All Process, No Paradox • 6

Posted on January 22, 2014 by Jon Awbrey

Re: R.J. Lipton • Anti-Social Networks
Re: Lou Kauffman • Iterants, Imaginaries, Matrices

Comments I made elsewhere about computer science and (anti-)social networks have a connection with the work in progress on this thread, so it may steal a march to append them here.

Comment 1

I have been interested for a long time now in using graphs to do logic.  For that you need more than positive links — negative relations are more generative than positive relations.  The logical situation is analogous to social networks where people can “unlike” or “¬like” other people, or website networks where the information at one node may contradict the information at another node.  In my pursuits it turns out that particular species of graph-theoretic “cacti” are much more useful than the garden variety trees and unsigned graphs.

Comment 2

For what it’s worth, here is my exposition of “painted cacti” and their application to propositional calculus.

A painted cactus is a rooted cactus with any number of symbols from a finite alphabet attached to each node.  In their ordinary logical interpretations these symbols (“paints”) stand for boolean variables.

Triangles are interesting in computational contexts because they arise in case-breakdown expressions.  In one of the common interpretations of cactus graphs, a rooted triangular lobe says the values of the two non-root nodes are logically inequivalent.

Resources

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Posted in Algorithms, Amphecks, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Lou Kauffman, Mathematics, Minimal Negation Operators, Painted Cacti, Paradox, Peirce, Process Thinking, Propositional Calculus, Spencer Brown, Systems, Time | Tagged , , , , , , , , , , , , , , , , , , , , , , | 10 Comments

All Process, No Paradox • 5

Posted on January 21, 2014 by Jon Awbrey

In the midst of this strife, whereat the halls of Ilúvatar shook and a tremor ran out into the silences yet unmoved, Ilúvatar arose a third time, and his face was terrible to behold.  Then he raised up both his hands, and in one chord, deeper than the Abyss, higher than the Firmament, piercing as the light of the eye of Ilúvatar, the Music ceased.

Tolkien • Ainulindalë

Re: Objects, Models, Theories • (1) • (2)
Re: Peirce List (1) (2) • Helmut Raulien (1) (2)

For continuity’s sake — as I try to recover my train of thought from the spin-offs of the solstice roundhouse — I’m recycling my replies to a comment by Helmut Raulien on the Peirce List which raised a host of questions about Peirce’s categories, logic, and semiotics in the light of Spencer Brown’s Laws of Form.

Comment 1

George Spencer Brown’s Laws of Form tends to be loved XOR hated by most folks, with few coming down in between.  I ran across the book early in my undergrad years, shortly after encountering C.S. Peirce, so I recognized the way it revived Peirce’s logical graphs, emphasizing the entitative interpretation of the abstract formal calculus immanent in Peirce’s “Alpha” graphs.  It took me a decade to gain a modicum of clarity about all that “imaginary truth value” and “re-entry” folderol.  I’ll say some things about that later on.

Comment 2

I mulled the matter over for a fair spell of days and nights and decided it wouldn’t be good to jump into the middle of the muddle about re-entry and imaginary truth values right off the bat, that it would be better in the long run to get a solid grip on what is going on with the propositional level of Peirce’s logical graphs and how Spencer Brown’s elaborations can be seen to manifest the same spirit of reasoning, if they are read the right way.  Toward that end I’ll append a list of resources to break the ice on this approach.

Resources

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Posted in Animata, C.S. Peirce, Category Theory, Cybernetics, Differential Logic, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Semiotics, Spencer Brown, Systems Theory, Tertium Quid, Time, Tolkien | Tagged , , , , , , , , , , , , , , , , , | 9 Comments

All Process, No Paradox • 4

Posted on January 20, 2014 by Jon Awbrey

Thus began the Days of the Bliss of Valinor;
and thus began also the Count of Time.

Tolkien • Quenta Silmarillion

While looking for something else on the web, I ran across an old note I had written in reply to an inquiry on the Conceptual Graphs List, and it seemed to express one of the points of the present thesis in a fairly clear fashion, so here’s the part I found fit to revive.

