[FOM] Permanent value revisited

[Harvey Friedman]

Lindauer 5/13/04 and Apostoli 5/14/04 have revisited prior postings I made
concerning the contrasting methodologies of mathematics, philosophy, and
foundations of mathematics, and the search for findings of permanent value.

Since it has been a long time since I have written about this, I searched
the Archives and found these four postings by Friedman.

[FOM] Re: Contrasting Methodologies
Harvey Friedman friedman at math.ohio-state.edu
Wed Oct 1 03:21:43 EDT 2003

[FOM] Re: Contrasting methodologies
Harvey Friedman friedman at math.ohio-state.edu
Fri Oct 3 14:07:35 EDT 2003

[FOM] Permanent Value?
Harvey Friedman friedman at math.ohio-state.edu
Sat Oct 4 23:58:45 EDT 2003

[FOM] Re: Example of beef: nonrigorous heuristics
Harvey Friedman friedman at math.ohio-state.edu
Sun Nov 16 00:24:50 EST 2003

The most relevant to the Lindauer and Apostoli postings is the third of the
above, Permanent Value?

But I also found item 11 from the last of these four that is certainly
relevant to this discussion. I attach the preceding item 10 for context.

10. Any substantial expansion of the scope of f.o.m. is likely to seriously
borrow from the existing development of f.o.m. If some initial insight is
made that does not seem to seriously borrow from existing f.o.m., then
existing f.o.m. will be used to substantially develop that initial insight.
I.e., I have full confidence that existing f.o.m. is the proper initial
segment of f.o.m. development.

11. Consequently, attempts at downgrading the status of f.o.m., as we
witness from time to time in rather determined form here on the FOM, are not
only entirely inappropriate, wrongheaded, and easily refuted, but are
completely counterproductive in light of 10.

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

I will attempt to briefly restate some contrasts between mathematics,
philosophy, and f.o.m. I will be quite brief and sketchy compared to the
four postings above.

1. The permanent value of core mathematical developments is apparent,
although general intellectual interest is usually limited or nonexistent.
Examples in the last five years might be

i) Poincare's Conjecture proof (being checked now);
ii) The prime numbers have arbitrarily long arithmetic progressions
(recently verified).

Going back a few more years, one could extend the list:

i) Fermat's Last Theorem
ii) Poincore's Conjecture
iii) Primes have arbitrarily long arithmetic progressions.

I picked three famous items (the third is rather new) that perhaps have the
most g.i.i. of recent core mathematical developments.

As you can see, it is awkward to talk about these things in terms of wider
intellectual significance, in the same way that you can with, e.g., Newton,
Einstein, Darwin, DNA, etc.

But the achievements are undeniably correct (will not be rejected later, as
is normal in almost any other subject), and undeniably clear, and undeniably
impressive. 

2. The g.i.i. of much of what philosophers write about is undeniable and
obvious. This is in contrast to the situation in core mathematics. (I should
also add that the g.i.i. of much of what philosophers write about is limited
or nonexistent). It is true that a fair amount of background discussion is
often needed to clarify just what these philosophical questions really mean
- or just what kinds of things are being sought after in a solution.

However, I have never met any philosopher who has maintained that definitive
contributions to the major philosophical problems are being made on any kind
of regular basis. Rather, they speak of incremental improvements in our
understanding of the merits and demerits of various positions, and the
creation of novel positions which have various new merits and demerits. This
is in exact accord with my own observations when listening to talks by
philosophers.

3. Foundations of mathematics, at its highest levels, spectacularly combines
the undeniably permanent value of core mathematical developments with the
general intellectual interest of philosophical discussion.

4. Fundamental methodological issues about philosophy remain rather
controversial, at least among philosophers.

My own view is rather hard nosed. The greatest achievements of philosophers
lie in their sometimes crucial roles in the emergence of
subjects-of-systematic-knowledge. The primary example in modern times is the
emergence of the subject-of-systematic-knowledge known as: foundations of
mathematics. Frege and various predecessors played crucial roles. Less clear
cases surround probabilistic and statistical reasoning.

This is not a pessimistic statement regarding the power of philosophical
thinking. This is because I am an eternal optimist regarding the emergence
of subjects-of-systematic-knowledge out of appropriately focused deep
philosophical thinking.

