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\begin_layout Title
Mereology and Postmodernism
\end_layout
\begin_layout Author
Linas Vepstas
\end_layout
\begin_layout Date
2 May 2020
\end_layout
\begin_layout Abstract
Almost all work on logic, symbolic reasoning, indeed, most of mathematics,
is built on the intuitive foundation of
\begin_inset Quotes eld
\end_inset
variables
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
functions
\begin_inset Quotes erd
\end_inset
that map between them.
There is an alternative viewpoint: that of
\begin_inset Quotes eld
\end_inset
connectors
\begin_inset Quotes erd
\end_inset
, which is developed extensively in the accompanying texts.
\end_layout
\begin_layout Abstract
This chapter focuses on how connectors have an impact on a number of philosophic
al arguments, ranging from mereology to postmodernism(!) This is a surprise
(to this author, a novice at philosophy.)
\end_layout
\begin_layout Section*
Philosophy
\end_layout
\begin_layout Standard
Here's a hint of how pervasively important the concept of variables and
values can be.
Some philosophers use these notions to anchor the idea of
\begin_inset Quotes eld
\end_inset
objects
\begin_inset Quotes erd
\end_inset
or
\begin_inset Quotes eld
\end_inset
things
\begin_inset Quotes erd
\end_inset
that
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\end_inset
exist
\begin_inset Quotes erd
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Quotes eld
\end_inset
An object is anything that can be the value of a variable, that is, anything
we can talk about using pronouns, that is, anything.
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key "Inwagen2002"
literal "false"
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This conception of an object then leads to confused discussions about the
identity of indiscernibles, and the importance of location (space-time)
in mereology.
The notion of connectors can be used as an alternate foundation for the
conception of
\begin_inset Quotes eld
\end_inset
objects
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
identity
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
location
\begin_inset Quotes erd
\end_inset
.
Specifically, one can instead define objects as
\begin_inset Quotes eld
\end_inset
things that can participate in relationships
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Subsection*
Relational algebras
\end_layout
\begin_layout Standard
In mathematics, a
\begin_inset Quotes eld
\end_inset
relational algebra
\begin_inset Quotes erd
\end_inset
is a system in which one can define collections of things, and make statements
about their relationships to one-another.
Examples of relations are
\begin_inset Quotes eld
\end_inset
is-a
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
has-a
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\end_inset
,
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\end_inset
part-of
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
Relations have several notable properties: they provide an ordering (something
may have a part which may have a part...) and thus a hierarchy.
Formally, such hierarchies are referred to as
\begin_inset Quotes eld
\end_inset
directed acyclic graphs
\begin_inset Quotes erd
\end_inset
or DAG's.
In general, any partial ordering can be represented with arrows – the directed
edges of the DAG.
That is, relations are usually asymmetric, but not always.
If B is a part of A, then A is not a part of B, at least, not in any convention
al sense.
But if A is-a B, then by convention, B is-a A.
One may write an entire book on the algebraic properties of relations;
and indeed, more than a few hundred books have been written on the topic,
to which the reader is invited to consult, for more information.
I recommend
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LatexCommand cite
key "Hodges1997"
literal "false"
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.
\end_layout
\begin_layout Standard
By definition (i.e.
so as not to confuse vocabulary), relations manifestly do
\emph on
not
\emph default
define functions or mappings or
\begin_inset Quotes eld
\end_inset
terms
\begin_inset Quotes erd
\end_inset
.
There is a broader algebraic system, called
\begin_inset Quotes eld
\end_inset
term algebra
\begin_inset Quotes erd
\end_inset
, which does supply these combinatoric elements; the relational algebras
are a fragment of term algebras.
\end_layout
\begin_layout Standard
Relational algebras find common deployment in computer science, where they
provide an explicit theory for databases (
\emph on
e.g.
\emph default
both SQL and no-SQL, and so on.) They are also central to dictionaries of
natural language, providing relational explanations aside from part-of-speech
tagging (
\emph on
e.g.
\emph default
\begin_inset Quotes eld
\end_inset
pitch•fork pĭch′fôrk″ n.
A large, long-handled fork with ...
\begin_inset Quotes erd
\end_inset
) In between these two extremes is the more general problem of knowledge
representation,
\emph on
i.e.
\emph default
of capturing knowledge in such a way that a computer can manipulate it,
that is, artificial intelligence, which is the ostensible topic of this
collection of essays.
\end_layout
\begin_layout Standard
Limiting oneself to a single part-whole relationship, one arrives at mereology
as a possible alternative to set theory as a foundation for mathematics.
