[FOM] Permanent value revisited

[Robbie Lindauer]

On May 14, 2004, at 10:28 PM, Harvey Friedman wrote:

> 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.

I think this was the reason I'd stopped following FOM.

> 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.

Why let the rats in when there are good mice here?  If something is 
easily refuted, say the supposition that choice, replacement, powerset 
and infinity are false, then why not present the proof.  That would be 
a remarkable thing.

> 1. The permanent value of core mathematical developments is apparent,

I guess I could have said more simply before "define value".  I had 
attempted to interpret your remarks apparently unsuccessfully.

> 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.

I'm unsure about this generally.  If you mean that it is there to prove 
the axiom of choice or replacement or power-set or something like that, 
that would be of general interest, but that's hardly what I've seen go 
on here.   That would also be spectacular.

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

Would you say that the difference between standard mathematics and 
intutionist mathematics is not fundamental and is not carried over from 
philosophical differences?  Are game-formalist, curry-formalists and 
(plain) formalists not "fundamentally" different ways of understanding 
mathematics?  If you mean that all mathematicians use numbers "in some 
sense" then this provides a way of contrasting mathematics with 
philosophy.

> The greatest achievements of philosophers

>  The primary example in modern times is the
> emergence of the subject-of-systematic-knowledge known as: foundations 
> of
> mathematics.

What about marxism or deconstruction?

>> 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 you say "an enduring value for FOM is proof" of the kinds you 
enumerated above, then wouldn't the excellent subject for study be the 
rigorous proofs of the axioms?   Perhaps a secondary subject might be a 
clarification of the underdefined notions in the system like "set" 
"truth" "arbitrary function", etc.

>> 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.

Lenin?  I would say that the development of the worldwide theory of 
capitalism has been expansive and impressive as well, but not as 
interesting as the development of the contrary theory.  So this would 
be the invention of the (modern) social sciences?  So if the global 
effect of capitalism and socialism has been a great contribution to 
human knowledge (and it has) then still the philosophers are pushing 
the limits.

Martin Luther King?

On the "professional philosopher" side - Wiggins, Kripke, Russel, 
Popper, Kuhn, Derrida, Sartre, Husserl, Levinas, Baudrillard, etc.

There have been plenty of philosophies - including particular points of 
knowledge - whose value is enduring - both life-philosophies and 
technical philosophies.  Do you really want to do a little history of 
20th century philosophy here?  We can say this for them, which is what 
we can say for the mathematicians - if their assumptions are true, 
usually their conclusions follow.

> Also are you particularly concerned with "theory of mathematical 
> knowledge?"
> If so, explain what this theory of mathematical knowledge looks like.

It would look, in fact, much like the history of the theory of 
mathematical knowledge does look like, and it would be properly called 
philosophy.  Zeno through Field and still trucking.

>> 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.

Say that the powerset axiom is false and there is no powerset of N.   
Then cantor's proof isn't a proof.  We have no proof of the powerset 
axiom (qua axiom) so we must give another explanation of why we think 
it - e.g. an account of our epistemic situation with regard to it and 
why that epistemic situation is in accord with our epistemic norms.  We 
may have to do this by giving an account of our epistemic norms, that 
will be giving an adequate epistemology.  Personally, I regard cantor's 
proof as a "proof-in-cantors-system" and await a proof of the axioms.  
That would seem like the most valuable "foundational" project that 
could be attempted that might be entertaining to you.

> I don't view astronomy as a political program.

You might view a particular pattern in astronomy as a political 
program.  For instance, if sending men and women to mars suddenly 
out-shadows the discovery of new stars and mapping the development of 
the universe as a major fundable goal of astronomy programs and all of 
a sudden astronomy was the second-best funded science department 
because the study of mars was suddenly very important because Mr. Bush 
wanted to get to mars right before the election.  This wouldn't 
necessarily make their findings false, but it might make it lopsided.

That would be an obvious way a science could become a political 
program.  There are un-obvious ways too - interpretations and 
philosophies could creep into them unchallenged making them ideologies 
of their own.  We would regard these sciences as non-sciences (say "The 
theory of Siva" or "The theory of creationism").  But if we don't 
challenge the un-obvious ways in which unfounded interpretations enter 
our own sciences, then we fail to fully appreciate the prejudice 
involved in our judgments.  This is where we get sciences like 
"trickle-down economics" which for some reason gets air time at 
Harvard, Yale and Stanford.

> F.o.m. is not a political program.

Apparently one immediate program is to distance itself from philosophy, 
make it sound less speculative.

