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Science Project Notebook/March 1990Roy LiskerPO Bx 243

Middletown, CT 06457860-346-6957

aberensh@lynx.neu.edu1. Destruction of Information and Causality in Quantum Theory2. Entropy Algebras3. Key Moment Formalism4. Conservation Mediates Equivalence5. Universal Limitation6. Jump Model for Light Propagation7. Observers in the "Now" Time8. Transportation Theory9. Topological Induction10. Boundary of Mathematics and Meta-Mathematics11. Pre-Math12. Indirect Proof of Indirect Proof13. 14. Information Breakdown15. Choice Function + Catastrophe Theory16. Entropy of Unknowing17. Entities Defined through Unsolvable Equations18. Algebra of Parenthood and Lineage19. Groupoid Project

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20. Misrepresentation Theory21. Characteristic Point Theory22. Functional Characters23. Summations on Cantor Permutations24. Co-Measurability25. Two Infinitesimals26. Periodizer; Equalizer; Renormalization27. Contractibility of Hilbert Space28. The Contractibility Conjecture29. K-Line Geometry30. R-Convex Sets31. Geomorphisms32. Multiple Adjoints33. Corkscrew Fractals34. Fractal Structure and Topology35. Integration over Fractals36. Fractal Dim Above =(?) Fractal Dim Below37.The Dilemmae

ffffffffffffff(1) Destruction of Information vs. Destruction

of Causality in Quantum Theory:

The customary descriptions of Quantum Theory speak of the intervention of the observer as destroying some aspect of the information sought: this interpretation appears too anthropocentric. It implies that an observer

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has some knowledge in advance of what he is looking for, at a moment in time that can be called the "Now" moment. It also implies that if there were no conscious observers there would be no quantum theory. How then would this explain spontaneous pair-particle creation, or zero-point energy, which do not depend on human observers?

Let us assume instead that 'conscious' observers exist in a "Now" time, which is different from the timeless 'when' of physics. The act of perception, by disturbing the observed, introduces an acausal factor in world evolution, and interjects the 'now' of consciousness into the 'when'.

ffffffffffffff2. Entropy Algebras

I am thinking, very generally, about every sort of notion associated with inaccessible knowledge, whether because of noise, non-computability, heat dissipation. The structure may or may not be "algebraic", in the sense of abstract algebra - group theory, etc. If so, we have an "entropy algebra".

Here is an example: Let R be the reals, and P(R) its power set. Cover P(R) with a 'dust cloud', allowing only 'bright objects' to shine through- i.e., distinguished sets of some sort. Replace the cloud by an 'entropy function'. The dust may be compared to complexity, noise, NP computability, or uncertainty ( quantum entropy, etc.)

The Carnot cycle is analogous to the Information Theory diagram. There is also an analogy to the diagrams

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that arise in Special Relativity in explaining the abolition of simultaneity. These diagrams contain the idea of the entropy algebra; they are geometrical and have group properties. We may even be speaking of a mathematical Category: feedback loops with loss of information.

Spectral definition of Entropy: The channel may be interpreted as an Operator, G,F,etc. Let the input message be S, the output S'. ThenG(S) = S'; F(S')=S''. G and F are forward and reverse channels for S. So FG(S) = S''. Define NORM = . The entropy minimizes the norm. The purpose of this research would be to draw together:(1) Information Theory(2) Carnot Cycle(3) Relativistic Clock Theory(4) Quantum Uncertainty(5) Inaccessibility Problems in Math, Logic and Science

ffffffffffffff3. Key Moment Formalism

Wish to examine the consequences of the complete abolition of isolated systems: An open-ended universe in which all world-lines are finite, being interrupted at "key moments" by forces and systems outside the boundaries. Begin with the principles : "All world-lines are finite"

" All systems are connected"(Math Notebook January 1987: pgs. 103-113)

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State variable of the system F(t,S): Sn+1 = (S, t1, t2 t3.....tn). The times were "key moments" at which the system S interacted with the outside world, the "perturbations". In QT these may have simply been observations of S! We make the assumption that it is not possible to know positions and velocities of S at all times t, but that it its possible to know:(I) S "at rest", at time t=0(ii) The forces perturbing S at times t1, t2 , t3....., tn . This corresponds to real physics and technology. "Knowing" all positions and velocities, even theoretically, has never corresponded to operational reality.

"Periodicity" loses meaning because unpredictable temporal inputs are not considered to be accidents but integral to Nature, indeed a law of Nature. (On page 107 of the Notebook of January 1987 we extend this scenario to that of a state variable that depends on a continuous temporal element, such as, for example, the integral of all past states:

In some sense, the 'vector dimension' of the state variable grows with time.EXAMPLE: BIRD POPULATIONS!: Birds one sees in a certain geographical region in August, are not the same birds that one sees in September. However, meaningful

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statements about the September birds depend on knowing something about the August birds, which are absent entirely if we happen to come to the woods only in September. The present does not supply enough information to know the future. This is closer to actual experience than the usual abstract theory.

