened yet over a 'long' period of time", and "go for it!"
> 
> Great. So from now on I recommend to accept and use NF. Justification:
> "inconsistencies should be easy
> and not take long to find, like Kunen's for ZFC + j:V into V, and this
> hasn't happened yet over a 'long' period of time", and "go for it!"
> 
Just yet another argument against just one of the natural stopping places in
the natural hierarchy (the level which essentially represents not stopping).

And, as usual, this argument has a perfectly respectable counterargument.

Also the perfectly respectable counterargument below has a perfectly
respectable countercounterarugument, and so forth.

For example:

NF has no reasonable correspondence with mathematical activity. Furthermore,
no one has given any reasonable story about why it is, or might be,
consistent. The standards for "reasonable" in the previous sentence are very
very low. In addition, the system is not known to be approachable from below
in any steadily coherent way. The large cardinals are stated uniformly with
elementary embeddings, where the domain and image get closer and closer to
each other. Etcetera.

And, no doubt, you can make up a countercounterargument, and I can make
another countercountercounterargument.

You illustrate my point very well. I wish to thank you.

Incidentally, I would win this back and forth argument easily, if we were to
evaluate it by the community of interested observors in mathematics and
philosophy. 

And you can obviously make the standard retort: science does not run on
votes. 

And then I can talk about the significance of the votes in this case, and
the reasons for these votes.

And we can continue to quite some time.

So my point is: this is an old fashioned and once productive way to do
f.o.m., but not any more - at least not generally.

In order for this kind of old fashioned f.o.m. to be productive, there has
to be something creative in it. Generally, this amounts to presenting a new
or modified view, together with a formalization of that, and then a
calculation as to where it fits in the apparently canonical hierarchy of
strengths. 

We should concentrate on the apparently canonical hierarchy of levels, fit
in new ideas for new levels, match with mathematical practice, use various
levels productively in new ways, etcetera.

Incidentally, I cannot tell whether you or Weaver accept the Gold Standard,
or wish to replace it with something else.

There is a critical mass of impredicative arguments being made and published
in normal and core mathematics that cannot be removed.

Harvey Friedman