An Epistemology of Partial Truths

S. Roof

Dec. 2002

ABSTRACT:
This paper will attempt to provide a novel understanding of what truth and knowledge consist of and how they relate to human thought. Aristotelian Logic is examined in the light of semiotic analysis and extended by introducing ideas taken from Chaos Theory. The identification of knowledge as limit cycles within attractor basins is presented. Most important of all, we provide a universal tool kit for thinking with depth. Implicit is a path toward understanding semantic ghosts and the anxiety inherent in man's immortal quest for the truth.

Throughtout history, man has attempted to establish truths that are absolute, objective, and not subject to the relative views and biases of individuals. In the search for truth, two main categories have arisen. One is the apriori or analytic truth of porpositional forms. The other is the empirical truth of facts about reality. While they appear to be quite different, they are in essence the same kind of truth. The distinction merely rests on the degree of abstraction of the objects of discourse that the said truths refer to.

A quick search on the web will convince one that many attempts are made to specify truths as belonging to various categories such as religious truth, poetic truth, political truth etc. as if they had a lock on their own brand of truth so that logical principles might not apply. Most of this categorization of truth is born of confusion. Identification of truth as belonging to a particular category of discourse is an attempt to evade the unavoidable consequences of manipulation by logic. The great fear I suppose is that sublime and sacred truth will be tarnished, torn apart or otherwise degraded into some trifle.

It is my opinion that it is more appropriate to distinguish knowledge from truth. The situation above rises when we say certain facts or conclusions are truths about such and such a thing. Instead we should say that we have a certain knowledge about such and such and this knowledge may be true or it may be false. If we keep this in mind when we make distinctions, then we can preserve 'knowing' as referring to an information gathering process which is subject to being validated or not.

I admit that this is a semantic distinction but it will be useful for the purpose of this paper to shed a little light on what truth and knowledge are in an epistemological sense. This paper will elucidate 'knowledge' as part of a multifacited ongoing process and at the same time refine 'truth' down to a solitary bare boned criteria of distinction making. After digesting this paper, one can then go back to his usual sloppy truisms of language with the confidence that he can turn on the laser light of mental discernment as needed. To investigate truth and knowledge we need to look at the origins of logic.

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The fundamentals of thought and logic rest on one very simple principle - the ability to make distinctions. If one is familiar with Bhuddism, he knows that the ancients were preimmanently aware that once a distinction is made, the many springs forth from the one. Gottfried Wilhelm Leibniz stated this as his Principle of the Identity of Indiscernibles. This principle says that no two different objects can have the same description. We may distinguish two things as having some discernible difference else they are one and the same thing, with just different labels perhaps. Discernability is just making a distinction. Let's look at the anatomy of a 'distinction'.

Distinction is a dual process that separates a unity into 'something that is' and 'something that isn't the something that is'. If one thinks of say a visual field, then the 'something that is' may be called the figure or object and the something that isn't may be called the ground. One can reverse the polarity of the duality, but it is always one side or other of the distinction that enjoys the status of being the positive focus. The other side is the 'not' or 'isn't'. The positive focus becomes embued with 'isness', being, or existence. It's opposite becomes identified as non-being or the not. If A 'is', then not-A is not A. Not-A is usually denoted as ~A. The fundamental principle of logic then is:

A is not equal to ~A

Note that we are not identifiying the 'not' with nothingness! The 'not' is merely the compliment to something and has as much existential truth as the something does. It does have the tendency to disappear from our awareness, though, as we focus in on the object of our distinction. This tendency has to be resisted at times and understood for what it is. The 'not' thus becomes a passive partner in any distinction and is just as important to the unity of the whole procrustean bed from which the something sprang forth by being distinguished.

The principle of distinction making, in one fell swoop both unifies and divides. The 'is' becomes a unity and the original unity is split in two. But what was that original unity? Well, in the language of logic, it is normally called 'The Universe of Discourse'. This establishes a context out of which the distinction is made. The classic visualization of this is the Venn Diagram as shown below.

