The Political Economy of Sets
H. de Bruijn
that ST suffered from (Russell's) paradoxes from the very beginning. This would
have assassinated any other kind of mathematical theory. It is remarkable that
Set Theory survived its shortcomings in the first place. Big surprise; it even
became the foundation "par exellance" whereupon Modern Mathematics is based.
From a rational point of view this must sound like a true miracle. It heavily
reminds to the way our Religions are still going strong, despite manifest lack
of scientific content. So there must be something out there which is even more
convincing than logic. Let's see what it is.
A satisfactory explanation can be found only IFF people dare to recognize that
Mathematics is a humble activity of human beings. This implies that mathematics
is bound to the historical and social restrictions in the first place. Yes (!),
I want you to get rid of the idea that Mathematics is independent of society.
How can somebody for example conceive the thought that the whole of mathematics
is made up from nothing else but Sets? This would be impossible if not society
itself had'nt adopted the shape of an "ungeheure Warensammlung" (unprecedented
collection of goods: Karl Marx in "Das Kapital"). Is it a coincidence that the
birth of Set Theory has its social analogue in the enormous accumulation of all
kinds of richness which marks the turn of the century? Is it a coincidence that
Georg Cantor's father himself was a merchant, so that his son became *very*
familiar with those huge "sets" in the storehouses of his family?
So we may conclude in the first place that the birth of Set Theory was inspired
by social circumstances. But this is not the end of the story. Even nowadays,
nobody can think of an idea which fits better the view of the Capitalist System
than Set Theory. (Go to a supermarket, and convince yourself!) This means that
no rational arguments can be used in order to deprive ST from its predominant
role in mathematics. Read my lips: I *don't want* to get rid of Set Theory as
a (less important) part of Mathematics.
Mathematical concepts originate and become important within the context of our
human society, with all its non-logic and non-scientific taboos.
But it is also thinkable that certain concepts will NOT originate in the given
social circumstances, simply because such new concepts would have unacceptable
economical and political (and personal) consequences. New ideas do not come,
essentially because, deep in our heart, we DON'T WANT them.
So maybe, yes, further progress in mathematics is *inhibited* by the way our
societies are organised ...
--
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354 | would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands. | Mathematics" (HdB). * for
* E-mail: rct...@dutrun.tudelft.nl ---| Fax: +31 15 78 37 87 ----* Blood
Allan Adler
I am pleased to see that Hans de Bruijn has decided to present his views
in a more organized form. As with many complicated ideas, when I am first
exposed to them, I feel as though I am walking into the middle of a
conversation already long in progress. Therefore, I hope Hans will not
mind if I ask some very stupid questions in order to try to get my bearings.
First of all, Hans, there are those who share your opposition to set theory
as the foundation of mathematics (without appealing to Karl Marx). For example,
some people feel that category theory is a more suitable foundation for
mathematics.
Second, would you be satisfied if set theory were abandoned as a foundation
but retained as a subordinate area of study ? Or is this merely the first
of a long list of demands ? If so, it is perhaps better to let us know now
what you really want.
Third, is this all that can be said of the relationship between Marx and
mathematics ? That Marx laid down the laws according to which economies
and societies work and that mathematics is a part of society and therefore
is subject to Marx's theories ? Is it not the case that Marx spent a lot of
time teaching himself calculus hoping to apply it to economics, that his
notebooks on the subject were published by the DDR, and that he shows no
special insight into mathematics per se ?
Incidentally, set theory is not the only area of controversy in mathematics.
For example, there was a controversy over the use of infinitesimals. How do you
feel about the use of infinitesimals ?
Then there are the teachings of Rudolf Steiner, who felt that there does not
exist a concept of number, but that each number is itself a concept. Where
do you stand on that issue ?
Then there are the followers of Lyndon Larouche who, like you, feel that
mathematics has political implications and ought to be politically correct.
Can you comment on some of their writings ?
Returning more to the mainstream, how do you feel about the Intuitionists, who
reject the law of the excluded middle ?
Since we are questioning everything, how do you feel about the very language
that we use to understand you and which you use to express yourself ? What
about the language Marx used to express himself ? Words and language are
themselves aspects of human activity and therefore, from your point of view,
subject to the grand vision of Marx. What have the capitalist imperialist
devils and their craven lackeys done to the language and how can we ever trust
what we express in words now that they have done it ? In particular, how can
we trust what you say ?