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

On 5/13/04 3:15 AM, "Robbie Lindauer" <robblin at thetip.org> wrote:

> Mr. Friedman asked genuinely, I think, whether or not Philosophers have
> made contributions to "enduring knowledge" in the sense of things known
> in the last five years comparable to the kinds of knowledge that
> mathematicians (and we'll assume he meant "pure mathematicians" for the
> moment) make regularly.

It would simplify matters if you explicitly agreed with my observation. As
you can see from the above, as well as from my earlier postings cited at the
beginning, I do not draw the inference that philosophical thinking is
unproductive.

> 1)  The underlying pragmatism that motivates Mr. Friedman's question is
> instructive simply because it betrays the notion of "enduring
> knowledge" that he's pushing as a standard of value.

I am simply commenting on "contrasting methodologies" of core mathematics,
philosophy, and f.o.m.

>If what is 
> valuable about knowledge is that it endures, then surely philosophers
> have made the greatest contributions to knowledge in every era.

Can you defend this statement? You need to be explicit about just what these
greatest contributions to knowledge are. In particular, restrict yourself to
1900-2000, and compare what the philosophers did in this period to what was
done in physics, engineering, mathematics, biology, etc.

>In our 
> modern era, in particular, we've seen the fall of this pragmatism of
> his to each of foundationalism, coherentism, anti-foundationalism and
> out-and-out relativism in the theory of knowledge.

I don't understand this assertion. Please explain. In particular "fall of
this pragmatism". 

>Thankfully, the 
> theory of knowledge has not found its way into a calculus book so few
> people are claiming that it really is a form of mathematics.

I don't know what you mean by "theory of knowledge". Are you talking about
what philosophers classify as "epistemology"?

Also are you particularly concerned with "theory of mathematical knowledge?"
If so, explain what this theory of mathematical knowledge looks like. Are
you simply talking about the various isms in philosophy of mathematics?

>Its 
> fruitfulness is to be measured not by how many theorems it proves, but
> rather by how many theorems may be seen as proved now that were seen as
> simply wanting of an adequate theory of knowledge, for instance.

Give us some examples of this.

>As it 
> stands, mathematics has no adequate theory of knowledge.  Recognition
> of this fact might lead to more interesting work on both sides (and
> certainly has).

Are you engaged in the appropriate philosophical thinking needed for the
emergence of a new subject-of-systematic-knowledge called "theory of
mathematical knowledge"?

> More interesting philosophically would be
> looking at why specific kinds of knowledge are deemed valuable (e.g.
> physics and computer science) and not, say, grammatology or
> communication theory?  Could deconstruction be mathematized the same
> way that information has been?

Are you engaged in the appropriate philosophical thinking needed for the
emergence of a new subject-of-systematic-knowledge called "theory of
mathematical knowledge"?
 
> 2)  Measuring philosophy's advance in grains of 5-years is unfair.

It appears that you agree with my assertions about contrasting
methodologies. 

> Thankfully, in the
> last five years there have been some nice pieces of work released which
> may not have created "permanent-value-as-knowledge" but certainly made
> an impact about what the options were - and that too is a kind of
> enduring knowledge.

It appears that you agree with my assertions about contrasting
methodologies. 
> 
> The desire to treat of foundations without addressing the philosophical
> importance and relations of mathematical issues to other sciences and
> arts is itself a philosophical program that falls under a political
> program.  

It is not possible to make major progress in a
subject-of-systematic-knowledge if one tries to treat too many issues at
once. For instance, it is difficult to make major progress in astronomy
while addressing the issues concerning the feasibility of sending men and
women to Mars. I don't view astronomy as a political program.

F.o.m. is not a political program. At its highest level, it has very clear
goals and addresses very clear issues which are of obvious general
intellectual interest. There is vast agreement about this in the case of the
work of Kurt Godel. Was Godel engaged in a political program?

>This program is fascinating in that it is an attempt to
> control or undermine (we're not able to tell which) the very inference
> schemas which we're allowed to call rational - that is it is an attempt
> to define rationality - and this program is thoroughly philosophical.

What program are you referring to? Perhaps too much time has passed since
the last postings you have made.

> It is also attempting to do it "outside the ring" that is outside of
> what is called philosophy - explicitly rejecting that name.  That it
> may also be a pragmatic attempt - to define rationality for a reason -
> is somewhat chilling.