The universe of discourse is denoted by U or the bounding box. This higher unity of course must be distinguished as part of some higher unity. This establishes distinction making as a recursive process i.e. every distinction we make rests upon a previous distinction already made. The above sentence may just express the most fundamental truth or absolute we can hope to realize. It is the first invariant of thought. Distinction making itself, though, is always a temporary and relative affair.

It took philosophers and mathematicians a long time to realize the recursive nature of thought. Formalized Logic was an attempt to establish invariant truths and recursion presents a huge problem. Many were not ready to accept that recursion itself is the fundamental invariant. I am quite sure that many gifted thinkers even today haven't awakened to this situation.

Normally logic is kept safe from recursion ad nauseum because the universe of discourse is simply identified and kept static. Simply staying within a fixed context as a universe of discourse, one can keep recursion at bay. Thought can then simply keep returning to its distinctions within that particular scope or context. There is implicit in this a tendency to forget this context though and stray. If one isn't aware that he has strayed into a different context, then error is possible. Even worse, if formulations or thoughts are made that do involve this self-referentiality of distinction making, recursion can create paradox. This leads us to the Principle of Contradiction.

The Principle of Contradiction states that a proposition is necessarily true if its negation entails a contradiction. The negation of A != ~A is to say that A = ~A but A is not ~A, hence we have a contradiction. This proves that A cannot be not-A. This obtuse way of stating the obvious is merely an illustration of the fact that a distinction, once made, implies an inviolable separation between a thing distinguished and it's compliment. This is the prime principle of Aristotelian Logic.

This principle is also called the Law of the Excluded Middle. This principle seems to wield a sharpness sharper than even Occam's Razor. There is no room for in between at all! It is an attempt at absolute universality. To violate it is to invite contradiction and confusion. It is the cornerstone of what mathematicians call Consistency. Without Consistency, mathematics is impossible.

Now the informed reader of all persuasions may balk at Aristotelian Logic. He may introduce notions of multivalued logic or even fuzzy sets as a refutation. No matter what system of thought or logic he comes up with, though, it will still boil down at some point to making singular distinctions which in turn are most precisely defined by Aristotelian Logic. It appears that in the final analysis, distinctions are distinctions no matter how we make them. Apparently truth is an entirely black-and-white affair.

I promise to help assuage any discomfort one feels at this point. If one relaxes a little and lets his thinking return to its normal mode, he can see that distinctions are indeed a question of degree. Our vision may be fuzzy or unfocused leading to a certain amount of uncertainty. The unity of the thing in focus at the moment may be in question. Our precise context may even be uncertain. Remember, every distinction relies on a previous distinction. The validity of our thought depends on what information we start with! Whoops, now we're faced with confusion, uncertainty, infinite regress, paradox and God knows what. Well it's not really that bad. Think of thought and logic as more of a fluid process and let intuition guide you. This is sort of like backing away from a problem and getting the big picture. Now refocus and let your distinctions become sharper. That's all it is - a process that has a certain degree of sharpness or specificity and we can vary this at will.

If our Eye of Horus becomes too diffuse, the symbols we are looking at will loose meaning and vanish into a haze of squiggles or marks. If we become too precise, our percepts will become so brittle that we will have to literally jump across the gulf of the excluded middle just to get to the other side of any simple equation. This will indeed require an act of faith that meaning lies on the other side and we don't forget what we were thinking about to begin with. Context is just as fragile or relative as is the object of our focus.

You may still be experiencing cognitive dissonance at this point. What is the truth here? Is truth absolute and Aristotelian or is it fuzzy. Aren't we in the middle of a horrendous paradox here? Well I have good news and bad news. The good news is that we can make another distinction here and resolve the paradox. The bad news (good to some though!) is that this new distinction will open the door to a whole other world of complexity.