I know you were only posting a sample of your philosophy, but as you release
future installments, I look forward to seeing you explore your own ideas and
their consequences in more depth.
Allan Adler
gh...@ms.uky.edu
William Ricker
If you're looking for an alternative foundation of mathematics....
The (in)famous John Horton Conway has spoken on, and I believe
published in a collection, a delightful alternate foundation based on
Game Theory, of all things. His construction of Natrual, Rational,
Algebraic, Irrational, Transcendental, and transinfinite numbers as
(rather peculiar) Games is very illuminating and quite simple. (I
think he also constructed a new class of numbers (hyper infinite?) and
found and "application" for them).
As I recall, he stated that his colleagues believed it could be
used as a complete foundation, and some were dabbling ocassionally at
rebuilding the key theorems using his game-theoretic number-theory as
basis. JHC did not recommend using GTNT as a professionaly
replacement for set theory as foundation, but (i'm working from loose
memory here) did recommend further study of it as a basis (a) for
pedagogic purposes, (b) for finding inspiration for hard problems, and
(c) for seeing strange new beauties of the existing structures.
Given time, I can probably unearth the reference from where-ever I
scribled it. Thanks for reminding me; it was a fun lecture. (At
local MAA section meeting, wtih JHC as featured Speaker while the
elections committee counted votes.) I could also dig out my notes,
but the multi- colored figures would translate to ASCII at best
laboriously.
--
/s/ Bill Ricker w...@wang.wang.com
"The Freedom of the Press belongs to those who own one."
*** Warning: This account is not authorized to express opinions. ***
Dan Hoey
>If you're looking for an alternative foundation of mathematics....
> The (in)famous John Horton Conway has spoken on, and I believe
>published in a collection, a delightful alternate foundation based on
>Game Theory, of all things....
Yes, but we should be careful to understand we are talking about
combinatorial games of perfect information. The topic ``game theory''
often refers to probabilistic games such as poker.
>Given time, I can probably unearth the reference from wherever I
>scribbled it.
It's been published as a wonderful little book, _On_Numbers_and_
_Games_ (Academic Press, 1977). I hope it's still available.
Despite the larger and more recent _Winning_Ways_ (by Berlekamp,
Conway, and Guy, Academic Press, 1982) there's a lot of good stuff
you'll only find in ONAG.
Dan
Chrystopher Lev Nehaniv
> Is it a coincidence that
>Georg Cantor's father himself was a merchant, so that his son became *very*
>familiar with those huge "sets" in the storehouses of his family?
>
>So we may conclude in the first place that the birth of Set Theory was inspired
>by social circumstances. But this is not the end of the story. Even nowadays,
>nobody can think of an idea which fits better the view of the Capitalist System
>than Set Theory. (Go to a supermarket, and convince yourself!) This means that
>no rational arguments can be used in order to deprive ST from its predominant
>role in mathematics. Read my lips: I *don't want* to get rid of Set Theory as
>a (less important) part of Mathematics.
>
>Mathematical concepts originate and become important within the context of our
>human society, with all its non-logic and non-scientific taboos.
>But it is also thinkable that certain concepts will NOT originate in the given
>social circumstances, simply because such new concepts would have unacceptable
>economical and political (and personal) consequences. New ideas do not come,
>essentially because, deep in our heart, we DON'T WANT them.
>So maybe, yes, further progress in mathematics is *inhibited* by the way our
>societies are organised ...
If set theory is a result of the capitalist distribution system of
wealth, what is responsible for category theory (which in some ways
is beginning to rival set theory in foundations)?
Is category theory the result of a obsession with the (arrows of)
trajectories
of German V2 rockets in the second world war? Which is when it
first appears. (cf. T. Pynchon, Gravity's Rainbow &
S. Mac Lane, Categories for the Working
Mathematician).
These arrows had such a profound impact on the lives of
individuals that they lost their objective physical meaning
and became somehow abstract entities with a life and importance
of their own.
Perhaps, as some have suggested, the appearance of
category theory with its (wonderfully
effective) disregard for 'inherent meaning' is not unrelated to
the appearance of deconstruction. All
this at some bizarre twist
of the Zeitgeist from what
gave rise to set theorists!
Question; Is it true that Hurewicz was the first to use
the notation of labelling a function by a letter
written over an arrow between two objects? This
must be the first act of taking arrows seriously
as mathematical objects.
--
C.L. Nehaniv (neh...@math.berkeley.edu) | " Things fall apart.
