[FOM] Re: Contrasting methodologies

[Harvey Friedman]

> Initial disclaimer: I apologise if I wrongly construe you as more optimistic
> for the philosophical power of FOM than indeed you are in the below, the
> only other alternate reading of your message I found construed philosophy as
> of no "permanent value." I chose the more acceptable of the two.

Trivial Point: we use FOM for the name of this e-mail list, and f.o.m. for
the foundations of mathematics. This convention was solidified by Steve
Simpson in the Wild Age of FOM.

>From time to time, Philosophy *leads* to things of great permanent value.
However, I don't know of a single case where this happened - i.e., the
process was completed - using **solely** the usual methodology of
Philosophy. 

Instead, in the examples I have in mind, there was at some point another
methodology of a decisively mathematical/scientific/technical nature.

When this crucial step(s) come into play, of a decisively
mathematical/scientific/technical nature, much is LOST from the point of
view of Philosophical methodology. Major portions of the philosophical work
that lead to the advances of permanent value reveal their contrasting lack
of permanent value by becoming obscure and highly controversial.

One extreme case of this is the Godel second incompleteness theorem. Here
are some features.

1. Almost universally accepted as of obvious permanent value.
2. Does not decisively answer a single major classical philosophical
question. E.g., does not prove or refute formalism, Platonism, realism, etc.
3. Decisively answers a focused question of great general intellectual
interest. I.e., can there be a foundation for mathematics with certain
properties? It (the work) provides the tools for decisive counterattacks
against skeptics about its obvious permanent value.
4. Raises more philosophical issues than it resolves. Does not resolve the
liar paradox. Does not provide any decisive understanding of the nature of
self reference in general contexts. Opens up attacks by realists against
formalists, pointing to the great extra power afforded by realist
viewpoints.
5. Almost every competent discussion in the philosophy of mathematics - and
many even in other parts of philosophy - at least mention it in some way.
6. Godel cites previous work on liar paradox, self reference. Such previous
work not "almost universally accepted as of obvious permanent value" on its
own. Usual claims that it is "of obvious permanent value" cite that it lead
to Godel's second incompleteness theorem.
7. Was achieved using a mathematical/scientific/technical methodology.
8. Much is LOST from the Philosophical point of view. Genuine self reference
entirely sidestepped. Issues about the nature and structure of real
languages completely ignored.

Another extreme case is Turing's development of his primitive model of
computation. 

>... However great the "permanent intellectual
> value" of the snail trail of FOM work which follows the philosophical issue,
> my contention was not that current FOM solutions to scepticism (such as
> complexity measures, finite work etc.) are deficient, but that FOM "full
> stop" is fundamentally impotent when it comes to answering the philosophical
> question. 

I take it that the philosophical question you are concerned about is

*what does it mean to follow a trivial rule like: given a finite string of
bits, append 1 at the end*

Or is the philosophical question you are concerned about is

*what is the relationship between this kind of simple-rule-following and an
objective unending sequence of natural numbers*

Or just what exact formulation do you want to give for this "philosophical
question"?

Godel's second incompleteness theorem ""full stop" is fundamentally impotent
when it comes to answering the philosophical question 'what is self
reference?', 'is formalism or Platonism 'true'', 'what is the nature of
mathematical knowledge', etc.". So what??

And if f.o.m. work is "fundamentally impotent", then tell me what is not
"fundamentally impotent"?

>I am not saying that research on this area of FOM is pointless,
> merely that any use it may have lies elsewhere than philosophy.

The f.o.m. work is a product of philosophical thinking.

A. WITHOUT advances of this sort (along the f.o.m. methodology), at least
from time to time, work of permanent value surrounding such issues appears
impossible.

B. WITHOUT the philosophical thinking, the f.o.m. style work of permanent
value surrounding such issues appears impossible.

>E.g. formal
> complexity measures may serve as useful psychological models of our
> intuitive rule building. (But fundamentally the best such models can hope to
> do is to use our intuitive practice to suggest extensions of that practice,
> rather than ever stepping outside the human psychological and linguistic
> make-up to give some scepticism beating justification.)