Let our population of bird species be given as P1 , P2 , P3 , ...Pk ,In a given month, M, the kinds of birds present in the woods will bePn1 ,.... Pns ,where v = ( n1 ,.....ns ) is the 'population vector', and is a function of time. Then our bird ecology function will beE = E( t, Pn1 ,.... Pns , v). If for a given i , ni = 0, this means that the birds have died out, not that they are not in the woods ( Distinction between 'vanished' and 'absent'). We end up with a matrix format in which absences are taken into account. In other words, to account for the presence or absence of various species of birds, we take into considerations such things as weather, water, ice, etc. Water would be an important ecological parameter. "Ice" would be present in December, but absent in May. Let M be a given month, v the bird population vector for M, and u the vector of relevant ecological parameters, ice, wind, temperature, available food, etc., u = ( l1 ,....lj ) . The bird

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population matrix for this month would be M=

, so that E = E(M,t) is the ecology function. Continuing on with the distinction between " a woods in which all the frogs have died", and "a woods within which no frogs have ever lived". We might say that some disease or poison or other lethal element has entered the woods in the former case, whearas in the latter the woods might be deemed healthy. Distinction between 5 cups, 4 filled and 1 empty, and 4 filled cups: Space is empty, not void

ffffffffffffff4. Conservation Mediates Equivalence

An idea derived from combining Emma Noether's Theorem, that 'Every Conservation Law implies a Symmetry', and my observation that 'Orthogonality is a Symmetry mediating an Equivalence'.

The application of an orthogonality twice cancels it. One can sometimes obtain, not the original object, but something equivalent to it: for example, the vector space and dual vector space. Lets compare the schemata:Equivalence:(1) (2) (3)

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Orthogonality:(1)(2)(3)

So the question I raise is "Does every conservation law bisect some equivalence?"Some examples, taken from pg. 9 of the October 1988 Notebook[1] We can say that systems D1 , D2 are "parallel" if they have the same state variable S at different points of Space-Time. ( Parallel world-lines).Consider the fibration of all of "phase-space-time" by these parallel trajectories. Orthogonal to this fibration is the normal vector field which has for its solution a 'fibration by Cauchy surfaces'.

Now the 'normal' to a given world line is its curvature vector. Can we identify this with the conservation of matter/energy. Now, the concept I'm having trouble with is "phase-spacetime": all positions, all momenta, under a Minkowski metric. If we ever manage to make sense of this, we can say that the conservation of matter-energy is 'orthogonal' to space-time.

ffffffffffffff5. Principle of Universal Limitation

from page 209 of the Mathematics Notebook of October,1988:

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We begin with the Aristotelian idea that matter, or substance, is inconceivable apart from extension. One cannot actually speak just of 10-15 grams: one must speak of 10-15 grams of matter occupying 10-10 cm3 of space . The matter/space combination is indivisible ( viz. Elliot Lieb on the stability of matter). This leads right to Lucretius/Democritus and the Atomic Theory. This also suggests a 'density restriction' is a universal constant, beyond which matter cannot be compressed. Once again we ought to refer to Elliot Lieb. This suggests that one ought to consider a whole list of limitations, in both directions, under some kind of metaphysical intuition, that "infinite entities" cannot exist in a real world:Examples:

1. Momentum , with limitation constants in both directions.

2. A maximum energy quantum as well as the minimum , h, that we already have. Since our energy formula is E = h , we therefore require an upper bound to . Energy can also be defined by E = mc2 , which puts an upper bound on m, perhaps even the mass of the entire universe.

A lower or minimum velocity: Nothing can move slower than s, or more quickly that c: < v < c . for all velocities. This is rather interesting: it ties in with the concept of zero-point energy, and therefore may be a good

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principle for RQT. It also eliminates another barrier between QT and SR, because it means that there are no absolute rest-frames in the real world, only in theory.

We are speaking of a metaphysical principle: It goes as follows:

: If we assume that all things are transient, as many philosophies do, then all things must decay in real time. This is only possible if they have finite extension or magnitude. Therefore, all cognitively or sensually perceptible entities are bounded both above and below.

Although the principle is metaphysical , it would not surprise me at all that, once physicists begin looking for these bounds, they will find many of them.

Corollaries:(1) No singularities ( which are absolute poles)(2) No two identical objects. This requires a

separate principle which is of course related.(3) There is no absolute rest(4) There is no absolute void, or empty space(5) There are upper and lower bounds on all

physical quantities: mass, velocity, position, momentum, energy, force, world lines, etc.