Dept. of Mathematics | It's scientific."
UC Berkeley, CA 94720 | -D. Byrne
William Mayne
>In article <15...@dutrun.UUCP> rct...@dutrun.UUCP (H. de Bruijn) writes:
>> Is it a coincidence that
>>Georg Cantor's father himself was a merchant, so that his son became *very*
>>familiar with those huge "sets" in the storehouses of his family?
>>
>>So we may conclude in the first place that the birth of Set Theory was inspired
>>by social circumstances. But this is not the end of the story. Even nowadays,
>>nobody can think of an idea which fits better the view of the Capitalist System
>
>If set theory is a result of the capitalist distribution system of
>wealth, what is responsible for category theory (which in some ways
>is beginning to rival set theory in foundations)?
Believe it or not, in the mid to late '70s (and maybe since) there was a
category theorist at SUNY Buffalo who was a Marxist and who saw some
connection between category theory and Marxism. I have this second
had from my advisor, who was a graduate student in computer science
there at that time. The name of the category theorist was William (Bill)
Lawvere, but I am not sure of the spelling. Pronounced Laveer.
10^3 pardons if I didn't get the name right and there is some non-Marxist
out there whose name is close enough to be confused with this. I wouldn't
want to raise the ghost of McCarthy.
Maybe this is enough of a reminder/reference for someone who knows
category theory to fill in some more of what may be an interesting
sidelight on mathematics and politics.
Bill Mayne (ma...@cs.fsu.edu)
Timothy J Biehler
(William Mayne) writes...
>Believe it or not, in the mid to late '70s (and maybe since) there was a
>category theorist at SUNY Buffalo who was a Marxist and who saw some
>connection between category theory and Marxism. I have this second
>had from my advisor, who was a graduate student in computer science
>there at that time. The name of the category theorist was William (Bill)
>Lawvere, but I am not sure of the spelling. Pronounced Laveer.
Lawvere is still here at SUNYAB. I have no idea as to the
truth or falsity of his reputed leftward leanings. If in
fact he does see some Marxist implications in category theory,
he isn't overly vocal about them now.
I don't know him very well ... he filled in for a couple of weeks
teaching an algebra course I was taking, but I didn't get to
know the guy. He is one of the "name" profs here ... he invented
some sort of variation on category theory called topos theory,
for which he is fairly famous, though personally I have no earthly
idea what a topos is or why they are so interesting.
>10^3 pardons if I didn't get the name right and there is some non-Marxist
>out there whose name is close enough to be confused with this. I wouldn't
>want to raise the ghost of McCarthy.
Well, I doubt the guy is gonna get in much trouble nowadays for
having Marxist ideas. Besides, even if one did want to foment
a proletarian revolution, one could certainly find better
vehicles for it than category theory. :-).
>Maybe this is enough of a reminder/reference for someone who knows
>category theory to fill in some more of what may be an interesting
>sidelight on mathematics and politics.
This I would like to hear. I'm afraid I don't understand
category thoery sufficiently well to ask Dr. Lawvere about
his political interpretations of it (at least without making
an ass of myself), but would be very interested in any remarks
or references people might have w/r/t mathematics and politics.
I personally don't see any clear or obvious means of transforming
arrows and objects into Marx and Lenin. But what do I know?
Tim
Allan Adler
It has been rumored that Lawvere invented topos theory. Actually, the term
topos was introduced in SGA4. One can generalize the notion of a topological
space by regarding the open sets as the objects of a category and the
inclusion maps as morphisms. One can then ask how to formulate the essential
features of a topological space in an arbitrary category. The solution
is stated in SGA4 and I don't want to go into details here, but basically
one wants to be able to talk about sheaves and when working with sheaves on
a topological space, one is always considering coverings of open sets
(and other sets) by collections of open sets. So one axiomatizes the
notion of a covering of an open set in a way that makes sense in an
arbitrary category. This is not a strict generalization of the notion
of topological space since two nonhomeomorphic spaces can have the
same category of open sets. But with some separation axioms on the
space, the space is determined by the category associated to it.
By axiomatizing the notion of a covering (or the related notion of a sieve)
one arrives at the definition of a topology on a category. A category with
a (Grothendieck) topology is called a site. The category of all sheaves of
sets on a site is called a topos, or rather, it is called a topos in SGA4.
