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\begin_body
\begin_layout Title
Meaning as Inverse Interpretation
\end_layout
\begin_layout Author
Linas Vepstas
\end_layout
\begin_layout Date
24 February 2019
\end_layout
\begin_layout Abstract
The extraction of meaning from observed reality can be thought of as an
inversion of the process of assigning meaning to axiomatic mathematical
systems of formal mathematical logic.
This essay formulates the analogy and explores it a little.
\end_layout
\begin_layout Section*
Introduction
\end_layout
\begin_layout Standard
In mathematical logic,
\begin_inset CommandInset citation
LatexCommand cite
key "Manin1977"
\end_inset
a collection of axioms, and the language that they form, can be studied
independently of the interpretation of their meaning.
This statement can be made precise: one defines what a
\begin_inset Quotes eld
\end_inset
language
\begin_inset Quotes erd
\end_inset
is, and separately, one provides a map from that language to a set.
That map is called the
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
; when it obeys a certain set of requirements regarding consistency, it
can be said to provide the
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
of the axioms.
\end_layout
\begin_layout Standard
In exploring the real world - in the activity of performing science, and
of discovery, one is faced with the inverse task: one has a collection
of evidence, and from this evidence, one attempts to tease out a minimal
explanation of the world, as a collection of rules, axioms and theories.
\end_layout
\begin_layout Standard
This essay provides a short definition of the formal mathematical concept
of an
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
, illustrates why it provides
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
, and then examines how one can find
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
in the evidence obtained from real life.
Effectively, the discovery of meaning is the inversion of the process of
assigning it: given some evidence collected from real life, we may not
know the meaning, but we can seek to find it by discovering the underlying
structure, and coupling the two with an interpretation.
\end_layout
\begin_layout Standard
This essay could devolve into a philosophical tract; rather, it attempts
to be just concrete enough, just grounded and practical enough that one
can imagine creating software that actually performs the described process.
\end_layout
\begin_layout Subsection*
Meaning
\end_layout
\begin_layout Standard
This subsection gives a quasi-formal definition of an interpretation.
It is quite brief, providing just enough detail to get the general idea
across, but playing fast and loose with important theorems.
The reader is urged to consult standard texts for a full and complete definitio
n.
\begin_inset CommandInset citation
LatexCommand cite
key "Manin1977,Dalen1980,Baader1998,Hodges1997"
\end_inset
\end_layout
\begin_layout Standard
One defines the concept of
\begin_inset Quotes eld
\end_inset
terms
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
relations
\begin_inset Quotes erd
\end_inset
.
A
\begin_inset Quotes eld
\end_inset
term
\begin_inset Quotes erd
\end_inset
is either some constant
\begin_inset Formula $c$
\end_inset
or some variable
\begin_inset Formula $x$
\end_inset
or some (uninterpreted) expression
\begin_inset Formula $f\left(t_{1},\cdots,t_{n}\right)$
\end_inset
where the
\begin_inset Formula $t_{k}$
\end_inset
are also terms.
In some given, fixed system, there may be zero, one or more constants,
and there may be zero or more function-expressions taking some number of
arguments.
A relation is an expression
\begin_inset Formula $R\left(t_{1},\cdots,t_{n}\right)$
\end_inset
that can be either true, or false.
\end_layout
\begin_layout Standard
One standard example of terms and relations is arithmetic: it has two function-e
xpressions (or operations):
\begin_inset Quotes eld
\end_inset
plus
\begin_inset Quotes erd
\end_inset
, and
\begin_inset Quotes eld
\end_inset
times
\begin_inset Quotes erd
\end_inset
, both taking two arguments, and two relations:
\begin_inset Quotes eld
\end_inset
equals
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
greater-than
\begin_inset Quotes erd
\end_inset
, and two constants:
\begin_inset Quotes eld
\end_inset
zero
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
one
\begin_inset Quotes erd
\end_inset
.
Another standard example is set theory, which has no operations, one constant
\begin_inset Quotes eld
\end_inset
the empty set
\begin_inset Quotes erd
\end_inset
, and two relations:
\begin_inset Quotes eld
\end_inset
equals
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
is member of
\begin_inset Quotes erd
\end_inset
.
