Dear Lubomir, Birger, Ken and all,

My apologies for the delay in getting back to you in response to questions
about my assertion there are significant benefits for design research in
translating  design theories into mathematical language.

I would add to my previous claim that such translation offers insights,
clarification and identification of faulty theory that are almost impossible
to identify using the approaches common in much of the literature of design.

Lubomir, what I wrote is not positivist. In philosophy of science terms it
is simple, straightforward and uncontroversial. In my posts to Ken and all.
I've been very clear about the sort of mathematical representation I have
proposed.  It is different from what Ken  and you seem to have inferred. I
pointed out there were two possible positions on creating mathematical
representations for  design theories.

The first position is a language translation. It is the exact equivalent of
translating  a design theory written in English into Finnish, Arabic,
German, Yiddish,  Yupic, or Archi. What  I proposed IS exactly that - to
convert  a design theory written in English into the language used in
mathematics (in exactly the same manner that one might convert it into the
language structures  of  Finnish, Greek  or Yupic ). I maintain there are
benefits in doing this for analysing  and testing the language-based aspects
of design theories and their implications as well as identifying
implications. This is neither positivist nor non-positivist.  It simply
offers an alternative basis for linguisitic, conceptual and reasoning
analyses that offers a better basis to identify  some of the linguistic
oversights in the English version . 

The second position would be to create a mathematical dynamic model that
represents the physical behaviour of the phenomena outlined by the word
version of a design theory. This is NOT what I proposed. It appears,
however, to be what some think  I proposed. 

One of the reasons for doing the language translation described in 1.  is
because we can't easily perceive errors in theories described in our own
language. An essential part of research is to continuously try to find
errors and gaps in theories and to find out the theoretical and real regions
they don't apply and the reasons they do not apply.  Finding these errors in
theory  is not easy nor straightforward due to human habits. We tend to
bridge the gaps mentally to make theories appear to work.  This is in the
same manner that underpins the problems of witness reliability. We can see
it practically in discussions about the current airline disaster. We ignore
uncritically under-justified generalities when  they seem like they should
apply.   There are of course methods of critical conceptual analysis for
making the familiar strange in conceptual terms. I used straight critical
conceptual  analysis in a previous post to phd-design analysing  and
identifying the weaknesses in Rittel and Weber's 10 points about what they
had called 'wicked problems'.

An alternative way to make the familiar strange in order to see the errors
in theories is to put them in a language that has different structures. In
doing that,  mathematics is particularly useful because its structures are
unusually well defined and hence there is less opportunity to blur over
unjustified aspects of a theory.

Let me give an example of the use of  the kind of language translation into
mathematics of the above position 1 to analyse a design theory by
translating it from English  into mathematical language.  Note in doing this
I've assumed a Popperian viewpoint that is  the opposite of  positivism.

Take the design theory claim:     'Wicked problems are not solveable' (part
of what Ken proposed earlier that I mathematically represent)

It is possible to translate this simply into  mathematical language along
the lines of the following. What follows translates the  above design theory
claim into the language of set theory. 

"There exists a set W that contains, and only contains, theoretical
entities, w, in which the boundaries of the abstract space containing, and
only containing, set W, and the members, w of set W, are defined by criteria
Cw (1… n), listed by Rittel and Weber as the characteristics of what they
call ‘wicked problems’. 
Each entity, w, in W, is assumed to be a functionally complete abstract
representation of a real situation z, from the universal set of real
situations, Z. The set Z contains and only contains all real situations
comprising real and virtual entities u, from the universal set of phenomena,
U.
Each real situation zi, defined as a ‘wicked problem’ situation by the
theoretical information in its corresponding theoretical representation wi,
also can be described by a different theoretical representation, dp, as a
design problem, that is, a member of a set DPr, which in turn is a member of
the super set DP. The super set DP contains and only contains, entities
identified as design problems appropriate to being addressed by design
professionals. The subset DPr of DP contains, and only contains, those dp
entities that describe real world situations for which a theoretical
solution entity s can be identified.
Each solution entity, s, is an abstract representation of a real
intervention, i, in which s is a member of the universal set S, of
solutions, and i is the corresponding member of the set I of real world
interventions. The boundaries of set I are defined by all real world
physical constraints Cphys (1…m); by theoretical constraints Cth (1…p), e.g.
laws, customs, habits, bounds of knowledge and theory; and by constraints of
human limitations Chuman (1…q), e.g. as a human being unable to identify or
understand or predict the behaviours of particular classes of solutions or
sub-sets of interventions. The boundaries of the abstract set S that
contains all possible theoretical solutions are also defined by the abstract
and real constraints and also by the nature of the entities it encloses
being solutions. 
The above theory claims  that for those members of  w in W, that describe
corresponding members z, from Z, consisting of combinations of entities u
from U, of which the members z are restricted to those in which each
theoretical entity  w can be also described by a corresponding theory entity
dp, which is a member of subset DPr and superset DP, THEN each of those
entities z do not have a corresponding entity, s, of the set S, on the basis
that it is implicitly claimed there does not exist a corresponding entity,
i, of the set I."

