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The penultimate paper of "Bit-string Physics: A Finite and Discrete Approach to Natural Philosophy" discusses an attempted revival of "Relation Arithmetic" with which Russell and Whitehead had planned to cap off their Principia Mathematica in its final volume.

Of Relation Arithmetic, Russel said:

"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. For this reason, if for no other, I am sorry that the theory of relation-arithmetic has been largely unnoticed."

-- " My Philosophical Development" by Bertrand Russell

An example of going astray in attempting to understand the empirical world is when people attempt to combine incommensurable quantities in their calculations, not understanding the structure of the relations between the quantities.

Ordinarily, programming languages treat units, as I/O formats for dimensions, as an afterthought -- independent of type checking. However, what if we saw numbers themselves as embodying relational structure, as intended by Russell, thereby unifying the notion of "type checking" with the notion of "number"? Might then the power of dimensional analysis be brought to bear, in a mathematically rigorous way, on the relatively ad hoc notions of "type", hence problematic areas such as the object relational impedance mismatch?

Anyway I'll proceed with something worth talking about when you said: "But maybe your quantum tabled resolution (or whatever I should call that) doesn't have any of the semantic and practical problems that Prolog and co. have. I'd be intrigued to hear about the nitty-gritty details in that case."

It is precisely the problem of time in relational programming systems that led me to investigate the work by the late Tom Etter there, and at HP reviving Russell and Whitehead's Relation Arithmetic via something he called Link Theory which is treats things like complex numbers (hence Hamiltonians and other dynamical constraint systems) as a particular symmetry on real-valued spinor matrices ala Macky.

If you do things in this way, you are less apt to "go astray in attempting to understand the empirical world" as Russell put it. To first order, think about it like this: What if you started not with pure, dimensionless numeric data types but with dimensional analysis that emerged from treating the columns of relation tables as dimensions of "relation numbers" to use Principia's terminology?

Anyway I'll proceed with something worth talking about when you said: "But maybe your quantum tabled resolution (or whatever I should call that) doesn't have any of the semantic and practical problems that Prolog and co. have. I'd be intrigued to hear about the nitty-gritty details in that case."

It is precisely the problem of time in relational programming systems that led me to investigate the work by the late Tom Etter there, and at HP reviving Russell and Whitehead's Relation Arithmetic via something he called Link Theory which is treats things like complex numbers (hence Hamiltonians and other dynamical constraint systems) as a particular symmetry on real-valued spinor matrices ala Macky.

If you do things in this way, you are less apt to "go astray in attempting to understand the empirical world" as Russell put it. To first order, think about it like this: What if you started not with pure, dimensionless numeric data types but with dimensional analysis that emerged from treating the columns of relation tables as dimensions of "relation numbers" to use Principia's terminology?

Anyway I'll proceed with something worth talking about when you said: "But maybe your quantum tabled resolution (or whatever I should call that) doesn't have any of the semantic and practical problems that Prolog and co. have. I'd be intrigued to hear about the nitty-gritty details in that case."

It is precisely the problem of time in relational programming systems that led me to investigate the work by the late Tom Etter there, and at HP reviving Russell and Whitehead's Relation Arithmetic via something he called Link Theory which is treats things like complex numbers (hence Hamiltonians and other dynamical constraint systems) as a particular symmetry on real-valued spinor matrices ala Macky.

If you do things in this way, you are less apt to "go astray in attempting to understand the empirical world" as Russell put it. To first order, think about it like this: What if you started not with pure, dimensionless numeric data types but with dimensional analysis that emerged from treating the columns of relation tables as dimensions of "relation numbers" to use Principia's terminology?

The most progress toward this end was made when I agreed to come aboard HP's eSpeak project (which didn't make any sense to me) so long as they let me pursue this vision for "Internet Chapter 2". This is as far as we got:

Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy [Hardcover] H. Pierre Noyes (Author), J. C. Van Den Berg (Author, Editor), J.C. van den Berg (Author), is the concluding book of the 27 volume "Series on Knots and Everything" and it, in turn, concludes with "Reflections on PSQM" which contains, on page 548:

"... Relation Arithmetic Revived , was written at Hewlett-Packard as part of the theoretical work that Jim Bowery and I were doing on the design of transactional languages for the Internet. It went much further in developing another idea briefly introduced in Structure Theory, which is that the theory of relations, and indeed the whole of mathematics, can be formulated in a language whose only primitive predicates are identity relations. This new work appears to have good long-range prospects for putting link theory on a deeper logical foundation, but that is outside the scope of the present paper."

