[p2p-research] Limits of mathematical modelling (was Re: Building Alliances )

Paul D. Fernhout pdfernhout at kurtz-fernhout.com
Mon Nov 9 06:53:40 CET 2009

J. Andrew Rogers wrote:
 > 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.

The "Axiom of Choice" seems to boil down to saying if you have a numeric 
interval you can pick a number from it. Almost a definitional argument. Like 
aspects of this conversation. :-)

Which also has become a digression from you squirming away from the main 
point, which is that there are alternative ways to put together resources to 
solve big problems. There's been a deafening silence in regard to the post I 
made about GNU/Linux's cost being in the billions of dollars to do by a 
centralized financial system. So, you're ignoring the existence proof, and 
putting up side issues.

Exhibit A:
"Concentrations of wealth are extraordinarily useful for real progress.
It is how most really great ideas get funded."

Exhibit B:
"It would be a death sentence for the development of many, many
innovations if one had to aggregate funds from a large number of

Now, you've qualified those with "most", and "many, many", but the subject 
of this  list is mostly peer production as exemplified by GNU/Linux and 
Wikipedia (or even the web itself seen as a totality) and things that flow 
around that (alternative economics, envisioning a better society).

So, you made the points above. Then you said:
Exhibit C:
"This planet is desperately short of people capable of usefully
contributing to non-trivial R&D efforts. A million average people with
no particular expertise cannot replace one extremely brilliant person
with deep expertise. If this was not true, "design by committee" would
not be pejorative."

And have proceeded to go on and say fairly derogatory things about people on 
this list (including myself) as well as this community in general. While 
ignoring that an essential aspect of p2p is how understanding emerges out of 
the community through a stigmergic and interactive process.

You also said, responding to what I wrote:
Exhibit D:
 > But, one could imagine that the people who started Google had just
 > published their idea, and further, that someway was found to adapt it,
 > in a "Folding at Home" distributed computer way to use idle CPU cycles
 > and a p2p meshwork. We might not need Google server farms at all to
 > do fast searches. But, there is not much incentive for people to spend
 > millions or billions developing that idea, even if it was better and
 > more secure and more democratic, because, in a captilist society,
 > where is the profit?
There are well-understood theoretical reasons why that won't work.
Centralization was required, not a design choice. No need to look for
the profit bogeyman when simple mathematics will do.

So, there you are justifying the current social order based on "simple 
mathematics" and "theoretical reasons" even when given a plausible 
alternative. Which is all starting to sound a lot like this:
   "The Mythology of Wealth"
"Justifications for elites and social hierarchy goes all the way back to the 
pharaohs. ..."

Then you said this:
Exhibit E:
I did research a few years ago on the mathematics of optimally
efficient and pervasively decentralized networks. This theoretical
area is very important because it is used to prove all sorts of things
about cooperative systems of independent agents. It is literally the
mathematics of dynamics of P2P systems in the abstract and very
challenging theoretically with many unanswered questions. One thing
that has surprised me is that it is never discussed here even though
it is very relevant.
   One of the critical theoretical problems of such systems is that there
are only two known strategies -- from two different theoretical
derivations -- for P2P system design that do not decay into
pathologically suboptimal equilibria. If you have a system that
necessarily cannot be constrained to those strategies, the system is
not stable in a well-functioning configuration.  The reason I was
researching it at all was because there was an interesting real-world
political policy problem that could not use either known strategy; I
never found a satisfactory solution, but I suspect one might not even
exist for that case.
   The problem is that the two robust strategies in literature (someone
may have come up with a new one in the last couple years, probably
not) are both effectively based on adaptive market-like pricing
mechanisms. You can solve the problem with strong centralization, but
that has its own problems in real systems e.g. single point of
   I don't have answers, but designing theoretically stable, strongly
decentralized P2P systems that do not have major provable pathologies
is very, very hard outside of some fairly narrow cases. I am very fond
of the idea of strongly decentralized P2P economies but I haven't seen
much in the way of a rigorous formulation of such a system here that
would be both robust and without significant pathologies.