(6)A Density limit, which prevents the formation of Black Holes, which are absolute singularities.

Perfection can exist only in:(a) Universal conservation laws(b) Ideas

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(c) Moral IntentionIn Summary, we want three postulates for our physics:

Postulate I: No infinite and no infinitesimal magnitudes

Postulate II: If any equation based on the conservation laws gives rise to an infinity, it must be 'renormalizable'

Postulate III: All limiting constants of nature may be derived from an equation of motion with a renormalizable singularity, and observational evidence.

ffffffffffffff6. Jump Model for Light Propagation

This construction lives in the boundary area between Special and General Relativity. If light is the limiting absolute velocity, then there must be a limiting acceleration. For, if we imagine some particle accelerating to the speed of light, its acceleration must become infinite to attain this.

One way to solve this is to work with a model whereby light jumps from point to point in space. Quanta are absorbed, then emitted, at a succession of points.

ffffffffffffff7. Observers in the Now Time

Science uses both a "when" time, and a "now" time. The "when time", t, is the dimension of physical science, and only speaks of before-after relations. The "now time", t, is lived or conscious time; it is the actual rather than the

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potential, theoretical or possible. The now-time gives meaning to the when-time, but all observations are made in the now-time. One could then possibly interpret the quantum theory as the "acausal eruption" of the 'now-time' into the ‘when-time' by the process of observation.

ffffffffffffff8. Topological Transportation Theory:

Our conceptions of environment, milieu, setting, neighborhood are much more complex today than they ever were in the past. Every means of communication or travel sets up its own neighborhood structure. For example, if we define a neighborhood of radius r, as 'the distance one can walk in r hours', then the world breaks up into disjoint components given by the continents. However, if a neighborhood is defined as the 'distance on can fly in r hours', then the whole world is connected. If a neighborhood is defined as 'the distance a typical ship can sail in r hours', then the world is a multiply-connected region on a sphere, bounded by the outlines of the islands and continents. Therefore we see that each mode of transportation is associated with its own topology.

When a variety of metrics are employed, one can define the 'distance' between two points A and B on the earth's surface as either:

(1) The shortest time it requires to go from A to B, using all methods, or combinations of methods, of transportation

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(2) The least expensive way of making the trip.Let us stay with (1), since the translation of "money"

into "time" is more intelligent than the converse. One can use the 'train' metric, the 'ship' metric and the 'plane' metric, to develop a 'global topology' based on shortest times. If T is a given amount of time, then we can encircle any point p on the globe, by a 'circle' comprising all those points that can be reached, using all available methods of transportation, in less than time T. This is a 'neighborhood'.

For example; since I haven't got a car, it requires 5 hours for me to get to Middletown from New Paltz. It only requires 2 hours to go to New York by train, although it is the same distance. Tivoli is as far from me as NYC, in terms of travel time; one is 30 miles away, the other is 90. Woodstock requires 1 hour, the amount of time it takes me to walk from Boughton Place to New Paltz. However, the bicycle takes half an hour, the bus 15 minutes. All these factors go into the 'neighborhood structure'. Defining a 'locality' as all place that can be accessed in less than 2 hours, my locality here in New Paltz includes Albany, New York, Tivoli, Woodstock, all the towns along the Metro North line including Yonkers ( it takes as long to go from downtown Manhattan to Jay's place, as it does to go from Poughkeepsie to his place. So that Tivoli, Rhinebeck, Poughkeepsie, Yonkers, NYC, Woodstock, Albany, etc., are in my locality, but Marlboro , NY is not, unless I can get a

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ride with someone, or I am lucky hitch-hiking on route 9. Hartford would be in this locality if there was a bus there from Poughkeepsie. So one sees how subjective this concept of locality is. If we define a 'region' by means of a 10-hour access period, then Paris is in my region; I don't know about San Francisco.

One can then turn from the topology of transportation, to that of communication. Today we have letter, telephone, fax, E-mail, telegram, etc. Since all but the letter are instantaneous, the natural metric is cost. Also, we can't combine them as we do travel.

ffffffffffffff9. Topological Induction:

Let Px be a proposition we wish to prove true for all x K, where K is a connected topological space. We start with

(i) A point O, on which P is true(ii) A guarantee: If P is true for any point y, then P is

true for some neighborhood N , including y, and(iii) If P is true on any set S, then it is true on its

closure THEOREM: P is true over the entire connected topological space KThis is topological induction. I don’t know how it can be proven, save by using itself as an axiom. It is therefore probably something like Peano induction, an independent postulate. Or it may be implicit in the very notion of

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connectedness. I’m in fact not sure that its true for all Hausdorff spaces, and may have to be added as an axiom. Consider, for example, the real line R, with a topology defined by half-opened intervals, all closed on the right (x,y] . In such a space , topological induction doesn’t hold. However, is it connected? No, for we can easily divide an open set, such as (0,1], which is open in this space, into disjoint open subsets. So I suspect that it is a property of connectedness.