This work was cut short when I insisted on retaining Dr. Etter for his mathematical specialty rather than "hiring all the H-1b's from India you want".

The most progress toward this end was made when I agreed to come aboard HP's eSpeak project (which didn't make any sense to me) so long as they let me pursue this vision for "Internet Chapter 2". This is as far as we got:

Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy [Hardcover] H. Pierre Noyes (Author), J. C. Van Den Berg (Author, Editor), J.C. van den Berg (Author), is the concluding book of the 27 volume "Series on Knots and Everything" and it, in turn, concludes with "Reflections on PSQM" which contains, on page 548:

"... Relation Arithmetic Revived , was written at Hewlett-Packard as part of the theoretical work that Jim Bowery and I were doing on the design of transactional languages for the Internet. It went much further in developing another idea briefly introduced in Structure Theory, which is that the theory of relations, and indeed the whole of mathematics, can be formulated in a language whose only primitive predicates are identity relations. This new work appears to have good long-range prospects for putting link theory on a deeper logical foundation, but that is outside the scope of the present paper."

This work was cut short when I insisted on retaining Dr. Etter for his mathematical specialty rather than "hiring all the H-1b's from India you want".

The most progress toward this end was made when I agreed to come aboard HP's eSpeak project (which didn't make any sense to me) so long as they let me pursue this vision for "Internet Chapter 2". This is as far as we got:

Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy [Hardcover] H. Pierre Noyes (Author), J. C. Van Den Berg (Author, Editor), J.C. van den Berg (Author), is the concluding book of the 27 volume "Series on Knots and Everything" and it, in turn, concludes with "Reflections on PSQM" which contains, on page 548:

"... Relation Arithmetic Revived , was written at Hewlett-Packard as part of the theoretical work that Jim Bowery and I were doing on the design of transactional languages for the Internet. It went much further in developing another idea briefly introduced in Structure Theory, which is that the theory of relations, and indeed the whole of mathematics, can be formulated in a language whose only primitive predicates are identity relations. This new work appears to have good long-range prospects for putting link theory on a deeper logical foundation, but that is outside the scope of the present paper."

This work was cut short when I insisted on retaining Dr. Etter for his mathematical specialty rather than "hiring all the H-1b's from India you want".

Whenever the government "picks winners" rather than letting nature pick winners, the technologists and therefore technology loses.

(Now that Cray is dead, according to the supercomputing FAQ, "The CCC intellectual property was purchased for a mere $250 thousand by Dasu, LLC - a corporation set up and (AFAIK) wholly owned by Mr. Hub Finkelstein, a Texas oilman. He's owned this stuff for five years and hasn't done anything with it.")

Secondly, as I've discussed before both operating system and database programming are awaiting the development of relations, most likely via the predicate calculus, as a foundation for software. Both are essentially parallel processing foundations for software.

This feeds into quantum computing quite nicely as well, as relations are not just inherently parallel, but are parallel in such a way that they precisely model quantum software.

The point I believe Etter is making is about the appropriate phrasing and location of those rules. He's attempting to deal with the notion of "thing" or ontology and how it relates to physics, which, I think, is quite important to the foundation of any attempt to model computation.

From a postscript to another of Etter's papers:

Logic, as currently taught, has three levels.

