Incestuous Mathematics
Dave L. Renfro
by E. R. Lorch [Amer. Math. Monthly 78 (1971), 748-762] and near
the bottom of page 756 Lorch writes
"In order to penetrate further into this subject it is
necessary to give an appropriate structure to T, the
set of all coherent topologies. As mentioned earlier,
this appropriate structure is itself a topology. This
circumstance, that a collection of topologies is
topologized, may seem a bit incestuous."
[[ Another example of Lorch's excellent prose occurs just
before the above quote: "... the reader is reminded of
the fact that sets which are of type F_sigma_delta_sigma
or G_delta_sigma_delta and not of lower type--with respect
to any of the classic topologies--are very thinly scattered
through the literature. In fact, looking for them is almost
like hunting for unicorns." ]]
Aside from Lorch's entertaining style, this got me thinking about
other examples of a similar nature.
Here are two examples that I can think of:
1. Given a set X, we can define a metric D on the collection of
all metrics M(X) on X by
D(d1, d2) = min{ 1, sup{ d1(x,y), d2(x,y): x,y in X} }.
There are at least half a dozen papers that study this space
of metrics. You can find a couple of these papers by doing a
title search using "metric space of metrics" (don't use quotes)
at either of these web pages:
http://www.ams.org/mrlookup
http://www.emis.de/ZMATH/en/full.html
2. Given a metric space X, there is a well-known way of defining
a metric on the collection K(X) of all nonemtpy compact subsets
of X (the Hausdorff metric). There are easily over a thousand
papers that make use of the Hausdorff hyperspace (as it is often
called). When I first learned about K(X) (20 years ago?), I
wondered if there were any interesting applications that make
use of K(K(X)) and higher K-iterations of X. A few years ago I
came across some applications of K(K(reals)) in function iteration
theory. I don't have the references now (I'm at home), but if
anyone is interested I can look them up later and post them.
[One example I know of is a talk that Andrew M. Bruckner gave
at a summer real analysis meeting a few years ago about some
joint work with Tim H. Steele in which K(K([0,1])) arises.]
Does anyone have some other interesting examples of "incestuous
mathematics"?
It's easy to find examples of what I'd call "weak incestuous
mathematics", where I'd characterize the relationship as being
at least as distant as second cousins, such as:
** The Zariski topology on the prime ideals of a commutative ring
with identity.
** The lattice of topologies in which compact T_2 spaces play
key role.
** Homotopy and homology groups of a topological space.
** Dual and double-dual of a vector space.
** The Jacobson density theorem for the ring of linear functions
from a vector space into itself.
But what about examples that are a bit more incestuous than these,
like items #1 and #2 above?
Dave L. Renfro
Bill Taylor
|> But what about examples that are a bit more incestuous than these,
|> like items #1 and #2 above?
In Bayesian statistics they consider probability distributions
of probability distributions. (That's by far my best one!)
Thw whole of ZF is based on sets of sets (and no other type of member!)
More particularly, ordinals are just all the preceeding ordinals.
Somewhat more disturbingly, as it gets very close to impredicativity and
paradoxes of ill-foundedness, there are things like the lambda calculus,
and other combinatorial systems, which work directly with functions.
Functions of what? Only of other functions, possibly including themselves!!
Then there is the universal Turing machine which can act like any Turing
at all. One might wonder what happens when you give it its own specifications?
What does it start off doing... ?
Math is full of such incest. It's all part of the fun. Family fun...
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Bill Taylor W.Ta...@math.canterbury.ac.nz
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William Elliot
"In order to penetrate further into this subject it is
necessary to give an appropriate structure to T, the
set of all coherent topologies. As mentioned earlier,
this appropriate structure is itself a topology. This
circumstance, that a collection of topologies is
topologized, may seem a bit incestuous."
_ But what about examples that are a bit more
_ incestuous than these, like items #1 and #2 above?
Of course there's P(S), P(P(S)), ...; functions, functions of functions
or functionals, functions of functionals, etc. But to avoid the
mundane...
The ordinal number of all ordinal numbers.
The cardinal number of all cardinal numbers.
The Borali-Forte paradox and it's big brother.
A = { x | x in x }; B = { x | x not in x }
The bunch of incestuous sets A, isn't a problem.
The bunch of non-incestuous sets B, is big problem.
What's the moral?
My favorite is the Loewenheim-Skolem paradox, every consistent first order
language has a countable model; even the uncountable is merely countable.
It's like condemning the imagination to perpetually countable inbreeding.
Even Godel numbering and self referent statements embody the paradox of
incest. But let's face it, what's more incestuous than consciousness
itself?
