[FOM] BANNING impredicative mathematics

[Harvey Friedman]

Weaver has still not responded to my question about ACA0 and predicativity.
My suggestion is that since it seems to be the case that

*everything in the "core" mathematics that Weaver is focusing on, that lies
within predicativity also lies within ACA0"

why not use the far far far simpler model of ACA0 than elaborate systems of
predicativity? Wouldn't ACA0 "fit mathematical practice better" than
predicativity? After all, if you are going to overshoot mathematical
practice by such a large margin - claiming that one is staying especially in
tune with mathematical practice - then why not overshoot it even more with
impredicative definitions? This would have the added advantage of being even
closer, because one does not have exceptions like

**the theory of closed sets of reals**

I had earlier discussed this work, in connection with trigonometric series,
that led to Cantor's development of set theory.

Consider 

i. Every uncountable closed set of real numbers has a perfect subset.
ii. Every uncountable closed set of real numbers is the union of a perfect
subset and a countable set.

We know that i is provably equivalent to ATR0 and ii is provably equivalent
to Pi11-CA0, all over RCA0.

How about this:

iii. Given two countable sets of reals, there is a continuous one-one
function from the first into the second, or from the second into the first.

We know that iii is provably equivalent to ATR0 over RCA0.

By the way, i,ii are still taught. Should we BAN the teaching of i and ii?
(You have repudiated a desire to BAN certain mathematical research, but have
not addressed the issues of BANNING in the classroom and thesis advising).

Or should be DISCOURAGE the teaching of i and ii? Perhaps we should
DISCOURAGE the teaching of ii but ENCOURAGE the teaching of i?

How about in a history of mathematics class? Do you have any recommendations
about that?

On 2/15/06 2:49 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> Harvey Friedman implied that I am out to "ban" all impredicative
> mathematics.  I forcefully denied this and insisted that nothing
> I have written suggests anything of the sort.

You state that impredicativity has no clear philosophical basis.

You wish to draw a distinction between this view and the suggestion that
impredicative mathematics should be banned.

In what sense do you support the continued development of mathematics that
you deem "has no clear philosophical basis?

I still regard you as promoting the idea that impredicativitive mathematics
should be banned.

However, perhaps you would prefer that I interpret this as

*impredicative mathematics should be condemned*

*impredicative mathematics is illegitimate mathematics*

Does the "fact" in your words, that impredicative mathematics "has no clear
philosophical basis" accompanied by any practical suggestions?

> Incredibly, Friedman neither apologized for misrepresenting
> me, nor quietly let the matter drop.  Instead, he posted a new
> message containing a farfetched argument which purports to show
> that I really am out to ban impredicative mathematics, despite
> the fact that I've never said this and indeed strongly denied
> it.  As if to shout me down, he has now begun writing BAN in
> all capitals.

Obviously you are free to adjust BAN to "has no clear philosophical basis".
Perhaps you want to strengthen this to "is philosophically incoherent", or
"mathematically illegitimate" or "nonsensical". You are obviously also free
to assert or deny that "people should not continue to teach or develop
mathematics that has no philosophical basis, or is philosophically
incoherent, or is mathematically illegitimate". Or do you not wish to make
any recommendation about that?

I earlier wrote

>> Do you want to BAN some of the work of that [unnamed] Fields
>> Medalist [the star of an impenetrable earlier anecdote] and his
>> colleagues?

Do you want to inform the relevant people that their work "has no clear
philosophical basis"? Or that it is "mathematically illegitimate" or is
"nonsensical"? That it shouldn't be taught to students? That they should not
supervise Ph.D.'s in it? That derogatory reviews should be written about
that work because it "has no clear philosophical basis?"

Or do you think that "having no clear philosophical basis" is not
derogatory? 

Do you want to DISCOURAGE these relevant people from pursuing this kind of
work?
> 
> Wait a minute, I think I *am* being BANNED!  After all, I want
> to think that my discussions of foundations will have some
> positive effect.  But now Friedman informs me that the effect
> is harmful.  The "practical effect" of what he says is to BAN
> me from debate!

I am not asserting that your presentations have on f.o.m. However, I don't
know just what positive effects you intend your presentations have on f.o.m.

We already know that predicativity is a limitation. We already know that a
large percentage of math fits in, or can be made to fit into predicativity.

Are you looking to make any recommendations for how to VALUE mathematics, or
what mathematics should be pursued?

> Enough farce.  What is my actual stand on impredicative mathematics?
> I agree with Paul Lorenzen (Differential and Integral, p. 37):
> 
> "Logically, there can be no objection to the erection of axiomatic
> theories (in this case axiomatic set theory).  Even if the axiom
> system of the theory is not known to be consistent, the deducing
> of theorems from the axioms is an unobjectionable preoccupation
> ... it is nobody's business here to permit or to forbid ... All
> anyone can do is to decide for himself what he wants to do."

So under this view, are you recommending anything?

I don't see any difference, from this pragmatic view, between the
development of mathematics formally in axiomatic set theory and the
development of mathematics formally in formal systems of predicativity. It
is just that the latter is well known to be weaker than the former.