Level 1 is the science of pure predication and proof. One of the great achievements of modern times is to have put this science into a form whose principles are as self-evident as 2+2 = 4. This happened in the latter part of the Nineteenth Century, and the people responsible, or at least those whose names we know, had modest aims, mostly having to do with fixing the serious bugs in the Aristotelian logic still taught in the schools of their time. What emerged, however, was a totally new branch of mathematics. This has come to be called the predicate calculus, and it has nailed down once and for all what is meant by logical proof. In essence, the predicate calculus is the grammar of the words AND, OR, NOT, SOME and ALL. What makes it so important is how deeply these rules are embedded in all of human thought. If you believe they are wrong, and you want to convince the rest of us to change them, you had better have awfully good reasons, for to do so would be like convincing us that 2+2 = 5.

The second level goes beyond those rules that govern predication in general to introduce a particular predicate, the equality predicate x=y. The axioms for equality are normally presented as belonging to logic, and Quine2 has argued very cogently that this is proper. Even if we agree with him, though, we must be clear that the equality axioms are independent of the axioms of "pure" logic at level 1. Level 2 thus has room for changes that don't bear on the rules for AND.

The third level, and here I am referring to what is found in most logic textbooks, is set theory. Quine3 argues, and again very cogently, that set theory does not really belong to logic, and here I agree with him. Though there have been various attempts to create a quantum set theory, I don't see how changes on level 3 can be fundamental enough to come to grips with complementarity.

This brings us back to level 2. What I am proposing is that we should found level 2 on a three-place predicate that I call sameness, rather than on the two place equality predicate x=y. My notation for this predicate is y(x=z), read "x is the same y as z". Examples: "2+2 is the same number as 2*2", "Mary is the same woman as Mrs. Smith", "Bill is the same man he was ten years ago", "The morning star is the same planet as the evening star", "Your car is the same color as mine" etc. The axioms of sameness are very simple:

Axiom 1. y(x=x)
Axiom 2. If y(x=z) then y(z=x)
Axiom 3. If y(x=z) and y(z=w) then y(x=w)

Logical analysis has shown that the sameness predicate has some remarkable properties. For instance, it can be "morphed" into any other predicate by applying suitable axioms; it's the Morpheus of predicates, so-to-speak. Another remarkable fact is that any axiom system can be translated into an axiom system in which sameness is the only predicate. All of this is beyond the scope of the present paper, however; the point here is to see how sameness handles complementarity.

Suppose we are given q and p such that q(x=z) means that x has the same position as z and p(x=z) means they have the same momentum. Now if we can measure position, then we can tell whether x has the same position as z, and similarly if we can measure momentum, we can tell whether x has the same momentum as z. Therefore if we can simultaneously measure both position AND momentum, we can tell whether x has the same position AND the same momentum as z. Thus the question arises as to whether this compound of two samenesses comes under the a

Quine, in his essay "The Scope of Logic" says the identity predicate x=y is inherent in the very idea of predicates.

Given the predicate P(x1,x2,x3,...xN), a "Quine identity" P(A=B) is the conjunction:

For all x2,x3,...xN ( P( A ,x2,x3,...xN) iff P( B ,x2,x3...xN) &
For all x1, x3,...xN ( P(x1, A ,x3,...xN) iff P(x1, B ,x3...xN) &
For all x1,x2, ...xN ( P(x1,x2, A ,...xN) iff P(x1,x2, B ...xN) &
...
For all x1,x2,x3,... ( P(x1,x2,x3,... A ) iff P(x1,x2,x3... B )

Tom Etter (yes, the author of Racter) posits three predicates about which nothing is presumed except that they are Quine identities:

Row(x=y)
Column(x=y)
Value(x=y)

Mathematics is now expressible with no further primitives.

FOR INSTANCE:

See The Expressive Power of Equality for a proof that three identities are sufficient to express Zermelo-Fraenkl set theory.

Yes it has.

See Relation Arithmetic Revivedand Structure Theory. These two papers were written as a result of Hewlett-Packard's E-Speak project's support of a continuation of work begun at Paul Allen's thinktank, Interval Research. These then led to an understanding of the importance of identity theory in performing logic with what we were calling "attributed assertions" aka digitally signed speech acts. After the E-Speak project terminated we continued work on identity theory with partial support from the Boundary Institute leading to a reformulation of the foundation of mathematical logic with The Expressive Power of Equality.

Yes it has.

