Sign Relational Manifolds • 5

Let me try to say in intuitive terms what I think is really going on here.

The problem we face is as old as the problem of other minds, or intersubjectivity, or even commensurability, and it naturally involves a whole slew of other old problems — reality and appearance, or reality and representation, not to mention the one and the many.  One way to sum up the question might be “conditions on the possibility of a mutually objective world”.

Working on what oftentimes seems like the tenuous assumption that there really is a real world causing the impressions in my mind and the impressions in yours — more generally speaking, that there really is a real world impressing itself in systematic measures on every frame of reference — we find ourselves pressed to give an account of the hypothetical unity beneath the manifest diversity — and how it is possible to discover the former in the latter.

Manifold theory proposes one type of solution to that host of problems.

Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Interoperability, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | Leave a comment

Sign Relational Manifolds • 4

Another set of notes I found on this theme strikes me as getting to the point more quickly and though they read a little rough in places I think it may be worth the effort to fill out their general line of approach.

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Sign Relational Manifolds • 3

I’m not sure when it was I first noticed the relationship between manifolds and semiotics but I distinctly recall the passage in Serge Lang’s Differential and Riemannian Manifolds which brought the triadic character of tangent vectors into high relief.  I copied out a set of excerpts highlighting the point and shared it with the Inquiry, Ontology, and Peirce lists.

Excerpts from Serge Lang, Differential and Riemannian Manifolds,
Springer‑Verlag, New York, NY, 1995.

Chapter 2.  Manifolds

Using the concepts and terminology from Lang’s text, I explained the connection between manifold theory and semiotics in the following way.

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Sign Relational Manifolds • 2

A sense of how manifolds are applied in practice may be gleaned from the set of excerpts linked below, from Doolin and Martin (1990), Introduction to Differential Geometry for Engineers, which I used in discussing differentiable manifolds with other participants in the IEEE Standard Upper Ontology Working Group.

What brought the concept of a manifold to mind in that context was a set of problems associated with perspectivity, relativity, and interoperability among multiple ontologies.  To my way of thinking, those are the very sorts of problems manifolds were invented to handle.

Reference

  • Doolin, Brian F., and Martin, Clyde F. (1990), Introduction to Differential Geometry for Engineers, Marcel Dekker, New York, NY.
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Sign Relational Manifolds • 1

Riemann’s concept of a manifold, especially as later developed, bears a close relationship to Peirce’s concept of a sign relation.

I will have to wait for my present train of thought to stop at a station before I can hop another but several recent discussions of geometry have brought this subject back to mind and I thought it might serve to drop off a few mail bags of related letters in anticipation of the next pass through this junction.

Here is a set of excerpts from Murray G. Murphey (1961), The Development of Peirce’s Philosophy, discussing Peirce’s reception of Riemann’s philosophy of geometry.

Later developments of the manifold concept, looking to applications on the one hand and theory on the other, are illustrated by excerpts in the next two posts.

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Landfail

Adrift on the oceans of memory
I touch on an isle of solidity
And try to plant my feet —
But only push off back to sea.

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Ask Meno Questions • Discussion 3

Re: Stephen Rose

In Aristotle’s De Anima or “On the Soul” there is a fine articulation of the universal join between the body and the soul, one so embedded in the marrow of our culture that it moves and shapes our thinking in ways we seldom recognize. Here is one place where I discussed the introductory sections of this earliest textbook in psychology.

Here is a salient excerpt from Aristotle’s text:

