Just a few points to clarify my response to Tom - and to nit pick with
Alasdair! :-)

> It seems to me that space syntax has to throw off its structuralist
> pretentions before it can really be taken seriously.  This doesn't just
> go for space syntax, but also geographers using GIS.  Both are eminently
> structuralist tools, and both have to be careful to understand their
> limitations.
>
> For example, look at these two sentences from Alan:
>
> > These theories are testable - falsifiable in Popper's terms - and
> > are at the centre research debate.
>
> > I content myself in the belief that space syntax papers are a model
> > of clarity compared to anything written by Lacan or Derrida or any
> > of their followers.
>
> (Surely *anything* is a model of clarity compared to Derrida?! :-)
>
> Usually at this stage I say "everyone knows Popper was wrong", at which
> point somebody pipes up to say that it does seem to make sense.  Exactly
> the trap, and incidentally why Derrida is unreadable.  So, to
> arbitrarily repeat Kuhn: no theory is testable because the grounds on
> which you test it are rooted within the paradigm of the theory.  It's
> actually much, much deeper than that: Godel's theorem threw theorems out
> of the window.  Grasping this is essential, because it seems much of
> post-structuralism is really just a reactionary response to the sudden
> realisation that we can't prove (or say) anything.

Not quite so - Godel threw the enterprise of constructing a single
consistent formalism in which all mathematics could be written and proven
out of the window. Godel himself proved a couple of theorems: 1. if formal
set theory is consistent then there exist theorems that can neither be
proven nor disproven; 2. there is no procedure that will prove set theory to
be consistent. Pretty devastating theorems to prove! But theorems just the
same. Theorems and theories persist and thrive in mathematics as in all the
sciences. Thank goodness for Godel - he showed that it was still worth
theorising, and that everything was not just a single logical construct to
be devloped in a purely mechanical fashion.

Now, what this shows is that there was somewhat less distance between Poper
and Godel than you might hold. Popper may have been wrong about some things,
but one contribution he made was to point out the flaw in Hume's reasoning.
The ultimate process of science does not lie in 'proof' of the correctness
of theories, rather in falsifying hypotheses. The effect of Godel's findings
was to bring mathematics into a much closer scheme with the rest of science.
Kuhn of course points out some problems in the 'culture' and human nature of
science, just as a good historian should, but his point was not that
Popper's scheme was wrong or logically flawed, just that the process of
science is not quite that perfect. Scientists (being human) spend a lot of
time trying to back up their favourite theories, and demolish those of
'competitors', thus paradigms tend to get demolished from outside. However,
they get demolished by being falsified, not by their replacement being
'proven' correct.

If your post structuralists are who I think they are, I suspect they havent
a clue about Godel anyway :-)

>
> Anyway, I digress.  These statements demonstrate that syntax is still a
> structuralist enterprise.  Even though the text of Alan's message does
> imply that there is no 'one' axial map it is surrounded by assertions of
> the opposite:
>
> > The reduction to fewest line maps by computer has been
> > demonstrated

Well - let me be clear - axial maps and all syntax representations are tools
applied by humans in all their frailty - there is no 'one' always useful
method of represntation - there are many - but my statement is true. In 1986
or so Stefan Czapski developed software that constructed these maps very
effectively - The interesting part of the story is that it led to us
dropping the 'fewest' clause in the rule, when we realised that the 'depth
minimising' clause was actually the more fundamental, and that the 'fewest'
rule would 'provably'lead to different outcomes from the same input map in
very specific cases. One aspect of this is that the 'fewest' rule was
unproblematic for organic settlments of the 'French hill village' type. It
is only in more regular urban forms that problems emerged, mainly as a
result of ties between same length lines in regular geometric forms. I guess
the most confusing part is that in the jargon people still call maps of this
sort 'fewest line maps' to distinguish them from 'all line maps'....

>
> > Some aspects of these - specifically those about how space can
> > be constructed are mathematical and certain
>
> > complete and mathematical protocols for constructing spatial
> > representations and for measuring their properties
>
> > The same goes for using space, it is 'lawful', and we use that
> > lawfullness
>

Lousy written english I'm afraid - I intended the 'it' to refer to space,
not its use. Space is lawful in the sense (for example) that if one divides
an alignment in its centre one maximises depth gain, if one divides it close
to an end one minimises depth gain. These are what I mean about 'how space
can be constructed', an it is the self consistencies of these processes that
I mean are 'mathematical and certain' (in deference to Godel let me say
'relatively setlf consistent' and 'relatively certain' - who knows what a
mathematician could prove!). Perhaps there are laws relating human use to
space, but it was not these that I was refering to, since most of these are
theory or hypothesis like in my book. I suspect that Alasdair's
structuralist accusation lies mainly on this poor expression on my part?

