Nick writes, in relevant part:
> I am, I think, a bit of what
> philosophers call an essentialist. In other words, I assume that when
> people use the same words for two things, it aint for nothing, that there is
> something underlying the surface that makes those two things the same. So,
> underlying all the uses of the word "entropy" is a common core, and ....
I'm not going to go anywhere near the mathematical question here. What
I want to do is challenge your "In other words" sentence (which, by the
way, I *hope* is not what philosophers would mean by calling you "an
essentialist").
One thing I have learned in the last three or four years, much of
which I have spent trawling through huge corpora of scholarly (and
less scholarly) writing, including Google Scholar (and just plain
Google Books), JStor, MUSE, PsycInfo, Mathematical Reviews, etc.,
is that "when people use the same words for two things",
it's distressingly common that it IS "for nothing", or nearly
nothing--either two or more different groups of scholars have
adopted a word from Common English into their own jargons, with
no (ac)knowledge(ment) of the other groups' jargon, or two or
more different groups of scholars have independently *coined*
a word (most usually from New Latin or New Greek roots that are
part of scholars' common store).
Actually, the first case of this that I really noticed was several
years before I got involved professionally. In a social newsgroup,
a linguist of my acquaintance happened to use the word "assonance".
And he used it WRONG. That is, he used it entirely inconsistently
with the meaning that it has had for eons in the theory of prosody,
and that every poet learns (essentially, assonance in prosody is
vowel harmony). When I challenged him on this, my friend said that
the word had been introduced to linguistics by the (very eminent,
now very dead) Yale linguist Dwight Bollinger. And he implied
that the linguists weren't about to change. Tant pis, said I.
Then I got involved in the Kitchen Seminar (FRIAMers, you can
ignore that; it's a note to Nick), and began to hear psychologists
(but not Nick!) use the phrase "dynamic system" (or occasionally
"dynamical system"). As a mathematician I knew what that phrase
meant, and they were WRONG.
After some years in the Kitchen, I began work on my book on
mathematical modeling for psychology; eventually I saw I
needed to write a chapter clarifying the uses of those phrases.
Three or four years of work on _The Varieties of Dynamic(al)
Experience_ later, I had accumulated *enormous* amounts of
textual evidence that there had been NO cross-pollination:
the two phrases arose entirely independently. (Then,
unfortunately, hapless psychologists and other "human
scientists" started appropriating [what they badly understood
of] the mathematical results that can be proved about
mathematicians' "dynamical systems" to draw ENTIRELY
UNSUBSTANTIATED conclusions about psychologists' "dynamic
systems".)
Most recently, I've been going through the same exercise
(again for a chapter, now not in a book of my own) for
"recursion" and "recursive". Again, I have accumulated
(and documented) enormous amounts of textual evidence
(from all those corpora); here is a brief outline of
the situation (with examples and all, the whole thing
is about 25 pages at the moment, interlarded with another
25 pages on "infinity" and topped off--I mean, bottomed
off--with 15 pages of references). Before the outline,
however, I will quote four practitioners of various
human sciences who have had cause to complain of the
present mess.
==a sociologist of law:==
In the context of causal analysis, as carried on in empirical research
(e.g. path analysis) nonrecursive models are employed, to denote the
case of mutual influencing of variables. When the autopoesis literature
speaks of recursive processes, it is presumably those nonrecursive
models of causal analysis that are meant. What a tower of Babel!
(Rottleuthner, 1988, p. 119)
==a physicist turned cognitive scientist (via LOGO):==
One is led to wonder if all authors are talking about and experimenting
with the same notion and, if not, what this notion could be. As it
happens, a careful reading shows that it is not so and that, unless a
very loose and rather useless definition of the term ["recursive"] is
assumed, it could be worthwhile to separate this confusing braid into
its constituent strands [...]. (Vitale, 1989, p. 253)
==an evolutionary linguist:==
Definitions of recursion found in the linguistics and computer science
literatures suffer from inconsistency and opacity (Kinsella, 2010, p.
179)
==a political scientist:==
The term `recursive´ [...] has multiple uses in the political science
literature. [... Political scientists should address] [t]he problem of
divergent meaning [...] through a survey of potential for reconciliation
or possible substitute terminology (Towne, 2010, p. 259) ===
Now, the outline.
o There are three distinct meanings of "recursive"/"recursion"--let me
abbreviate that to R/R from now on--in mathematics. The oldest one
describes so-called "recurrence relations" (like the one that defines
the Fibonnaci sequence: F1=1, F2=1, Fn = Fn-1 + Fn-2). The next oldest,
dating only from last century, is the one used in mathematical logic;
it's derived from the oldest but it's much more general ("recursive
functions"). There's an entirely UNrelated one used in a minor branch
of dynamical systems theory (which has had no influence outside of a
very small circle), apparently named because of a connection to
"recurrence" in colloquial English (think "Poincare section" if
that helps).
o The oldest mathematical sense has spawned a meaning that
started in economics and then spread (it's the one that
Rottleuthner was talking about); mathematically, it corresponds
to upper-triangular matrices (coding causalities).
o The next-oldest has spawned the present, barely coherent
(cf. Kinsella), use of R/R in linguistics and linguistics-inspired
social science. *Some* of Seymour Papert's--and, thence, the LOGO
community's--uses of R/R come from this tradition (one of his two
Ph.D.s is, after all, in mathematics).
o Another sense of R/R comes from Piaget (with a nod towards
Poincare). *The rest* of Seymour Papert's--and, thence, the LOGO
community's--uses of R/R come from this tradition (his second
Ph.D., in Psychology, was supervised by Piaget). Piaget, I am
afraid, is responsible for a great deal of muddle on this subject.
