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

[J. Andrew Rogers]

On Sat, Nov 7, 2009 at 1:35 PM, Paul D. Fernhout
<pdfernhout at kurtz-fernhout.com> wrote:
> Only in the sense that all Turing-complete systems are somehow equivalent.
> They may indeed have very different sets of information and representational
> structures associated with them. That's the point in having them -- each can
> get at different answers in different ways depending on what is most
> efficient (minimal energy use or minimal mass or maximum speed) for the
> current context.


They are the same structure. The bottom line is that the encoding of
algorithmic information *must* be lossy, not because mathematics
requires it but because the properties of our universe does. Obviously
predictions/decisions will be different in some contexts depending on
the information you discard.


> If you talk about "ideal", everything biological we see in the universe is
> shaped, to the best of our knowledge, by evolutionary constraints like
> durability, power consumption, response time. Intelligence simply does not
> exist apart from evolutionary processes that shape it.


More accurately, biological intelligence is shaped by evolution, both
in terms of the hardware it had to work with and the environmental
constraints that selected some parts of the phase space out of the
gene pool.  Intelligence in the abstract has no requirement to conform
to such restrictions.


> Theoretically, heavier than air flight was impossible too, or so said many
> people. Even when they saw birds and balloons.


>From the above we can conclude that such people were committing an
elementary logical flaw, or you are conflating an opinion about
practicality with possibility.  Either way, this is not an argument.


> They are all about networks, right? What's the difference? A network of
> volunteer people or a network of volunteer search nodes? :-)


They are different in that they are using different protocols that are
required to make different guarantees.  You know, those all-important
parameters that are absolutely necessary for the mathematical
analysis. It would be highly ironic for you to commit the same exact
failure of reason that you (rightly) accuse economists of committing.


> You're talking math. I'm talking the history of science.


Science is inductive.  It is essentially required to be wrong insofar
as it "being wrong" does not violate mathematics. Additionally, there
are formal methods for ordering a set of hypotheses by probability of
correctness, with "rational" generally being defined as making
decisions based on the highest probability hypothesis.

This is informally referred to as Occam's Razor, though 99% of the
people that refer to Occam's Razor don't actually understand what it
asserts in a strong, formal sense.


> Besides, mathematical proofs have been found to have errors. Wiles first
> attempt at proving Fermat's conjecture was found to have a big error.
> Happens all the time, especially in complex things hardly anyone understands
> about stuff not easy to look at experimentally.


Sure, there are errors in mathematical proofs on occasion, as with any
other human endeavour, but not as many as you think.

Pretty much any theorem you are likely to refer to has not only been
examined by countless people for hundreds of years, but has been
re-examined by computers in recent decades that are much more thorough
and less prone to mistakes.  Indeed, a big meta-topic with regard to
mathematics is the fact that computers are increasingly far better at
proving theorems than humans are.


> For another example, mainstream economists have all sorts of wonderful math
> to explain the economy and guide public policy, but it is almost all harmful
> and based on controversial or clueless assumptions. But, it's easy to
> intimidate people by saying things are "mathematical". So what? What are the
> assumptions? What are the values? What are the emotions? What in the
> rationale for a choice of reasoning tool or approach? What are the limits of
> the tool? And so on...


And at no point in the above was the mathematics incorrect.  If
someone states that mathematics proves the Red Sox are going to win
the World Series this year, you would be an idiot to believe them even
if their execution of the mathematics was perfect.

You can prove you got the right answer, but that doesn't mean you were
asking the right question.  Fortunately, we can also prove that there
are many answers that are impossible to get such that the question
doesn't matter.

The problem I am seeing raised here is that even if you fixed the
questions so that they accurately represented what you wanted to know,
you still do not like the answer.  That is not a problem of
mathematics.


>> What is possible in the unknown areas of science is
>> bounded by what is allowed in mathematics.
>
> And if something did not fit, we'd just invent some new math. What does that
> prove?


Wrong. It doesn't work like that. All mathematics is strictly derived
from a set of axioms that we've been using for ages. You can't change
or discard a part of mathematics you don't like; even the most trivial
change would destroy the whole system. Adding or deleting an axiom
does not have the consequences you seem to be imagining. Mind you, the
axioms we generally use are not absolute truth but they are extremely
effective in practice.

I mean, you can argue that the axioms of mathematics are completely
wrong and therefore the theorems derived from them, but then you would
be in the position of discarding literally *everything* you know and
starting from scratch, right down to invalidating basic counting and
arithmetic.  Most people find mathematics to uncannily effective for
simple day-to-day tasks.


> And every time someone has pretended that reality doesn't apply to some
> aspect of something previously proved mathematically, disaster has ensued.


I reiterate my point: there has never been an instance of reality
violating a mathematical proof. Not once in the entire history of
civilization. If you think such an example exists please feel free to
provide evidence.  There are examples of people using math
incorrectly, but that doesn't reflect on the math and the nice thing
about math is that rigorous review exposes such cases in a
straightforward manner.

There are entire books devoted to the topic of the eerily robust
correctness of mathematics as it applies to the real world.


> You might say, but the assumptions underlaying economics were wrong. Or the
> math was too simplistic. OK, but that's part of my point.  Which mathematics
> with which assumptions? To whose benefit? And who pays the costs?


Who cares?  This is essentially the equivalent of a typo on a form
that causes a bureaucracy to do the wrong thing.  Whose fault is it
that you didn't double check what you entered on the form?


> Mathematics is a huge field, and classically does not even include
> individual based modeling, because if it does, mathematics is just
> computing. And as I see it, mathematics *is* essentially a subset of
> computing.


That is not a correct perception of mathematics.  The computing parts
are mostly subsumed under the very large tent of algorithmic
information theory, but there is an awful lot not under that umbrella.


> Just about all computational models are made for some purpose. They are
> simplifications of reality to some end (economic models usually to make
> someone rich in ration units). So, they are not the same as reality. They
> are a reflection of part of it in some way, perhaps.


Huh?  That doesn't make any sense.  You apparently have no idea what a
computational model is?


> Besides, in general, the way your argument is leading, it is ignoring chaos
> theory. In practice, we don't have ideal information. We make
> approximations. That's real practical discrete mathematics.


You would benefit greatly from actually studying this topic rather
than just talking about it. The idea of computational models as rigid,
axiomatic, binary machines went away several decades ago.

In fact, the kinds of computational models we use to reverse engineer
and predict the behavior of particular humans are essentially based
completely on non-axiomatic term logic.  Or in other words,
computational models that pervasively have no concept of certainty or
absolute correctness at their most fundamental levels. So what is your
objection again?

Mathematics has no difficulty with fuzziness, chaos, or uncertainty.
It would be pretty useless otherwise, and we have not always had these
maths since someone had to fill in the blanks at some point. For
example, almost the entire body of mathematics on intelligence was
formalized within the last ten years, and the required mathematics
mostly did not exist twenty years ago. The nice thing is that once you
have solved it, it really doesn't change unlike science.


-- 
J. Andrew Rogers
realityminer.blogspot.com