Examples can be found everywhere in mathematics: groups, rings, fields,
algebras and so on.
\end_layout
\begin_layout Standard
Formulas are obtained by combining relations with logical connectives:
\begin_inset Quotes eld
\end_inset
and
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
or
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
not
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
for all
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
there exists
\begin_inset Quotes erd
\end_inset
.
This collection of terms, relations and logical connectives, and their
various combinations forms a
\begin_inset Quotes eld
\end_inset
language
\begin_inset Quotes erd
\end_inset
: a collection of symbols arranged in various ways with respect to one-another.
But what is the
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
of that language? What do all of the symbols mean?
\end_layout
\begin_layout Standard
The canonical answer is that an interpretation provides
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
to the above formal symbolic expressions.
It makes a language concrete by assigning real-world items to each of the
symbols, each of the expressions obtained by combining these various elements.
It connects the abstract to the real.
\end_layout
\begin_layout Standard
An interpretation is a function
\begin_inset Formula $\phi$
\end_inset
that maps constants, terms and relations to elements of some set
\begin_inset Formula $M$
\end_inset
.
For example, the set
\begin_inset Formula $M$
\end_inset
could consist of an apple, pear and banana, and of all of the different
ways that they might be arranged on a table-top.
In this example, one might consider the axioms that define the permutation
group
\begin_inset Formula $S_{3}$
\end_inset
, which contains abstract expressions such as
\begin_inset Formula $a^{2}b$
\end_inset
and
\begin_inset Formula $e$
\end_inset
and
\begin_inset Formula $ba$
\end_inset
and so on.
What do
\begin_inset Formula $a^{2}b$
\end_inset
and
\begin_inset Formula $e$
\end_inset
and
\begin_inset Formula $ba$
\end_inset
mean? Acting on the set of arrangements of an apple, pear and banana on
a tabletop, it becomes clear: the element
\begin_inset Formula $e$
\end_inset
does nothing at all, the element
\begin_inset Formula $b$
\end_inset
swaps a pair of fruit, and the element
\begin_inset Formula $a$
\end_inset
exchanges all three pieces of fruit.
The equality relation expression as well: the world in which we applied
only
\begin_inset Formula $e$
\end_inset
comes out just the same as the world where we applied
\begin_inset Formula $b^{2}$
\end_inset
.
The
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
of the permutation group
\begin_inset Formula $S_{3}$
\end_inset
is the re-arrangement of three items.
\end_layout
\begin_layout Subsection*
Interpretation
\end_layout
\begin_layout Standard
A quick sketch of the more formal definition of an interpretation is in
order.
An interpretation of a constant
\begin_inset Formula $c$
\end_inset
is some element
\begin_inset Formula $\phi\left(c\right)$
\end_inset
of the set
\begin_inset Formula $M$
\end_inset
.
(In real life, this
\begin_inset Quotes eld
\end_inset
element
\begin_inset Quotes erd
\end_inset
might be a single object, a movement or action, a color or attribute; nothing
is presupposed).
In casual speech, the constants
\begin_inset Formula $c$
\end_inset
are the
\begin_inset Quotes eld
\end_inset
names
\begin_inset Quotes erd
\end_inset
of real-life things
\begin_inset Formula $\phi\left(c\right)$
\end_inset
.
We call something
\begin_inset Formula $c$
\end_inset
whenever we see, in the real world, an object
\begin_inset Formula $\phi\left(c\right)$
\end_inset
.
\end_layout
\begin_layout Standard
The interpretation of an
\begin_inset Formula $n$
\end_inset
-ary term
\begin_inset Formula $f\left(t_{1},\cdots,t_{n}\right)$
\end_inset
is a function
\begin_inset Formula $\phi\left(f\right):M\times\cdots\times M\to M$
\end_inset
, that is, it is some function defined on the Cartesian product of
\begin_inset Formula $n$
\end_inset
copies of
\begin_inset Formula $M$
\end_inset
, returning a value in
\begin_inset Formula $M$
\end_inset
.
Here, by
\begin_inset Quotes eld
\end_inset
function
\begin_inset Quotes erd
\end_inset
we mean the usual thing from school: a function that has a domain and a
range: takes objects as inputs and generates an output.
