Any approach to AGI needs a mathematical metaphysics. The most
widely-accepted such metaphysics at present is the Turing Machine's
foundation for Algorithmic Information Theory. While some Theories of
Everything argue against the implied causal structure's unidirectional
"arrow-of-time" it is not unreasonable to attempt to elaborate the Turing
Machine approach to AGI. In that elaboration, what might be thought of as
a "standard library" must be constructed. In so-doing, "the relational
dimensions of the empirical world", represented as "number" with implied
*dimensionality* must fall out quite early and naturally as applicable to
physical dimensions (length, mass, etc.) or we are on the wrong track. In
what follows I summarize a lifetime of professional support of, if not my
own work on programming languages toward this end. This is not a complete
picture, as the metaphysical assumption about time's arrow, implied by the
Turing machine, precludes modeling the deeper relational structures from
which it time, itself, may-yet emerge. Geometric algebra may-yet place
Turing's metaphysical assumption about time in its proper perspective as
emergent from a mathematical metaphysics "standard library" that
better-compresses our empirical observations. In that process of
philosophical discovery, I fully expect that AIT's other-half, sequential
decision theory, will find its utility function specified and, by
implication, provide mathematical structures for "awareness", "qualia",
etc., if not "consciousness".
First I'm going to make a few radical assertions:
- A real-world relation is best-regarded as a random variable. Think of
measurement. This is consistent with SQL's default allowance of duplicate
rows in an extension. These count tables represent the probability
distribution of the random variable. Each relationship (row) in an
extension is, therefore, best thought of as a single measurement, or case.
The duplicate row counts are therefore case-counts. A probability density
function results from simply dividing each case's count by the total counts
in the relation.
- The properties of a measurement (say, time and distance) are the
dimensions of the measurement and these correspond to the columns of the
extension.
- Any measurement can be thought of as a low-dimensional selected
projection of the empirical world: the universe. The universal extension
has a single row -- a row with as many dimensions as the entire history of
the universe has properties: We might call this row "That which is The
Case."
Now, accepting all of that (which philosophers may well argue against --
particularly if they don't like Descartes, etc.):
In order for the random variable to have meaning, its dimensions must have
counts, just as do its duplicate rows. For instance, we might think of a
relation whose composite dimension is velocity, with columns: time and
distance. Although there might be meaning to a physical dimension of
time*distance (time^(+1) * distance^(+1)) that is not the physical
dimension we call "velocity". To obtain a velocity relation, we need
distance/time which is time^(-1) * distance^(+1). Note that these terms
commute because multiplication (like 'and') commutes. Column order is
meaningless, just as is row order meaningless since addition (like 'or')
also commutes.
Now consider the relational dimension of energy where we join the velocity
relation with a mass relation and assign column counts thus: time^(-2) *
distance^(+2) * mass^(+1).
Note that thus far, I have not talked about "units", nor of "types". First
a down-to-earth comment about "units": It is important to regard "units"
as I/O formats (or "representations") with isomorphic transformations
between them (1:1 correspondence between a distance measurement in inches
and one in feet). Second is a more philosophical comment about "types" vs
"dimensions" that gets to the heart of what I believe is a huge mistake in
the foundation of computer science dating to Russell and Whitehead's
Principia Mathematica:
PM's type theory (and elaborations/variations thereof) is the current
foundation of computer science. Russell used it to develop Relation
Arithmetic. In "My Philosophical Development", of Principia Mathematica
Part IV "Relation Arithmetic", Bertrand Russell laments:
*"I think relation-arithmetic important, not only as an interesting
generalization, but because it supplies a symbolic technique required for
dealing with structure. It has seemed to me that those who are not familiar
with mathematical logic find great difficulty in understanding what is
meant by 'structure', and, owing to this difficulty, are apt to go astray
in attempting to understand the empirical world [emphasis JAB]. For this
reason, if for no other, I am sorry that the theory of relation-arithmetic
has been largely unnoticed."*
However, the ultimate project of Principia Mathematica was directed at "the
empirical world" in the conclusion of PM: Part VI "Quantity". "Quantity"
consists of 3 sections the last of which, section "C", is about
"Measurement" in terms of a generalization of the concept of number
(section "A"), to include units of measurement (mass, length, time, etc.)
as commensurable (dimensioned) quantities ("B" "Vector-Families").