In case the site has only one object and only one morphism, the associated
topos is the category of sets and functions, which for this reason is
called the "punctual topos". One can try to interpret set theoretic notions
in a general topos by interpreting them in terms the values which the
sheaves assume on objects, these values being sets. If one does so, one
finds that one is dealing with systems that satisfy some of the axioms
of set theory (perhaps formulated arrow theoretically) but not others.
This suggests that one can view a topos as being a kind of model of
set theory but with different axioms than the usual Zermelo-Frankel
axioms.
Lawvere (let's say, I really am not sure) wrote down the arrow theoretic
axioms satisfied in a topos which made them seem like models of set theory.
These axioms actually describe a categories more general than the topos
defined by Grothendieck. Lawvere appropriated the terminology "topos" and
used it to describe his more general categories. This creates a certain
amount of confusion, similar to that caused by changing the definition
of scheme to mean what prescheme formerly meant. If someone uses the
term "topos", you really do not know what they mean unless they say something
else. Strictly speaking, the term for what Lawvere's school calls a topos
is "elementary topos", while the term used by Lawvere's school for
what Grothendieck calls a topos is a "Grothendieck topos".
I am writing the above comments not to provide an introduction to topos,
nor to explain how it is that people propose to use category theory as
a foundation for mathematics, but to address the point of another poster
who suggested that Lawvere had invented the concept of a topos.
I should emphasize that I am not an expert on topos of either sort and it is
better to refer to books and articles on the subject than on my sketchy
account. I apologize for any inaccuracies in the above account and would
be happy to have them brought to my attention.
For an introduction to the notion of an elementary topos, there is a nice
book by Goldblatt, published by North Holland, called Topoi: The Categorical
Imperative. Incidentally, Goldblatt thinks topoi is the plural of topos,
instead of topos or toposes.
Allan Adler
gh...@ms.uky.edu
Nico Verwer
>Believe it or not, in the mid to late '70s (and maybe since) there was a
>category theorist at SUNY Buffalo who was a Marxist and who saw some
>connection between category theory and Marxism. I have this second
>had from my advisor, who was a graduate student in computer science
>there at that time. The name of the category theorist was William (Bill)
>Lawvere, but I am not sure of the spelling. Pronounced Laveer.
In {Logic Colloquium '73} there is an article ``Continuously Variable Sets;
Algebraic Geometry = Geometric Logic'' By F. William Lawvere, who was then at
the University of Perugia, Italy.
I shall summarize the introduction.
The (elementary) theory of topoi is a basis for the study of continuously
variable structures, as classical set theory is a basis for the study of
constant structures.
There are representations of variable structure in set theory, ``But there is
an analogy here with the notion of variable quantity, a notion which was taken
quite seriously by the founders of analysis and which has not been
`eliminated' by set theory any more than continuity has been eliminated by the
`arithmetization of analysis' (which is just that and not analysis itself).''
In the period before set theory, Friedrich Engels, Marx's friend and close
collaborator, writes in {Anti-D\"uhring}, in the section on Quantity and
Quality, considers the advance from constant quantities to variable
quantities the mathematical expression of the advance from metaphysics to
dialectics.
``The characterization of motion as the presence of the same body in two
places at the same time is only an irresolvable contradiction if we ignore
that the metaphysical opposition between points and neighbourhoods ... is not
maintained in the practice even of mathematics. As Lenin affirmed in his
Conspectus of Hegel's Lectures on the History of Philosophy ... it is that
characterization of motion which correctly expresses the continuity of time
and space, ...''
Unfortunately I don't understand most of the rest of the article -- I am a
computer scientist, and someone gave it to me because I am a marxist, not
because I am particularly interested in continuously variable sets.
In his introduction, Lawvere points at one of the most important and least
understood tools of marxist theory, the dialectic.
The dialectic has been used by many `marxists' to `prove' the validity of
marxism, and the inevitability of revolution.
It has often been seen as a _law_ which everything must obey, rather than a
_model_ or a _method_ of explaining things. Engels himself has introduced some
of the confusion in his ``dialectics of nature'', which in crude
interpretations has become very mechanical.
Lenin has written some words on it, but I've never read them.
Trotsky writes on the dialectic in his philosophical notebooks, which were
published for the first time last year.
I find it impossible to explain in a quick and dirty way what dialectics is
exactly (well, what it is in my view).
As you can see from the words of Lawvere, dialectics is a _logic_of_change_,
a way to describe and understand continuously developing processes.