Its specific and concrete, whereas the uninterpreted term
\begin_inset Formula $f\left(t_{1},\cdots,t_{n}\right)$
\end_inset
had no such constraints: it wasn't a function at all, just a (meaningless)
arrangement of the the typographical symbols
\begin_inset Formula $f,t,1,n,(,)$
\end_inset
on the written page.
That the typography of it just happens to resemble that of a function is
already a powerful nod to the importance of notation and visual resemblance
to understanding...
and also to misunderstanding.
\end_layout
\begin_layout Standard
The interpretation of a relation
\begin_inset Formula $R\left(t_{1},\cdots,t_{n}\right)$
\end_inset
is a subset
\begin_inset Formula $\phi\left(R\right)$
\end_inset
of
\begin_inset Formula $M\times\cdots\times M$
\end_inset
.
That is, the
\begin_inset Formula $n$
\end_inset
-tuple
\begin_inset Formula $\left\langle \phi\left(t_{1}\right),\cdots,\phi\left(t_{n}\right)\right\rangle \in\phi\left(R\right)$
\end_inset
if and only if
\begin_inset Formula $R\left(t_{1},\cdots,t_{n}\right)$
\end_inset
is true.
This is just the usual correspondence between set-membership, and indicator
functions.
So a relation
\begin_inset Formula $R$
\end_inset
is some intensive predicate; it's interpretation is some extensive set
of objects.
It is easiest to consider a unary predicate first, i.e.
\begin_inset Formula $n=1$
\end_inset
so that
\begin_inset Formula $R(t)$
\end_inset
is either true or false for any given
\begin_inset Formula $t$
\end_inset
.
Given the set
\begin_inset Formula $\phi\left(R\right)=\left\{ \mbox{cat},\mbox{dog},\mbox{mouse}\right\} $
\end_inset
we might conclude that
\begin_inset Formula $R\left(t\right)$
\end_inset
is true whenever
\begin_inset Formula $\phi\left(t\right)$
\end_inset
is a mammal.
(or perhaps if
\begin_inset Formula $\phi\left(t\right)$
\end_inset
is furry...)
\end_layout
\begin_layout Standard
Terms can also be variables.
The interpretation of variables is more abstract and difficult to understand;
it is picked up again in a later section.
It is, however, the central theme to this essay: the interpretation of
variables is how one actually
\begin_inset Quotes eld
\end_inset
learns
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Subsection*
Metaphysics
\end_layout
\begin_layout Standard
The above is a broader, and looser sketch of an
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
than what is canonically found in books on mathematics.
There's nothing particularly wrong with the formulas provided; rather,
its the examples.
Fruit and animals are never used as examples; whereas integers and sets
are.
The goal here is different: it is to illustrate that the formal, narrow
mathematical concept of
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
has a broader application than just mathematical logic.
\end_layout
\begin_layout Standard
But, I can almost hear you object: if the goal is to make analogies between
real life and mathematics, there are many ways that this can be done.
So, the interpretation
\begin_inset Formula $\phi$
\end_inset
behaves a lot like a homomorphism, why not talk about homomorphisms, instead?
But also,
\begin_inset Formula $\phi$
\end_inset
behaves a lot like an action, so why not talk about actions? The answer
is perhaps a let-down: the interpretation
\begin_inset Formula $\phi$
\end_inset
is a lot closer to what we really want, than a homomorphism or action is.
\end_layout
\begin_layout Standard
What do we really want? We want a correspondence between an internal mode
of thought and perceived external experience, coupled in such a way that
the internal system adequately reflects or describes the external world:
predicting it, as it were, allowing the external world to be manipulated
based on deductions performed on the internal system.
\end_layout
\begin_layout Standard
But why mathematical logic? Here the answer is stronger.
Mathematical logic is a theory of symbols, and how they work together.
Much of (but not all of) human endeavor involves a relationship with the
symbolic world: for communication, for science, for deducing evidence in
murder-mystery thrillers.
The symbolic domain is part of human existence, and vitally important for
science; its not going away.
Mathematical logic has a well-articulated, deep and broad theory, with
a giant treasure-chest of useful results that can be used (here, I am thinking
broadly: proof theory, model theory, category theory and type theory all
being sub-disciplines of mathematical logic).