The above representation in mathematical language is useful in design
research terms in several ways:

1. It Identifies the multiple different theoretical entity types that are
intrinsic to  the above design theory claim but are hidden and overlooked by
its presentation in English language

2. It identifies the bounds of the variety of entities within each of the
above intrinsic entity types.

3. It identifies the criteria that create those bounds 

4. It identifies the specifics of the relationships between entities, sets
of entities, bounds and criteria

5. It identifies and provides the basis for testing the above theory claim
in terms of the epistemological integrity of the entities, entity types,
sets,  relationships, criteria - as well as by exclusion (i.e does the
theory set representing the claim exist, and in practical terms (by
reversing and using null hypothesis)

6. It identifies and provides a basis for testing the criteria (including
Rittel & Weber's characteristics of 'wicked problems') in terms of their
internal and external epistemological validity and completeness. I.e. do
they match with the epistemological characteristics of the entities to which
they refer, are they internally consistent, do they provide a set of
complete bounds for set W (and by implication appropriate bounds for the
other sets, is there redundancy etc?

7. It enables easier and more explicit  careful searching for implicit
tautologies, unjustified assumptions, internal contradictions etc

Incidentally, the above translation reveals at least one internal
contradiction in Rittel and Weber's  theory claim.

To maintain all aspects of this analysis in a single place, I have
deliberately refrained from removing the prior posts. I have however trimmed
them of address details.

Best wishes,
Terry

---
Dr Terence Love
PhD(UWA), BA(Hons) Engin. PGCEd, FDRS, AMIMechE, MISI
Director,
Love Services Pty Ltd
PO Box 226, Quinns Rocks
Western Australia 6030
Tel: +61 (0)4 3497 5848
Fax:+61 (0)8 9305 7629
[log in to unmask] 
--

==
Dear Terry,

I was going to ask you a similar question, but Birger got ahead. Show me one
design theory that can be mathematized. But before that, we have to settle
on what is design, what is design theory, how a theory can be expressed in
mathematical terms, and so forth. 

It seems to me that you enter very boldly into a philosophy of science
minefield and on top of that, you make very extreme statements. If you say
that some design theories can be mathematically expressed, it might be
easier to defend such a position. 

I am a bit astonished that you take such an extreme positivist view. What
you say is not new and not unheard of, but people try to stay away of it for
several decades. Is there are resurgence of Positivism? My observations are
that the Positivism has "softened" a lot under the pressure of the
renaissance of the humanistic paradigms. What is going on here? 

Best,

Lubomir

==
Dear Terry
Please show us one mathematically expressed design theory.
 
Best regards

Birger Sevaldson 

==
Hi, Ken,
Again,  you are conflating the two different ideas. you can't make a silk
purse out of a sows ear. . .  If the original word version of a theory
doesn't provide a complete functional  model of all aspects of design, then
how would representing that same theory in mathematical terms do so?

E.g. '"Wicked problems" have a large number of solutions' can be transformed
into mathematical notation into ' The number of entities in the solution set
of the set of problems bounded by the following criteria tends towards a
large number'.
Why would I expect the mathematical representation of this  to be a working
model of design activity? I don't get your logic. Goedel's is a different
argument.

You could think  of it slightly differently. If you  gave me a word-based
description of a motorcycle you had seen; the word-based descriptions could
be equivalently be created as a visual-representation, as a sketch of the
motorcycle;  or the words  or sketch could be represented  (as above) in
mathematical propositions. I'm at a loss as to why you think the words,
sketch or mathematical representation  would  be a working motorbike.
descriptions result in

Rittel & Weber's  wicked problem 'theory' is a their proposal for a
collection of boundaries on the characteristics of the concept they have
called 'wicked problem'. In that sense it has the characteristics of a
definition not a theory.  As you say, it's a set of theory-related
(theoretical) propositions. Propositions, though. In spite of their nature
as parts of a definition, most of them don't define anything. The most that
can be said is they are partial pointers. In some cases they are unrelated.
I'm happy to critique  Rittel and Weber's propositions, andnd they are easy
enough to represent in set terminology. It's value, however, will probably
more that it will demonstrate they ain't a theory, nor do they make sense
except perhaps as a political statement, or a proposal  for avoiding
responsibility.