Wouldn’t it be interesting, however, if it could not be derived from connectedness alone? Example: We will show, in R2 , that if (x,y) are the Cartesian coordinates of any point, then x2 + y2 ≥ 2xy

(i) Its true for (0,0)(ii) Suppose its true for some point x*,y*. We let d

be a small quantity, and compute (A) (x*+d)2 + (y*+d)2 = x*2 + y*2 + 2d2 + 2d(x*+y*), and (B) 2(x*+d)(y*+d) = 2x*y* +2d2 + 2d(x*+y*) ≤ (A) . The hypothesis is therefore true in any closed neighborhood around (x,y) and is therefore true over all of R.

ffffffffffffff10. Boundary Of Math and Meta-Math

This is a ‘peculiar’ problem: One must find a good definition of ‘mathematics’ before one can define ‘meta-mathematics’. Many examples are present:

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Definition I: Mathematics is what mathematicians do. If this is the definition, then meta-mathematics is a branch of mathematics, which is, in fact the way most universities deal with it.

Definition II: Mathematics is a collection of techniques for solving problems that arise in connection with numbers and relations. Once again, meta-mathematics is a part of mathematics.

Definition III: Meta-mathematics is a branch of applied mathematics, in which the area of application is mathematics itself. Thus, even as ‘mathematical physics’ is mathematics applied to physics, so ‘meta-mathematics’ could also be considered ‘ mathematical mathematics. However, this is not totally satisfactory, since the subject of ‘metaphysics’ is not the same as ‘mathematical physics’, so we would expect that ‘meta-mathematics’ would not be the same as ‘mathematical mathematics’. Here are examples. It impresses me that the following statements are , in the best sense of the word, ‘meta-mathematical’

(i) Theorem X is ‘trivial’(ii) The Phythagorean Theorem is ‘deep’(iii) Arithmetic arises from the pure synthetic apriori

intuition of time. ( Brouwer, Leibniz, Kant. etc. )whereas Gödels 2 Theorems would really seem to

belong to a subject which we can call “mathematical mathematics”. The whole of mathematics is ‘arithmetized’,

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( which means that we assume both Zermelo-Fraenkel and Peano), and a contradiction derived.

ffffffffffffff11. Pre-Mathematics

This would be a subject which relates to ‘mathematics’ as ‘mathematics’ does to ‘meta-mathematics’. For example, statements like “2 Apples” , “2 Houses”, from which the idea of “2” has not yet been extracted

ffffffffffffff12. Indirect Proof of Indirect Proof

For example: can one prove that it is impossible to prove that √2 is irrational by direct means? Must this proof be an indirect proof? Can that be proven? Doesn’t indirect proof always imply an infinite chain of indirect proofs, and is this not a possible criticism of the Axiom of Choice?

ffffffffffffff13. Is the set well-defined?

Apparently not: here are examples of the difficulties one runs into: Suppose that S is, in fact, a properly defined set according to the criteria of Russell-Whitehead/Zermelo-Fraenkel. Let A = {S}, the set containing S as its single element. Since A is a proper set, it follows that A is a member of S. That is , S is not in S, but the set containing S is in S. This is clearly not permitted; for example, the descending chain condition is violated. We need an axiom which, is general, disallows sets whose prior existence is

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present in their definition. We see this in a much simpler example: Define G by

(a) {0} G(b) {G} G

That is G = { 0 , {G}} Clearly this is not well-defined. The question still remains, if we exclude all these improper sets, if S is well-defined.

ffffffffffffff[14] Information Breakdown

First we will describe what we mean by ‘information breakdown’. Then we will list the different kinds. Then we will see if these can be described by chaos theory and catastrophe theory.

We will try to express these via Shannon Diagrams: CODER Channel 1 > DECODER Channel 2 > CODER

[I] Information Overload (Chaos)This happens when the buildup of information is too

fast for the loop to process it effectively. [II] Interception-Confirmation Paradoxes,

(Infinite Regress). Message sent by Coder must be confirmed by Decoder, who must confirm confirmation, etc...

[III] Misinterpretation. The feedback loop of even a small misinterpretation grows to huge proportions until there is a total breakdown in communication.

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The “confirmation infinite regress” is related to the “misinformation breakdown”. A sends a message to B, B is supposed to send back the same message to demonstrate that he has received it. If we compound noise in Channel I with Noise in Channel II, the message returned to A may have little resemblance to the one originally emitted.