What program are you referring to? Perhaps too much time has passed since
the last postings you have made.
 
> The concern for ethics, ontology and epistemology is important - more
> important than choosing between ZF and NF or giving a proof-procedure
> for a specific kind of mathematical problem (even though that last
> might be very useful for someone some day).

The concern for the human race, war, and disease, poverty, birth rates, life
expectancies, etc., is much more important to many more people than issues
in the foundations of mathematics. So what?

> Philosophers shouldn't
> want to prove things from axioms, but rather find or deconstruct ways
> of living that are good - that is to make rational decisions about our
> actual lives based on that bogey, again, the truth.  The connection
> between mathematics and ethics and epistemology is less obvious than in
> ontology where several attempts have been made to reconcile the two
> already - but they are just as interesting.

Are you engaged in the appropriate philosophical thinking needed for the
emergence of a new subject-of-systematic-knowledge?
 
> Mr. Friedman himself has thought it important to address
> epistemological issues in this forum - his certainty machines, aliens
> and crystal balls.  The arguments there follow the method and tone of
> philosophy - if that was mathematics then "anything goes".

I was deliberately attempting to engage in the appropriate philosophical
thinking needed for the emergence of a new subject-of-systematic-knowledge.
Are you interested in joining in?

You quote me as follows:

> Friedman:
> 
>> 1'. Philosophers do not normally engage in the quest for
> "permanent-value-as-knowledge", hoping, at most, that it will lead to
> such
> at some future date.
> 
> Do you suggest that we pick up the last 5 issues of the Journal of
> Philosophy and evaluate the articles strictly in terms of
> 
> permanent-value-as-knowledge"?
> 
> What conclusions would you draw?
> 
> ____
> 
You have a long reply, but you declined to give me any specific examples
from the last 5 issues of the Journal of Philosophy, and evaluate it in
these terms. 
> 
> So the conclusion I draw - to answer you question directly - is that
> the issues that philosophers attempt to grapple with are more difficult
> than those that mathematicians do and so take much longer to evaluate.

It is trivial to state mathematical problems that one does not expect to see
much progress on for, say, thousands of years. That is more pessimistic than
what philosophers generally say about their philosophical problems.

If mathematicians used such problems as their benchmark for success, then
they would have a much much slower record than you say philosophers have.

However, mathematicians are very good at breaking problems up into doable
problems, and also just recognizing doable problems. This allows for rather
steady production of findings of permanent value.

I should also add that many very well known philosophers I know admit that
they find even rather basic mathematics extremely difficult to understand
appropriately. I trust that you don't have this difficulty.

>  Occasionally 
> philosophers are mathematicians and vice versa - like you.

I am not properly classified as a mathematician or as a philosopher. I work
in the interface between mathematics and philosophy, in a field called
foundations of mathematics. I hope to make contributions to foundations of
other subjects.

>But in 
> particular, when you attempt to answer ontological problems about "Are
> any things of the kind being studied by this group of people and are
> their methods of justification adequate?", you tend to be doing
> philosophy.  

This is the kind of question is also addressed systematically in f.o.m.,
with the high standards of a subject-of-systematic-knowledge.

>And when the attempts to decide that external question
> becomes important to the science itself (say in the debate between
> intuitionism and platonism and logicism or something) it may have some
> of the interesting side-effects that you've been hoping for by studing
> Foundations in your sense.

Everything in high level f.o.m. has "interesting side-effects" in this
sense.
> 

> What I claimed was that your statement that "mathematicians don't
> normally engage in philosophical thinking" is not true - here you are
> doing it.  

I am not properly classified as a mathematician or as a philosopher. I work
in the interface between mathematics and philosophy, in a field called
foundations of mathematics. I hope to make contributions to foundations of
other subjects.

On 5/14/04 12:49 PM, "Peter John Apostoli" <apostoli at cs.toronto.edu> wrote:

>> Mr. Friedman asked genuinely, I think, whether or not Philosophers have
>> made contributions to "enduring knowledge" in the sense of things known
>> in the last five years comparable to the kinds of knowledge that
>> mathematicians (and we'll assume he meant "pure mathematicians" for the
>> moment) make regularly.
> 
> I too missed this interesting post from Prof. Dr. Friedman. I'd like to
> point out the limits of formalized axiomatic foundations of mathematics by
> comparing ZF with an alternative approach to the set theoretic foundations
> of mathematics based upon pure semantics. The later approach offers a
> degree of scientific unification undreamed of in Prof. Friedman's
> philosophy.\

Are you engaged in the appropriate philosophical thinking needed for the
emergence of a new subject-of-systematic-knowledge?

