[FOM] Math Practice/Beta(N)

Harvey Friedman friedman at math.ohio-state.edu
Thu Feb 23 23:01:52 EST 2006

On 2/23/06 7:22 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

>> Weaver asserts that predicativity can do more functional analysis
>> than the PRA level, but apparently concedes that the difference
>> is not substantial.
> The difference is substantial.  I'm not sure how you got "not
> substantial" out of my message
> http://www.cs.nyu.edu/pipermail/fom/2006-February/010005.html
> Do you think the difference is not substantial?

There is nothing there that looks very substantial. I also don't see any
argument that what you mention there is of logical strength beyond PRA. So
as far as I can tell, might it all be formalizable in convenient
conservative extensions of PRA or PA?

In fact, is there any evidence whatsoever that any of these can't easily be
done in a system like ACA0? This is a well known conservative extension of

Then Weaver quotes me as follows:

> "Weaver advocates predicativism because it accords well with
> normal mathematical practice."

Weaver appropriately retracts this quotation of me in his later posting
> No.  I claim that predicativism is special because it has a clear
> philosophical basis and impredicative mathematics does not.

I have been saying: Predicativism does not match mathematical practice
particularly well, nor does it have any special standing philosophically or

> because impredicative mathematics is justified on platonic grounds
> which do not hold up under scrutiny.

It holds up quite well under scrutiny. Do you see any problem with it?

The only problem you have mentioned in all of your postings is simply that
it isn't predicative.

FINITIST: The completed totality of natural numbers does not hold up under

ULTRAFINIST: The notion of arbitrary natural number does not hold up under

LIMITED SET THEORIST: The notion of arbitrary ordinal does not hold up under

PREDICATIVIST: The notion of arbitrary set of natural numbers does not hold
up under scrutiny.

I can go on with finer positions. Obviously there is nothing special about

> *The fact that predicativism
> accords well with normal practice is merely additional evidence
> that it is special.*

This statement is why I have made the effort to refute your "accord well".

> The fact that
> what came out matches predicativism much better than platonism
> seems significant.

This is either false or grossly misleading.

In particular, I ask all subscribers:

Which fits mathematical practice in French style algebraic geometry better:
set theory or predicativity?

Also, as I write this, I have been uncovering a very large terrain of uses
of impredicative arguments in COUNTBLE CORE ALGEBRA! I have to admit that

*I am very surprised by just how much flagrant impredicative reasoning
abounds among these core mathematicians in dealing with countable and even
finitely generated structures. They don't seem to care about removing any of
it. It is only slowly becoming clear what can be removed and what cannot be
removed. I have been beginning to be successful in getting them interested
in removability issues.*

PROFESSOR WEAVER: how does THAT accord with predicativity? You will have to
expand predicativity in order to deal with this. You will expand it right
into substantial impredicative definitions.

You have already hinted at an expansion of predicativity via the

**Pi11 comprehension axiom scheme**

which has been roundly rejected as predicative by the community of

This includes the work of that mysterious Fields Medalist, but apparently
goes much further, as I had discussions with a core mathematician in my
Department today. 

I intend to report on this later.
> Friedman seems to concede that impredicative systems can only be
> justified on platonic grounds.

I have been arguing that the very notion of Platonism here is so unclear
that it applies to the mere existence of the empty set or even the natural
number 0. 

Also, there is absolutely no convincing argument to this day that even the
kind of Platonism that Weaver is referring to, has any problems, or is in
any way shape or form incoherent. In fact, the mathematical community has
clearly accepted strong impredicativity as perfectly coherent and perfectly
valid for over 80 years.

>Instead he challenges my position
> by claiming that predicativism also rests on platonism.  This
> argument seems to me not very convincing.

You can't argue convincingly against "0 is Platonism". Try. You will have to
define Platonism just to start. Your argument won't be any more or less
convincing than untold numbers of other old fashioned f.o.m. type arguments.

>You justify, say, Z_2 in
> terms of the existence of an abstract metaphysical world of sets
> in which the axioms hold.

FINITSTI: You justify PA in terms of the existence of an abstract
metaphysical world of numbers in which the axioms obviously hold.

