Changes

===2.5.  Differential Geometry===

===2.5.  Differential Geometry===

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential df of a function f : X → R.  Everyone begins with the idea that df ought to be a locally linear approximation dfu(v) or df(u,v) to the difference function Dfu(v) = Df(u,v) = f(u+v) – f(u).  In this conception it is understood that "local" means in the vicinity of the point u and that "linear" is meant with respect to the variable v.

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential df of a function f: X -> R.  Everyone begins with the idea that df ought to be a locally linear approximation dfu(v) or df(u,v) to the difference function Dfu(v) = Df(u,v) = f(u+v) - f(u).  In this conception it is understood that "local" means in the vicinity of the point u and that "linear" is meant with respect to the variable v.

====2.5.1.  Local Stress and Linear Trend====

====2.5.1.  Local Stress and Linear Trend====

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, e.g. (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, e.g. (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

=====2.5.1.1.  Analytic View=====

=====2.5.1.1.  Analytic View=====

The first group, more analytic, strives to get intrinsic definitions of everything, defining tangent vectors primarily as equivalence classes of curves through points of phase space.  This posture is conditioned to the spare frame of physical theory and is constrained by the ready equation of physics with ante-metaphysics.  In short they regard physics as a practical study that is prior to any a priori.  Physics should exert itself to save the phenomena and forget the rest.  The dynamic manifold is the realm of phenomena, the locus of all knowable reality and the focus of all actual knowledge.  Beyond this, even attributes like velocity and momentum are epiphenomenal, derivative scores attached to a system's dynamic point from measurements made at other points.

The first group, more analytic, strives to get intrinsic definitions of everything, defining tangent vectors primarily as equivalence classes of curves through points of phase space.  This posture is conditioned to the spare frame of physical theory and is constrained by the ready equation of physics with ante-metaphysics.  In short they regard physics as a practical study that is prior to any a priori.  Physics should exert itself to save the phenomena and forget the rest.  The dynamic manifold is the realm of phenomena, the locus of all knowable reality and the focus of all actual knowledge.  Beyond this, even attributes like velocity and momentum are epiphenomenal, derivative scores attached to a system's dynamic point from measurements made at other points.

This incurs an empire of further systems of ranking and outranking, teams and leagues and legions of commissioners, all to compare and umpire these ratings.  When these circumspect systems are not sufficiently circumscribed to converge on a fixed point or a limiting universal system, it seems as though chaos has broken out.  The faith of this sect that the world is a fair game for observation and intelligence seems dissipated by divergences of this sort.  It wrecks their hope of order in phenomena, dooms what they deem a fit domain, a single rule of order that commands the manifold to appear as it does.  To share the universe with several realities, to countenance a real diversity?  It ruins the very idea they most favor of a cosmos, one that favors them.

This incurs an empire of further systems of ranking and outranking, teams and leagues and legions of commissioners, all to compare and umpire these ratings.  When these circumspect systems are not sufficiently circumscribed to converge on a fixed point or a limiting universal system, it seems as though chaos has broken out.  The faith of this sect that the world is a fair game for observation and intelligence seems dissipated by divergences of this sort.  It wrecks their hope of order in phenomena, dooms what they deem a fit domain, a single rule of order that commands the manifold to appear as it does.  To share the universe with several realities, to countenance a real diversity?  It ruins the very idea they most favor of a cosmos, one that favors them.

=====2.5.1.2.  Algebraic View=====

=====2.5.1.2.  Algebraic View=====

The second group, more algebraic, accepts the comforts of an embedding vector space with a less severe attitude, one that belays and belies the species of anxiety that worries the other group.  They do not show the same phenomenal anguish about the uncertain multiplicity or empty void of outer spaces.  Given this trust in something outside of phenomena, they permit themselves on principle the luxury of relating differential concepts to operators with linear and derivation properties.  This tendency, ranging from pious optimism to animistic hedonism in its mathematical persuasions, demands less agnosticism about the reality of exterior constructs.  Its pragmatic hope allows room for the imagination of supervening prospects, without demanding that these promontory contexts be uniquely placed or set in concrete.