Once again, you say there are reasons, or there is a literature, and do not 
explain them, and do not cite anything specific, and then proceed to make a 
broad sweeping claim that, essentially, everything people are doing in P2P, 
like Wikipedia, Debian GNU/Linux, Apache,  and the rest of the web, and so 
on, can't possibly work on theoretical grounds. Well, you use wors like 
"decay into pathologically suboptimal equilibria" but that seems to me what 
you are trying to say, that it won't work. Even when it does. And then 
proceed to tell us we should be studying math to prove they work. Well, math 
is nice. Math may be useful. But, I don't see the connection between math 
and your points about how P2P will never fly, like Lord Kelvin said heavier 
than air vehicles will never fly, even when there were birds flying all 
around him.

Or, maybe I truly just don't understand what you are trying to accomplish here?

 >> 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.

This is a beautiful sentiment. But, it is not true. :-) Or rather, it is 
only true when you define science differently than math. If math and science 
are considered as their historic literatures and a bunch of people working 
together to build on that literature as a social process, then there is no 
difference. Both involve an accumulation of facts and ideas and examples. 
You hand wave away mistakes in proofs, or unproductive unpublished 
explorations in math as not important, but they happen all the time. So, is 
math as a human endeavor really than different than science in that sense?

The fact that people think heavier than air aircraft can fly now, and have 
theories about it, does not invalidate that the literature exists relating 
to statements with rationales for why they could never fly. Some things 
about flying still remain controversial arguments, like which is more 
important, the kite effect or the Bernoulli effect; otherwise, some ask, how 
could aircraft fly upside down?

You have made statements about math and P2P, as in the exhibits above. You 
offer no evidence to back them up. You don't even say what "the problem" is. 
I don't see that as making a solid point. I'm happy to read and listen to 
what you have to say about this -- if you said it. :-) Rather than just 
saying, we don't understand your handwaving. Mathematics is not the same as 
bullying with confident sounding terminology. If you have a truly deep love 
of mathematics you wished to share with us on the list, then by all means, 
we could all benefit from that.

Maybe being financially rewarded for doing math has destroyed your love of 
the subject? :-(

In which case, please don't blame the people who are trying to help fix the 
larger social situation if you are becoming alienated from that which you 
have obviously long cared about:
   "Studies Find Reward Often No Motivator: Creativity and intrinsic 
interest diminish if task is done for gain"

>> 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.

True. Probably one reason I did not do so well in a PhD program in 
Operations Research and Statistics at Princeton. :-)

In the end, I did learn to have more appreciation for networks and 
centralization (and now cite Manuel de Landa as penance. :-)

On the other hand, I was right that their mathematical models, with all 
sorts of proofs and lots of fancy handwaving, focusing on "optimizing" to 
maximize short-term profits were extremely risky to our civilization, and 
now, twenty years later, many trillions of dollars have been forked over to 
the bankers to paper that over, and tens of millions of people are without 
income, we face all sorts of other instabilities from wars over oil profits 
and such, and many other aspects of our society are failing, and you can 
thank people like at Princeton University for all that. "Picking up nickels 
in front of a steamroller", indeed:

No hard feelings though: :-)
   "Post-Scarcity Princeton, or, Reading between the lines of PAW for 
prospective Princeton students, or, the Health Risks of Heart Disease "

Even though everyone else got the decades of health insurance, nice offices, 
students learning from then, pensions, nice walkable surroundings full of 
happy people, and so on. All while they were helping set up one of the 
biggest swindles in history and potentially setting the stage for even more 
huge wars. All based on "math" and "certainty", using language much like you 
have been using.

Again, "math" is really a meaningless term. It's almost like saying you 
solve problems with chalk on a chalkboard. What are the assumptions? What 
are the values? What tools have been chosen and what are their limits and 
areas of applicability? What things are uncertain? What are the unknown 

Now, you yourself may understand that. But the language you use here does 
not seem to reflect that. Again, the exhibits above.

>> 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.

Again, it shows the limits of people trying to build what might seem like 
the simplest of mathematical models -- some water molecules interacting.

What do you mean by "validity of mathematics"? Again, from what I've been 
reading here, it seems like you are defining "valid" and "mathematics" as 
essentially the same thing. If it's invalid, it can't be mathematical. If 
it's valid, it is or will be part of mathematics.

Maybe the issue is that I read an ellipsed "the" in front of mathematics?
So, I read "The validity of *the* mathematics is not dependent on better 
measuring tools or the next CPU upgrade." Because, as I see it, that's what 
one can talk about. If you want to talk about "mathematics" then we have to 
talk about a social enterprise. If you talk about "the mathematics" than we 
need to talk about assumptions, values, choice of tools, and so on.