It stands on three basic points:
* The unity of opposites. This tries to describe the development of processes
in terms of constituents which influence the development in different
directions. It looks at _counteracting_tendencies_ (called thesis and
antithesis) which are part of, and contained within this process. One should
identify these tendencies in order to understand the development.
* The change of quantity into quality
Processes develop gradually, or continuously. That is, they undergo
_quantitative_ changes. At some point, however, a _qualitative_ change occurs,
because of this continuous quantitative change. For example: Adding protons
and neutrons to an atomic nucleus is a quantitative change. This leads to
qualitative changes in the chemical features. Another example: Take a bowl of
water and heat it. Going from 98 to 99 degrees celcius, and then to 100 is a
quantitative change, leading to a qualitative one (the water starts boiling).
* The formation of a _synthesis_ from the _thesis_ and _antithesis_.
The process develops through the struggle between thesis and antithesis. In
this development, both components change their character, to form a new
synthesis. (I'll keep examples for some future posting).
The criticism of many marxists (without the quotes) is, that mathematical
logic (which they identify with aritotelian logic) is static, and
non-dialectic. But I think they forget the development of mathematics since
the beginning of this century.
Once I studied _catastrophe_theory_, which some net.readers may know about. It
fits perfectly with the above description of dialectics.
To come back to the point of the poster who started this thread, I think it is
not a coincidence that catastrophe theory was popular in the early seventies,
when Lawvere made references to marxism in his papers. The student's and
worker's revolts of the late sixties were still echoing at the universities at
that time. In the late seventies people became disillusioned because of the
lost struggle of the late sixties, and progressive theories became much less
popular.
I think that it is again no coincidence that nowadays fractal geometry is used
to describe the same things that catastrophe theory was used for. Fractal
geometry is a typical example of a static theory, concerned with outward
features of processes. You can make nice pictures with it, though :-) .
Well, I could go on, but I suppose that most people have pressed `N' already.
I hope I have added some evidence to the statement that mathematics _is_ a
social activity, influenced by social circumstances. After all, as Marx said,
``People make their own history, but not under the circumstances that they
choose themselves.''
Regards,
--
Nico Verwer | ni...@cs.ruu.nl
Dept. of Computer Science, University of Utrecht | phone: +31 30 533921
p.o. box 80.089, 3508 TB Utrecht, The Netherlands | fax: +31 30 513791
Allan Adler
Nico Verwer has kindly provided us with a quotation from Lawvere's 1973
article giving some aspects of Lawvere's views of the relationship
between topos theory and Marxism. Nico then disqualifies himself from
commenting on the mathematics since he is a computer scientist and does
not know mathematics. Instead his interest in the matter at hand is that
he is a marxist and proceeds to give us a "quick and dirty" explanation
of dielectics. He reports that he has studied catastrophe theory and
informs us that it is no coincidence that it appeared after the frustration
of the 70's and that the appearance of fractals in our own time is likewise
no coincidence.
I am not a student of Marxism or of dielectic materialism, but I have read
a little philosophy and have some vague impressions of what dielectics are
all about. I would like to begin with some ignorant comments on this concept.
Kant's Prolegomena to Any Future Metaphysics is an interesting work from a
number of standpoints including his discovery of the "synthetic a priori".
Towards the end of the essay he applies this concept in an enlightening
discussion of antinomies (i.e. contradictions) and that is what
I would like to focus on. Kant distinguishes between what he calls
"mathematical antinomies " and "dynamic antinomies". A mathematical
antinomy is a flat contradiction, such as 1=0. A dynamic antinomy appears
to be a contradiction but on closer examination of the issues involved
is actually seen not to be one. I do not remember the examples Kant gives
of dynamic antinomies, but if I were to invent one on the spot, the
apparent contradiction between the assertions "light is a particle" and
"light is a wave" is resolved by the introduction of quantum mechanics.
Hence the wave-particle antinomy is a dynamic antinomy.