The second group, more algebraic, accepts the comforts of an embedding vector space with a less severe attitude, one that belays and belies the species of anxiety that worries the other group.  They do not show the same phenomenal anguish about the uncertain multiplicity or empty void of outer spaces.  Given this trust in something outside of phenomena, they permit themselves on principle the luxury of relating differential concepts to operators with linear and derivation properties.  This tendency, ranging from pious optimism to animistic hedonism in its mathematical persuasions, demands less agnosticism about the reality of exterior constructs.  Its pragmatic hope allows room for the imagination of supervening prospects, without demanding that these promontory contexts be uniquely placed or set in concrete.

=====2.5.1.3.  Compromise=====

=====2.5.1.3.  Compromise=====

In attempting to negotiate between these two philosophies, I have arrived at the following compromise.  On the one hand, the circumstance that provides a natural context for a manifold of observable action does not automatically exclude all possibility of other contexts being equally natural.  On the other hand, it may happen that a surface is so bent in cusps and knots, or otherwise so intrinsically formed, that it places mathematical constraints on the class of spaces it can possibly inhabit.

In attempting to negotiate between these two philosophies, I have arrived at the following compromise.  On the one hand, the circumstance that provides a natural context for a manifold of observable action does not automatically exclude all possibility of other contexts being equally natural.  On the other hand, it may happen that a surface is so bent in cusps and knots, or otherwise so intrinsically formed, that it places mathematical constraints on the class of spaces it can possibly inhabit.

Thus a manifold can embody information that bears on the notion of a larger reality.  By dint of this interpretation the form of the manifold becomes the symbol of its implicated unity.  But what I think I can fathom seems patent enough, that the chances of these two alternatives, plurality and singularity, together make a bet that is a toss up and open to test with each new shape of manifold encountered.  It is likely that the outcome, if at all decidable, falls in accord with no general law but is subject to proof on a case by case basis.

Thus a manifold can embody information that bears on the notion of a larger reality.  By dint of this interpretation the form of the manifold becomes the symbol of its implicated unity.  But what I think I can fathom seems patent enough, that the chances of these two alternatives, plurality and singularity, together make a bet that is a toss up and open to test with each new shape of manifold encountered.  It is likely that the outcome, if at all decidable, falls in accord with no general law but is subject to proof on a case by case basis.

====2.5.2.  Prospects for a Differential Logic====

====2.5.2.  Prospects for a Differential Logic====

Pragmatically, the "proper" form of a differential logic is likely to be regulated by the purposes to which it is intended to be put, or determined by the uses to which it is actually, eventually, and suitably put.  With my current level of uncertainty about what will eventually work out, I have to be guided by my general intention of using this logic to describe the dynamics of inquiry and intelligence in systematic terms.  For this purpose it seems only that many different types of "fiber bundles" or systems of "spaces at points" will have to be contemplated.

Pragmatically, the "proper" form of a differential logic is likely to be regulated by the purposes to which it is intended to be put, or determined by the uses to which it is actually, eventually, and suitably put.  With my current level of uncertainty about what will eventually work out, I have to be guided by my general intention of using this logic to describe the dynamics of inquiry and intelligence in systematic terms.  For this purpose it seems only that many different types of "fiber bundles" or systems of "spaces at points" will have to be contemplated.

Although the limited framework of propositional calculus seems to rule out this higher level of generality, the exigencies of computation on symbolic expressions have the effect of bringing in this level of arbitration by another route.  Even though we use the same alphabet for the joint basis of coordinates and differentials at each point of the manifold, one of our intended applications is to the states of interpreting systems, and there is nothing a priori to determine such a program to interpret these symbols in the same way at every moment.  Thus, the arbitrariness of local reference frames that concerns us in physical dynamics, that makes the arbitrage or negotiation of transition maps between charts (qua markets) such a profitable enterprise, raises its head again in computational dynamics as a relativity of interpretation to the actual state of a running interpretive program.

Although the limited framework of propositional calculus seems to rule out this higher level of generality, the exigencies of computation on symbolic expressions have the effect of bringing in this level of arbitration by another route.  Even though we use the same alphabet for the joint basis of coordinates and differentials at each point of the manifold, one of our intended applications is to the states of interpreting systems, and there is nothing a priori to determine such a program to interpret these symbols in the same way at every moment.  Thus, the arbitrariness of local reference frames that concerns us in physical dynamics, that makes the arbitrage or negotiation of transition maps between charts (qua markets) such a profitable enterprise, raises its head again in computational dynamics as a relativity of interpretation to the actual state of a running interpretive program.

===2.6.  Reprise===

===2.6.  Reprise===