Time, Topology, Differential Logic • 5

Posted on November 19, 2016 by Jon Awbrey

And the founder, having shod a plough with a brazen ploughshare, and having yoked to it a bull and a cow, himself drove a deep furrow round the boundary lines, while those who followed after him had to turn the clods, which the plough threw up, inwards towards the city, and suffer no clod to lie turned outwards.

Plutarch • Life of Romulus

Re: Peirce List Discussion • ET • ET • JBD • JA • JA

Our inquiry now calls on the rudiments of topology, for which I turn to J.L. Kelley.

Chapter 1. Topological Spaces

1.1. Topologies and Neighborhoods

A topology is a family \mathcal{T} of sets which satisfies the two conditions:  the intersection of any two members of \mathcal{T} is a member of \mathcal{T}, and the union of the members of each subfamily of \mathcal{T} is a member of \mathcal{T}.  The set X = \bigcup \{ U : U \in \mathcal{T} \} is necessarily a member of \mathcal{T} because \mathcal{T} is a subfamily of itself, and every member of \mathcal{T} is a subset of {X}.  The set {X} is called the space of the topology \mathcal{T} and \mathcal{T} is a topology for {X}.  The pair (X, \mathcal{T}) is a topological space.  When no confusion seems possible we may forget to mention the topology and write “{X} is a topological space.”  We shall be explicit in cases where precision is necessary (for example if we are considering two different topologies for the same set {X}).

The members of the topology \mathcal{T} are called open relative to \mathcal{T}, or \mathcal{T}-open, or if only one topology is under consideration, simply open sets.  The space {X} of the topology is always open, and the void set is always open because it is the union of the members of the void family.  These may be the only open sets, for the family whose only members are {X} and the void set is a topology for {X}.  This is not a very interesting topology, but it occurs frequently enough to deserve a name;  it is called the indiscrete (or trivial) topology for {X}, and (X, \mathcal{T}) is then an indiscrete topological space.  At the other extreme is the family of all subsets of {X}, which is the discrete topology for {X} (then (X, \mathcal{T}) is a discrete topological space).  If \mathcal{T} is the discrete topology, then every subset of the space is open.  (Kelley, p. 37).

References

  • Kelley, J.L. (1955), General Topology, Van Nostrand Reinhold, New York, NY.
  • Plutarch, “Romulus”, in Plutarch’s Lives : Volume 1, Bernadotte Perrin (trans.), Loeb Classical Library, William Heinemann, London, UK, 1914.

Resources

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Time, Topology, Differential Logic • 4

Posted on November 15, 2016 by Jon Awbrey

Re: Peirce List Discussion • JFS • JA

JA:
Trying to understand inquiry and semiosis in general as temporal processes is one of the things that forced me to develop differential logic as an extension of propositional logic, for which I naturally turned to Peirce’s logical graphs as a starting point.
JFS:
Yes, that’s another path to explore.  For any version of logic, it’s important to determine what kinds of problems it can express and what solutions it can facilitate.  What useful stories or Gedanken experiments can you explain in terms of it?

A first try at answering this question might well begin by reflecting on the analogous question in the quantitative realm:

  • What kinds of situations does the differential and integral calculus serve to describe and what kinds of solutions does it help to facilitate?

Differential logic is simply the qualitative analogue of the differential and integral calculus.  Both are called upon as we pass from the description of static situations to dealing with changes, differences, and transformations among multiple situations, those that occur in different modes of being or through different points in time.

References

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Time, Topology, Differential Logic • 3

Posted on November 14, 2016 by Jon Awbrey

Re: Peirce List Discussion • JA

Worldly events are interfering with my concentration quite a bit this week, perhaps for days to come, but it does help to immerse myself in work.  I am starting a blog series to follow out the present train of thought and trace whatever dialogue, internal or external, may ensue.  Plus, the blog medium will give me better formatting if we get any further into the math.

By the way, there are a number of Facebook pages I devoted to these subjects, for anyone who makes use of that environment:

References

Posted in C.S. Peirce, Change, Differential Logic, Dynamical Systems, Inquiry, Inquiry Driven Systems, Logic, Logical Graphs, Mathematics, Peirce, Propositional Calculus, Semiotics, Systems Theory, Time, Topology | Tagged , , , , , , , , , , , , , , | 9 Comments

Time, Topology, Differential Logic • 2

Posted on November 13, 2016 by Jon Awbrey

Re: Peirce List Discussion • JBD • JA • JA

Topology is the most general study of geometric space.  It is critical here to get beyond the “popular” accounts and learn the basics from a real math book.  A classic introduction is General Topology by J.L. Kelley but there are lots of equally good choices out there.