  1. The theories of the soul (psyche) handed down by our predecessors have been sufficiently discussed; now let us start afresh, as it were, and try to determine (diorisai) what the soul is, and what definition (logos) of it will be most comprehensive (koinotatos).
  2. We describe one class of existing things as substance (ousia), and this we subdivide into three: (1) matter (hyle), which in itself is not an individual thing, (2) shape (morphe) or form (eidos), in virtue of which individuality is directly attributed, and (3) the compound of the two.
  3. Matter is potentiality (dynamis), while form is realization or actuality (entelecheia), and the word actuality is used in two senses, illustrated by the possession of knowledge (episteme) and the exercise of it (theorein).
  4. Bodies (somata) seem to be pre-eminently substances, and most particularly those which are of natural origin (physica), for these are the sources (archai) from which the rest are derived.
  5. But of natural bodies some have life (zoe) and some have not; by life we mean the capacity for self-sustenance, growth, and decay.
  6. Every natural body (soma physikon), then, which possesses life must be substance, and substance of the compound type (synthete).
  7. But since it is a body of a definite kind, viz., having life, the body (soma) cannot be soul (psyche), for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, i.e., as matter.
  8. So the soul must be substance in the sense of being the form of a natural body, which potentially has life. And substance in this sense is actuality.
  9. The soul, then, is the actuality of the kind of body we have described. But actuality has two senses, analogous to the possession of knowledge and the exercise of it.
  10. Clearly (phaneron) actuality in our present sense is analogous to the possession of knowledge; for both sleep (hypnos) and waking (egregorsis) depend upon the presence of the soul, and waking is analogous to the exercise of knowledge, sleep to its possession (echein) but not its exercise (energein).
  11. Now in one and the same person the possession of knowledge comes first.
  12. The soul may therefore be defined as the first actuality of a natural body potentially possessing life; and such will be any body which possesses organs (organikon).
  13. (The parts of plants are organs too, though very simple ones: e.g., the leaf protects the pericarp, and the pericarp protects the seed; the roots are analogous to the mouth, for both these absorb food.)
  14. If then one is to find a definition which will apply to every soul, it will be “the first actuality of a natural body possessed of organs”.
  15. So one need no more ask (zetein) whether body and soul are one than whether the wax (keros) and the impression (schema) it receives are one, or in general whether the matter of each thing is the same as that of which it is the matter; for admitting that the terms unity and being are used in many senses, the paramount (kyrios) sense is that of actuality.
  16. We have, then, given a general definition of what the soul is: it is substance in the sense of formula (logos), i.e., the essence of such-and-such a body.
  17. Suppose that an implement (organon), e.g. an axe, were a natural body; the substance of the axe would be that which makes it an axe, and this would be its soul; suppose this removed, and it would no longer be an axe, except equivocally. As it is, it remains an axe, because it is not of this kind of body that the soul is the essence or formula, but only of a certain kind of natural body which has in itself a principle of movement and rest.
  18. We must, however, investigate our definition in relation to the parts of the body.
  19. If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye.
  20. Now we must apply what we have found true of the part to the whole living body. For the same relation must hold good of the whole of sensation to the whole sentient body qua sentient as obtains between their respective parts.
  21. That which has the capacity to live is not the body which has lost its soul, but that which possesses its soul; so seed and fruit are potentially bodies of this kind.
  22. The waking state is actuality in the same sense as the cutting of the axe or the seeing of the eye, while the soul is actuality in the same sense as the faculty of the eye for seeing, or of the implement for doing its work.
  23. The body is that which exists potentially; but just as the pupil and the faculty of seeing make an eye, so in the other case the soul and body make a living creature.
  24. It is quite clear, then, that neither the soul nor certain parts of it, if it has parts, can be separated from the body; for in some cases the actuality belongs to the parts themselves. Not but what there is nothing to prevent some parts being separated, because they are not actualities of any body.
  25. It is also uncertain (adelon) whether the soul as an actuality bears the same relation to the body as the sailor (ploter) to the ship (ploion).
  26. This must suffice as an attempt to determine in rough outline the nature of the soul.

Aristotle, “On The Soul”, in Aristotle, Volume 8, W.S. Hett (trans.), William Heinemann, London, UK, 1936, 1986.

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Ask Meno Questions • Discussion 2

Re: Victoria N. Alexander

I tend to favor that general way of speaking, having been brought up on classical cybernetics, optimal control, and systems theory, where we think of a system as passing through its state space, controlled by its objective to optimize a specific objective function defined on its space of states.  When it comes to choosing intuition-aiding names for the various components of systems and regions of state spaces, those details can be fit to the particular type of system in view.

In the case of creatures like ourselves, it is possible to regard our conscious contents as representing our body’s theory of what is likely to deserve our mind’s attention.  If that theory is a good one, then we may live to be conscious of another day.

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Ask Meno Questions • Discussion 1

Re: Victoria N. Alexander

I think you are stating several important insights.

The word object in pragmatism and pragmatic semiotics has a much wider range of meanings than the extremely reductive sense of a “compact physical object”.  Anyone wishing to explore the richness of its meaning could hardly do better than to sample the senses of the Greek pragma that imbue the Latin-derived object.

There is reason to think that the sense of the word object that means objective, purpose, target, intention, goal, end, aim, and so on is more fundamental than the more restrictive sense of a compact physical object.  That is in fact one of the most critical insights that comes down to us from long lines of physical theory and also from the traditions known as “process thinking”, suggesting that our concepts of physical objects are derivative in relation to our concepts of process, since they arise from our ability to discover “invariants under transformations”, that is, the formal constructs that are preserved by the operations or processes that transform the states of a system.

As a general rule, we should avoid language that confuses signs and objects.  In particular, referring to mental representations as “ideal objects” is just asking for trouble, and that on several counts, including the risk of confusing mental ideas with Platonic ideas.  Language like that brings all the confusions of conceptualism, nominalism, and psychologism down on our heads.

Given that Peirce’s critique of Cartesian philosophy is of a piece with the rest of his thought, it does not seem wise to backslide on this score and reinfect semiotics with the dualisms that Peirce was so persistent in rooting out.  For example, so far as the mind/body dualism goes, Peirce regarded the body as one of the first objects that a developing being would naturally “construct” from the flux of experience, that is, construe or conceptualize as an object from the impressions available in the stream of awareness.  There is a potential misunderstanding that needs to be avoided here.  It is not that noticing a dualism is a bad thing — where one is operative it cannot be denied.  The important thing is — by the time one notices a dualism, one has already become a third party, a mediator, a synthetic operator, and so one must recognize that more than two components are already in play.

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Ask Meno Questions • Chrysalis

Chrysalis

Memories of being held
      In closely knit spheres
And guided beyond the orbits
      Of childhood fears
Entrusted with a word
      That rustles in a breath
And warrants respect for
      The not yet beautiful

In Honor of My Parents’ Golden Wedding Anniversary
Jon Awbrey, Amherst, Massachusetts, March 21, 1996

“That is a chrysalis”, she said, when I showed her that funny-looking thing on the leaf. In that moment I, the thing, and the word were one. I learned a word, I wrapped it around a new-found thing, and I listened in wonder to my mother’s story of what it was and what it would be.

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