> With the exception of the last statement, no, no and thrice no.  Even
> with the last statement: yes if we are structuralist, but we *cannot be*
> structuralist, we know too much.  Is using space lawful?  Okay, let me
> concede one point: I believe we should be entering a phase of
> "re-structuralism", i.e., a non-naive structuralism which understands
> the implications of post-structuralism.  Yes, I believe that there are
> probably "laws" governing how a space is used --- as Alan says, we are
> currently working on agent based models to investigate further.
>
> So, what does this have to do with what an axial line is?  Well, the
> point is, pragmatism and empiricism (those of you who know me will also
> know I'm a keen adherent to van Fraassen).  An axial line does not need
> a mathematical definition.

Well, I'm afraid I disagree here. The procedure is both pragmatic and
empirical, but axial maps are not arbitary. They reflect somthing about the
world, but their value lies in that somthing being relatively 'reproducible'
by other researchers faced with the same data. I am not quite sure what
Alasdair means by a 'mathematical definition', but it essential to note that
results are reproducible, and when people draw axial maps badly we find that
their explanatory power is reduced. This is an interesting empirical
finding. Why should a map drawn one way correlate better with observed
movement patterns for instance than one drawn another, and why should this
finding be repeated? Why is the axial map so powerful? Open questions thus
far - many proposed answers but still open questions....

>
> Firstly note: 1. drawing them inherently involves many decisions about
> what is and is not important to the social usage of a space (as Alan
> rightly says, 2m walls and so on). 2. it's mathematically impossible to
> generate a (single) fewest line axial map for an arbitrary map. (VGA
> does solve the second problem).

See my note above about 'fewest', for those outside the field the problem is
that jargon is also historical - the rules changed but the names used for
the maps did not. The original rule used to generate axial maps was somthing
like 'draw the fewest and longest lines that cross all convex spaces and
make all rings of circulation' - the process of automating this rule by
Stefan Czapski in the mid 80's led us to invent the all line map, the
overlapping convex space map and to alter the rule used by human researchers
to 'draw the set of longest lines that cross all convex spaces and make all
rings of circulation whilst minimising the depth between any pair of
lines' - Stefan's software automated the production of these maps, and
although there is no mathematical proof that the software works consistently
for any 'arbitary' input map, it was good at producing maps that a trained
human would agree with.

>
> Now, does this matter?  The answer is actually "no".  Precisely because
> it is impossible to create a consistent formalism *in any scientific
> enterprise*.  I suspect the trouble to Tom Dine's friends is that with
> most science they cannot see this impossibility because it is hidden
> from them in men in white coats who call themselves "experts" and hence
> they must be above suspicion.  But they can see the impossibility with
> syntax because (a) it so obviously involves a human step (b) it's done
> by architects not scientists or engineers.

Alasdair appears to be arguing that "all science is impossible (and syntax
is no different) so that's all right then." Science is littered with
consistent formalisms. The point is that there may be no *single* consistent
formalism that has all the answers. Godel proved that consistency for set
theory was undecidable. In doing this he just put maths in the same boat as
the rest of science, and helped destroy some myths about the aims of maths
at the same time. The aim of science for me is better explanations, not
necessarily a fully consistent 'grand unified theory of everything'.
Explanations can be useful even if isolated from other explanations.
>
> To Tom: ask your friends to consider a CFD (Computational Fluid
> Dynamics) analysis of a building.  Start with the small problems: CFD
> involves breaking a space into aggregation units to make it computable.
> So how do you decide how to break the space up into blocks?  Then move
> onto the large ones: in built environments CFD is used to model airflows
> (Yes, I know) Is air a fluid? (No) so can you apply C *fluid* D to air?
> (Usually I find: don't know).  The answer is: yes you can apply CFD to
> airflows in buildings, *so long as you know the limitations of the
> technique*.  If you want a direct comparison with "what is an axial
> line", call the aggregation cube by some technical sounding name, and
> then get them to define one.
>
> So what is an axial line?  I would suggest as difficult to define as any
> term you try to define... (I took an MSc in artificial intelligence, now
> what's that?!)
>
> Best wishes,
>
> Alasdair
>