Ok, so what's the deal? The deal is - "the process of making distinctions is a process". I'm not trying to be cute here. The distinctions of Aristotelian logic are merely the idealized attractors to which the ongoing process of our logic apparatus tends toward or is attracted to. Another way of saying this is that the Aristotelian Venn Diagram is a paradigm of awareness in general. Any precise and specific distinction objectified for all to see is thus a specific instantiated generalization of some ongoing and non-terminating process within some context. That is why the object A in the Venn diagram is a set (circle) and not just a point. Distinction 'A' really represents what is called an attractor basin.

Points inside the circle of 'A' are said to be included in or belong to 'A'. Inclusion is the corrallary or complement of Exclusion. Now logic demands that there has to be a definitive clear-cut way to discern whether a point is included or excluded. This is where process enters the picture. Our distinction making apparatus, if you will, has to apply some criterion to make a distinction to begin with. This is true even if A is just a single point. In this fundamental case, the criterion is identifed as the distinction itself. We shall see though that this degenerate case is never achievable in practice and probably not in theory either. It seems impossible to even conceive of a single point without any divisible properties residing in a singular indivisible context. When the process comes to explicate representation ala diagram or symbol, the whole multiplicitous cat is already way out of the bag. Sharp distinctions require repeated invocations of some criteria that as a process converges to some degree of clarity or not. The formal symbolism merely states the generalized dynamic of this process. The generalized dynamic of any distinction we can make is an abstract attractor basin for some on-going process of knowing.

Thus we see that mathematicians now identify set theory as fundamental to logic. Conventional set theory is static though and leads to paradox when it becomes self-referential. Recall that every distinction is self-referential in a sense because it presupposes a previous distinction. The degree of self referentiality determines the amount of meaningful content in a distinction. One can see this by realizing that the distinction of making distinctions is totally self-referential and conveys the minimal amount of information any distinction can convey. At the other extreme is some clearly unique and distinguishable unity like the letter 'A' that conveys perhaps a huge body of knowledge labeled as 'A'.

Well since we mentioned information, just how much info does the minimal fully self-referential distinction mentioned above convey? Well so far we have mentioned several distinct entities resolved from any one distinction we can make. Unity it appears is at minimum three. We mentioned A, ~A, and U. Actually we mentioned a fourth entity that is the most crucial of all - a criterion that lies implicit within the distinction. This criterion is identified as the particular functionality of the distinction making process that led up to the final attractive distinction.

Now one can see that ~A is always implicitly defined given U and A, so we can lay it aside as complimentary and always available if needed because ~A is simply what's left if we take A out of U. This is why we can conveniently ignore the background in any distinction we make, so long as we dont forget our original context U!

Ignoring ~A as redundant information, puts us back to having three aspects of the minimal distinction again so it would be good to label everything and see where we stand. Let us call the unity of any inherently-triune distinction-making process by the name 'knowledge'. Any item of knowledge thus is seen as the generalized static attractor-limit version of a selective criterion applied as ongoing dynamic process. The process itself might be identified as thinking, distinction making, perception, or even simple awareness itself. Anthing less can be identified as deep sleep, coma, or perhaps transcendental oneness. That is a subject for future debate.

To continue with our labeling, we can identify the something or object distinguished ('A' ) as content. We shall label the context or Object of discourse as form. Why this word has been chosen will become apparent later. This still leaves us the need for a nifty label to dub onto our functional criterion implicit in the distinction making process. We shall simply label it function.

We are now in a position to loosely wrap up a definition of what knowledge is.

Knowledge = {Form,Function,Content}

The order as shown implies form is prior to content and that content is the product of an intervening function acting on form.

Now we can also visualize 'function' as really the critical aspect (from a functional point of view eh eh!) and redraw our Venn diagram in a new way. A simple relational diagram provided below highlights our meaning.

All the above aspects of knowing should be familiar terms to everyone, but I am not aware of anyone who has formalized the relations as being the singular foundation for knowing. They all of course are found separately in works of philosophy, mathematics etc. but the triune relation appears from my research on the web to be lacking as a unified interpretation. The only pages I found using all three words together were mostly by computer software and database firms. These guys being on the cutting edge of knowledge commerce naturally have to know their stuff, so this is not surprising.