Anyway, that's another reason we are talking at cross-purposes here. It's 
almost like you had said "The science of p2p is flawed, as I know from 
studying it" and we said, "Where is the evidence? Can you cite anything? 
What specifically do you object to?" and then you start talking about how 
wonderful the scientific method is for arriving eventually at some truths. 
Well, sure, math and science are nice (although, not the only ways to relate 
to the universe or other people). But, moving right along, where is the math 
about what specific problem that you are saying invalidates what specific 
aspect of what we are talking about here?

I'm not saying math is not important. It is. You're right to make the point 
that not much math is discussed here and we may all benefit from some of 
that. I'd agree with that. But that seems to be the starting point for 
contributions by you or others, not the ending point of some argument about 
P2P like you make above.

What are the P2P questions you think math could help solve? And can you give 
real life examples of the problems? You point in the above Exhibit E above 
to "pathologically suboptimal equilibria". Can you give a real life example 
of that? Does Debian GNU/Linux exhibit that somehow?

 >> 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.

Good point. If you have a computational view of the universe, which I am 
inclined to myself. Still, they do represent different paradigms. Different 
types of models.

Just because you can model chemistry as physics does not meant that 
chemistry is physics. They use different models for various practical reasons.

A joke on this I worked out on this a while back. :-)

All philosophy is sociology.
All sociology is psychology.
All psychology is biology.
All biology is chemistry.
All chemistry is physics.
All physics is mathematics.
All mathematics is philosophy.
(See the first point. :-)

 >> 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".

There is so much talkie-talk at the edge of math that yes, it can be "wrong" 
even within the bounds of math. Again, Andrew Wiles' first proof of Fermat's 
conjecture was wrong. I'm even suspicious of the current one, because it is 
so long and convoluted. :-)

Still, right or wrong, Andrew Wiles was doing math when he worked on that 
proof. Glorious, courageous math (even hiding away in his attic for seven 
years because he knew all the other mathematicians would laugh at him, call 
him crazy, and maybe lock him up in an attic somewhere. :-)

And, from another angle, if Andrew Wiles had to hide his work for almost a 
decade, was he not, at least in a social sense, doing "wrong" math?

The same thing happens in science. Some breakthrough Nobel-prize winning 
"high temperature" superconductor work at IBM Research was done on the sly 
because the higher ups did not think it worthwhile. They don't mention that 
here though:
   "IBM's Zurich Research Lab celebrates its 50th anniversary "
"Also in 1986, ZRL scientists K. Alex Müller and J. Georg Bednorz discovered 
high-temperature superconductivity, for which they received the physics 
Nobel prize for following year."

Really, they risked their jobs messing with that stuff. It was not what they 
were supposed to be working on. Had they been found out before repeatable 
success -- most likely fired for insubordination, I think.

Chalk on a blackboard is not wrong. It simply is. :-) But, what you can 
write on a blackboard in your office is socially constructed and socially 

 > Applications of mathematics can be wrong, but
 > you don't invalidate mathematics just because someone forgets to carry
 > the one when balancing their checkbook.

Good point.

On the other hand, you've also defined away most of the interesting issue 
about math. Because, much of how mathematical models are used is exactly 
like balancing a checkbook. You have assumption (initial balance, accuracy 
of inclusion of all the right deposits and deductions), values (what you 
decide to spend money on), and choice of logical reasoning tool (double 
entry method? Single entry? Paper? Spreadsheet? Online?). Big questions like 
"cash flow" may never enter your mind, staring at a checkbook. After all, 
it's money, and you're tracking it. What else could there be to know about 
cash? Also, things just out of the blue, like hyperinflation, can render the 
whole activity meaningless.

 > If you don't like mathematics,
 > you don't have to use it.

Well, if computation underlies the universe, and math is a subset of 
computation, I can't avoid using it as long as I'm in this universe. :-)

 > Invent your own mathematics if you wish, it
 > is certainly allowed.

And people do that all the time, in the sense of creating both new ways to 
model things, and using old ways to build models about new things.

Again, you're trying to make this an issue of "mathematics" while I'm 
sticking a "the" in there, to talk about "the mathematics" of something.