Or are you merely trying to say that a very large portion of "normal" or
"core" mathematics can be done in predicative systems?

This is already well known. The known exceptional cases, which keep growing
in number, are of course of great interest for f.o.m.

On 2/15/06 4:36 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:
 
> There are strong philosophical reasons for adopting a
> predicativist foundational stance, which I have discussed
> in some detail in a number of messages I've posted on this
> list over the past several months.

There aren't any strong reasons.

Anybody can routinely take any level E, and another higher level E', write a
story that depicts something arguably around level E, and declare

1. E is "philosophically coherent".
2. E' is "not philosophically coherent".
3. Justify 2 by asserting that E' is not reducible to E.

This does not provide any justification whatsoever for assertions 1 and 2.
It also does not provide any "strong philosophical reason" for asserting
that E is "good" and E' is "bad".

All that you have is that E is E and E' is E', and E' is not reducible to E.
That is all. Nothing more. Nothing less.
 
> Much of what I've had
> to say involved criticism

CRITICISM. Any practical suggestions emanating from this CRITICISM? Banning,
discouraging, rejecting, negatively reviewing, ... ?

>of the idea that there objectively
> exists a unique well-defined metaphysical world of sets.

What basis for criticism do you have? Obviously, anybody can just assert

*I hereby criticize the idea that there is an objective such and such*

Is that what constitutes criticism? Merely a statement that one "hereby
criticizes"? 

>If
> one accepts such a notion then predicativist critiques have
> little force.  

I don't accept or reject statements like "there is an objective uniquely
defined whatever...". I do not accept any statement of philosophical
incoherence of impredicative definitions, which is much much weaker.

There is obviously no philosophical incoherence whatsoever here. That
doesn't address any issues of absolute truth, etcetera.

It just means that I see nothing incoherent about it.

>But one then has to deal with the classical
> set-theoretic paradoxes --- in particular, if this supposed
> universe is completely well-defined then presumably we should
> be able to talk about the set of all entities lying in it,

This does not follow. We have all come to see that the notion of "set" has
long been clarified to stay very clear of this naïve conception.

I agree with the sense of Martin Davis that there are breakthroughs to come
in our understanding of the paradoxes. This I try to work on.

> and that seems to lead directly to contradiction.  A related
> difficulty is that this kind of platonistic view on its face
> seems to justify unrestricted comprehension, again leading to
> paradox.  

We have long clarified set in order to avoid unrestricted comprehension.

>One can try to avoid these difficulties by conceiving
> of the set-theoretic universe as an incomplete entity which is
> in some sort of "perpetual, atemporal process of becoming"
> (Maddy).  The retort would then be that this notion is simply
> incoherent.

It is obviously not incoherent.
 
> The preceding arguments are indirect: if one accepts platonism
> then various difficulties arise, therefore doubt is supposed to
> be cast on the platonistic approach.

By the same token, we can cast doubt on all of science.

> It is also possible to
> directly criticize the belief that ...

It is perfectly possible to directly criticize any belief that ... , in the
sense that one can write

*I hereby criticize the belief that ...*

Not only is that easy to do (all you have to do is write down these words in
front as I did above), but this is also an essential component of academic
freedom. 

> Anyone who accepts a platonic foundational stance can comfortably
> dismiss the predicativist contention that sets must

MUST. What do you recommend for mathematicians who violate your imperative?
What consequences do violations have?

>be "built up
> from below".  

>After all, if sets are simply "there" then we have
> no particular reason to believe that they all must be reachable
> from below, definable in a non-circular manner, etc.

You are already making a case for one of my points. That there appears to be
nothing whatsoever incoherent about impredicative methods.

I think that you can now agree that impredicative methods are perfectly
"philosophical coherent". So then what is the disagreement all about?

> Harvey Friedman blandly asserts that "claims that predicativity
> has some special place in the robust hierarchy of logical strengths
> ranging from EFA through j:V into V are unjustified."  Can he give
> a compelling justification for any impredicative system that does
> not assume a platonistic conception of sets?

I am not in the business of making compelling justifications, period. I
don't know quite what you mean by "compelling". You might mean a

*justification so strong that it makes someone who rejects the thing being
justified 'cognitively crippled'*

Under this notion of "compelling", certainly even very weak fragments of
Peano Arithmetic do not have compelling justifications. In fact, under such
a definition, only some unusually extreme form of absolute finitism, perhaps
where the number of objects in the universe is at most 100, might be OK.

> In his previous message in this thread he claims that ZFC can be
> justified by "extrapolation from finite set theory".  That is a
> good example of an explanation that I would not consider compelling.

I did not use the word "compelling". I was trying to say that it seems to be
what is actually behind the great comfort level that most mathematicians
have with ZFC. 

> He also writes "There is also a clear philosophical basis for the
> impredicative comprehension axiom scheme" but does not tell us
> what it is.

Clear, but it does not meet the standard of compelling that I stated above.
You have already given it when you wrote "After all, ..." above.

Harvey Friedman