>You justify predicativism in terms of
> marks on paper.  Calling the latter "platonism" is a bit of a
> stretch.  

FINITIST: The completed totality is Platonistic.

ULTRAFINIST: Arbitrary finite is Platonistic.

PLATONIST: What's wrong with Platonism? There is no problem.

>Key difference: sets are supposed to be *unique,
> canonical* abstract objects.

SOME SET THEORIST: Sets form out of a fundamental equivalence relation of
having the same elements.

FINITIST: Natural numbers form out of a fundamental equivalence relation of
having the same count.

> Marks on paper may be imagined
> objects, but they're not really abstract.

SET THEORIST: The equivalence relation is abstract. Without the underlying
equivalence relation, you can't do anything with various marks on various

>And there is no
> requirement of uniqueness; indeed, belief in the intelligibility
> of *any* omega structure is enough.  "Marks on paper" is just a
> picturesque description.

SOME SET THEORISTS: There is no requirement of uniqueness in sets. One can
just deal with the nonextensional notion with many equivalent copies. But
then in order to simplify things, one brings in extensional equality.

FINITIST: There is no requirement of uniqueness of strings. One can just
deal with the underlying strings with many equivalent copies. But then in
order to simplify things, one brings in "having the same count".

FINITIST: There is no omega structure. Besides, what is a structure?
> If you're going to say that I have to be a platonist in order to
> be a predicativist, you have to say that one cannot believe in
> the intelligibility of the concept "omega structure" without
> being a platonist.  Does anyone believe this?

You can find someone who believes in practically any given thing, and that
person can argue for his/her point of view as well as you can for yours.
Remember Hilary Putnam's paper: Philosophy of Mathematics: why nothing

On 2/23/06 7:23 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

Friedman wrote:

>> beta(N) is the space of ultrafilters on N. Hence in predicativity,
>> beta(N) does not have any nontrivial elements.
>> So how is beta(N) to be handled in predicativity?
> Huh?  That's exactly backwards.  Predicatively, ultrafilters
> over N are treated as proper classes.  So beta N does not exist
> as an object but individual (nontrivial) ultrafilters do.

Beta(N) is therefore a space of proper classes. This does not handle
mappings from beta(N) into itself and subsets of beta(N), which is all an
integral part of the theory and use of beta(N).

E.g., see 

Andreas Blass (and coauthor) Finite Preimages Under the Natural Map from
beta(N x N) to (beta N)x(beta N), joint with Gugu Moche (Topology
Proceedings 26 (2001-2002) 407-432)

available from http://www.math.lsa.umich.edu/~ablass/set.html

IN ADDITION, there are awkward problems in dealing with even an individual
ultrafilter as a proper class.

For example, if you take any standard system for predicativity then it will
not even be provable that

*some proper class is a nonpincipal ultrafilter*

In order to derive this statement, you have to add some assertion in the
predicative theory to the effect that

*the sets of integers are the sets of integers in some L(lambda)*

and even this might not work.

In fact, it appears that the situation is rather delicate, and I am writing
a separate post on the science of this.

> A quick way to see this is by identifying ultrafilters over N
> with homomorphisms from l^infinity into the scalars.  As I
> explained in a previous post, the dual of l^infinity does not
> exist predicatively but individual linear functionals do exist.

Give us some predicative examples of nontrivial linear functionals. If the
identification with ultrafilters is appropriate, then you would be giving an
example of a nonprincipal ultrafilter on N. How are you going to do that?

Put it another way: there are 2^c ultrafilters on N. Are there 2^c
individual objects you are going to handle predicatively?

> See the previous message about accord with mathematical practice.
> To the extent that beta N really has been centrally used in core
> mathematics my point about "exact fit" is weakened.  To the extent
> that uses of beta N are actually inessential my point is strengthened.

The finitists and people sitting at various levels between finitism and
predicativity make the same point. Again, nothing special about

> Most core mathematicians I know tend to regard beta N as a highly
> pathological object that is better avoided.
Beta(N) apparently is a preferred vehicle for published papers proving
several really interesting theorems about partitions of N into finitely many
pieces. This people use beta(N) with great pride. You will have to come to
grips with this phenomena. See Blass's posting

Harvey Friedman

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