Time Representation

A point of view arising from fundamental physical considerations makes the concept of Process more fundamental than the concept of Time, since references to a time parameter are simply references to a process taken as standard, in other words, a clock.

We can always develop another “naive physics”, natural language “tense logic”, or implicit psychological theory of time, and maybe that’s all we need in particular settings, but if we push for a deeper logical analysis of timed processes themselves then we need a logical framework capable of dealing with relations between systems which undergo changes in their properties, as described by logical statements.

That is the impulse motivating Differential Logic.  As it turns out, Peirce’s way of doing logic, especially in graphical form, is naturally adapted to dealing with change and difference in logical form.

Resources

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Posted in Abstraction, C.S. Peirce, Conceptual Graphs, Cybernetics, Differential Logic, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Process Thinking, Spencer Brown, Systems Theory, Time, Tolkien | Tagged , , , , , , , , , , , , , , , , | 9 Comments

All Process, No Paradox • 3

Posted on January 17, 2014 by Jon Awbrey

Consider what effects that might conceivably
have practical bearings you conceive the
objects of your conception to have.  Then,
your conception of those effects is the
whole of your conception of the object.

Charles S. Peirce • “Issues of Pragmaticism”

Re: Peirce List • Paul Eduardo

A riddle is a description of something, typically in metaphorical, oblique, and very partial terms, from which the respondent must abduce the identity of the thing described.  One of the interesting things about Gollum’s riddle is the pragmatic way he describes the object of his conception in terms of its effects on the contents of a whole universe of discourse.  If we weren’t at hazard for being devoured ourselves, we’d be at leisure to sit down and work out a logical analysis of those effects.  There are a few fine points we’d have to settle, like when he says this thing devours all things — Does it devour itself or other things only?

I meant to write more, but it’s later than I thought it would be by now …

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Posted in Animata, C.S. Peirce, Change, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Pragmatic Maxim, Process Thinking, Spencer Brown, Systems, Time, Tolkien | Tagged , , , , , , , , , , , , , , , , , | 11 Comments

All Process, No Paradox • 2

Posted on January 15, 2014 by Jon Awbrey

These are the forms of time, which imitates eternity and revolves according to a law of number.

Plato • Timaeus

Re: Lou Kauffman • Iterants, Imaginaries, Matrices

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of inquiry Charles Sanders Peirce and George Spencer Brown pursued into graphical syntaxes for logic, just last month opened a blog and his very first post touched on perennial questions of logic and time — Logos and Chronos — puzzling the wits of everyone who has thought about them for as long as anyone can remember.  Just locally and recently these questions have arisen in the following contexts:

Kauffman’s treatment of logic, paradox, time, and imaginary truth values led me to make the following comments I think are very close to what I’d been struggling to say before.

Let me get some notational matters out of the way before continuing.

I use \mathbb{B} for a generic 2-point set, usually \{ 0, 1 \} and typically but not always interpreted for logic so that 0 = \mathrm{false} and 1 = \mathrm{true}.  I use “teletype” parentheses \texttt{(} \ldots \texttt{)} for negation, so that \texttt{(} x \texttt{)} = \lnot x for x ~\text{in}~ \mathbb{B}.  Later on I’ll be using teletype format lists \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} for minimal negation operators.

As long as we’re reading x as a boolean variable (x \in \mathbb{B}) the equation x = \texttt{(} x \texttt{)} is not paradoxical but simply false.  As an algebraic structure \mathbb{B} can be extended in many ways but it remains a separate question what sort of application, if any, such extensions might have to the normative science of logic.

On the other hand, the assignment statement x := \texttt{(} x \texttt{)} makes perfect sense in computational contexts.  The effect of the assignment operation on the value of the variable x is commonly expressed in time series notation as x' = \texttt{(} x \texttt{)} and the same change is expressed even more succinctly by defining \mathrm{d}x = x' - x and writing \mathrm{d}x = 1.