It seems like we've got at least there mathematics running around here:
* "mathematics" as you want to use it, which is the dross of everything that 
is self-consistently valid.
* "the mathematics" relating to a specific thing people want to model
* "mathematics as a social enterprise" which actually is a stigmergic P2P 
thing, as I reflect on it just now, like all academic things involving 
publishing and indirect collaboration through a literature.

In that sense, I guess, if I were to be accept your earlier point on the 
impossibility of p2p, especially considering most early famous 
mathematicians did math as a hobby or for spiritual or aesthetic reasons 
(not for a paycheck), then mathematics can not exist, because the p2p social 
processes surrounding it for the last few millennia long ago should "decay 
into pathologically suboptimal equilibria".

So, as a proof, working together, we've just proved mathematics does not 
exist. :-) Wow. We make a great team. :-)

But, mathematics does seem to exist. A contradiction!

So, with a contradiction in hand, we can prove anything:
"If you ring a contradiction into a theory (however subtly) you can prove 
anything you like symbolically."

So, this means, we've just proved by formal logic that P2P works! :-)

 > What it boils down is that you don't like the fundamental processes of
 > mathematics.

I don't know about that. You seem to be the one who is not willing to 
acknowledge the social and iterative dimensions of the activity?

 > It seems you are looking for a way to falsify a theorem
 > that does not comport with a preconceived notion,

No need for that now. I've got a contradiction. Now I can prove anything I 
like, and there is nothing anyone can do mathematically to stop me. :-)

Hmmm. Maybe I should use that contradiction to prove I have a million US 
dollars? :-)

Step one: One of: "I have a Million dollars" and "I don't have a million 
dollars" is true.

Now, how does the rest of all that go again? :-) Ah, it would be too easy to 
make money that way. Not enough fun. :-)

 > 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.

Again, we're talking past each other here. I'm making very specific points 
about mathematics as a social enterprise and mathematical models used by 
people for specific purposes. You keep shifting this to be a referendum on 
whether chalk works on chalkboards. Sure, it works. Now what are we going to 
draw on the chalk board and what will it prove?

 >> 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.

Ah, see, here comes magic words. "Intractable". "High degree of certainty in 
any case many times". "Cheaper".

Or, in other words:
   "How To Speak Hedgie: What hedge-fund managers mean when they talk about 
"In these days of market volatility, hedge-fund managers and executives at 
all types of money management firms have been forced to explain why their 
funds are shutting down, losing money hand over fist, and freezing 
investors' funds. When they do so, however, they frequently lapse into a 
strange euphemistic dialect. And so we thought it would be helpful to 
provide a handy Hedgie-English glossary."

A translation table:

"Intractible" == "Mathematics can't solve it in practice, even it can in 
theory if we have complete control over all matter and energy in an infinite 
number of universes."

"High degree of certainty in any case many times" == "Wrong a lot"

"Cheaper" == "We only do what we can do easily, and leave the hard bits for 

Or, in other words, that's all why chemistry is not physics, and biology is 
not chemistry, and so on. And why mathematics only gets you so far, given 
incorrect assumptions, disagreements over moral values, and the limitations 
of various tools.

 > 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.

Completely *interrelated* things. Along with the rest of human knowledge.

 >> Science is not "inductive"...
 > Huh?  At its core, that is all science is.

"Induction, also known as inductive reasoning or inductive logic, is a type 
of reasoning that involves moving from a set of specific facts to a general 
conclusion. ... Criticism of inductive reasoning Inductive reasoning has 
been attacked several times. Historically, David Hume denied its logical 
admissibility. Sextus Empiricus questioned how the truth of the Universals 
can be established by examining some of the particulars. Examining all the 
particulars is difficult as they are infinite in number.[2] During the 
twentieth century, thinkers such as Karl Popper and David Miller have 
disputed the existence, necessity and validity of any inductive reasoning, 
including probabilistic (Bayesian) reasoning.[3]"

But in any case, you clipped that point. Here it is again in its rhetorical 
entirety: "Science is not "inductive"; science is a social enterprise, and 
many scientists spend a lot of time thinking inductively."

And I stand by that. Again, you're trying to make this IMHO a referendum on 
whether validity (math) is valid. :-)

This particular choice of quote bothers me a lot, because it's like you're 
intentionally avoiding thinking about my point of math and science as social 
enterprises (and fairly p2p ones at that).

Normally, I might not care much. But in this case, it ties in understanding 
the whole p2p issue which is also itself a social enterprise issue. So, your 
choice here to miss my point, fits in with trying to skip around the bigger 
issue of peers sharing socially and using handwaving and mathematical jargon 
to denigrate p2p and the people on this list.