Many of these results are described as algorithms, with various degrees
of efficiency: parsers and solvers coming in a rainbow of different forms.
\end_layout
\begin_layout Standard
Perhaps there is a better way of thinking about and talking about the correspond
ence between abstract understanding and real-world evidence.
For right now, though, the concept of
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
seems adequate.
In particular, the previous section is trying to set up the next section:
a claim that one can invert this process, and discover interpretations,
and meaning, algorithmically, automatically, by machine.
\end_layout
\begin_layout Subsection*
Science
\end_layout
\begin_layout Standard
One naive way of understanding science is that it is the process of obtaining
formal, axiomatic systems that have interpretations in physical reality.
To oversimplify: one observes how gravity makes things fall down, and then
deduces
\begin_inset Formula $\vec{F}=m\vec{a}$
\end_inset
.
What is the meaning of
\begin_inset Formula $\vec{F}=m\vec{a}$
\end_inset
? It means
\begin_inset Quotes eld
\end_inset
things fall down
\begin_inset Quotes erd
\end_inset
.
How do we know that is what it means? We know in two different ways.
First, it has a very large collection of interpretations: one where
\begin_inset Formula $m$
\end_inset
is an apple, another where
\begin_inset Formula $m$
\end_inset
is a canon-ball, and yet another where its a planet.
The second way is that
\begin_inset Formula $\vec{F}=m\vec{a}$
\end_inset
can be combined with the logical connectives
\begin_inset Quotes eld
\end_inset
and
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
or
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
for all
\begin_inset Quotes erd
\end_inset
,
\begin_inset Quotes eld
\end_inset
there exists
\begin_inset Quotes erd
\end_inset
, as well as other formulas an symbol manipulation techniques to obtain
a very large collection of related formulas, each of which also has an
interpretation that is consistent with logical reasoning applied to the
formulas.
That is, the
\begin_inset Quotes eld
\end_inset
weight of evidence
\begin_inset Quotes erd
\end_inset
is multi-faceted: not just that there is a large body of evidence, but
that this evidence can be reverse-mapped to the theory, and to the symbolic
articulations of the theory.
\end_layout
\begin_layout Standard
The claim of this essay is that the formal understanding of what an interpretati
on is, together with the informal explanation provided above, is just enough
to be able to design and implement an actual, functional machine that extracts
\begin_inset Quotes eld
\end_inset
meaning
\begin_inset Quotes erd
\end_inset
from
\begin_inset Quotes eld
\end_inset
observation
\begin_inset Quotes erd
\end_inset
.
In short,
\begin_inset Quotes eld
\end_inset
machine learning
\begin_inset Quotes erd
\end_inset
.
But this first requires an examination of how variables are interpreted.
\end_layout
\begin_layout Subsection*
The Interpretation of Variables
\end_layout
\begin_layout Standard
In general, terms can be of the form
\begin_inset Formula $f\left(x,y,\cdots,z\right)$
\end_inset
containing variables
\begin_inset Formula $x,y,\cdots,z$
\end_inset
and likewise relations
\begin_inset Formula $R\left(x,y,\cdots,z\right)$
\end_inset
.
These two have to be given an interpretation: what do they mean? In colloquial
terms, its not hard: one imagines that
\begin_inset Formula $x,y,\cdots,z$
\end_inset
stands for the set of all of those things that are appropriate for the
term, or make the relation true.
It takes some work to convert this informal understanding into a formal
statement.
\end_layout
\begin_layout Standard
One interprets not a single variable at a time, but all of them, at once,
assigning to each variable
\begin_inset Formula $x$
\end_inset
an element
\begin_inset Formula $M$
\end_inset
.
The assignment is effectively
\begin_inset Quotes eld
\end_inset
random
\begin_inset Quotes erd
\end_inset
.
In itself, it has no particular meaning.
One defines the space
\begin_inset Formula $\overline{M}$
\end_inset
as the space of all possible variable assignments.
Thus, a single point
\begin_inset Formula $\xi\in\overline{M}$
\end_inset
corresponds to an assignment of elements in
\begin_inset Formula $M$
\end_inset
for
\begin_inset Formula $x,y,\cdots,z$
\end_inset
.