See Relation Arithmetic Revivedand Structure Theory. These two papers were written as a result of Hewlett-Packard's E-Speak project's support of a continuation of work begun at Paul Allen's thinktank, Interval Research. These then led to an understanding of the importance of identity theory in performing logic with what we were calling "attributed assertions" aka digitally signed speech acts. After the E-Speak project terminated we continued work on identity theory with partial support from the Boundary Institute leading to a reformulation of the foundation of mathematical logic with The Expressive Power of Equality.

Yes it has.

See Relation Arithmetic Revivedand Structure Theory. These two papers were written as a result of Hewlett-Packard's E-Speak project's support of a continuation of work begun at Paul Allen's thinktank, Interval Research. These then led to an understanding of the importance of identity theory in performing logic with what we were calling "attributed assertions" aka digitally signed speech acts. After the E-Speak project terminated we continued work on identity theory with partial support from the Boundary Institute leading to a reformulation of the foundation of mathematical logic with The Expressive Power of Equality.

INTRODUCTION: COUNTING SHEEP.

Once upon a time there was a sheep farmer who had ten small barns, in each of which he kept five sheep. When asked how many sheep he had altogether, he replied "many", for people in those days counted on their fingers, and no one had ever thought of counting beyond ten.

Every morning he would drive his sheep over the hill and through the woods to their pasture, where they assembled in five fields, ten sheep to a field. The farmer, who was of a reflective bent, saw here a curious and beautiful law of nature: "Ten barns each with five sheep, and then five fields each with ten sheep!" Unfortunately this law did not always hold, and when the wolves howled on the hill at night, it failed quite often. The farmer had an explanation for this: "The howling of the wolves greatly upsets my sheep, and the laws of nature, like the laws of man, are often disobeyed when agitated spirits prevail".

The farmer realized that to make his law universal he would have to modify it thus: "When tranquillity reigns, ten of five turn into five of ten."
We today who know arithmetic would say that the farmer's law, though true enough in his particular situation, isn't a very good law by scientific standards. It needs to be "factored" into two laws, the first being the simple and very general law that xy = yx and the second a more complicated and specialized law having to do with sheep and wolves. The farmer was indeed aware that xy = yx, at least in the case of 5 and 10, but what he could not see is that the essential condition for xy to be yx has nothing to do with sheep or wolves or tranquillity but is simply that the total number of sheep remain constant. One reason he couldn't see this is that he lacked any conception of the total number of his sheep; that's because in those days there were no numbers beyond ten, just "many".

There are three morals to this tale. The first is that it's not enough just to ask whether a law is right or wrong - we should also ask whether it gets to the point. The second is more subtle: If the point escapes us, maybe it's because we lack the raw materials of thought needed to even conceive of it. The third is not subtle at all: learn to count!

We have learned to count beyond ten sheep and even beyond three dimensions, but we still are under a very stifling conceptual limitation in not being able to count beyond the two types of phenomena that we call classical and quantum. This paper will set these two among many more. It will do this by teaching us some new ways to count cases, such as how to keep counting when the count goes below zero! This will provide us with the raw materials for thought we need to clearly see some crucial points that quantum philosophy has so far missed, notably the significance, or rather the insignificance, of the wave function, and the essentially acausal nature of quantum processes.

If you read the relation arithmetic paper you'll see there are 3 equality axioms: row, column and value. These three axioms correspond to the "triple stores" that are fundamental not only to RDF but to XML databases generally (as well as other flexible-schema stores such as LDAP). There are a couple of big differences however: Whereas most formal systems start with the idea of "ontology" or "things with names and definitions", relation arithmetic makes possible mathematics without ontology and the idea of negative presence (see phenomenology for the underlying philosophy). Both of these are a radical advance, although the mathematics without ontology is perhaps the more fundamental of the two, as subsequent work has shown.

First thing that came to my mind. Just after S11, a string of researchers around the world found a large deviation from the norm of sources of randomness which they were observing.

A sec while I google this. There for example, a study which suggests 1)that the world is linked as Gaia, 2) there was forknowledge of the events, as the curve of randomness deviated from it's baseline about 2 hours before the strikes.