It is my observation that philosophers get so lost in refining their distinctions that they inevitably ignore content. Mathematicians are prone to do this also. Most of the success in mathematical theory comes by generalizing to the point that the content is stripped out and all that remains are the symbols themselves. This is a useful artifice for sharpening distinctions but has a cost ultimately. To be useful, knowledge has to have meaning that relates to the real world. Recently mathematicians have recognized the limitations of static set theory and have indeed invented what is called Category Theory to incorporate the dynamic and relational aspects of knowing.

One thing should be emphasized here. The three aspects form, function, and content cannot be separated from each other. We can of course distinguish and talk about any one of the three but we can never completely strip the other relational components out of the equation. They are a trinity somewhat like Quarks of Physics. Also note that I haven't even talked about whether we are referring to real attributes of reality or just mind itself. After getting used to this way of looking at knowledge, one eventually sees that the distinction of inside-outside, mind-matter etc. are beside the point but still can be engaged in freely. It just depends on what your Universe of discourse is.

Our discussion has suddenly gotten quite abstract, so it behooves us to look at abstraction now. What is abstraction, or more precisely, what is the concrete vs. abstract distinction? Well it should be obvious that abstractions are abstracted from the concrete. Now it is tempting to identify the 'concrete' as content, but wait a second. Form is apriori to the content, so content emerges from form as the distinction is made. Therefore it is form that is concrete and content is abstracted from it. The concrete (form) provides the bounding box or framework in which to lay the abstract (content). A letter (with it's message) is more abstract than the envelope (bounding box). This insight threw me at first as it was unsuspected. Part of the reason was that we tend to think of things like sand,rock, i.e. particulate matter as concrete not abstract. When we turn on ther mental laser, though, we recall that every distinction (abstraction) depends on a previous distinction, which itself is more concrete. Our very perception of particulate matter occurs only after our distinction making apparatus in the retina, optic nerve etc. have already distinguished a lot of A's and ~A's. To reiterate, we reason from the concrete to the abstract. Any A abstracted from U is a generalization also.

This opens the question of what is the ultimate concrete level. If our perception of rocks, pebbles etc. is abstract what is the real concrete? The real concrete is a formless visual (since we're talking visual attributes here) field before light(A) and dark(~A) etc have been distinguished. Thus we see that the procrustean bed of concrete reality is just a formless void of sorts, not nothingness remember, but just undistinguished except perhaps as simple protean awareness. One only has to take a sufficient amount of psychotropic drugs to experience this directly as his more abstract neural cogitations short circuit.

Where does that put physical theories like quantum physics? At the extremely abstract level of course. Is there a limit to abstraction? Yes and No. Yes if one is talking about individual distinctions of content made from preceeding distinctions. One ultimately reaches a point of indivisibility or indiscernability. The abstraction making process itself is extensibly infinite however, because abstractions can be combined and new distinctions made about the combinations.

What does function have to do with abstraction? Well, it is the most abstract aspect of all. In can be implicit or explicit. Implicit functionality is so abstract we have a tendency to forget it's there because, recall that content is distinguished from form as a matter of focus on A as opposed to ~A. Eventually one must face the fact that the knower and known are one and the same thing, but the knower still seems to identify himself as the functor. The knower (function) identifies content as the thing known. Form is the context (previous known) for knowing the current known (content). The knower(function) can even know himself as content to some degree. Thinking about thinking will put one in this state quite nicely.

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I have discovered some exciting new ideas of late having to do with a new kind of attractor and it's probable function in human thought, but since I have not received any response at all to my recent invitation to dialogue on this subject, I will put that topic on hold. I do wish to investigate the attractor issue further in this paper. To do so will require a bit of diversion into linguistics and symbolic systems. We also need to get a better grip on the idea of truth also.

