[p2p-research] Building Alliances (limits of mathematical modeling)
Paul D. Fernhout
pdfernhout at kurtz-fernhout.com
Mon Nov 9 00:52:26 CET 2009
How about the work of Kurt Goedel? :-)
http://en.wikipedia.org/wiki/Kurt_G%C3%B6del
"Gödel is best known for his two incompleteness theorems, published in 1931
when he was 25 years of age, one year after finishing his doctorate at the
University of Vienna. The more famous incompleteness theorem states that for
any self-consistent recursive axiomatic system powerful enough to describe
the arithmetic of the natural numbers (Peano arithmetic), there are true
propositions about the naturals that cannot be proved from the axioms. To
prove this theorem, Gödel developed a technique now known as Gödel
numbering, which codes formal expressions as natural numbers."
See also:
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem
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? :-)
http://en.wikipedia.org/wiki/Parallel_%28geometry%29
"""
While in Euclidean geometry two geodesics can either intersect or be
parallel, in general and in hyperbolic space in particular there are three
possibilities. Two geodesics can be either:
1. intersecting: they intersect in a common point in the plane
2. parallel: they do not intersect in the plane, but do in the limit to
infinity
3. ultra parallel: they do not even intersect in the limit to infinity
In the literature ultra parallel geodesics are often called parallel.
Geodesics intersecting at infinity are then called limit geodesics.
"""
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.
While I agree there are aspects of math that do not seem formalizable as
computing (because they are mostly thought stuff and sometimes handwaving),
mathematics as a whole is more like a bunch of software packages. People
build new packages to do new things. They sometimes find bugs in old
packages. They more often find bugs in new software that is being written on
top of old software packages. People get packages working pretty well and
then tend to forget about them, until something happens and they break. So,
we think using a system of geometry where parallel lines never meet, until
suddenly non-Euclidian geometry becomes important.
Once you move beyond the basics, and actually use math to do something, the
complexities and uncertainties and assumptions pile up pretty quickly.
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. :-)
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? :-)
And that's even ignoring the more profound social statement by Muriel
Rukeyser, poet that: "The universe is made of stories, not atoms." :-)
Essentially, I suggest the field of math is the continuum from first assumed
principles through data acquisition to the output of the models and beyond
as a social enterprise (and thus many of your points, while technically
true, are irrelevant), and you are defining "math" as the part of that large
enterprise which is true. :-) Well, I can't disagree with you that math is
true if you've already assumed the point which is under discussion. :-)
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. But it seems to be
that where math goes wrong, given people with computers have gotten so good
at numerical calculations and formal logic, is in hazier areas like
assumptions, scaling, problem definition, data, and so on, as I mentioned.
Another way to look at this is the term "emergent properties". So what if we
do have perfect models of quantum wave functions for the basic elements? We
don't have any practical way of using that to make large predictions about
the universe without infinitely large computers and lots of data
(approaching or exceeding the size of the universe). So, the math in that
case is useless in many cases, and we resort to other simplifications
instead. So, we can't even model a couple of water molecules interacting at
the quantum level, but we fudge it instead and move on.
Anyway, perhaps to make progress on this, we would need to talk about the
definition of "mathematics". Since obviously, we seem to be talking about
different things. :-)
A starting point:
http://en.wikipedia.org/wiki/Mathematics
"""
Mathematics is the study of quantity, structure, space, and change.
Mathematicians seek out patterns, formulate new conjectures, and establish
truth by rigorous deduction from appropriately chosen axioms and definitions.
There is debate over whether mathematical objects such as numbers and
points exist naturally or are human creations. The mathematician Benjamin
Peirce called mathematics "the science that draws necessary conclusions".
Albert Einstein, on the other hand, stated that "as far as the laws of
mathematics refer to reality, they are not certain; and as far as they are
certain, they do not refer to reality."
Through the use of abstraction and logical reasoning, mathematics evolved
from counting, calculation, measurement, and the systematic study of the
shapes and motions of physical objects. Practical mathematics has been a
human activity for as far back as written records exist. Rigorous arguments
first appeared in Greek mathematics, most notably in Euclid's Elements.
Mathematics continued to develop, in fitful bursts, until the Renaissance,
when mathematical innovations interacted with new scientific discoveries,
leading to an acceleration in research that continues to the present day.
Today, mathematics is used throughout the world as an essential tool in
many fields, including natural science, engineering, medicine, and the
social sciences. Applied mathematics, the branch of mathematics concerned
with application of mathematical knowledge to other fields, inspires and
makes use of new mathematical discoveries and sometimes leads to the
development of entirely new disciplines. Mathematicians also engage in pure
mathematics, or mathematics for its own sake, without having any application
in mind, although practical applications for what began as pure mathematics
are often discovered later.
"""
Although, I personally feel the Wikipedia definition leaves out a stronger
relationship with computation, since the study of "change" in practice does
not really concern that (it's more like calculus that is meant there) and
computer science and discrete math as disciplines don't even in practice
extend to what I mean. It's sort of a necker cube paradigm shift to see math
under computation and programming instead of computation and programming
under math. :-)
http://en.wikipedia.org/wiki/Necker_cube
On that page on mathematics is this: "Common misconceptions: Mathematics is
not a closed intellectual system, in which everything has already been
worked out. There is no shortage of open problems. Every month,
mathematicians publish many thousands of papers that embody new discoveries
in the field."
Science is not "inductive"; science is a social enterprise, and many
scientists spend a lot of time thinking inductively. In any case, for
modeling reality, including p2p, we make simplifications and we choose our
tools and our assumptions, and do that in a social context. So, the results
are imprecise, but may still be useful for survival as individuals or a
species or a biosphere or noosphere. And sometimes we come up with new
paradigms more appropriate to new conditions. 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. The math you say you saw or did may well be correct within
its assumptions, but that's like saying you have a program that runs on
Java's JVM that prints "P2P peer production does not work". :-) It doesn't
really tell me anything that lets me evaluate if the program is suitable for
some specific purpose or that its conclusions were not just straightforward
implications of the assumptions.
--Paul Fernhout
http://www.pdfernhout.net/
J. Andrew Rogers wrote:
> 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.
>
>
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