The
origin of empathy and love from infinity is usually sensed as a
metaphor. However, one can show that the mathematical concept of
infinity generates properties and entities, which can be interpreted
as a set-theory equivalent of those emotions, and thus as a model of
them. Furthermore no finite mathematical structure can imply those
properties and entities, and infinity is necessary for them.
The
talk addresses the formal model of emotion recognition.
Completeness
and incompleteness are sharply distinguished as to finiteness, but
not as to infinity. For infinity is both universal (thus complete)
and open (thus incomplete) suggesting a special class of phenomena,
which combine universality and openness in a way inaccessible from
finiteness. The mathematical models of empathy and love are founded
on the nonempty intersection of completeness and incompleteness
possible as to infinity.
An
infinite set “A” and an arbitrary nonempty set “B” are given
so that their intersection is empty: Then one can constitute their
union, which will be an infinite set, too, and which will content the
initial infinite set as a true subset. There exists that one-to-one
mapping by means of the axiom of choice in general, which generates a
one-to-one image “BA”
of the arbitrary external set “B” into “A” so that “BA”
is a true subset of “A”. “BA”
will be designated as an “empathic image” of “B” into “A”
and the relation of “A” to “B”, empathy “EAB”.
The necessary and sufficient condition for the existence of the
empathic image “BA”
and empathy “EAB”
is “A” to be infinite if the axiom of choice is granted. It,
being equivalent to a well-ordering in “A”, can be interpreted as
time in a philosophical and mathematical sense. Consequently time as
well as infinity can be discussed as intimately linked to the ability
both of empathy and love.
If “B” is also
infinite, another empathic image “AB”
corresponding to “BA”
arises in turn. Furthermore “A” and “B” can restrict mutually
their degrees of freedom if both intersections “A∩BA”
and “B∩AB”
are nonempty. “BA”
and “B” can be described jointly by one and the same Hilbert
space “H1”,
and “AB”
and “A” can by “H2”
as well. However these two Hilbert spaces “H1”
and “H2”
are not identical if and only if “A∩BA”
and “B∩AB”
are nonempty. Then the common Hilbert space of the system of “A”
and “B” cannot be factorized to “H1”
and “H2”,
which defines that “A” and “B” are entangled. Consequently
two or more entangled quantum computers might model “love” in
that formalized sense. The resulting restriction of the degrees of
freedom both of “A” and “B” is equivalent to a force of
attraction between “A” and “B” in any given moment of “time”
being defined above as a well-ordering always guaranteed for the
axiom of choice.
That
construction is a negative result in the theory of AI since it shows
that no Turing machine can model those emotions or it “hangs up”
if it “tries” anyway. That set-theory model of empathy allows the
Gödel incompleteness theorems to be interpreted in another way.
There
is a direct link between those models of emotions and the universal
pattern recognition expected from any intellect: A quantum computer,
being able to accomplish infinite calculations in principle in
relation to an external “user”, can as recognize any pattern as
feel both “love” and “empathy” in a sense formalized above.
The
study can be continued to an empathic theory of perception on the
ground of the above model of empathy.
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