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Bertrand Russel and others were profoundly disturbed by the aura of self-reference and inevitable paradox looming over set theory and the foundations of human knowledge. The Hilbert Program was doomed as eventually demonstrated by Goedel and his infamous theorem about consistency and completeness in symbolic systems.

One way to look at the problem is to realize that Aristotelian distinctions clarify logic but sacrifice completeness because of the absolute demand for consistency. So long as we keep our distinctions fuzzy and dont look too close all is well and we just muddle along without going very far. When we demand absolute precision, however, we are faced with dealing with all those prior distinctions upon which any one distinction is based. This infinite regress into reductium ad absurdum is impossible to encapsulate in any finite list of propositions. Any finite distinction thus becomes incomplete. The only recourse mathematicians have had was to rely on axioms or logical atoms upon which a synthetic system could be built from the ground upward. Axioms fixate the context in which they are embedded. This solves the problem in the short term, but if the system becomes complex enough, it is possible to directly encode self-referential statements. This lets the monster out of the cage again.

The Predicate Calculus manages to keep content in the picture by explicitly predicating the extent and nature of free vs. bound variables within propositions. In doing so though, it was the same very enabling factor for composing self-referentiality directly. Recall that any content is derivative of a previous content i.e. the current context was previously a content derived from a prior context. Apriori fixed contexts nip the bud at the bottom, but not at the top.

Another way of looking at the problem involves looking at levels of abstraction. This involves linguistics and the semantics of truth so lets dig in.

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We use language to communicate, express, command etc. by way of sentences. A complete sentence seems to encapsulate some unity of thought or feeling. It has to be well formed of course or the meaning will be corrupt, ambiguous, contradictory or just plain absent. Sentence fragments can trigger a semantic response, but to qualify as a complete conveyance of a 'thought', a sentence has to be complete.

Now there are several types of sentence but only one type can be said to be true or false. We have declarative, interrogative, exclammatory, and command sentences. Only declarative statements can be evaluated for truth. Now we may get information even from a sentence fragment or single word, but to get truth we need this particular kind of sentence, because only it provides a statement of fact which can either be checked against reality or tested for internal consistency and validity.

Truth is loosely a yes/no answer to whether or not a statement corresponds to the facts. What are the facts? The facts are the content generated by some distinction. The statement is about those facts and may or may not be true i.e. valid.

Look at the following example:

Let our universe of discourse be natural numbers N = { 0,1,2, ... }

"Five is a number between three and eight." is a sentence which is either true or false.

It is actually true because the fact is that 5 is 2 more than 3 and it is 3 less than 8. Five lies in the sequence 3,4,5,6,7,8. This is an example of empirical truth. A specific assertion has been made which can be checked against the facts. The sentence in mathematical terms is called a Proposition.

Now we can generalize the form of this proposition and call it a propositional form. We can do this by just using a blank instead of the number 5. We now have:

_______ is a number between 3 and 8.

Any number (within the Universe of Natural Numbers) can be inserted in the blank, and the proposition tested for truth. Now more specifically we can identify the blank as a bounding box or context provided as a Universe of Discourse. That is why I chose the word form in the original discussion of the trinity of distinction. Whenever we fill out a form such as input boxes on the internet etc. we are essentially putting content into a box. The bounding box is usually labeled for us so that we make sure we are aware of the context or Universe of Discourse (U in our Venn diagram).

The body of a propositional form also has functionality but no content (except that of inherent necessity of representation). Content is only there when we fill in the form. We thus have a generalized proposition which can be replaced with a more concise form as below:

x is a number between 3 and 8.

We have replaced the blank with what is called a variable. Variables are nifty space-saving inventions that allow us to temporarily ignore content and focus on form and function. It is also a way of moving from empirical to propositional truth. Just remember, a direct proposition can only be tested for truth if it really does have content. If we move on to higher order propositions, we can make statements like x + x = 2x. Such statements are statements of equivalence and are actually statements about statements. They can indeed be true or false even though we are using variables. The truth of the facts is embedded in the equivalence of structure of the statements and therefore its internal consistency and validity is testable. If true, such statements may be classed as invariant. They are also called tautologies. There can be a great deal of information in a tautology but only one truth! Mathematics and Science are primarily engaged in the search for invariants.