But one also arrives the the notion of
\begin_inset Quotes eld
\end_inset
gunk
\begin_inset Quotes erd
\end_inset
in philosophy (that, roughly speaking, infinite recursion is possible,
when considering the structure of physical reality), as well as a number
of puzzles dating to Ancient Greece (the Ship of Theseus: what happens
when parts are replaced? The Statue and the Lump of Clay: what happens
when parts are rearranged?).
That there are such puzzles is perhaps endemic to the desire to apply part-whol
e relationships, and the adduced DAG partial orders as a fundamental ground
on which to build philosophical discussions.
it also perhaps suggests that a search be undertaken for some other foundationa
l principle.
This text offers up
\begin_inset Quotes eld
\end_inset
connectors
\begin_inset Quotes erd
\end_inset
as the alternative.
\end_layout
\begin_layout Subsection*
Mereology
\end_layout
\begin_layout Standard
The paradoxes of mereology mentioned just above: the difficulty of defining
part-whole relationships in a consistent manner, the problem of the existence
or non-existence of
\begin_inset Quotes eld
\end_inset
gunk
\begin_inset Quotes erd
\end_inset
, and even the puzzles from Ancient Greece (the Ship of Theseus,
\emph on
etc
\emph default
.) One proposed solution to these riddles is the notion of
\begin_inset Quotes eld
\end_inset
restricted composition
\begin_inset Quotes erd
\end_inset
(see Wikipedia
\begin_inset CommandInset citation
LatexCommand cite
key "WP-Mereology"
literal "false"
\end_inset
).
The idea is that
\begin_inset Quotes eld
\end_inset
things
\begin_inset Quotes erd
\end_inset
are still composed of
\begin_inset Quotes eld
\end_inset
other things
\begin_inset Quotes erd
\end_inset
, but the manner in which these can be composed is restricted.
The body of ideas is very nearly identical to the concept of seeds and
sheaves introduced in the earlier chapter.
\end_layout
\begin_layout Standard
To recap, here is a seed:
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Think of it as a burr, the black dot in the center being the seed; the white
circles being unattached connectors.
The connectors may be typed, so that they attach much as jigsaw-puzzle
pieces.
To make it painfully obvious, some figures:
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\begin_layout Standard
The explicit labels in the figures above have no particular meaning for
this text; rather, keep in mind the general notion of jigsaw-puzzle pieces
for the remainder of this text.
\end_layout
\begin_layout Standard
Then, quoting directly from the Wikipedia article on mereology, we can can
learn that the notions of restricted compositionality include:
\end_layout
\begin_layout Description
Contact — that a complex object Y is composed of X's if and only if the
X's are in contact.
\emph on
viz.
\emph default
that seeds may attach only if they are sufficiently close to one-another
to become connectable.
\end_layout
\begin_layout Description
Fastenation — the X's compose a complex Y if and only if the X's are fastened.
\emph on
viz.
\emph default
seeds attach only if the connectors allow mating.
Once mated, they are fastened together.
\end_layout
\begin_layout Description
Cohesion — the X's compose a complex Y if and only if the X's cohere (cannot
be pulled apart or moved in relation to each other without breaking).
The point of connectors is that they are not meant to be pulled apart,
once mated.
Insofar as the abstract definition of a sheaf given in earlier sections
failed to appeal to and force any order of the connectors, or to the length
of the edges, the concept of
\begin_inset Quotes eld
\end_inset
movement
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\end_inset
remains incomplete.
\begin_inset Foot
status open
\begin_layout Plain Layout
This can be repaired in a variety of ways.
In practical applications, it is convenient to order the edges, and to
give them weights.
These weights can sometimes act as distances, e.g.
in
\begin_inset Formula $N$
\end_inset
-dimensional space.
Furthermore, the branch of mathematics known as
\begin_inset Quotes eld
\end_inset
algebraic topology
\begin_inset Quotes erd
\end_inset
is very highly developed, and can provide very explicit and concrete statements
about the structure of space.
For example, the dimensionality of the embedding of the network is given
by the degree of the largest simplicial complex in the network.
\end_layout
\end_inset
\end_layout
\begin_layout Description
Fusion — the X's compose a complex Y if and only if the X's are fused (fusion
is when the X's are joined together such that there is no boundary).
Connectors, once connected, serve no further purpose, and are discarded.
What mattered was the bond that was made.
In this sense, the bonds are the fusion of connectors.