I have not read Hegel, per se. When I was in high school, I read the
introduction to the Modern Library edition of a selection of Hegel's works
and it explained Hegel in terms that made sense to me then and which I think
I still remember after 30 years. After I read Kant, what I had read about Hegel
made even more sense. Based on this, it seems to me that what Hegel means
by a dielectic is essentially the resolution of what Kant calls dynamic
antinomies. But whereas Kant was apparently content to take his antinomies
one at a time, Hegel wanted to pass to the limit, so to speak. Let us say
that a philosopher (e.g. Hegel) wants to formulate hs philosophy. He states
it as best he can. This inevitably has contradictions in it, but by closer
examination of the issues involved, these are resolved and a new statement
emerges. This, however, also has contradictions in it which must in turn
be resolved. Again, one looks more deeply at the concepts and arrives at
a new resolution. Thus one obtains a sequence of formulations, each of
which is obtained by resolving the apparent contradictions of the one before
it. The induction step is called "synthesis", the apparent contradiction
the "thesis" and "antithesis". Hegel wished to pass to the limit and thereby
arrive at the Absolute Idea. (Hegel apparently did not consider the possibility
that one might have to do transfinite induction :-) ). Up to this point, one
can kind of see that this might make some sense in principle, but then Hegel
really takes off and mixes up his concept of the Absolute Idea with what
appears to be his nationalism: he conceives of the Absolute Idea as moving
over the various parts of the globe and settling for a time here and there
and favoring one or another nation with it. He mentions where it has been
in the past (India ? I forget) but concludes that it then moved to Germany
where it has remained ever since. Lest we be too shocked, let us also remember
that Hegel also considered it obvious that there were only 7 (?) planets.
The bizarre step of taking logical inquiry (i.e. the resolution of dynamic
antinomies) to be a metaphor for the progress of history was therefore
apparently taken by Hegel. Marxists have embraced this concept and regard
human struggles as the process by which one arrives at a synthesis from
an antithesis. Thus the struggles of peoples throughout history are perceived
as the iterations of Hegel's construction of the Absolute Idea and this is
referred to as "the historical process" which in the limit converges to the
Millenium.
I should mention that there is a wide spectrum of people who consider
themselves Marxists, just as there is a wide spectrum of sects claiming to be
Christians, and as nearly as I can tell, the various Marxist splinter groups
hate each other more than they hate the capitalists. Therefore, I will not
attempt to formulate conclusions such as whether the limit described above
occurs at a finite point in time or at infinity. Marxists also have their own
problems of educating people in Marxism and in order to teach people who do not
have the prerequisites or the time to acquire them, they often water down their
own concepts. It is in this light that I regard the three principles of
Marxism as set down by Nico Verwer, to wit:
(1) The unity of opposites
(2) The change of quantity into quality
(3) The formation of the synthesis from a thesis and an antithesis.
His formulation makes it difficult to distinguish Marxism from Aesthetic
Realism. In case you haven't heard of the Aesthetic Realism people, they
are the followers of an obscure poet, now deceased, named Eli Siegel who won
a prize for a collection of poems called Hot Afternoons Have Been in Montana
and containing a short poem in which he espouses the point of view that beauty
consists in the aesthetic juxtaposition of opposites. He founded the Aesthetic
Realism Society and also used his ideas about opposites as a kind of
psychological theory in which people are encouraged to regard their conflicts
as somehow interesting and aesthetic. They feel that Mister Siegel, as they
invariably refer to him, was the greatest person who ever walked the earth
and they spent a lot of time picketing in front of the Metropolitan Museum of
Art in NYC because it was across from the apartment of the editor of the New
York Times who impudently refused to publish the good news about Aesthetic
Realism that they claim the public certainly would want to hear. One could
write a book about these kooks and I hope someone does as they also seem to
have political ambitions and some of them were working, e.g, in the Board of
Education of the City of New York last time I checked.
At this point, it is perhaps useful to remember that Marx wrote about
economics. Marxism is not merely distilled and diluted Hegelian philosophy.
Marx attempted to describe the forces that are actually at work in the
struggles of people. He introduced concepts that influence the study of all
contemporary social sciences. But the scope of his concepts is often
exaggerated by people who claim to represent them, just as the scope of
Christian doctrine was exaggerated by those who censored Galileo.
As for Nico's claims that "it is no coincidence" that certain mathematical
theories arise at certain social epochs, this is your typical conspiratorial
view of history and many people who consider themselves Marxists just love
that kind of nonsense. It is nonsense because it commits the "Post Hoc
Ergo Propter Hoc" Fallacy and because it depends on the most subjective
assessments and definitions of the phenomena being observed.
If indeed it is no coincidence, let him explain in detail what is the
mechanism.