To continue, we can now more concisely label our above propositional form as proposition C which is dependent on x as:

C(x) = 3 < x < 8

I choose the label C because I want to identify it as a conditional. If the proposition C is true we can say that it's conditional truth has been satisfied. It is conditional because it depends on which particular number we submit as x. The number 2 would render the proposition false.

Now we are able to take a huge leap and jump straight into set theory. We can now define a set A that consists of all the numbers between 3 and 8. We have:

A = { x N : C(x) is true } or explicitly in this case

A = { x N : 3 < x < 8 }

This is read as "A is the set such that for every x in N, if C(x) is true, then x belongs to A. The conditional expression C establishes a truth criterion. It be identifed as function. 'xN' can be identified as form. A itself is identified as content.

Now we are in a position to give a formal description of our original Aristotelian Venn diagram of basic logic. Letting '1' stand for true and '0' stand for false, we have:

A = { x U : A(x) = 1 }
~A = { x U : A(x) = 0}

Ok, let's take a breather and refresh our understanding of what we're doing and where we're headed with all this lingo. We originally established the fundamental core of logic and knowing as a distinction making process. We have just now achieved a concise way of saying the same thing in symbols. A singular sharp functional criterion of truth, repeatedly applied to some Universe of Discourse enables the distinction or abstraction of content. We may loosely identify the knower as the function, content as the known, and form as the context. If knowledge is the thing known, it is identified as abstraction. The reader must realize that while he is a knower in this case also, he is a meta-knower focusing on the whole trinity as content with epistemology as his context. Oh, and dont forget that we are still on the road to understanding what knowledge has to do with attractor basins. I promise to get there eventually.

So what is truth? Truth is just the result of any criterion of distinction. What are facts? Facts are just the content of any distinction made. Now we can and frequently do just identify truth with content and say that the truth is what the facts are. This is especially common when we are dealing with so-called empirical truths.

Recall that the singular direct and specific proposition "5 is between 3 and 8" was an example of empirical truth. It was testable against the facts (although admittedly this was an obvious set of facts to most). One may argue that it is not really an empirical truth because the facts are not in the real world, yet still it gives an easy example of the same sort of process. The facts of empirical truths are usually determined by our own minds. The mind is in essence function here. The facts of Propositional truths on the other hand are determined by explicit symbols and functions. Again, as I said at the beginning of this paper, the distinction is really more of subject matter (context, form) and degree of abstraction than anything else. Actually I think it more proper to reserve propositional truth as designating second order equivalence testing to arrive at invariants, but we shall see that even complex empirical truths end up having this structure also, so the distinction is still moot.

The proposition above had a singular truth and the set A in this case would have only one point. This is no problem. The generalization into set A emphasizes that every truth is but a recursive cycling that unifies into some distinguishable content. Even a singular proposition, if very complex, can cause the mind to engage in distinction making that circles repreatedly until eventually and hopefully it settles into some graspable result. Set theory and Boolean Algebra generalize our most primitive thought process into proceedures which can be implemented in machinary and therefore are extendable as models of real-world systems behaviour. Keep this in mind and you will go far, thinker.

Empirical truth is usually encountered at a high level of abstraction and involves many many distinctions, but in general we may identify such situations as involving distinctions born of language and distinctions perceived directly in the real world. The language distinctions form a model of structural facts which can be equivalence tested against real world facts. This is similar to the example above of an invariant. The statement "It is raining" is tested by going outside and seeing if it is raining. If the language model is equivalent to the factual model, we have established an invariant that works all the time anywhere we can go outside and verify the weather. We again traditionaly mis-identify truth with content and simply say, it really is raining or "It is raining" is a true statement.