Yet, other than *314:
*"Relational real numbers are useful in applying measurement by means of
real numbers to vector-families, since it is convenient to have real
numbers of the same type as ratios."*
I see nothing in Part VI that references anything like "relation numbers"
as defined in Part IV.
Before I get into a resolution strategy, I want to add one final issue that
is key to understanding relational structure in terms of measurement:
Any value that we assign to a cell in a table has what is called
"measurement error". Note, I'm talking here not of a relation (table) nor
of a relationship (row), but of a relata (cell value) of that
relationship. Take, for instance, a table of velocities with time and
distance columns. Each case (row, or relationship between measured
properties) has two measurements for that case: a measured distance and a
measured time. What we call "measurement error" is an estimate of the
probability distribution that would prospectively obtain with repeated
measurements of the same conditions. In other words, assigning measurement
error, or "fuzziness", is best thought of as imputing missing data -- those
prospective measurements just mentioned. In any rigorous attempt to deal
with the fuzziness of the real world, it is important to keep in mind the
relational structure of the measurements so that propagation of measurement
error is understood in terms of relational composition (aka 'JOIN' to use
database jargon).
Now to proceed to the resolution strategy:
Late in Russell's life he admitted he regretted Type Theory, stating it was
the most arbitrary thing he and Whitehead did and that it was more of a
stopgap than a theory.
As it turns out, Russell admitted this because he was relieved and
delighted he lived long enough to see the matter resolved in the late 1960s
book titled "The Laws of Form" by G. Spencer Brown. The resolution was to
include what logicians think of as "paradox" as a, if not the, primary
foundation of mathematical logic:
Russell's Paradox (The set of all sets that don't contain themselves as
members.) which motivated PM's Type Theory, is only one form of this
protean "paradox". The most Laconian form is:
"This sentence is false."
The resolution provided in GSB's LoF was to introduce the the square root
of -1 as primary in mathematical logic. This is otherwise known as the
imaginary identity 'i' found throughout all of dynamical systems theory.
Dynamical systems are about changes. In relational database terms, these
are updates. Relational updates are addition and subtraction of rows.
Under the notion of row-as-relationships-as-case, subtraction entails
negative case counts.
Interestingly, negative case counts permit the emergence of something
called Link Theory which Paul Allen's think tank, Interval Research
supported until its demise, at which point I supported it at HP's "Internet
Chapter II" project aka "eSpeak" until _its_ demise, at which point
Federico Faggin (co-founder of Intel's microprocessor division) underwrote
its final support at Boundary Institute.
Link Theory utilized negative case counts to provide a relational
description of physics including the core of quantum mechanics -- and was
therefore of interest in the quantum computing field. This is due to the
fact that quantum measurement involves projection (as do all measurements
-- see my prior invocation of "That which is The Case.") that included not
only ordinary probabilities, but also what are called "probability
amplitudes". Quantum probability amplitudes have complex values on the
unit circle of the complex plane. Complex values have imaginary components,
Link theory accommodated QM's imaginary components with a particular
symmetry used by George W. Mackey in his 1963 book "Mathematical
Foundations of Quantum Mechanics" representing 'i' as a 2x2 spinor matrix:
0 1
-1 0
See Appendix A of "Link Theory -- From Logic to Quantum Physics
<https://web.archive.org/web/20150925004918/boundaryinstitute.org/bi/physics.htm>
".
The -1 in this spinor corresponds to the negative case counts required for
relational structure to encompass quantum measurement.
Federico Faggin supported this work because hardware design languages
needed a formal theory other than conventional logic to model digital
circuits with feedback (ie: memory, state change, etc.). George Spencer
Brown developed his mathematics as a result of inventing minimal circuits
in the early days of the transistor -- and found he was working with
imaginary logic values.
So, tying this all together to address the original point: It would appear
that the computer science notion of "type" is not only ill-founded --
leading to all manner of confusion regarding "the empirical world" (in
Russell's apt descriptive phrase) but is recognized as being ill-founded by
its founder!
My assertion is that the notion of "type" is rescued by the notion of
"unit" and that "abstract type" is rescued by the notion of "dimension"
within the relational paradigm. That this might be the case should be no
surprise as the natural sciences (particularly physics) most rigorously
address "the empirical world".
Once we accept the framework of dimensionality as relational structure, we
can see, further, the potential for new modes of schema analysis based on
the scientific discipline of dimensional analysis.
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Artificial General Intelligence List: AGI
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