Returning to the matter of dielectics and Nico's claim that Marxists
feel that mathematics does not allow for them, let us view the concept
of dielectic in its true perspective. It is not the case that all
contradictions can be resolved by a dielectical process. Kant distinguished
mathematical antinomies from dynamical antinomies. Dielectics have to do with
dynamic antinomies. The fact is that there are mathematical antinomies and
there are dynamic antinomies. Mathematics is concerned with the former
and Marxists are concerned with the latter. There is no real contradiction
between the two. Thus, the apparent contradiction between Marxism and
mathematics is a dynamic antinomy in the sense of Kant and it is herein
resolved by what someone might want to call a dielectical process.
Allan Adler
gh...@ms.uky.edu
Keith Ramsay
In article <1991Apr22.1...@mailer.cc.fsu.edu>, William Mayne writes:
> [...] there was a category theorist at SUNY Buffalo who was a
> Marxist and who saw some connection between category theory and
> Marxism.
I once read that not only is category theory the counterpart in the
philosophy of mathematics to dialectical materialism, but Platonism to
Catholicism, Intuitionism to Protestantism, and Formalism to Atheism.
:-) I think it was an after-the-talk remark at some logic colloquium.
--
Keith Ramsay
ram...@zariski.harvard.edu
Nico Verwer
Let me first apologize for putting so little of mathematics in this posting.
Apparently the word "marxism" which I included in my previous posting sounded
the alarm at Allan Adler's office. I think some of his criticism is not quite
correct, so I'll reply here.
Some things I'll say do have to do with mathematics (and philosophy).
In <1991Apr24....@ms.uky.edu> gh...@ms.uky.edu (Allan Adler) writes:
>[about Hegel taking dialectics to the "limit"]
>The bizarre step of taking logical inquiry (i.e. the resolution of dynamic
>antinomies) to be a metaphor for the progress of history was therefore
>apparently taken by Hegel. Marxists have embraced this concept and regard
>human struggles as the process by which one arrives at a synthesis from
>an antithesis.
That is exactly what Marx did _away_ with. Marx was one of the "young
Hegelians", who did not take the "limit" of dialectics. After that he went
further, and revised the concepts of dialectics completely, keeping most of
Hegel's terminology. One of the points where he breakds with Hegel is, that he
does _not_ see dialectics as a universal metphor, but as a method of studying
processes. That is what I explained in the posting you refer to.
A Hegelian style of mechanical dialectics was re-introduced under stalinism in
the thirties, to justify Stalin's counter-revolution. Stalinism abuses the
marxist terminology for its own ends, presenting repression of workers as
"workers liberation", etcetera. The stalinists were not the first to regress
to a mechanical form of dialectics. Some twenty years earlier,
social-democratic writers like Kautsky had done the same. He took the
dialectic as a metaphor, and "proved" that social change was inevitable, so
that one could just sit down and wait for it.
>Marxists [...] often water down their
>own concepts. It is in this light that I regard the three principles of
>Marxism as set down by Nico Verwer, to wit:
> (1) The unity of opposites
> (2) The change of quantity into quality
> (3) The formation of the synthesis from a thesis and an antithesis.
Please read my posting! I put these forward as three central concepts of
dialectics, not of marxism!
>As for Nico's claims that "it is no coincidence" that certain mathematical
>theories arise at certain social epochs, this is your typical conspiratorial
>view of history
I don't claim there is some sort of mechanical link between what happens in
the outside world and what happens in mathematics. But I do think that the
former influences the latter. Mathematical concepts are usually invented when
there is a need for them.
The ancient Greeks could have invented infinitesimal calculus, but they had no
use for it. In the period of early industrialization, mechanics was needed to
construct new devices, meand of transport, etc. It was at that time that
infinitesimal calculus was developed. This is what I mean by "no coincidence".
I think the same holds for catastrophe theory. In the early seventies, many
people, especially at the universities, were interested in social change.
Therefore they were interested in theories that could explain why things
change, and how the forces that are at work can be described.
When the political tide turned, people started to use theories of a purely
_descriptive_ nature, like fractal geometry, to analyse processes that others
had used catastrophe theory for.
Note: I do not object to fractals per se. All I argue is that you cannot just
say: "Hey, I don't understand this process. Therefore I'll call it chaotic,
and through fractal geometry I'll understand it", which is a charicature, but
not unlike the method of many sociologists and biologists. There may be areas
where fractals are an excellent tool.
>Returning to the matter of dielectics and Nico's claim that Marxists
>feel that mathematics does not allow for them
Again, you have not read my posting very well. I said that many marxists feel
this, but I am certainly not one of them.