The classic example of the confirmation paradox is that of the two halves of an army camped on a hill, with the enemy army in the valley. The army on the hill wants to coordinate a mutual strategy, which will only work if they agree on the time to attack, and the messengers sent through the valley are not intercepted.

Misinterpretation complications readily mix with confirmation regresses: because I don’t know if you know that I know that you know that....., therefore a misunderstanding arises which gets worse, until we come together, say, and realize that you did know that I knew that you know that......

Say I write a book, which is controversial. Somebody else, who is hostile to me and my ideas, comes out with a rebuttal. Since I know that reading this rebuttal will anger me, I deliberately don’t read it. He believes that I’ve read it and goes out of his way to avoid me because he thinks I’m angry with him. When he’s learned that I didn’t read it, he’s angry with me because he wasted so much time avoiding me, and so on....If these scenarios become too complicated, then the overload breakdown may take place.

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Accumulated misunderstandings between nations, with their confirmation regresses and misinterpretations, lead to a situation of confusion which is so large that the overload breakdown occurs, which may lead to a war.

Information overload is the simplest one to deal with. Information “backs up” into the channel, until it blocks the receiver itself and further transmission is impossible.

Misinterpretation breakdown is your simple Shakespearean Comedy of Errors: The hospital of the father of a new born child wants to send a message “Child born deaf”. What is transmitted is “Child born dead.” He sends back a reply, “Does she know?” referring to the mother, which is erroneously transmitted as “Does he know?”, which the hospital interprets to mean, “Does the child know that its deaf?”. And so on . One could actually write an amusing comedy based on two slightly defective channels that, within a short time, create a mushroom cloud of total disinformation.

Other categories of information breakdown are probably combinations of these. For example, where do we put the non-sequitur?This, also, takes various forms ( Much of this discussion reappears later in the LQT PQT paper. )

(a) Nonsense(b) Threats ( Nonsense in which, however, the

meaning is clear: I don’t want to communicate with you.)(c) Multiple meanings (Art)

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ffffffffffffff[15] Catastrophe Theory and the Axiom of

ChoiceThis attempts to bring together the notions of ‘equally

likely events”, the choice function of set theory, and the singularities of Catastrophe Theory.

Singularities to begin with: These are of three kinds(1) Discontinuous “jumps”( Fold Catastrophe)(2) Infinities, ( or sometimes “zeros”, as in Morse

Theory)(3) Multiple values or branchings

How are these related to the construction of logical choice functions. The interesting paper of Smale, Shub and Blum has some good ideas.

(a) Rules for selecting values of R that terminate at some point. One then “jumps” to another value and begins again. This is essentially the construction at work in the formation of the Hamel basis.

(b) The Box Paradox. Trying to count a set without knowing in advance that it is countable, is already a paradox

(c) Axiom of Choice, Continuum Hypothesis(d) Non-computable numbers.

These are only suggestions of large areas to explore. On catastrophe surfaces, one has a “choice” to make at a given moment, a set of alternatives which may be discrete or continuous, with drastic consequences attached to each

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choice. However, if the choices are “equally likely”, then it may be impossible, logically, to construct a choice function.This needs lots of thinking, which will be well worth the effort.

ffffffffffffff[16] Knowledge Entropy

Teaching is an energy transfer process. This is particularly true in creative activities such as the arts and the theoretical sciences. Teaching which extends from one generation to the next invariably leads to a definite ‘dissipation’ of the knowledge of the previous generation. One need only recall how often one has gone to a library to find some acknowledged classic of, say, the 19th century, only to discover that it is not to be found in any library catalogue save the Library of Congress, the British Museum, etc. Yet in its own day it was a best seller read all over the world, ( I’m still looking for William Paley’s interesting writings on “Design in Nature”.)

Indeed, even the process of transferring ideas from “brain” to “paper” brings about some dissipation. There is at the same time, a ’feedback loop’, whereby the appearance of the idea on the written paper will concentrate the mind on its features.

It may be possible to measure these forms of ‘idea dissipation’, and define an unbreachable entropy barrier between, say, the 19th and 20th centuries, or all other

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centuries, etc... This goes contrary to the popular notion that the accumulation of knowledge is an ‘entropy reversing process’, related to something we call “progress”.

It is interesting to note that some astronomers now believe that a primitive form of micro-organism may have existed on Mars 3.6 billion years ago. Life did not emerge on earth until a mere 500 million years ago, in the Cambrian era. This gave Mars plenty of time to evolve sentient beings, high civilizations, large accumulations of knowledge, comprehensive theories of nature...and then die away completely. One billion years, twice the time of life on Earth, would have been sufficient for this. This possibility undercuts our notions that what we are doing on Earth today constitutes “progress” in some absolute sense.