\begin_inset Foot
status open
\begin_layout Plain Layout
This is reminiscent of Derrida's statement that
\begin_inset Quotes eld
\end_inset
deconstruction is not analysis
\begin_inset Quotes erd
\end_inset
.
One cannot break up meaning into atomic parts; there are no self-sufficient
atomic parts of meaning.
Meaning derives only from the fusion of words into text and into language.
\end_layout
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\begin_layout Description
Brutal
\begin_inset space \space{}
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Composition — "It's just the way things are." There is no true, nontrivial,
and finitely long answer.
\end_layout
\begin_layout Standard
The last point is perhaps the most interesting notion coming out of restricted
composition.
It is not an ingredient, it is rather a consequence.
If one abandons the explicit DAGs of partial orders, one then also abandons
the notion of logical predicates and truth assignments.
One no longer arranges logical connectives into a tree structure, or even
into Horn clauses.
The network, rather, represents things
\begin_inset Quotes eld
\end_inset
as they are
\begin_inset Quotes erd
\end_inset
.
This is not to deny predicate logic, or Boolean satisfiability: these are
tremendously useful concepts.
Rather, this is to avoid the symbol grounding problem (ref Wikipedia, again).
By discarding the arrow from
\begin_inset Quotes eld
\end_inset
signifier
\begin_inset Quotes erd
\end_inset
to
\begin_inset Quotes eld
\end_inset
significand
\begin_inset Quotes erd
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, one can no longer peer into the gaping chasm of the question
\begin_inset Quotes eld
\end_inset
what does this symbol mean?
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Subsection*
Postmodernism
\end_layout
\begin_layout Standard
These last observations are very much in line with theories of postmodernism
and deconstruction evident in the works of Derrida,
\begin_inset Foot
status open
\begin_layout Plain Layout
The différance of Derrida can be seen as a rejection of the linguistic equivalen
t of mereological nihilism: there is no infinite regression of looking up
the meanings of words in a dictionary.
\end_layout
\end_inset
Deleuze, Lyotard.
There is no difference between appearance and true form; the difference
is undecidable.
The constructed network of connections and relations is all there is.
\begin_inset Foot
status open
\begin_layout Plain Layout
In this sense, it is also a rejection of Searle's
\begin_inset Quotes eld
\end_inset
Chinese Rom
\begin_inset Quotes erd
\end_inset
: there is no additional meaning beyond the vacant activity of the mindless
drones performing algorithmic actions.
That is all that there is, nothing more.
For Searle to pretend that there is something more to meaning, language
and translation is effectively an appeal to the existence of some sort
of magical metaphysical quintessence of language that simply is not there.
Words have meaning insofar as they related to each other; a bad translation
rendered by a mindless Chinese Room is nothing more than a bad translation.
The Chinese Room neithr demonstrates the existence of intelligence, nor
the absence of it.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
In the remaining chapters, the sheaf construction will be used as a tool
to create A(G)I representations of reality.
Whether the constructed network is an accurate representation of reality
is undecidable, and this is true even in a narrow formal sense.
Famously, it is known that the question of whether two different presentations
of a (group-theoretical) group refer to the same group is undecidable,
in the computational sense: there is no algorithm, guaranteed to terminate
in finite time, that is capable of making this decision.
This is the undecidability problem in group theory.
That this result carries over into the fullness of a network built from
a sheaf should come as no surprise: the presentation of a group is little
more than a certain collection of labeled vertexes and edges (the group
elements and the group operations) and decidability requires the graph-rewritin
g of these into a different form.
There are no graph rewrite rules that can always bring two different presentati
ons into normal form.
Oddly enough, such
\begin_inset Quotes eld
\end_inset
received wisdom
\begin_inset Quotes erd
\end_inset
from mathematics can be used as a foundational cornerstone for post-modernism.
This is unexpected.
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status open
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Despite this, sheaves are still somehow fundamentally
\begin_inset Quotes eld
\end_inset
structuralist
\begin_inset Quotes erd
\end_inset
, not post-structuralist.
The bonds are relational, in the end.
There is always still a predicate: either an edge exists, or it doesn't.
To every graph there is a corresponding adjacency matrix, populated with
zeroes and ones.
Fortunately, one can wriggle ones way out if this, by assigning numerical
weights to edges, making some probable and some improbable.
Whether true and false can be replaced by a subobject classifier in this
context is perhaps a step too far, but perhaps also an interesting step.
\end_layout
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\begin_layout Standard
Much of postmodernism is concerned with linguistics.
This is more directly attacked in the subsequent chapters.
\end_layout
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