--
Recursive functions, if not cannibalistic, are incestuous. The worst
being the natural ordinals generated as {}, {{}}, {{},{{}}}, ...
{{}, {{}}, {{},{{}}}, ... },
{ {{}, {{}}, {{},{{}}}, ...}, {}, {{}}, {{},{{}}} ... }, ...
Which start out with the snake swallowing it's tail, or actually, the
inverse of that process. Extensions or generalizations of this incestuous
process, are the constructible universe and the pure universe of sets
without objects.
Such incesuous sets as {}, {{}}, {{{}}}, ... are allowed.
What about I = ...{{{}}}... ?
Does such an infinitely incesuous set exists?
Is the existence of I indepenent of ZFC axioms?
What about J = {{{... ...}}} ?
Does such an infinitely incesuous set exists? No, it's decreeded a
cardinal offense by the axiom of regularity or foundations.
for all x,(for all y in x, some z in x, z in y) ==> x = {})
for all x /= {}, some y in x, for all z in x, z not in y
Even more offensive is J_a = {{{... a ...}}} for it's conceptual defiance.
Moral? Tho, an infinitely incestuous history gets you executed, it's
possible to contemplate an infinitely incestuous future, perplexing the
orthodox. For example, as I /= nulset, what is an element of I?
Does I = I_a = ...{{{a}}}... = I_b ?
Indeed I = I_{} = I_{{}} = ... So when does I_a /= I_b ?
So again, to answer that, what's an element of I_a, I_b ?
--
[[ Another example of Lorch's excellent prose occurs just
before the above quote: "... the reader is reminded of
the fact that sets which are of type F_sigma_delta_sigma
or G_delta_sigma_delta and not of lower type--with respect
to any of the classic topologies--are very thinly scattered
through the literature. In fact, looking for them is almost
like hunting for unicorns." ]]
Unicorns are nice to find, most useful indeed, for when riding a unicorn,
who could ever be caught on the horns of a dilemma?
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Bill Taylor
|> being the natural ordinals generated as {}, {{}}, {{},{{}}}, ...
|> {{}, {{}}, {{},{{}}}, ... },
|> { {{}, {{}}, {{},{{}}}, ...}, {}, {{}}, {{},{{}}} ... }, ...
|>
|> Which start out with the snake swallowing it's tail, or actually, the
|> inverse of that process.
Not a bad analogy! Or image, anyway. The snake swallowing its own tail,
and also two snakes swallowing each other's tail, are classic icons in
various folklore and legendaria.
But the opposite isn't - the old sages missed a trick there! Of course,
it couldn't be snakes, with their egg-laying tendencies. But say a mammalian
monster of similar physiology. We could have two of them each giving birth
to the other, or, if long and narrow enough, one giving birth to itself!
That's a really kool & kyewt image, I feel.
Of course, more abstractly, it's pretty much what most religions do about
cosmogeny, though they tend not to admit it openly, except maybe for Hinduism.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Buddhism: The religion of obscurity.
Christianity: The religion of guilt.
Hinduism: The religion of backwardness.
Islam: The religion of violence.
Judaiism: The religion of exclusivity.
------------------------------------------------------------------------------
Michele Dondi
wrote:
>Does anyone have some other interesting examples of "incestuous
>mathematics"?
What about the whole Theory of Categories? :-)
Michele
--
Liberta' va cercando, ch'e' si' cara,
Come sa chi per lei vita rifiuta.
[Dante Alighieri, Purg. I, 71-72]
I am my own country - United States Confederate of Me!
[Pennywise, "My own country"]
Kevin Foltinek
> Does anyone have some other interesting examples of "incestuous
> mathematics"?
The set of monoids is a monoid under the direct product. Let the
trivial monoid be E; given two monoids A, B, their direct product AxB
is a monoid; ExA, AxE, and A are isomorphic; associativity is obvious.
(Similarly the set of semigroups is a semigroup.)
I don't know if this is "interesting".
Kevin.
Jesse F. Hughes
I guess you don't mean it when you say "set" of monoids?
--
Jesse Hughes
"LOL. How arrogant you are. Now when you realize that I DID prove
Goldbach's conjecture and that I proved Fermat's Last Theorem as well,
how are you going to feel then?" -- James Harris
Kevin Foltinek
> Kevin Foltinek <folt...@math.utexas.edu> writes:
>
> > The set of monoids [snip]
>
> I guess you don't mean it when you say "set" of monoids?
To be honest, I don't know if I mean it or not. I know that calling
too many things a "set" is troublesome, but I don't know if this
trouble arises when considering all monoids.
If it is troublesome, though, then is the collection of all monoids,
while not a set, still a monoid? (In other words, is it necessary
that a monoid be a set?)
Kevin.