## Time, Topology, Differential Logic • 6

Re: Peirce List Discussion • ETJFSJAJBD

Let me see if I can get back in the saddle on this topic, the dormitive virtues of tryptophan and a few pounds added notwithstanding.

I was addressing the following question from Jeffrey Brian Downard:

I wanted to see if anyone have might suggestions for thinking about the analogy between:

1. mathematical models of the differentiation of spaces starting with a vague continuum of undifferentiated dimensions and trending towards spaces having determinate dimensions
2. models for logic involving similar sorts of dimensions?

How might we understand processes of differentiation of dimensions in the case of logic?

By way of review, here are my blog posts on the discussion so far:

• Time, Topology, Differential Logic • (1)(2)(3)(4)(5)

We can now get back to preparing the ground required to tackle Jeff’s question.

## Peirce and Democracy • 1

Re: Peirce List Discussion • GRJA

In my mind the connection between Peirce and Democracy has long revolved around the concept of representation.

Representation in its semiotic sense has to do with signs that represent pragmatic objects to agents and communities of interpretation.

Representation in its political sense has to do with forms of government that address the res publica, the public concern, through elected representatives who represent, hopefully, the good will and the best information of the public at large in their stations at the rudders of the ship of state.  Here the twin senses of representation converge on the common root meaning of the words cybernetics and government.

I have written a lot about this twofold sense of representation over the years but weeks of watching “The Death of a Nation” on TV have left me too dispirited to say any more on the subject.

I did happen on a recent blog post that seems to fit here:

The question for our day remains —

• What are the forces that distort our representations of what’s observed, what’s expected, and what’s intended?

## Time, Topology, Differential Logic • 5

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 • ETETJBDJAJA

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.

## Time, Topology, Differential Logic • 4

Re: Peirce List Discussion • JFSJA

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.

## Time, Topology, Differential Logic • 3

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:

## Time, Topology, Differential Logic • 2

Re: Peirce List Discussion • JBDJAJA

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.

## Time, Topology, Differential Logic • 1

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

Walt Whitman • Leaves of Grass

Re: Peirce List Discussion • ETJFSJA

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 2)