But I would not separate "dynamic antinomies" from "mathematical antinomies".
I think of mathematics as just a tool to describe and explore things in the
real world. Therefore, I do not think of mathematics as a world of its own,
and "mathematical antinomies" are only useful insofar as they can describe
antinomies in the real world. If a mathematical antinomy cannot model a
dynamic antinomy, you should not use it for that purpose.
But here we run into another controversy in mathematics, namely the debate
between the intuitionists (Brouwer) and the classical logicians (Hilbert).
Allan Adler
Nico Verwer defends his original posting by suggesting that I am somehow
offended by Marxism. One will note from my posting that I gave due credit to
the importance of Marx's ideas in contemporary social sciences. Where I differ
from Nico Verwer is that I do not exaggerate the scope of Marx's ideas.
Nico Verwer believes that certain mathematics arises at certain times.
That can mean various things, e.g.:
(1) There is a some vague relationship between what kinds of ideas are
possible and the progress of history.
(2) A particular idea arose at a particular time in response to a particular
event.
(3) Certain kinds of mathematics are emphasized in response to certain specific
economic needs.
I am inclined to disagree with both (1) and (2), but I feel it is important to
note that they qualitatively different. Even if one were to grant that (1)
is true, that does not mean that it can be used with the precision needed
to make statements of the form (2). As for (2) and (3), I distinguish between
the occurrence of specific ideas and the economic viability of those who study
a particular branch of mathematics. I believe that (3) is obviously true: when
there is funding for a certain kind of research (e.g. converting lead into
gold), there will be people who will fill the economic niche that is created.
It does not mean that they solve the problem, which seems to be the assumption
implicit in (2).
As nearly as I can tell from reading Nico Verwer's modification of his earlier
comments, he does not distinguish between (2) and (3), nor does he distinguish
between (1) being true in principle and his own subjective ability to make
statements of the form (2).
It is Nico Verwer's assertions of the form (2) that bother me, not his
pretensions to Marxism.
Allan Adler
gh...@ms.uky.edu
Patrick J. Fleury
Lawvere is more than fairly famous. He is one of the world leaders
in category theory - or, as Serge Lang refers to it, The Theory of Abstract
Nonsense. (Insert many smilies.) His thesis, written for Sammy Eilenberg at
Columbia, went under the jaw-breaking name of _Functorial Semantics of
Algebraic Theories_ and was a major contribution to category theory. It
was published in the Proceedings of the National Academy of Science, quite
an unusual happening.
As for his "leftward leanings," - if anybody cares - his paper
in the Proceedings of The International Mathematical Congress, (I think
it was '72 but it might have been '68), has one reference - Mao's Little
Red Book. (Many more smilies) Check it out if you can find the book.
As for connections to Marxism, I know less than the previous
poster. er.
Hans Huttel
(Patrick J. Fleury) writes:
[ stuff about Bill Lawvere deleted ]
> As for his "leftward leanings," - if anybody cares - his paper
>in the Proceedings of The International Mathematical Congress, (I think
>it was '72 but it might have been '68), has one reference - Mao's Little
>Red Book. (Many more smilies) Check it out if you can find the book.
>
> As for connections to Marxism, I know less than the previous
>poster. er.
Another example: Anders Kock is one of the big names in categorical
logic (and a Dane, like yours truly); his chapter on the subject in
the Handbook of Mathematical Logic (co-authored by F. Reyes) should be
well-known. Kock is also an active member of the (very) small
so-called Danish Communist Party of Marxist-Leninists, a Stalinist
splinter group that has/has had close ties with Albania. At Aarhus
University he would sometimes sell books on Albania, the Party etc.
from a stand in the refectory. As a result, I sometimes wonder if the
notion of hyperdoctrine was an example of Stalin-inspired
categoryspeak.
Who knows - perhaps the notion of duality is inspired by the Stalinist
version of Hegelian dialectics ? :-)
Hans
P.S. I am a leftist (as I confessed in sci.logic last year) but
certainly neither a Stalinist nor a category theorist.
--
Hans H\"{u}ttel, Office 1603 JANET: ha...@uk.ac.ed.lfcs
Lab. for Foundations of Comp. Sci. UUCP: ..!mcvax!ukc!lfcs!hans
JCMB, University of Edinburgh ARPA: hans%lfcs.e...@nsfnet-relay.ac.uk
Edinburgh EH9 3JZ, SCOTLAND This is _not_ a clever quote from a song.