> There is vast agreement about this in the case of the
> work of Kurt Godel. Was Godel engaged in a political program?

I don't know what Godel's political/religious intentions were.  I've 
read that he considered himself a variety of philosopher and have read 
some of his religious statements.

>> 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.

In the case of mathematics, we've had various attempts from both sides 
to rework ontology, the theory of reference, logic and epistemology in 
order to account for our mathematics.  Jourdain says "logic had to be 
sharpened".  This is so because, for some reason, mathematics is 
important for philosophy qua "synthetic a priori knowledge" and 
"necessary truth" and "actually infinite" and such.  But also because 
it seems that philosophy wants at least some of mathematics in order to 
do its job -  proof and discovery of non-mathematical synthetic a 
priori necessary truths.  It is wedded to mathematics in a way.  In 
order to prove something we think we must be able to enumerate the 
premises, for instance.

Recently you made a list of topics that you don't want to discuss on 
FOM - "what is a set?" "what is truth?" etc.  Fair enough, there are 
other forums.  But to provide "foundations of mathematics" without 
addressing these is like agreeing to buy a mobile home.  If truth isn't 
an important foundational issue for mathematics (e.g. what's the 
difference between "aleph-0 < aleph-1" and "Santa Claus is bigger than 
Mrs. Claus"?) then nothing is.  That is, it becomes a directed 
discussion for true believers.  That's fine, as I said, there are other 
forums.  But it also takes the OOMF out of FOM.

>> 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.

There are many political and economic programs served by the intense 
study of the automation of thought-like-processes, mathematical 
theorems with computer-science and engineering applications and proof 
methods as well as by the dictation of specific methods of thinking as 
normative laws for people as a kind of thought control.

> 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?

They don't teach geography in grade school anymore.  There's a reason - 
it's not an important job skill.  There are other skills that are not 
taught in school anymore, logic and rhetoric aren't, for instance, 
while calculus is.  Isn't it odd that calculus is so much more 
important than logic and rhetoric?  I think people need logic and 
rhetoric more than they need calculus "in their real lives".

My cousin studied math at UCLA.  He didn't ever take a logic course 
except as the introduction to his analysis textbook.  The logic to 
which he was introduced was the one that is appropriate for the 
development of set-theory - because that's what the course was about.  
That there are other logics was never mentioned.  Obviously had he 
found it worth studying at a higher level it would have come up.  But 
why do these things only come up later?  Because there is a dominating 
ideology.  Why is there a dominating ideology?  You would think it was 
because there was proof that it was true - but when you look at the 
foundations of mathematics, you find exactly the opposite to be the 
case.

>> 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?

Our first encounter was over the term "certainty".  You're 
certainty-producing hyper-computer and proof-checker sounds fantastic, 
but instills in me not the same kind of certainty that I have with your 
basic 1 + 1 = 2, but rather a kind of generalized numbness having 
worked with merely actual computers.  We have a disagreement about the 
term "certainty" and I don't think the methods of FOM are appropriate 
for resolving our disagreements, rather philosophy is needed.

As for speculation about what crystal balls might accomplish, it's 
entertaining but leaves me still in philosophy-land wondering what the 
crystal ball must be in order for it to decide all our questions for 
us.

I suspect that speculations of that kind will lead you down the 
never-to-be-resolved-with-all-parties road, but interesting in the same 
way that nous is interesting and speculations about split-brains and 
such.

> 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.

It depends.  If, say, you consider, for instance, cantor's proof a 
"proof" and not a "proof-in-cantors-system" then it's agreed that 
mathematicians move much more quickly than philosophers.  On the other 
hand, if you think that that question - e.g. whether or not it should 
be considered a proof simpliciter or a proof-in-the-system is something 
about which philosophers are optimistic about then it's somewhat 
doubtful.

> I am not properly classified as a mathematician or as a philosopher.


>> 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.

I expect that it will have then either the enduring value or the speed 
problem.  Perhaps they are related.

>> 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.

Unfortunately my skeptical ears prick up when I hear someone say 
they're not a philosopher but they study the same things that 
philosophers do (and have) for years.  "I'm a philosopher, but not THAT 
kind of philosopher."  We could call physicists "Natural Philosophers" 
and would be right to do so if they were speculative in their tone - 
e.g. if they were trying to resolve which was a better interpretation 
of quantum mechanics, realism or formalism.  Sometimes they do.  When 
they do, they tend to separate it out, if they're rigorous, as the part 
that isn't supported by any evidence.  In "the study of pure 
structure", the question of realism and formalism is nearly the same 
except that we must make do without the evidence altogether.

Best wishes,

Robbie Lindauer