The advance of knowledge is painted in main-stream contemporary writing as a process of ‘energy concentration’, rather than dissipation. This is clearly only an article of faith, since there is certainly an “entropy of teaching” barrier which prevents us from absorbing all the knowledge present in previous ages.

ffffffffffffff[17] Entities defined indirectly through

problems without solutionsThis is a long-standing pre-occupations. My paper

“Inconceivable Numbers” is all about this. The paradigm is

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of course the ‘invention’ of i through the insoluble equations x2 = -1.

In the most general terms, we consider any algebraic structure Q, a group, algebra, even a category, with elements ej and a “covering algebra” A of forms obtained by concatenation of indeterminate terms over the operations of the structure. Then, if f is a member of A, ej some element of Q, some of the equations f(x1, x2,......xk) = ej , will not have solutions; so we ask, when is it possible to adjoin vectors v, built from elements outside of Q, such that f(v) = ej will be the solution. “Number generating machines”. For example, in arithmetic, one can write down the equation xy +yx = 0, and use this to derive the quaternions.

ffffffffffffff [18] Algebra of Kinship Structure

It is certain that there is a school of anthropologists which have applied Graph Theory to work out the lattices used in the description of kinship structures, incest taboos, fratries, moities, etc. However, I ought to work on it a bit to see if I turn up some original way of doing this.

ffffffffffffff19. Groupoid Project

This has already become my 150 page paper on Non-Linear Algebra

ffffffffffffff20. Misrepresentation Theory.

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I have already begun a paper on this (8/96).Its’ become bogged down for the following reasons: (1) I don’t know enough about differential equations (2) Some of my results turned out to elementary (3) There are some good ideas in it, but I don’t know how these relate to what is already being done. Conclusion: Keep working on it.

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21. Characteristic Point TheoryBegin with the following observation. Let f be define on the complex plane, (domain and range), and analytic. Let P = { p1, p2, p3,... pn,... } be a pre-determined sequence of complex numbers, and suppose that we know, for every n, that f (n) (pn) = 0 ; that is, the p’s are nth order critical points of f. Then we claim that knowing the values of the p’s completely determines f.General outline of theorem: We can let a0 = 1 without lose of generality, and write:

So, we can systematically eliminate all the ai, for i>1, up to an, using Kramer’s rule. This gives an equation of the form

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where R is the remainder that must be shown to converge to zero. If it does, then we derive:

, and likewise for all the other coefficients. Even without the proof that R goes to zero, this shows that if f exists, it will be unique and given by this expression. If we then derive the ai in this fashion and show that the result has a non-vanishing radius of convergence, then the remainder must go to zero.

Want to generalize this to the “characteristic points” of any shape, which are enough, given the boundary conditions, to completely determine the shape itself. Clearly this goes back to the ideas in “Causal Algebras”.

Here is a possible list:(1) All critical points (n) (z) = 0 , all n(2) Points at which curvature = 0. ( These are

critical points of the second derivative)(3) Fixed points under iteration ( Dynamical

Systems)(4) Iterations on the curvature function(5) Critical points of the arc-length function(6) Roots of the integral ( places where total area

= 0)In general, we assemble points { p j } and, as in

physics, look at those functions which are completely determined by their values. This expands the notion of

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causality, which in its most simplistic form, is based on the derivatives at a single point, and nothing else.

The characteristic point description, however, corresponds much better to our actual experience as living beings, who tend to register only the “exceptional events”.

ffffffffffffff22. Functional Characters

This relates to, yet is somewhat different from characteristic point theory. We start from the following interesting observation: Taylor’s Series are local, Fourier Series are global! The evaluation of the coefficients of the Fourier Series requires that we take integrals over their entire period, which is, effectively, their range. Taylor Series’ are based on derivatives taken at a single point.

Since there would be little purpose in using a Fourier series to calculate the values of a function, which must all be known in advance in order to evaluate the integrals, its usefulness must lie in the representation of formal structural features of a global character.

For example, the formula requires only that we know the value of n, and the interval of definition x [a,b].It is the form of f which is then used in the evaluation of specific values. For the Taylor series, however, a collection of coefficients associated with a single point is all that is required.

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[Note: this suggests some interesting possibilities for generalizations of analytic extension, which takes place on the boundaries of the circles of convergence of Taylors series; what is the analogue for Fourier series?]

Suppose we have a real function f, about which we know only the values of a list of points { (xn, yn) } . The Fourier series cannot help us, but this information may be sufficient to give us the complete Taylor’s’ series. However, let f be

(1) real, analytic in an interval 0<x<p(2) periodic , of period p.Suppose that the points A= { (xn, yn) } all fall in this

periodic interval. Then there must be set of limit points { j} of the points of A. If a is such a limit point, then it may be possible to derive all the derivatives of f at this point , and thus generate the MacLaurin series around it. Under what conditions on A can we do this? Nothing comparable exists for the Fourier coefficients.

ffffffffffffff23 Summations on Cantor Permutations

This kind of problem is treated in my fragment of a paper on “Misrepresentations.”

