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 How are these rescued or reformulated or ?? as dimensions? I have noted that the 4 truth values identified by Kauffmann/ Collins https://arxiv.org/abs/1905.12891 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 ;) 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-Mc53e1db7bdc7ec105323cca9 Delivery options: https://agi.topicbox.com/groups/agi/subscription
