[p2p-research] Building Alliances (limits of mathematical modeling)

[J. Andrew Rogers]

On Sun, Nov 8, 2009 at 3:52 PM, Paul D. Fernhout
<pdfernhout at kurtz-fernhout.com> wrote:
> You seem to be using the term mathematics below like it was one unified
> whole, inclusive also of physics, and one where all the conclusions flowed
> from a few basic unarguable assumptions (like, do parallel lines touch at
> infinity? :-)


On the contrary, I was fairly explicit that the axioms of mathematics
are arbitrary, selected for simplicity and power rather than any
universal truth. See, for example, the Axiom of Choice which was
controversial at first but which has become generally accepted because
it could be used to prove a large number of important theorems that we
could not prove without it.


> There was a time when all that was not understood about geometrical
> possibilities. What mathematical issues now are the same? We may think we
> understand them only because we do not see the other possibilities, as you
> suggested elsewhere, some assumption made decades ago that gets propagated
> through the mainstream thought on some subject.


You are conflating deductive axiomatic systems with inductive
non-axiomatic systems; algorithm design is an engineering discipline
even though you can (sometimes) prove the properties of a particular
design.  In mathematics, a theorem is strictly proven from a set of
axioms.  There is no "think we understand", it is either proven
absolutely or not. There is plenty of mathematics that is not
well-understood in a formal sense. The job of a mathematician is
nominally to fill in those holes so that mathematics can move on to
the next hypothesis.

While new patterns and relationships may be found, they do not and
cannot invalidate anything that has already been proven.  We may get
new knowledge, but it never destroys or invalidates old knowledge in
this context.  Math is not science.


> You seem to be doing a standard mathematicians trick here. :-) That trick is
> to take any messy and interesting part of the problem and define it as
> outside the scope of the area of study. Or, alternatively, the trick is
> doing some handwaving that because we have some guesses about quantum wave
> functions, the simulation of large universes are left as an exercise to the
> reader, but are proof that mathematics we now know covers everything going
> on in the universe. :-)


You have some very strange ideas about what mathematics is and how it
functions.


> So, we have books on, as you say, "the eerily robust correctness of
> mathematics as it applies to the real world", but the fact is, even with our
> best supercomputers, it is my understanding (from a few years back) we still
> can't 100% accurately model how a few molecules of water interact at the
> quantum level. How are those two statements reconcileable, other than to
> state that people who like math are often willing to look the other way? :-)


Your example above, modeling molecular interactions, has nothing to do
with mathematics nor does it say anything about mathematics. Your
assumptions about these things are sufficiently wrong that you are not
making much sense. The validity of mathematics is not dependent on
better measuring tools or the next CPU upgrade.


> And that's even ignoring the more profound social statement by Muriel
> Rukeyser, poet that: "The universe is made of stories, not atoms." :-)


Meh. The universe is made of information, which includes both stories
*and* atoms.


> Or, another trick is to say, math can't be wrong, because if something is
> wrong, it isn't math. Well, it's hard to argue with that.


Again, you are betraying a deep misunderstanding of the basics of what
mathematics is.  Math isn't "wrong" within the context of the accepted
axioms, it simply "is".  Applications of mathematics can be wrong, but
you don't invalidate mathematics just because someone forgets to carry
the one when balancing their checkbook. If you don't like mathematics,
you don't have to use it.  Invent your own mathematics if you wish, it
is certainly allowed.

What it boils down is that you don't like the fundamental processes of
mathematics. It seems you are looking for a way to falsify a theorem
that does not comport with a preconceived notion, but the only way to
do that is to discard conventional mathematics *entirely* and start
with a new set of axioms of your choosing.  That seems like too much
work (true) so you are trying to find a way to selectively edit
mathematics to your liking.


> Another way to look at this is the term "emergent properties".


The term "emergent properties" is used in the *application* of
mathematics for systems where the properties of the system have not
been formally proven because it would require too much work or because
no one built a proper non-statistical model.  It isn't all laziness,
sometimes the proof would be intractable and we can measure the
properties of the system inductively to a high degree of certainty in
any case many times.  Inductive tests aren't "proof" in the absolute
sense, but they are much, much cheaper. Just look at the amount of
work required to formally prove a small piece of software and you'd
know why.

A lot of laypersons think "emergent systems" or "emergent properties"
is a codeword for "magic", but it really isn't.  In fact, computer
science has mostly stopped using the term "emergent" to describe
systems because it gave too many people the wrong idea.


> So, we can't even model a couple of water molecules interacting at
> the quantum level, but we fudge it instead and move on.


That is science, not math. Completely unrelated things.


> Science is not "inductive"...


Huh?  At its core, that is all science is.


> So, when you, earlier make
> sweeping statements about the "stability" of p2p networks and so on, by the
> way not citing any specific literature, well, that's why I take them with a
> grain of salt.


This email was routed using a protocol proven using the mathematics
you are taking "with a grain of salt".

That's fine, it is not as though you are qualified to make such
determinations anyway. It is telling that you choose to ignore the
math instead of working on the open questions in math related to P2P
that could help make it robust and viable.

Really, if I have one complaint it is this. When faced with hard facts
that contradict preferences, the reaction here seems to be knee-jerk
denial of reality.  Instead of doing something constructive like
understanding the limitations well enough to work around them, we'll
pretend the limitations don't even exist.


-- 
J. Andrew Rogers
realityminer.blogspot.com