o Yet another sense of R/R, used in human ecology,
anthropology, political science, sociology, and educational
theory sprang--somehow--out of cybernetics and General
Systems Theory (even though none of the early cyberneticists
like von Neumann, Shannon, and Weiner, and none of the early
GS people like Bertallanfy and Rapoport, ever seem to have
used the word AT ALL, except for a couple of times in early
papers of von Neumann where he was using it in the oldest
mathematical meaning). It really seems that Bateson pulled
the word out of the air (that is, out of his no doubt rigorous
classical education) at some point, and it spread from him,
in a (typically) incoherent fashion, and apparently mostly
by word of mouth--he didn't commit either word to print until
the year before his death, though his biographer Harries-Jones
has seen a notebook in which Bateson recorded using the word
in a lecture in 1975. (Harries-Jones's title for the biography,
_A recursive vision: Ecological understanding and Gregory
Bateson_, is, in my opinion, irredeemably tendentious, and
a perfect example of muddle.) Insofar as Bateson ever tries
to actually *define* R/R, it's here:
==
[T]here seem to be two species of recursiveness, of somewhat different nature,
of which the
first goes back to Norbert Wiener and is well-known: the "feedback" that is
perhaps the best
known feature of the whole cybernetic syndrome. The point is that
self-corrective and quasi
purposive systems necessarily and always have the characteristic that causal
trains within the
system are themselves circular. [...] The second type of recursiveness has been
proposed by
Varela and Maturana. These theoreticians discuss the case in which some
property of a whole is
fed back into the system, producing a somewhat different type of
recursiveness[...]. We live in
a universe in which causal trains endure, survive through time, only if they
are recursive.
(Bateson, 1977, p. 220)
===
Needless to say, Wiener never called feedback (or anything
else) "recursive", and it's a real stretch to connect the
mathematics of feedback to mathematical notions of R/R.
Nor did Varela and Maturana EVER use R/R (in print at least)
before 1977; they instead coined "autopoeisis", which again,
insofar as it can be mathematicized, is not mathematical
R/R. (Later Maturana does use "recursive".)
o An Australian economic geographer named Walmsley somehow came up
with a notion of R/R c. 1972; until and unless he answers my e-mail
(pending now for several months, so I'm not holding my breath), I can
only assume, from references he cites, that he somehow came up with
his idea by combining General Systems Theory (though the word doesn't
appear there) with Piaget. Given that he states in one place that
"Shopping is a form of recursive behavior", you won't be surprised
that his idea--whatever it may be--appears entirely unrelated to
mathematical (or linguistic) R/R. In any case, he doesn't seem to
have inspired any followers.
o A sociologist named Scheff starts using the *words* "recursive"
and "recursion" c. 2005, for ideas (either his or others'; see
below) that were around starting in 1967.
==Scheff (2005):===
In one of my own earlier articles (Scheff 1967), I proposed a model of
consensus that has a
recursive quality like the one that runs through Goffman's frame analysis.
[...] As it happened,
Goffman (1969) pursued a similar idea in some parts of his book on strategic
interaction. [...]
[A] similar treatment can be found in a book by the Russian mathematician
Lefebvre (1977), The
Structure of Awareness. [...]I wonder whether Lefebvre came up with the idea of
reflexive mutual
awareness independently of my model. He cites Laing, Phillipson, and Lee
(1966), a brief work
devoted to a recursive model of mutual awareness that preceded Lefebvre´s book
(1977).
However, he also cites his own earliest work on recursive awareness, an article
(1965) that
precedes the Laing, Phillipson, and Lee book.
It is possible that Lefebvre´s work was based on my (1967) model of
recursive awareness, even
though the evidence is only circumstantial. As Laing, Phillipson, and Lee
(1966) indicate,
their book developed from my presentation of the model in Laing's seminar in
1964. Since there
were some 20 persons there, Lefebvre could have heard about the seminar from
one of those, or
indirectly by way of others in contact with a seminar member.
===
However, despite all the heavy lifting involved in Scheff's
name-dropping, the words "recursive" and "recursion" appear
nowhere in the cited works by Laing, Phillipson & Lee (1966),
Scheff (1967), or Goffman (1969). Lefebvre (1977, but not 1965)
does use "recursive" in the two major mathematical senses, and
even quotes Chomsky (although I think it likely--I haven't been
able to get the Russian originals of Lefebvre--that all that
was introduced by his translator, Rapaport of GS fame).
Rather, Laing, Phillipson & Lee, Scheff, and Goffman
consistently use the words "reflexive", "reflection", and
"reflexivity". These are glossed by Scheff in a variety of ways:
"recursive awareness", "mutual awareness" (harkening back to
Goffman´s signature phrase, "mutual consideration"), "not
only understanding the other, but also understanding that
one is understood, and vice versa", "not only a first-level
agreement, but, when necessary, second and higher levels of
understanding that there is an agreement", etc. Sheesh.
o Finally (thank you for the reference, Nick), Peter Lipton
and Nick Thompson published an article in 1988 titled "Comparative
psychology and the recursive structure of filter explanations."
It's a great article, but the sense in which it uses "recursive"
(Lipton's coinage) is unrelated to any of the other senses
(nor has it been taken up since, as far as I can tell).
[Here endeth the outline.]
The "common core", if there is one, is nothing more than
the collocation of the morphemes "re-" and "-cur-", of which
the former is still very productive in English, while the
latter is (at most New) Latin and no longer productive at
all; semantically, this makes the meaning of that common
core approximately "RUN AGAIN", which I submit is AT BEST a
trivial commonality of the various different uses, and (as
far as I understand some of the woolier uses, which is not
that far) not a commonality AT ALL of the entire set.
If that be essence, make the least of it!
Lee Rudolph
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