Now suppose we are observing the time evolution of a system X with a boolean state variable x : X \to \mathbb{B} and what we observe is the following time series.

\begin{array}{c|c} t & x \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ 5 & 1 \\ 6 & 0 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ \ldots & \ldots \end{array}

Computing the first differences we get:

\begin{array}{c|cc} t & x & \mathrm{d}x \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \\ 5 & 1 & 1 \\ 6 & 0 & 1 \\ 7 & 1 & 1 \\ 8 & 0 & 1 \\ 9 & 1 & 1 \\ \ldots & \ldots & \ldots \end{array}

Computing the second differences we get:

\begin{array}{c|cccc} t & x & \mathrm{d}x & \mathrm{d}^2 x & \ldots \\ \hline 0 & 0 & 1 & 0 & \ldots \\ 1 & 1 & 1 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 3 & 1 & 1 & 0 & \ldots \\ 4 & 0 & 1 & 0 & \ldots \\ 5 & 1 & 1 & 0 & \ldots \\ 6 & 0 & 1 & 0 & \ldots \\ 7 & 1 & 1 & 0 & \ldots \\ 8 & 0 & 1 & 0 & \ldots \\ 9 & 1 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}

This leads to thinking of the system X as having an extended state (x, \mathrm{d}x, \mathrm{d}^2 x, \ldots, \mathrm{d}^k x), and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule x' = \texttt{(} x \texttt{)} can now be expressed by the rule \mathrm{d}x = 1.

The following article has a few more examples along these lines.

Resources

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Posted in Animata, Boolean Functions, C.S. Peirce, Cybernetics, Differential Logic, Discrete Dynamics, Laws of Form, Logic, Logical Graphs, Lou Kauffman, Mathematics, Paradox, Peirce, Plato, Process, Spencer Brown, Timaeus, Time | Tagged , , , , , , , , , , , , , , , , , | 11 Comments

All Process, No Paradox • 1

Posted on January 13, 2014 by Jon Awbrey


This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.

Tolkien • The Hobbit

Talking about time is a waste of time.  Time is merely an abstraction from process and what is needed are better languages and better pictures for describing process in all its variety.  In the sciences the big breakthrough in describing process came with the differential and integral calculus, that made it possible to shuttle between quantitative measures of state and quantitative measures of change.  But every inquiry into a new phenomenon begins with the slimmest grasp of its qualitative features and labors long and hard to reach as far as a tentative logical description.  What can avail us in the mean time, still tuning up before the first measure, to reason about change in qualitative terms?

Et sic deinceps … (So it begins …)

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Posted in Animata, C.S. Peirce, Change, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Process Thinking, Spencer Brown, Systems Theory, Time, Tolkien | Tagged , , , , , , , , , , , , , , , , , | 11 Comments

Precursors Of Category Theory • 3

Posted on January 3, 2014 by Jon Awbrey

Act only according to that maxim by which you can at the same time will that it should become a universal law.

Immanuel Kant (1785)

Precursors Of Category Theory

Peirce

Cued by Kant’s idea on the function of concepts in general, Peirce locates his categories on the highest level of abstraction affording a meaningful measure of traction in practice, reserving judgment on the absolute unity of perfect ambiguity and the numerous dualisms which taken together may well converge on the same conception as Peirce’s trinity.

Selection 1

§1.  This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it.  (CP 1.545).

§2.  This theory gives rise to a conception of gradation among those conceptions which are universal.  For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied;  and so on.  (CP 1.546).

C.S. Peirce, “On a New List of Categories” (1867)

Selection 2

I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.

That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of.  We thus think of the thought-sign itself, making it the object of another thought-sign.

Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions.  Does this series proceed endlessly?  I think not.  What then are the characters of its different members?

My thoughts on this subject are not yet harvested.  I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:  Actuality, Possibility, Destiny (or Freedom from Destiny).

On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.  (CP 4.549).

C.S. Peirce, “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546 (1906), CP 4.530–572.

The first thing to extract from this passage is that Peirce’s Categories, or “Predicaments”, are predicates of predicates.  Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the logic of second intentions, or what is handled very roughly by second order logic in contemporary parlance, and continuing onward through higher intentions, or higher order logic and type theory.

Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel.  The names he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like quality, reaction, and symbolization to maximally abstract terms like firstness, secondness, and thirdness.  Taken in full generality, k-ness may be understood as referring to those properties all k-adic relations have in common.  Peirce’s distinctive claim is that a type hierarchy of three levels is generative of all we need in logic.

Part of the justification for Peirce’s claim that three categories are necessary and sufficient appears to arise from mathematical facts about the reducibility of k-adic relations.  With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates.  With regard to sufficiency, all higher arity k-adic relations can be analyzed in terms of triadic and lower arity relations.

Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 8 Comments

Precursors Of Category Theory • 2

Posted on December 30, 2013 by Jon Awbrey

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists …

☙ Marcel Proust

Precursors Of Category Theory

When it comes to looking for the continuities of the category concept across different systems and systematizers, we don’t expect to find their kinship in the names or numbers of categories, since those are legion and their divisions deployed on widely different planes of abstraction, but in their common function.

Aristotle

Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different.  For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named.  For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different.  For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding.  Thus a man and an ox are called animals.  The name is the same in both cases;  so also the statement of essence.  For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.

— Aristotle, Categories, 1.1a1–12.

Translator’s Note.  “Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture.  We have no ambiguous noun.  However, we use the word ‘living’ of portraits to mean ‘true to life’.”

In the logic of Aristotle categories are adjuncts to reasoning designed to resolve ambiguities and thus to prepare equivocal signs, otherwise recalcitrant to being ruled by logic, for the application of logical laws.  The example of ζωον illustrates the fact that we don’t need categories to make generalizations so much as we need them to control generalizations, to reign in abstractions and analogies that are stretched too far.

References

  • Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  • Karpeles, Eric (2008), Paintings in Proust, Thames and Hudson, London, UK.
Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 6 Comments

Precursors Of Category Theory • 1

Posted on December 20, 2013 by Jon Awbrey

A few years back I began a sketch on the Precursors of Category Theory, aiming to trace the continuities of the category concept from Aristotle, thorough Kant and Peirce, Hilbert and Ackermann, to contemporary mathematical use.  Perhaps a few will find these notes of use in the current context.

Precursors Of Category Theory

Preamble

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers:  “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.

— Saunders Mac Lane, Categories for the Working Mathematician, 29–30.

Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 10 Comments

Alpha Now, Omega Later • 7

Posted on December 19, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Theorems From Physics?

In a way, the relation between “physics space” and “information space” is one of the topics I address in my work on inquiry driven systems.  Here is a pertinent place in medias res, from which point one may happily sample both backwards and forwards:

Prospects for Inquiry Driven Systems • Unabridgements

Part of my task in the projected work is to make a bridge, in theory and practice, from simple physical systems to the more complex systems, also physical but in which new orders of features have become salient, that begin to exhibit what is recognized as intelligence.  At the moment it seems that a good way to do this is to anchor the knowledge component of intelligent systems in the tangent and co-tangent spaces that are founded on the base space of a dynamic manifold.  This means finding a place for knowledge representations in the residual part of the initial factorization.  This leads to a consideration of the questions:  What makes the difference between these supposedly different factors of the total manifold?  What properties mark the distinction as commonly intended?

From a naturalistic perspective everything falls equally under the prospective heading of physis, signifying nothing more than the first inklings of natural process, though not everything is necessarily best explained in detail by those fragments of natural law which are currently known to us.  So it falls to any science that pretends to draw a distinction between the more and the less basic physics to describe it within nature and without trying to get around nature.  In this context the question may now be rephrased:  What natural terms distinguish every system’s basic processes from the kinds of coping processes that support and crown the intelligent system’s personal copy of the world?  What protocols attach to the sorting and binding of these two different books of nature?  What colophon can impress the reader with a need to read them?  What instinct can motivate a basis for needing to know?

Previous Discussion

Posted in C.S. Peirce, Differential Logic, Dynamical Systems, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Propositional Calculus, Semiotics, Sign Relations, Spencer Brown | Tagged , , , , , , , , , , , | 4 Comments