Now obviously it may be raining in one spot and not in another. It probably is raining all the time at least somewhere and not raining all the time also. Thus we see that unless the Universe of discourse (form,context) is very clearly distinguished, the truth testing will have problems. The test itself will have problems of precision and accuracy. Suppose we want to make our test for rain more objective by putting a rain gauge in the yard. We have to establish a criterion for a minimal amount of water collected to make a determination and the gauge has to be precise enough to distinguish this amount. Accuracy will be affected also if we put our gauge under a tree. Then it will give incorrect readings even if very precise. Empirical truths can become very messy indeed. The truth distinction will only be as precise as our criterion or model of criteria is. In addition the validity(accuracy) will depend on how well we apply (consistency) our abstraction of real world facts to the model.

Precision has to do with sharpness of distinction and will determine if our content is exact or fuzzy. Accuracy has to do with consistency. Final validity (the real truth?) finally rests on a number of complex affairs and is dependent on our particular definition of what we define our functional distinction to be to begin with. Truth is in a sense what we look for. I am of course mis-identifying truth as content here. The point of all this is to show that truth is simply the result of functional distinctions regardless of whether abstract and mental or concrete and exisiting in the physical world. Invariance is the real truth if one wants to seek for ultimates.

The truth of invariance is found only in the same finite and imperfect distinctions made by function. Function involves time, process, and relation. Invariance is discovered by finding equivalences of structural relations between form and content and between model and reality. External reality itself is born of this same process of distinction making. The real real Reality is just the sum total of everything and contains all these goings on both within thinkers and without them. Reality is thinking too. It is progressively becomming more abstract in toto almost by definition. Chris Lanagan (Chris Langan - CTMU - Cognitive-Theoretic Model of the Universe ) claims indeed that expansion of the Universe is really a result of the fact that Universe is self-referential. Only the invariances emergent in this process remain changeless. These are the only absolutes. If one wants to become a Platonist, by all means do so, but dont forget to check out all the scenery along the way.

I have not explained how it is that language and symbols enables us to do this type of distinction making, but that is a more advanced aspect of semiotics and we leave that to another paper. This paper is more focused on the basic form inherent in all low level process and how it relates to the real world. Distinctions occur naturally in the real world as well as in our minds. The same trinity of form-function-content is universal. Gravity sorts out sedimentary layers according to this same game. The invariant laws of nature in general conform to this principle. As said at the beginning, recursive distinction making is the mother of all invariants. We, mind, and nature are all part of the same one reality, and knowledge if real at all is everywhere.

Finally we are ready to look at non-linear set dynamics and attractors. They put a whole other light on the discussion of truth, knowledge, and reality.

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JULIA SET BASINS OF ATTRACTION

Let us confine our Universe of Discourse to numbers in the complex plane Z.

Define a set J = { z Z : C(z) = true } where

C(z) is a function of some point zZ. It returns a truth value 1 or 0. If it returns 1, z belongs to J.

Define C(z) = {
if the absolute value of the kth iterate of z -> z*z is less than 1, then C(z) is true, else C(z) is false }

Note that C(z) involves a recursively defined function that executes (or must be applied) k times. The resulting set we get by applying the above propositional form is a unit circle centered on the origin. It makes a nice clean example for A in our Venn diagram type of picture.

Now what happens when we actually perform the set extraction and look at each iterate of z -> z*z. If z starts inside the unit circle, the iterates spiral progressively toward the center. This is easy to see because any number less than 1.0 times itself is smaller. If z starts outside the unit circle the iterates spiral outward. If z starts on the circle it simply rotates exactly around the circle. The first case is attractive, the second case is repulsive. The last case is indifferent.

In general, complex multiplication involves a rotation and a scaling, hence the spiral nature of the orbits. Orbits or trajectories are just successive iterates of feeding z back into the same equation that produced it. Now complex addition is pure translation. What happens if we introduce a small additive factor 'c'? Our functional part of the proposition will now be z -> z*z + c.

The behaviour now is incredible. For small c, the circle starts to crinkle up and become wavy. Eventually the nice boundary of the set becomes fractal and looks like a very ragged coastline. We have essentially what is called a Julia set.