In formulas,
\begin_inset Formula $\phi\left(x\right)\left(\xi\right)=\xi\left(x\right)\in M$
\end_inset
, so that
\begin_inset Formula $\phi\left(x\right):\overline{M}\to M$
\end_inset
is a function that maps a point in
\begin_inset Formula $\overline{M}$
\end_inset
to an element of
\begin_inset Formula $M$
\end_inset
, and (by currying)
\begin_inset Formula $\phi:V\times\overline{M}\to M$
\end_inset
takes a variable, and some point in
\begin_inset Formula $\overline{M}$
\end_inset
, and produces and element in the
\begin_inset Quotes eld
\end_inset
real world
\begin_inset Quotes erd
\end_inset
\begin_inset Formula $M$
\end_inset
.
That is, the interpretation of a variable is a function ranging over variable
assignments.
\end_layout
\begin_layout Standard
This is a bit painfully abstract, but it allows for a natural way of examining
statements such as
\begin_inset Quotes eld
\end_inset
if
\begin_inset Formula $x$
\end_inset
is furry, then
\begin_inset Formula $x$
\end_inset
is an animal
\begin_inset Quotes erd
\end_inset
.
We can try to test if this statement is generically true by replacing
\begin_inset Formula $x$
\end_inset
with an instance of every possible object in the real world.
If a formula
\begin_inset Formula $p\left(x\right)$
\end_inset
remains true as
\begin_inset Formula $\xi\in\overline{M}$
\end_inset
is varied, i.e.
if
\begin_inset Formula $\phi\left(p\right)\left(\phi\left(x\right)\left(\xi\right)\right)=\phi\left(p\right)\left(\xi\left(x\right)\right)$
\end_inset
remains true for all
\begin_inset Formula $\xi\in\overline{M}$
\end_inset
, then one says that
\begin_inset Formula $p\left(x\right)$
\end_inset
is a
\begin_inset Formula $\phi$
\end_inset
-true statement.
The goal of learning is to discover
\begin_inset Formula $\phi$
\end_inset
-true statements.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
In mathematics, when an interpretation
\begin_inset Formula $\phi$
\end_inset
of a collection of formulas
\begin_inset Formula $\mathcal{E}$
\end_inset
renders all of the formulas
\begin_inset Formula $\phi$
\end_inset
-true, then
\begin_inset Formula $\phi$
\end_inset
is called a
\begin_inset Quotes eld
\end_inset
model
\begin_inset Quotes erd
\end_inset
of
\begin_inset Formula $\mathcal{E}$
\end_inset
.
This is perversely backwards from the meaning of the word
\begin_inset Quotes eld
\end_inset
model
\begin_inset Quotes erd
\end_inset
in engineering and process control.
In the latter case, one has the evidence
\begin_inset Formula $\phi\left(\mathcal{E}\right)$
\end_inset
of real life, but doesn't really know what either
\begin_inset Formula $\phi$
\end_inset
or
\begin_inset Formula $\mathcal{E}$
\end_inset
are.
When some
\begin_inset Formula $\phi$
\end_inset
and
\begin_inset Formula $\mathcal{E}$
\end_inset
are found that approximate reality, these are called a
\begin_inset Quotes eld
\end_inset
model
\begin_inset Quotes erd
\end_inset
of reality.
Examples include a thermostat, which attempts to model the temperature
inside a house.
It cannot ever know the true temperature (drafty windows, cold basements),
but it can form some reasonable model of what this might be, and then act
to turn on the heat (or cooling).
Likewise, the casual use of the word
\begin_inset Quotes eld
\end_inset
interpretation
\begin_inset Quotes erd
\end_inset
can mean exactly the opposite of the defintioin given here.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Revising the example might make this clearer.
Consider instead the proposition
\begin_inset Formula $p\left(x\right)=$
\end_inset
\begin_inset Quotes eld
\end_inset
if
\begin_inset Formula $x$
\end_inset
is
\begin_inset Formula $c_{f}$
\end_inset
then
\begin_inset Formula $x$
\end_inset
is a
\begin_inset Formula $c_{a}$
\end_inset
\begin_inset Quotes erd
\end_inset
.
The symbols
\begin_inset Formula $c_{f}$
\end_inset
and
\begin_inset Formula $c_{a}$
\end_inset
are (uninterpreted) constants.
