img notes

Sha Xin Wei • 415-725-3152 • xinwei@otter.stanford.edu

Spring Session •  1995

Alice Rayner joined the discussion with Judy Dolan, Michelle Baldonado-Wang, Decker Walker and Bon Horn.   We summarized the themes the img has visited over the past quarter or so.

There's been a fruitful tension between the practical orientation and the theoretical.   The practical typically expresses itself using terms like "practical" and "design."    The theoretical doesn't position itself as antipodal to "practical" concerns so much as pay more attention to context, structure, interpretation and ideology.

The main themes of the img include:

Semantics -- Categories, context, speech acts
Structure -- topology
Modeling intent (theory of action) vs structure (architecture)

Case studies include:

Visual symbol systems
Hypermedia

The img has enjoyed several currents of discussion.    Starting with a general interest in media and how we ascribe meaning, we decided to examine some typical approaches to understanding the world via categories.   George Lakoff's Women, Fire and Dangerous Things provided a stimulating survey of how we categorize physical and symbolic worlds.   Even in the "vulgar" HCI -- such as graphical UI layout, icon systems, and dialog design,  Lakoff, Rosch et al. have much to teach us about how humans create and understand categories.   This typology includes attribute-based sets, Wittgensteinian family resemblances, radial categories, fuzzy sets, generative categories.   The physical basis for meaning is predicated on the claimed gestalt comprehension of basic-level categories.   The point that a scaled set of catgeories need not by reducible to primitives is a crucial one that we cannot overrestimate:   almost all first-order GUI design is focused on finding primitive elements (WIMP, direct manipulation, etc.).   If Lakoff et al. are right, such design studies or surveys may be quixotic.

 Somewhat unsatisfied by the relatively undeveloped founding of semantics upon basic body categories. we began our trek toward other studies of metaphor and discussions of meaning.  We started with a reading of Derrida's  Signature, Event Context.
 
 Michelle gave a precis of Austin's and Searle's contributions to the philosophy of language in the form of speech act theory, against which Derrida was writing his critical essay.   There were many generally useful points in that essay, including a consideration of what features characterized writing in most fundamentally -- iterability.
 
 Almost all second-order UI research is concerned with language-based notions, such as defining a protocol for inter-communication between software agents who represent human desires, or a formal description of document structures that will capture all machine-encodable knowledge, or formal languages to express user interaction.    If Derrida is at least partially correct, if for example, context can never be saturable and hence can never completely determine meaning of writing in the most general sense, we should approach even this more sophisticated HCI design research with some caution.
 
 Following this line of discussion, we delve deeper into the study of communication and theories of social interaction.   This complemented some readings in Winograd & Davis's Winter 1994 course on Phenomenology, Computers and Design, by Reddy, Heidegger, Saussure, and Iser which were more concerned with communication, reading, language, interpretation of text.   
 
 Here in the img, we decided to essay systems theory, as represented by N. Luhmann's Improbablity of Communication.    Luhmann's work, which reflected T. Parson's study of generalized theories of action, approaches human systems as inter-operating functions rather than merely a given set of identified categories (boxes and statistics on boxes).   This seems to offer a more nuanced way to interpret the world.    At this level, it seemed difficult for us to apply Luhmann's work to the issues of interactive media, though the program was, at this point, to understand human communication and human interaction.    Ben gave a good introduction which situated Luhmann relative to cybernetics, Parsons, and related work.
 
I suggested that we may as well look carefully at the underlying metaphors which are drawn from dynamical systems, which in turn draws from topology and (the qualitative theory of) differential equations.    Another source of  Luhmann's Improbablity of Communication is (information theory) statistical mechanics, which has deep links with dynamical systems.

We can venture in several directions: 

1.  Structures or Models of Interactive Media: Topology and Dynamical Systems

2.  Architecture and urban design:  C. Alexander, Mike Davis, Mike Sorkin et al.

3.  Visual Systems or  Languages

Summer Session • 6 July 1995

Question of identity in traversing the Oedipal cycle...

While discussing hypertext, we visited again the standard picture of a graph

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maybe in new media there is no circle possible? 

XW: in new media, closure becomes problematic.
 (as noted in \(Hyper/Text\)/Theory)

XW: 


    As architects move into cyberspace -- what are the politics?

Topology • 13 July 1995

Why should we be interested in topology?