If they do, then the quote from the following text will lead to 21st century science (that really should have been 20th century science but for some rather unfortunate concepts born of the Continental -- primarily German/Swiss -- physicists):

Thus we find that the concept of linking, which before led us immediately to the heart of quantum mechanics, has now led us immediately to the heart of relativity!

Out takes from Process System and Causality and "Reflections" on same.

Most discussions of the meaning of quantum mechanics these days seem to be about the problem of the "collapse of the wave function." In link theory this problem simply vanishes, since there is no wave function to collapse. Imagine if the Eighteenth Century caloric were still hanging around as the official theory of heat: we'd be chronically plagued by ever more complicated theories explaining the collapse of the "caloric field" when you measure an atom's energy. What a relief to get away from the spell of such nonsense!

...

This large-number explanation of quantum mechanics raises two basic questions: Large numbers of what? and Must we buy it?

The answer to the first question is implicit in the above discussion, but needs to be said simply: The things we count large numbers of are cases. Simple arithmetic reveals that the core quantum laws, in a generalized form, are features of any probabilistic system whatsoever. Von Neumann's formulation of the Born probability rule prob(P) = trace(PS) holds at every connection between the parts of such a system, and the dynamical rule S'T = TS governs every part that is connected at two places.

I brought up caloric to draw a parallel between our present situation and the situation in physics when it was discovered that the laws governing heat could be interpreted as statistical laws of atomic motion. However, there is a big difference. In the case of heat, the statistical theory sat on top of the Newtonian theory of motion, whereas in our case there is no underlying empirical theory at all. Probability theory is just the arithmetic of case counting, so the generalized quantum laws are like xy = yx in that their truth is assured, the only empirical issue being where and when they apply.

The answer to the second question is no, we don't have to. However, the same can be said about the arithmetical explanation of five fields with ten sheep each. It's logically possible that when true tranquillity reigns, the gods always make sure that every field contains ten sheep (presumably the age of true tranquillity is long since past). It's also logically possible that the non-local "guide wave" explanation of quantum phenomena is the right one. With both sheep and quantum, the arithmetical explanation makes so much more sense that it would be most malicious of the gods to reject it just to save our old habits of thought.

...

We'll see that there is another reason to prefer the arithmetical explanation, which is that, as our discussion of Markov processes suggests, it also applies to classical things like computers. This at last enables us to make sense of quantum measurement, which has always been a great mystery. Quantum and classical now stand revealed as two "shapes" made of the same stuff, so there is nothing more mysterious about their both being parts of the same process than there is about round wheels and square windows both being parts of the same car. The radical path also leads to a good Kantian solution of Hume's problem, which is that of finding causality in the order of succession, and we'll see that the choice between acausal and causal/classical thinking is to some extent a choice of analytical method, like the choice between polar and rectilinear coordinates.

...

Boost theorem. u = (v+v')/(1+vv'), i.e., taking the velocity of light be 1, the velocities of linked binary variables satisfy the relativistic addition law.

Proof: Let p and q be the probabilities of HEADS and TAILS for V, and similarly let p' and q' for V'. Then v = p-q and v' = p'-q', and from the definition of linking one can quickly verify that u = (pp'-qq')/ (pp'+qq'). Thus we must show that (pp'-qq')/(pp'+qq') = (p-q+p'-q')/ (1-(p-q)(p'-q'). Now in fact these two expressions are not identical as they stand, but only become identical when we bring in the additional fact that probabilities add up to one, i.e. p+q = p'+q' = 1. The easiest way to take these conditions into account is to note that v = (p-q)/(p+q) and v' = (p'-q')/(p'+q') and substitute these expressions for v and v' in (v+v')/(1+vv'); the resulting expression then reduces to (pp'-qq')/(pp'+qq'). QED.

Applied to observer and object, the boost law implies the Lorenz transformation.

Thus we find that the concept of linking, which before led us immediately to the heart of quantum mechanics, has now led us immediately to the heart of relativity!

