On Sat, Sep 11, 2021 at 9:01 AM Ben Goertzel via AGI <[email protected]> wrote:
> James, > > *** > 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". > > *** > > I don't get what you mean by " "abstract type" is rescued by the > notion of "dimension" within the relational paradigm". ? > > Take e.g. a complex dependent type as expressed in Idris, or a > probabilistic type as expressed in > > https://arxiv.org/abs/1602.06420 My point is that *all* "type theories", as currently conceived, are ill founded because they aren't grounded in the requirement that they express an arithmetic of *observation* which, in multiplicity, become quantities called *measurements* represented as *numbers* that carry with them *dimensions* -- what Russell was trying to get at with "relation numbers" -- grounded in the relational dimensions of the empirical world. Physics has had to explore this *metaphysics *for the obvious reason that the axioms of physical theory are observations -- these observations qualify as axioms because they are the "givens" of the formal systems of physics. There is something profound about this reversal. By "reversal" I mean we ordinarily think of "induction" as producing "axioms" from observations so as to produce a computer program (the Kolmogorov Complexity program). This reversal relates to the way dynamical systems theories provide time-reversible laws of nature via complex valued functions. Taking this from another angle: In model theory terminology, there must be an *interpretation* for any *meaningful* "type theory" -- where "meaningful interpretation" is what model theory is (supposedly) all about: a model. I say "supposedly" because what we usually see in model theory is ungrounded interpretations -- interpreting one theory in terms of another theory. All this beating-around-the-bush provides employment for mathematicians but gets us nowhere. How are these rescued or reformulated or ?? as dimensions? Think about it like this: A testing suite for any formal system purporting relevance is that dimensionless numbers must be a special case of the default case: dimensioned numbers aka measurements. Moreover, its arithmetic must do inherent "type checking" by asking whether two numbers that are being added/subtracted have the same dimensions (and conserve their case counts) and whether the result of any multiplication/division results in the expected addition/subtraction of their dimensional exponents. I have noted that the 4 truth values identified by Kauffmann/ Collins > > https://arxiv.org/abs/1905.12891 Yes, that approach is getting closer to the metaphysics I'm talking about. > map into Patterson's Constructible Duality (paraconsistent) logic's > truth values, and that logic maps into a pair of Heyting algebras > which means that expressions in CD logic correspond to pairs of > programs in dependently typed languages without continuation. In > this sense LoF expressions are isomorphic to pairs of dependently > typed expressions, but I'm not sure who is rescuing whom from what ;) > Again, appealing to my notion of a formal systems "test suite": Such a test suite -- such a "set of requirements" -- for a formal system "rescues" the formal system from mental masterbation in the same way that a programmer's test suite or set of requirements rescues the programmer. > > ben > > On Fri, Sep 10, 2021 at 8:39 AM James Bowery <[email protected]> wrote: > > > > 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". > > > > 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. > > Artificial General Intelligence List / AGI / see discussions + > participants + delivery options Permalink > > -- > Ben Goertzel, PhD > http://goertzel.org > > “He not busy being born is busy dying" -- Bob Dylan ------------------------------------------ Artificial General Intelligence List: AGI Permalink: https://agi.topicbox.com/groups/agi/Ta03542805b689301-M84ca090c527bd72f916b73a0 Delivery options: https://agi.topicbox.com/groups/agi/subscription