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24. Co-Measurability

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Measures on fractals. I’ve done some investigations of this, and I know that a lot of dynamical systems research is on this kind of thing.

ffffffffffffff25. Two Infinitesimals

It may make more sense, logically, to interpret an infinitesimal as a kind of set of vanishing measure , or whose closure has vanishing measure . One could then speak of measurable infinitesimals, and non-measurable infinitesimals!

ffffffffffffff26. Periodizers

Let f be a function with infinitely many roots:

Define h indirectly by the equation f( x+h(x) ) = f(x) . h maps the arc AC into CE into ....... It resembles renormalization, particular if we make the same construction on the derivative f’(x+h(x)) = f’(x), which carries critical points into critical points. For example if the range of f is infinite, the curvature is restricted, and

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must go through cycles which can be expressed in this way. Since h(x) is a generalization of the period, we can call it the “periodizer”. In fact, we have such things asf(x+h(x) + h(x+h(x)))= f(x+h(x)) = f(x),Periodic functions that have the special property that h(x) = constant.

ffffffffffffff27. Contractibility of Hilbert Space

We are looking at the contractibility properties of H . There are two kinds of contraction: Start with a nesting of homeomorphic topological spaces:

, all resulting from a single iterating homeomorphism, .(1) Pointwise contraction, defined by

where S0 is the space. (2) Set-wise contraction . These are quite different, as we know: the orbit of every single point in S might converge to 0, while S itself might be reproduced in every iteration, so that B = {0}, but A = {S}.Here are some regions of Hilbert Space. Let v be a generic vector

. What are the properties of these regions?

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(1) R:

(2) U:

(3) T: All 3 are contractible: (1) is obvious (2) requires use of the shift operator

Region (3) is homeomorphic to Region (2). There is a difference: under a contraction (3) contracts to a point; but under the shift operator, (2) contracts to the null set. To show that (2) and (3) are homeomorphic, observe that if we replace a1 by some arbitrary < 1 in the defining equation for (2) , we obtain a “shell” of (3). Letting go to 0, we end up with the boundary of (3).

ffffffffffffff28. The Contractibility Conjecture

This has already been the basis of a research project at SUNY New Paltz. After we proved the basic result, it was discovered that it had been done and published in the 40’s. The conjecture was: Given any infinite subset A of the line, it contains a proper subset B, homeomorphic to A. It was proven that this is true if A is countable, but that the

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axiom of choice could be used to construct a very weird set that did not have this property.

ffffffffffffff29. K-Line Geometry

Start with any familiar shape, such as

. The “V-Shape” for example, formed by two half lines intersecting in a vertex. We examine the complete list of all the intersection properties of this shape. These become the “axioms” of the “V-shape geometry”. We then remove these axioms from their origin and examine all the other interesting models that can be derived from them. Sticking with this example, let us say that the fundamental ‘K-line’ of our geometry will be this V-shape, always with the same angle for each wedge.For example, how many points are sufficient to determine a wedge? The answer depends on the value of the angle q. Suppose we have 3 non-collinear points. How many wedges can we draw including all three of them?

If q is acute then, save for some exception cases, such as when one or more of the angles of the triple is already equal to , 3 points will in general determine 4 K-lines. However, it may happen that some of the angles of the

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triangle formed by the 3 points, will be less than . This will change the number. Strictly speaking, there are 3 really different cases: q = 60˚ , q < 60˚ , q > 60˚.

In this ‘wedge geometry’ there are 24 species of intersection, and to axiomatize the geometry we must use a vocabulary including terms like ‘vertex’, ‘arms’, inside-outside, semi-coincidence, parallel and skew.

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30 r-Convex SetsWe will say that the compact connected ( not necessarily simply) closed region A is “r-convex”, if, for any two points a and b A, there is a circular arc of radius r which connects them and lies entirely inside A.

ffffffffffffff31 Geomorphisms

The paradigm: consider the surface of a cone K given

by . Take a small section T of this surface, say a small trapezoid. As long as one does not cross the boundaries of T, the geometry of the conical-geodesic segments in T is identical to that of the hyperbolae obtained by cutting K with planes. So we will say that the conical geodesics in this region are ‘geomorphic’ to the hyperbolae. Likewise, if both systems of segments are

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projected down to the x,y plane, then the straight line segments have the same geometry as the projections of the conical geodesics, provided with stay inside T’, the projection of T. Here is another example: Lines in the plane may be represented by these two families of curves:

(i) ax+by = 1(ii) y = kx

If we make the transformation , these become systems of hyperbolae ax+b/y = 1 , and kxy = 1. The geometry of these systems is indistinguishable from Euclidean geometry, except that the point (0,0) has been pushed off the plane to (0,∞ ). So we must adjoin a line (k,∞) to replace the line (k,0).