One of the reasons I like Kelley is that it has an appendix on set theory where I got my first real taste of axiomatic set theory.  I posted excerpts from the appendix and the main text to several discussion groups early in the present millennium and I archived copies at the following locations:

These are raw text copies right now but I’m in the process of segmenting them for ease of study and retrieving Internet Archive links for the discussion pages no longer live on the web.

Another good text I recall on topology is Munkres.  I imagine there are newer editions still in print.

References

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Time, Topology, Differential Logic • 1

Posted on November 11, 2016 by Jon Awbrey

The clock indicates the moment . . . . but what does
eternity indicate?

Walt Whitman • Leaves of Grass

Re: Peirce List • ET • JFS • JA

Trying to understand inquiry in particular and semiosis in general as temporal processes is one of the forces that drove me to develop differential logic as an extension of propositional logic, for which I naturally turned to Peirce’s logical graphs as a starting point.

It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time.  A time variable is a reference to a clock — a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process.  This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others.  (Reference 3)

References

  1. Differential Logic
  2. Differential Propositional Calculus
  3. Differential Logic and Dynamic Systems
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Pragmatic Maxims • 4

Posted on November 8, 2016 by Jon Awbrey

Re: Peirce List • (1) • (2)

I haven’t been able to do more than randomly sample the doings on the Peirce List for the last half year, being deeply immersed in other Peirce work that I hope to report on one of these days, but Jon Alan Schmidt’s remarks on the Pragmatic Maxim drew me back in for a bit.  He recited one of the places where Peirce declares the role of the Pragmatic Maxim in giving a rule to abduction, a point often missed by many of the most careful commentators.  There are aspects of the Pragmatic Maxim that come more naturally to engineers, workers in the applied sciences, therapeutic professions, and other practical categories than they do to specialists in spectator philosophies.

But I have lost track of that direction for the moment.  No doubt the occasion will arise again …

Reference

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Pragmatic Maxims • 3

Posted on November 5, 2016 by Jon Awbrey

Re: Peirce List • Jerry Rhee

Inquiry begins in doubt and aims for belief but the rush to get from doubt to belief and achieve mental peace can cause us to short the integrated circuits of inquiry that we need to compute better answers.

For one thing, we sometimes operate under the influence of fixed ideas and hidden assumptions that keep us from seeing the sense of fairly plain advice, and here I would simply recommend reading those versions of the Pragmatic Maxim again and again and trying to triangulate the points to which they point.

For another thing, not everything in logic is an argument.  A well-developed formal system will have:  (1) Primitives, the undefined terms that acquire meaning from their place in the whole system rather than from explicit definitions, (2) Definitions, that connect derived terms to primitives, (3) Axioms, propositions taken to be true for the sake of the theorems that can be derived from them by means of certain (4) Inference Rules.

But that’s just the formal underpinnings — there’s all sorts of informal heuristics, regulative principles, rules of thumb that go toward sustaining any system of significant practical use, and that’s where bits of practical advice like the Maxim in question come into play.

Reference

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Pragmatic Maxims • 2

Posted on November 4, 2016 by Jon Awbrey

Re: Peirce List • Jerry Rhee

I tend to think more in relative terms than absolute terms, so I would not expect to find an absolute best formulation of any core principle in philosophy, science, or even math.  But taken relative to specific interpreters and objectives we frequently find that symbolic expressions of meaningful principles can be improved almost indefinitely.

Reference

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Pragmatic Maxims • 1

Posted on November 4, 2016 by Jon Awbrey

Re: Peirce List

Here is a set of variations on the Pragmatic Maxim that I collected a number of years ago, along with some commentary of my own as I last left it.  As I understand them, they all say essentially the same thing, merely differing in emphasis, point of view, or rhetorical style as befit the moment’s audience or occasion.

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{ Information = Comprehension × Extension } • Comment 6

Posted on October 28, 2016 by Jon Awbrey

Note.  This is a placeholder, to be developed later.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

Figure 2. Disjunctive Term u, Taken as Subject

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

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