There are many phenomena in the world which involve continuous deformations or transmutations.   Kneading clay.  The dissipation of smoke.   The parting of water around your hand as you dip it into the river.  Saussure's "infinite" paradigmatic absence and its implications for an open or unbounded semantics.  The play of interpretation or emotional response as a performance evolves.   General intelligence.   IQ  is a perfect example of the Measurement Problem: the tendency to identify some complex phenomenon to some (often a single) parameter just in order to be able to say something quantifiable about it.   This modeling problem pervades all social sciences, but is actually much broader.   In any case, it's often unclear even what are the dimensions of a problem.   Is general intelligence measured along a single real axis, along five, 100?   Are dimensioning and measurement the only ways to precisely reason about something?

 There are phenomena or substances or things or qualities that we try ot measure at our peril.   This is because of practical or interpretive and analytical problems.   Nonetheless, topology and allied branches of mathematics show that there's much that we can say, precisely, without resorting to a metrization or dimensionalization of a phenomenon.

Why should we be interested  in dynamical systems?

Larry asked if time and extension were somewhow intrinsic in my example of kneading clay.    No.   Just because we're creatures who perform actions in time, it was most natural for me to take a single piece of clay and deform it over a period of time.   The same deformation could have been parametrized over some arbitrary parameter, for example along a trail in space.   In fact, I could have presented a synchronic array of clay blobs, each presenting a different deformation of the first.   So topology can be discussed independent of measurement (metric), dimension, time.

But what about a "systems" of "functions" or "fields" that evolve in "time"?    Dynamical systems theory deals with abstractions of such phenomena.   But in order to develop a language, we'll first turn to point set topology.

In these notes we're going to assume that you know arithmetic and have some notion of sets.
Reference: Jnich, Topology, Springer-Verlag.

What is topology?

Topology may be described, loosely, as the study of what remains after we erase color, mass, and most other physical qualities, after we erase even such apparently fundamental qualities as dimension, extent and all other quantities that we can ordinarily measure.    Our atomic notions will be set (space), point, and subset.   Beyond these basic notions, one of the weakest features we can study is the notion of the neighborhood of a point.   What are some properties that we would like neighborhoods to have?

1.  If  a subset U contains a neighborhood of x, then U is a neighborhood of x.
2.  A neighborhood of x always contains x.     The full space X is a neighborhood of all its points.
3.  U and V are neighborhoods of x, then U ⋂ V is a neighborhood of x.
4. each neighborhood of x contains a neighborhood of x which is a neighborhood of all its points.

We start with a set X and define what we call a topological space, < X, Ο >.

To define a topological space based on a set X, we just have to define which of all the subsets of X we will call open.    There is a catch however: the family of open sets, Ο, must satisfy some minimum standards of behavior.

1.  Any union of open sets is open.
2.  The intersection of two open sets is open.
3.  X and  ϕ  are open.

Why "open"?   This comes from looking at an interval.   Every point in an open interval Ω  has a small open set around it entirely contained within Ω.

What's not an open set?   Well, we say that a set is "closed" if its complement is open.  A set plus its boundary is closed.  If you remove its boundary, then you can never reach the "edge."

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Sets can be far wilder than these simple intervals -- connected subsets of the real line.    For example, let X = (R^2)  and take as a subset the rational points inside a unit square.

Clear[UnitQ,UnitRationals];\[RawReturn]UnitQ[{a_,b_}]:=a<=b;\[RawReturn]UnitRationals[n_] := Module[{pp,qp},\[RawReturn]\[RawTab]\[RawTab]pp=Flatten[Outer[List,Range[n],Range[n]],1];qp=Union[\[RawReturn]\[RawTab]\[RawTab]\[RawTab]Apply[Divide,Select[pp,UnitQ],1]\[RawReturn]\[RawTab]\[RawTab]\[RawTab]];\[RawReturn]\[RawTab]\[RawTab]pp = Outer[List,qp, qp];\[RawReturn]\[RawTab]\[RawTab]pp =  Point/@ N[pp];\[RawReturn]\[RawTab]\[RawTab]Show [Graphics[pp]];\[RawReturn]pp\[RawReturn]]\[RawReturn]

Examples of topological spaces

1  The set of all people whose ages are between  α   and  β.
Ie.         α  <  age(p)   < β

2  The set of all paths taken by smoke particles from a candle flame.

3  A lump of particolored clay.

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`(,`_P

Metric spaces

Maybe it's not surprising that mathematicians arrived at point set topology by abstracting from more familiar structures such as metric spaces, spaces for which we can define a distance and thus quantify nearness or neighborhood.    (R^n) and R   are particular examples of metric spaces.

A metric is a function d(x,y) that measures two points, x and y,   such that

1. positive definite: d(x,y) ≥ 0, d(x,y) = 0 iff x = y
2. symmetric:  d(x,y)  = d(y,x)
3. triangle inequality:  d(a,c) ≤  d(a,b) + d(b,c)

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What's a metric?   Ordinary distance on a plane or on a globe are metrics.   The international telephone fee system is not symmetric between cities -- it costs more to call from Venice to San Francisco than from San Francisco to Venice -- so it is not a metric.