What might they mean? Well, if
\begin_inset Formula $\phi\left(c_{f}\right)=\mbox{"furry"}$
\end_inset
and
\begin_inset Formula $\phi\left(c_{a}\right)=\mbox{"animal"}$
\end_inset
then we are on to something, because when we interpret
\begin_inset Formula $\phi\left(x\right)$
\end_inset
as a the set of all possible things that could be
\begin_inset Formula $x$
\end_inset
, we can then examine if
\begin_inset Formula $p\left(x\right)$
\end_inset
is true (or not) (or rather, more precisely, if
\begin_inset Formula $\phi\left(p\right)\left(\phi\left(x\right)\right)$
\end_inset
is true or not, as it varies over all of
\begin_inset Formula $\overline{M}$
\end_inset
).
\end_layout
\begin_layout Standard
This now provides a criterion for determining whether an interpretation
is real-world-reasonable, or not.
So,
\begin_inset Formula $\phi\left(c_{f}\right)=\mbox{"furry"}$
\end_inset
and
\begin_inset Formula $\phi\left(c_{a}\right)=\mbox{"animal"}$
\end_inset
seems to work, but
\begin_inset Formula $\phi\left(c_{f}\right)=\mbox{"slimey"}$
\end_inset
and
\begin_inset Formula $\phi\left(c_{a}\right)=\mbox{"wooden"}$
\end_inset
does not.
\end_layout
\begin_layout Subsection*
Machine Learning
\end_layout
\begin_layout Standard
The point of this machinery is that it provides just enough specificity
to imagine that it could be implemented as a machine learning algorithm.
There is no actual machine learning algorithm presented; nor is there any
claim that it might be efficient, fast or practical.
You are welcome to tell me that doing the above will take longer than the
age of the universe when implemented on an IBM-386 PC.
Rather, it sketches what one must attempt to do, if one wishes to develop
a symbolic understanding of reality.
\end_layout
\begin_layout Standard
Nothing presented her so far is in any way astounding, taken from the point
of view of machine learning.
A standard route in machine learning is to formulate some hypothesis
\begin_inset Formula $R\left(x,y,\cdots,z\right)$
\end_inset
, and then crawl over a dataset of
\begin_inset Formula $x,y,\cdots,z$
\end_inset
to see if it holds.
This is typically CPU-intensive, and most hypothesis do not hold.
The holy grail of machine learning is to be able to generate accurate hypothesi
s from the get-go, and then to evaluate them quickly.
\end_layout
\begin_layout Standard
But what it does show is that it is possible, at least in principle, to
formulate the hypothesis
\begin_inset Quotes eld
\end_inset
if
\begin_inset Formula $x$
\end_inset
is
\begin_inset Formula $c_{f}$
\end_inset
then
\begin_inset Formula $x$
\end_inset
is a
\begin_inset Formula $c_{a}$
\end_inset
\begin_inset Quotes erd
\end_inset
together with an interpretation
\begin_inset Formula $\phi\left(c_{f}\right)=\mbox{"furry"}$
\end_inset
and
\begin_inset Formula $\phi\left(c_{a}\right)=\mbox{"animal"}$
\end_inset
and then run around, look at things, and see if it is true.
If one can confirm this hypothesis, and a handful of others, then one is
well on the way of developing an understanding of the real world, in such
a way that one can navigate in it, without getting killed.
\end_layout
\begin_layout Subsection*
Conclusion
\end_layout
\begin_layout Standard
The point here, so far, is perhaps a more philosophical one, after all:
a defense of the idea that learning is all about the extraction of patterns
from the observation of nature.
But there is more: the extraction can be done in such a way that a symbolic
structure emerges.
The symbols can be arranged into terms and relations, in a concrete enough
that they can be compounded with logic connectives to obtain formulas,
and that those formulas can then be used to make predictions about reality.
We've got tools and theories and systems that know how to manipulate symbols,
and to do so reliably: again: the human race is deeply entangled with the
processing of symbols.
The point here is that symbols can be extracted from observations of nature;
once they are, we are home-free.
\end_layout
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "/home/linas/src/novamente/src/learn-git/learn-lang-diary/lang"
options "plain"
\end_inset
\end_layout
\end_body
\end_document