There is still a lot of work to be done to relate the above theorem to the concept of "probability space" based on separability. One approach here may be to interpret "time lines" as binary Markov chains from which the LEFT-RIGHT variables are abstracted statistically. 1x1 space-time would then be the indefinite process that results from linking these velocity variables in an unspecified collection of such chains. Notice the formal resemblance here to our construction of complex amplitudes, which also resulted from linking an indefinite set of processes via a binary phase variable.

The question arises whether this resemblance is more than just an analogy. Could it be that at some fundamental level, the phase particle and the "velocity particle" are one and the same? Let's briefly consider where this would lead. Since in (complex) Minkowski space boosts are rotations of the complex plane, this identity would make the relativity of amplitude phase into a generalization of the relativity of motion.

Even more important for the science of the future is that the conjugation symmetry of the phase particle would become the symmetry of v and -v, which is the symmetry that results from reversing object and observer.

Given the importance of computer modeling in today's science, it's hardly an exaggeration to say that, for most scientists, to explain something means to describe it in a way that could in principle be turned into a real-time computer simulation. This belief, which I'll call computerism, usually does not rise to the level of an explicit statement; it's just one of those things that "goes without saying". It's a funny thing about things that go without saying, though, which is that when you actually say them carefully, and then take a close look at what you have said, they sometimes turn out to be wrong!

Is computerism wrong? That's not something I'll take sides on here. However, I have observed that many people hold onto computerism simply because they can't imagine any other possibility. Here is where a proper understanding of Markov processes makes a big difference. It turns out that computers are only a tiny island in the vast sea of formal possibilities encompassed by the general concept of a Markov process. The quantum is another tiny island.

As mentioned, there are also hybrid forms that belong to neither island. The important point is that by no stretch of imagination can the encompassing expanse of Markovian forms be located on Computer Island alone. Quantum structures can't be located there, even quantum computers can't be located there, and most of the remaining expanse isn't even in sight.

Which brings us to the future of science. Physical science grew up in close collaboration with engineering, and for the most part shares with engineering a view of the world as something to be taken apart into functional units. To this the engineer adds the art of reassembling functional units into useful functional wholes; this is called technology. The abstract skeleton of a functional part is a transition matrix, also sometimes called a transfer function, representing the functional dependence of a set of outputs on a set of inputs. In the deterministic or "causal" case, the actual values of the outputs are a function of the values of the inputs, while in the more general case it is only the probabilities of these values that are a function of the inputs. The generality of engineering consists in its being to able to use a small variety of functional parts and design principles to assemble a large variety of useful complex structures.

Here is where I see the broader significance of PSCQM. I believe its chief accomplishment was to mathematically extend the basic conception of lawful change that underlies current scientific practice. This extended lawfulness retains Markovian separability, but no longer requires that we separate things into functional parts. To put it another way, it no longer requires that the internal variables be inputs connected to outputs. The links between parts, and even between past and future, can now have a two-way information flow. This is easy to say, and it turns out to be rather easy to formulate mathematically, but it also turns out to be very hard to digest. Indeed, most of the work since PSCQM has involved trying to digest it. We have studied numerous examples, which provided numerous surprises, and a lot of work has 5 gone into grounding the mathematics at a more fundamental level - we'll come to this in the next section.

Major changes in science are foreshadowed by movements in the culture at large. A variety of cultural movements in modern times, ranging from the counterculture of Woodstock to the arcane isms of Continental philosophy, share a strong discontent with the technocratic narrowness of science as it stands. The broad message here is that nature, including human nature, has many ways of being besides using things. A world that is nothing but functionality is a world fit only to be used. The world of the engineer is an abstraction geared to a particular mode of activity, not the world we live in.

But the world of the engineer is also an enormous intellectual achievement, and there is the problem. It is romantic folly to think that throwing away this achievement would return us to some imagined idyllic state of nature. I would like to think that PSQM offers a hint of a less foolish path. It clearly describes radical alternatives to functional composition that are none-theless accessible to the engineer's mathematical tools. It also shows how these can simply explain some of the more puzzling laws of physics. This is certainly not The Answer, but it does offer hope that there may be ways to steer the intellectual power of science into a better partnership with our real human nature.