The idea is the following: imagine that we are dropped in a desert and begin to survey it. The desert is finite, but we don’t know how large it is. We start to triangulate it, but until we can gain some knowledge of a larger area that includes the desert, we don’t know if our lines are straight lines, hyperbolae or conical geodesics! Straightness is not intrinsic to any finite portion of the plane. In fact, from what we have seen, if a single line is removed (0,k), we don’t know if we’re using hyperbolae or lines over the entire plane. This is a step beyond Poincare’s “Conventionalism” Even Euclidean lines are a convention, they may be drawn in several ways!

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32 .Multiple AdjointnessParadigm: Intersect these three curves in 4-space and solve for x, y

(a) xyuv = A(b) x+y+u+v = B(c) x2 + y2 + u2 + v2 = C. Clearly I

mistranscribed what I have in the notes from my original notebook. This is a shame, because I worked very hard to get this result, and now I must dig back to find the original notebooks and reconstruct it. Anyway, we can eliminate u, v and parametrize a curve S in the x,y plane. It has the following property: If (a,b) is on S, then there are two other numbers, c and d, such that all these points are on S: (a,b), (a,c), (a,d), (b,c), (b,d), (c,d),

(b,a), (c,a),(d,a), (c,b), (d,b), (d,c)Definitions of “weal adjointness” and “strong adjointness”. Let R be a relation over a set A, with these properties:

(i) Let a0 be any element of A, and D = { ak }, k = 1,2,....n , the set of elements of A such that a0Rak . Then we also have aiRaj for all i,j = 1,2,..n, i≠j. We will call this property “ nth order weak adjointness”. There seems to be a problem with this definition, because if we have discovered n elements a0 , a1 , ....a n-1 , then an need not be unique. So we have a further condition that we may call the (m,n) condition.

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Given m terms of the set D, then these are sufficient to determine the remaining n+1-m terms . These definitions can be clarified with examples:

(1) Straight lines l, m,....in R2, all going through

origin . Let our relation be . One of the angles between l and m is 60 °. This is “3-adjoint”: Given lines l and m, then there is a unique n, such that nRl and nRm.

(2) Perpendicularity of axes in R3. Given l, there is an infinite set from which to choose m, but then only one choice for n.

There are numerous “adjoint relationships” associated with quaternions:

This is a (1,16) adjointness relation(3) “Adjoint triples” of quaternions , r,s,t, with the property that r.s=ts.t = r , t.r=s Research product involves putting concept of adjointness, which is a generalization of symmetry, on a solid basis.

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ffffffffffffff33. Corkscrew Fractals

The helix, straight line and circle all share the property of uniformity: Given a segment of any of these, it is not possible to identify the part of the curve from which they derived: uniform linear translations are possible along them. Essentially, the helices are members of the isometry group of 3-space.

They can be used in the construction of 3 dimensional fractals in the same way that straight line segments in the plane are used in the construction of planar fractals. Around a helix C1, one can wrap a helix of smaller pitch C2 that bends uniformly around it : from the vantage of someone on C1, C2 looks the same at whatever point he is placed.

Around C2 we construct C3 in the same way, with a pitch in the same proportion k as C2 to C1. We continue this infinitely, giving us a corkscrew fractal as the end result, C. We then remove all the helices used in making the construction of C. What are its properties?

ffffffffffffff 34. Fractal Structure and Topology

This simple observation is that the “law of fractal structure” is identical to one of the 3 Axioms of Topology: Given a neighborhood Np, there exists a neighborhood Mp properly contained in it. Any foundation with this self-reducing axiom admits of fractal structures.

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ffffffffffffff35. Integration over Fractals.

Fractals have dimensionality, therefore can be treated as “amounts” of specific “magnitudes”, just like area, length, etc. Therefore it should be possible to develop a theory of integration of forms like , where has a fractal dimension between 1 and 2.

ffffffffffffff36. Fractal Dimension Above = Fractal

Dimension Below?Given a fractal J in R2 produced by removing plane regions, such that J has some dimension properly between 1 and 2, is it always possible to create J by starting with an arc of dimension 1, and “complicating” it?

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37. The “Dilemmae”(a) The notorious bus paradox(b) The post office factors

(1) clerk/customer ratio(2) The intersections of the weaving line

(c) The Eurorail dilemma(d) The bicycle dilemma ( ‘uphill’, ‘downhill’, ‘work’

vs. ‘seeing’)ffffffffffffff

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