So, it seems to me at this point like you are being willfully evasive on 
this issue of the functioning of social networks in general. But, that's 
what this list is about in a deep sense (at least, social networks doing 
peer production in a not strictly hierarchical way).

>> 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".

Oh, come on. People probably hacked the code together and some 
mathematicians came around later and started talking about it. :-)

But that is a pereniall issue with engineering vs. science/math. Engineers 
do the impossible. Then scientists using math or whatever deny it, and then 
eventually grudgingly explain it, and likely figure out some generalities. 
The engineers then take those generalities and make even more amazing stuff 
instead of flying by the seat of their pants. And somewhere along the line, 
the engineers do the impossible again, and the whole thing starts over. :-)

And sometimes, the cycle goes the other way. A theoretician says, I think 
this new thing should be theoretically possible (antimatter? lasers?) and 
engineers set out to make it.

 > That's fine, it is not as though you are qualified to make such
 > determinations anyway.

Seriously, you probably know little about me. Why say that? What does it 
accomplish? Even let's say everything I've said is stupid. Maybe I'm having 
a bad day. Or a bad week. Or bad year. Why make such a sweeping generalization?

 > 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.

Ah, at least here we have a "the". :-) "The math". :-) That's good.

Still, let me translate that without the "the": "P2P is fragile and 
non-viable, unless you do lots of math first".

OK, let's assume that was true. Then how do you explain Wikipedia? Debian? 
Apache? The Web? Email? Twitter? Facebook? Science? Math? :-) And so on?

You're sounding here, to me, like a mathematics faculty person berating a 
grad student. :-( But, maybe, admittedly, that's just projection from my own 
psychological baggage. :-)

Another way to look at this is, if you, say, define "good thinking" as 
"math", then whenever people think in some productive way it is "math".

Still, sometimes it seems to me like there really is some sort of "cult of 
math" that has dominated much of engineering and science in academia. It's 
understandable because people get good at some branch or twig on the 
mathematics tree, learning to use some mathematical tools really well, and 
they like to use it and tell others about it. It's only human. But it's also 
convenient, because if engineers and scientists were encouraged (or even 
just allowed) to think about assumptions, values, and the limits of their 
tools, the world might be a much nicer place -- though likely less 
profitable in the short-term for a very few.

Again though, you make a sweeping claim with no specifics. No list of open 
questions. No literature references. No web references. Just blanket 
condemnation at this point. And no attempt to help people who have been 
willing to spend a significant amount of their time reading what you wrote 
and replying to it.

I'm not writing this to disagree with you, so much as to help you (and 
others) see some of these issues more clearly (at least, clearly from my own 
hazy point of view. :-)

 > Really, if I have one complaint it is this. When faced with hard facts
 > that contradict preferences,

Have you introduced even one citeable (linked to any literature) hard fact 
into this discussion?

Oh, there's probably one somewhere. Maybe even a few. But not as regards 
your main points, at least that I recall.

 > the reaction here seems to be knee-jerk denial of reality.

Well, reality is a slippery topic. :-) Especially if it is a simulation. :-)

But, as I see it, we are raising questions about assumptions, values, and 
tools, including in regard to what you write here, and you are ignoring all 

 > Instead of doing something constructive like
 > understanding the limitations well enough to work around them, we'll
 > pretend the limitations don't even exist.

Well, what are the limitations of mathematical inquiry as a social process? :-)

And how has P2P influenced the development of mathematics?

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly 
French) 20th-century mathematicians wrote a series of books presenting an 
exposition of modern advanced mathematics, beginning in 1935. With the goal 
of founding all of mathematics on set theory, the group strove for rigour 
and generality. Their work led to the discovery of several concepts and 
terminologies still discussed.
   While Nicolas Bourbaki is an invented personage, the Bourbaki group is 
officially known as the Association des collaborateurs de Nicolas Bourbaki 
(Association of Collaborators of Nicolas Bourbaki), which has an office at 
the École Normale Supérieure in Paris.

Sounds P2P-ish to me, involving peer production to build a mathematical 
commons. :-)

What really happened there?

I'm not saying it's a great example of modern P2P, but it's a starting point 
for thinking about it as far as P2P and mathematics as a social process.

--Paul Fernhout

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