A metric (distance function) on a
 space Y defines a topology on Y.    Just take as basic open sets the open balls defined as
\[RawReturn](Subscript[B, r])(x) = {y  ϵ Y | d(x,y) < r}\[RawReturn]
and take all possible unions, and all countable intersections of open balls.

Though metric is inspired by the ordinary notion of distance, a metric can be pretty unusual..
 It may be bounded.   In fact, given any metric d(), you can define a new metric by
 d'(x,y) = (( d(x,y))/(1+  d(x,y))).   Note that d'  is also a metric, but it's always  ≤ 1.  
 
Moreover a legal but weird metric is one given by:

d(x,y) =  (Subscript[{, 1  x  ≠ y]^(0  x = y))

Notice that we haven't talked about dimensions at all!
We haven't even got enough structure to define vector spaces and stuff of  linear algebra.
Given a topological space, how do we tell that it may be a vector space?

Topological properties

What's a topological property?   Something that can be defined solely in terms of open sets.

connectedness

A set  Ω   is connected  if it cannot be written as the union of two disjoint open sets:  Ω = A

Path-connected => Connected  (prove by contrapositive)

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D`&R``"@`(,`_P

But Connected does not imply Path-connected.   Here's an example.   Take the union of the  segment between -1 and 1 on the vertical axis with the   sin[(1/x)]  curve

Plot[Sin[1/x], {x,0,1}]

Look at this curve carefully.  You see that it wiggles infinitely and arbitrarily more densely as we approach the vertical axis.

Plot[Sin[1/x], {x,0.1,1}]

Plot[Sin[1/x], {x,0.01,0.1}]

separability

A topological space is Hausdorff  if any two points a,b  can be separated by two disjoint open sets.

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`(,`_P

Most of the common topological spaces are Hausdorff .  But consider two different spaces of foliations (flow lines):

Why should we care?    Hausdorff separation helps us avoid some unpleasant pathologies.   For example,  consider a limit.    Every convergent sequence in a Hausdorff space has a unique limit point.

limit

A sequence of points (Subscript[p, 1])   (Subscript[p, 2])  (Subscript[p, 3]) ... converges to a point  p, if every open set around p contains points in the sequence { (Subscript[p, i ])}.

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compactness

Arguably the most useful elementary topological property is compactness:   A subset Ω  of X is compact if every  covering by open sets   ⋃ (Subscript[V, α])   ⊃  Ω  has a finite subcover  
(Subscript[V, 1]) ⋃ (Subscript[V, 2]) ⋃ ... (Subscript[V, n]) ⊃  Ω.

One of the most important theorems in topology is the fact that a real-valued continuous function on a compact set always achieves its maximum and minimum.   Maybe you can try to imagine why this might be true.

Continuous functions

Now that we have defined topological spaces, we can define functions between them. We usually denote this by

	f :  X  -> Y

where f is the function, X is the domain space and Y is the destination space.

 Kneading clay is a transformation -- a function which takes each particle of clay and sends it to another position in the clay body.  In this case, X = Y = the clay lump.

The age() function which assigns a number to each human is a function from X = the set of people to Y = the set of positive numbers.   With respect to some appropriately defined topology on the set of people, the  age() function is continuous.

What is an appropriate topology for the space of people?

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What we're trying to capture is the notion that closeness should be preserved by a continuous function.    Think about the fact that  toddlers have certain features in common, approximately (height, weight, verbal dexterity).   How do we define "toddlers"?   By saying, for example, all kids between 2 and 5 years of age.   But this is  (age^(-1))((2,5)) -- the inverse image under the age() function of (2,5).

Here's a definition that works: A  function f:X->Y  is continuous iff (f^(-1 ))(open set in X)  is an open set in Y.

Why inverse map?   Why not use the criterion "f(open) is open"?

Consider the function f: R -> R that looks like:

Clear[f];\[RawReturn]f[x_] := x+1  /; x <= -1;\[RawReturn]f[x_] := 0  /; ((x > -1) && (x < 1));\[RawReturn]f[x_] := x - 1  /; x >= 1;\[RawReturn]Plot[ f[x], {x, -2,2}, PlotRange -> All]

Consider the image of f[(1/2, 3/2)].   Is it open?

This is exactly the class of functions that preserves topological properties.   For example, a continuous map of a connected set Ω is connected.   (If not, the consider  (f^(-1))(A)  ⋃  (f^(-1))(B) where  f(Ω)  = A ⋃ B   is a disjoint union ).

Other examples of continuous functions

1  Homotopy (aka morphing)