[FOM] The Gold Standard

[Harvey Friedman]

On 2/17/06 12:47 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> What I'd really prefer is that you interpret it as "impredicative
> mathematics has no clear philosophical basis".
> 
I wrote:
 
>> We now agree that any talk of banning is silly.
> 
I think I have previously asked whether you accept the current GOLD STANDARD
for mathematical proof, apparently used by all reputable Mathematics
Journals, of ZFC?

If impredicative mathematics has "no clear philosophical basis" then why
should ZFC be the GOLD STANDARD? Is this intellectually honest or advisable?
If not, what do you recommend be done about it?

Is it in any sense important that a mathematical development have a "clear
philosophical basis? Or is this perhaps entirely unimportant? Is it even
desirable?

What do you make of all of this unabashed highly impredicative work, done on
a daily basis by fancy core mathematicians, inside countable algebras, that
I mentioned, in and around one of my two papers in Simpson's Reverse
Mathematics volume of papers?

************************

In any case, your claim that

"impredicative mathematics has no clear philosophical basis" whereas
"predicative mathematics has a clear philosophical basis"

is not defensible, and represents a misuse of the word "philosophical".

Consider the following levels indicated roughly as follows: (there is
nothing inclusive about them, and nothing unique about the justifications).

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!"

ZFC + medium large cardinals. Justification: coherent consequences, inner
models, and "inconsistencies should be trivial and not take long to find,
like Kunen's for ZFC + j:V into V, and this hasn't happened over a 'long'
period of time".

ZFC + small large cardinals. Justification: reflection principles.

ZFC. GOLD STANDARD (rightly or wrongly). Justification: extrapolation from
finite set theory.

ZC. Justification: weaker extrapolation from finite set theory, tempered by
what appears in mathematical practice without coding (direct translation
into set theory without coding).

Z_2. Justification: realist view of the real number system.

Predicativity. Justification: N is absolute, subsets of N form only by
mental constructions, and then analyze mental constructions of subsets of N.

PA. Justification: N is absolute and inductively created, and all "real"
objects are finite.

ACA0. Extension of PA analogous to NBG over ZF. General extension process
akin to "classes from sets". Just a useful adaptation of PA.

PRA. Justification: Sequential processes on incompleted totality.

EFA. Believed (conjectured) to be exactly what you need to support finite
core mathematics.

Ultrafinitist systems. Justification: Arbitrary natural numbers has no clear
philosophical basis. The relevant systems are not yet standardized. There
are emerging several levels of ultrafinitism.

****************************

One can easily adopt one of these and complain in various ways about the
others. You can trivially point to one and say that the higher ones "have no
philosophical basis", and the lower ones "are too weak to support green and
purple mathematics".

They all have good stories, and they all have good complaints against them.

In fact, the most cumbersome of all of the above is predicativity.

Harvey Friedman