My main f.o.m. like suggestions are by no means appropriately characterized
in terms of "formal complexity measures". E.g., I was talking about the deep
intellectual work needed to

*get a computer to follow simple rules when appropriately prompted by a
human to do so*

This is obviously relevant to any deep understanding of issues surrounding
any philosophical analysis of the corresponding situation of

**getting one human intelligence to follow simple rules when appropriately
prompted by another human intelligence to do so**

I suspect that 

***the deep 'cart before the horse' issues that had to be fleshed out in
order to build a real (even very low level) computer system based on the
stored program idea***

are intimately related to the philosophical analysis. Obviously the
researchers involved are disjoint. E.g., Wittgenstein was not helping out
von Neumann and various design engineers at Princeton(?) back then, to my
knowledge, in that Golden Age.

> You write:
> (*)"Give a theory which tells us why certain "machine instructions" are
> elemental, and others are not."
> To which I would respond, like a well trained Wittgensteinian, because they
> concur with our linguistic use.

Not bad. Now do something with that idea. Provide a calculus for determining
which ones concur with linguistic use and which ones do not. Go well beyond
W. 
 
> For me the most convincing argument for rule-scepticism, and for its
> ultimate invulnerability to any amount of formal treatment,

I know of nothing invulnerable to formal treatment.

E.g., here you might endeavor to *prove* that under your understanding of
"rule-scepticism", rule-scepticism is "ultimately invulnerable to any amount
of formal treatment".

This is an example of a "proof" that nothing is invulnerable to formal
treatment. If it was invulnerable to formal treatment, then try to prove
that it is invulnerable to formal treatment, formally.

Of course, this may also be impossible. Then try to prove that, formally.

This may even iterate. Try to prove the iteration.

I think it highly unlikely that all of these projects would simply sit there
and be impossible. Certainly there is not the slightest reason to suspect
that.

>is found not in
> Wittgenstein but in Goodman. My Turing machine functions which behave
> differently depending on the number of instructions already executed are
> directly parallel to Goodman's "grue" (meaning "green before time t and blue
> after time t" for the mathematicians on this list). The initial response is
> always to invoke the extra complexity of "grue's" definition, but this
> thought is soon dismissed upon the realisation that green and blue can just
> as easily be defined in terms of "grue" and "bleen" than the other way
> round. Goodman's ultimate (and I believe correct) conclusion, is that
> projecting "green" not "grue" ("1111111..." not "11..1000..") is right
> simply because it is what we do.

An informal form of some symmetry argument...

And how does this relate to the fact that we have no problems now with
computers misbehaving on trivial tasks?
 
> I doubt this reply will be overly well received due to your perfectly
> reasonable:
> "I thought I made it clear that I was not attempting to engage in a
> discussion of skepticism directly"
> My justification for it, is that wittingly or unwittingly, you have
> persistently made remarks that a philosopher would be hard pressed to not
> construe as "engaging in a discussion of scepticism", both directly e.g. (*)
> above and the sentence I quoted at the start of the last e-mail, and
> indirectly in your vague notion of the "permanent intellectual value" of
> FOM.

I agree that I have formulated f.o.m. projects in response to this
discussion that attack this kind of skepticism.

However, that is only because those f.o.m. projects are EASIER to formulate
and are closer to what has already been done, drawing on fundamental
advances in the computer revolution.

However, there are also f.o.m. projects of potentially permanent value that
DEFEND this kind of skepticism. They seem to be harder and more novel to
carry out.

E.g., define an associated "skepticism attack" on a well defined large
family of rule following formalisms. Prove that any rule following formalism
satisfying certain conditions always has a skepticism attack satisfying
certain conditions.
> 
> To finish on a note more closely related to the subject line, there are only
> "contrasting methodologies" between FOM and philosophy as long as one does
> not attempt to do the other's job.
> 
The specific relationship I see between f.o.m. and philosophy (under normal
methodologies) is presented in A and B above.

Harvey Friedman