Julia sets have some rather strange properties. Technically a Julia is defined as the closure of all the periodic repeller points of f, but we dont want to get into the technical aspects. We want to simply look at the behaviour, so dont be too concerned about understanding the precise definition. An easier way to say it is that a Julia is the boundary of all the attractors of f, where f is just z -> z*z + c.

What is an attractor? No one can say. Consistent mathematical definitions are lacking. Attractors are implicate tendencies. An attractor basin is the set of points whose orbits remain bound to the attractor. With a Julia set, for any point outside an attractor basin, the basin acts as a repeller. There are any number of attractors for a Julia set though, and the net boundary of all these attractors is the set itself. It has a strange property. If we take any point on the boundary that belongs to a given attractor basin and draw an arbitrarily small neighborhood about it, there will in this neighborhood be at least one point that belongs to each and every one of the other attractor basins. This is what makes the boundary fractal. It is also a prime example of non-locality mixed in with locality. Repeller orbits warp and woof infinitely close to Attractor orbits. This is the very meaning of sensitivity to initial conditions i.e The Butterfly Effect.

There are several types of behaviour associated with dynamic orbits. Stepwise repeated rotations and scalings introduce periodicity which can be exact or not, stable, indifferent, spiralling inward or out. Translations enable or destroy periodicity and introduce strangeness. The net result is that we have these broad classes of behaviour:

1) Super attractive - very quick attraction to a fixed point

2) attractive - periodic and bounded

3) indifferent

4) repulsive - unbounded

5) strange - periodic about two or more attractors

A Julia set can be connected but not equit-continuous. Sometimes it is not even connected and it literally is fractal dust. The boundary is the point of distinction, though, that decides whether points lie inside or outside the set. The interior is attractive and that is where content born of distinctions goes. The boundary is the edge of judgement. Judgements can be chaotic.

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The discussion above should make it clear that sometimes the truth of propositions is not always clear cut. Truth is identified as the boundary of distinction between A and ~A. This exposes the fallacy of the Law of Contradiction and The Excluded Middle, even when using precise Aristotelian standards. This boundary under some functions of distinction is chaotic. This principle applies to both mental constructs and physical realities. Truth is the judgemental edge of the sword of distinction. Truth is forever temporary and relative to whatever process is wielding the sword. Knowledge is an accretion product of this process - the bounty if you like. It also is relative and impermanent. Only invariances remain changeless. Invariances occur as implicate order. They can sometimes be revealed explicity if the process is super-attractive. This an exception rather than the rule, however. Super-attractive process is often considered a degenerate case. The search for knowledge and truth thus resolves into the search for the attractive and often complex structure of invariants within ongoing process. It is an endless journey full of sudden surprises and unexpected results.

If mathematicians and Scientists did develop a Theory of Everything and code it into a set of symbols, then someday it could survive only as a totally meaningless undecipherable record of graven images on some tablet. Only a living breathing community of active minds can keep the link going. We thus have good news and bad news. Truth and Knowledge are no longer as mysterious as they might have been, but there are real limits to our ability to know. A new tolerance and appreciation of other minds comes with this understanding. Metaphysical ghosts are no longer necessary to explain most things, but the door is still open to new levels of reality and who knows. Religious and political truths will likely always stand in the context of solicitation, but perhaps solicitation to join a community of like minds is not so bad.

LAWS OF EPISTEMOLOGY

I. Knowledge is recursive. corollary:Everything is recursive

II. Truth is a function of distinctions made.

III. The relation between form, function, and content is the implicate unifying model of reality.

IV. Knowledge accretes to implicate order.

V. Truth can be superstable, stable, periodic, chaotic, or strange.

VI. Any single truth is partial and incomplete.

It only rains on Wednesday, except when its Monday.

APPENDIX

PLAY THE GAME YOURSELF. PICK ANY SYSTEM AND SORT IT OUT

S.Roof, Dec. 2002