 
> But first, and as a side remark, we should keep in mind that
> mathematical work done within the framework of ZF has only as much
> "lasting value" as the framework itself.

All modern mathematics is done within the framework of ZFC (with minor
exceptions). However, since the "lasting value" of ZFC is obvious and
enormous, I can't argue with your statement.

>As no one knows whether that
> framework in even deductively consistent, how much "lasting intellectual
> value" this work has may still be an open question.

1. All modern mathematics is, or can easily be, done within tiny fragments
of ZFC, where consistency is regarded as particularly unproblematic. In
fact, an enormous amount can be done in very weak fragments of Peano
Arithmetic, so those worried about PA, for some reason, need not worry about
these very weak fragments.

2. The lasting intellectual value of ZFC does not rest on removing all
conceivable doubts among all reasonable people about whether or not it is
consistent. 

> ZF is not a model of set theory.

ZFC is a formal system, and obviously not a model. This is clear to any
student of mathematical logic.

>It is a set of axioms which allow
> sustantive questions of set theoretic truth to be replaced by linguistic
> question regarding formal derivability. As such, it is a theory of
> "representations", rather than a theory of set-like objects (sets).

The first statement is idiosyncratic. The second is incoherent. Please
explain what you are talking about.

>Due to
> deductive incompleteness, ZF falls far short of specifying the fine
> grained structure of the set theoretic universe that an adequate ontology
> is required to give.

This is misleading. For the purposes of practical foundations of actual
mathematics in the current published literature, ZFC not only doesn't fall
short, it is actually overkill!

I had to work for several decades to give any examples where ZFC, or even
ZF, falls short, with regard to mathematically interesting propositions of a
normal concrete nature.

Perhaps you are not talking about issues related to mathematical practice.
Explain "adequate ontology" here.

> We speak of "the" set theoretic universe because
> philosophers have developed a canonical model for naive set theory. The
> model, constructed using at most 4th order arithmetic, is the unique
> solution to Russell's paradox.

>The model is canonical in the sense that it
> is the canonical model of an associated (non Kripkean) modal logic. In
> particular, it contains every logically possible set as an element, that
> is, every logical possibility is represented.
> 
> However, this solution to
> Russell's paradox requires that the universe of sets be "granulated" under
> the set-theoretic indiscernibility relation. In other words, Russell's
> paradox is tanamount to a proof that the set theoretic continuum has a
> granular structure. Sets come surrounded by a Halo or granule of
> infinitesimally close ("modal") counterparts. This granular structure of
> sets directly relates the canonical universe to the areas of infinitesimal
> analysis, rough set theory, quantum computing and renormalization theory.
> The granularity of the set theoretic continuum is a mathematically
> demonstrable truth (first established in Cocchiarella's formal systems
> T,T* from the denial of Russell's contradiction), which asserts that the
> continuum has a fundamental "Planck length". It is upon this discrete
> basis that the phenomenal continuum (the characteristic smoothness of the
> continuum) emerges as a large-scale approximation. By Goedel's
> incompletness theorem, this fine grained detail could never be derived
> from an axiomatic set theory such as ZF.
> 
> This model does more than provide
> a categorical consistent conception of set. It unifies set theory with
> modal logic, the foundations of physics, the theory of rough sets and
> rough set approaches to informatics and engineering science more
> generally. Such theoretical unification is unheard of in mainstream FOM
> but is the earmark of successful science these days (when traditional
> academic divisions between the sciences are eroding).
> 
> Therefore, the canonical universe of sets is a better scientific
> explanation of sethood that ZF. And its still an open question whether ZF
> provides any explanations at all.
> 
> How's that for lasting value in the FOM?
> 

Sounds much too far fetched to form anything like a coherent foundation for
mathematics, like ZFC and fragments of ZFC do so well.

However, if you are really serious, you should explicitly state the
"canonical model for naïve set theory" right here on the FOM list, and
indicate how mathematics is straightforwardly formalized under this
approach. 

Harvey Friedman