But, lets take it from the top on  Rittel and Weber's 10 comments on what
they called  'wicked problems'
1. Means  there's no definition here so what follows in the remaining 9
items is irrelevant. Non-sequitor.
2.  Is simply confused language.  *Activities* have stopping rules.
Creating solutions to problems have rules and boundaries. *Problems*,
however, are simply problems. They don't go about stopping or starting.
Non-sequitor.
3. Means there is an undefined (good-bad) spectrum of optimisation criteria.
Weird -  of course solutions are not 'true or false. Do you know ANY
*solutions* that are true or false, except in the realm of binary logic?.
4. Contradicts 3.  In 3, Rittel and Weber have already said they are
defining the success of a solution to a 'wicked problem'  on the criteria of
an undefined spectrum of good-bad. Regardless of it being undefined, they
have indicated they are testing solutions against a criteria  - which
contradicts 4.
5. Apart from the faulty logic (confusion between creating the solution and
the solution itself) , its true, less goes at getting things right means
each attempt is more significant. This isn't really a major theory item.
6. Means there are lots of possible solutions (i.e. the set of possible
solutions tends towards a large number) - the second part of 6 doesn't seem
to make sense.
7. Each problem is different (is that unusual?) 8. All problems result from
the outcomes of other problems. Is this unusual?
all events result from other events.
9. Its confused language with faulty logic but it seems to mean ' The
outcomes resulting from prior events (that if R&W had been trying to  create
a solution to them, they would have called a wicked problem) can be
explained in lots of ways.  Is this not obvious?
10. The professional identifying the solutions  is responsible. This is
clearly untrue for most planners and designers. I've yet to see the costs of
social problems from poor designs being charged against planners personal
bank accounts.

So? Rittel and Webers collection  of sentences and comments about wicked
problems is  a theory? It doesn't look like it to me. It doesn't have
predictive power, it doesn't define, and it isn't coherent. It had value at
the time to draw attention to some characteristics  of planning decisions,
but a theory  - no.

Warm regards,
Terry

==

Dear Terry,

Thanks for your reply.

Before offering the example you request, I'll state that there is no
straw-man argument in my notes. If a theory meets the criteria you propose,
it will permit a complete mathematically-based functional model of all
aspects of design.

You make this statement about ALL theories in any discipline. Since the set
of ALL design theories must incorporate the sub-set of those theories that
model ANY design activity, the sub-set of ALL design theories will afford us
a functional, mathematically rigorous model of all aspects of design. As I
wrote, I do not think this is possible.

You wrote,

-snip-

1. All theories (any discipline) can be wholly and exactly represented as
mathematical functions (this is the set of ALL theories regardless of valid,
useful or not). The relevant mathematical functions are typically found in
complex non-linear multivariable multidimensional spaces.

2. There are significant benefits in terms of validity and usefulness if
theories are in the set represented by continuous well-behaved mathematical
function.

3. If a theory cannot be represented by a well-behaved continuous
mathematical function, this is an indication that the phenomena being
theorised about needs representing in a different theoretical manner
typically by more than one theory.

4. A simple test for whether a theory is represented by a well-behaved
mathematical function (and hence is a 'good' theory) is whether the
phenomena and the mathematical theories are free from discontinuities or
singularities.

5. The existence of discontinuities and singularities in the mathematical
space field  that represents a theory indicates the need to represent
phenomena on each side of the discontinuity or at the point of the
singularity differently. That is, it indicates that what was previously one
theory actually requires several theories of which the minimum number is
(n+m+1) where n is the number of discontinuities and m is the number of
singularities.

6. The above applies to a single characteristic of a phenomenon. Where
discontinuities and singularities occur at the same point in respect to
multiple characteristics of the same phenomenon this indicates the presence
of (n+m+1) different phenomena and the need for (n+m+1) different vector
space fields (bodies of theory) to describe them).

Conceptual analysis, the work of Foucault and other theorists in realms of
sociology, communications and design theory can each be seen as a sub-sets
of the above.

-snip-

Now, as requested, I will give an example of a design theory that you feel
cannot be represented mathematically, at least not in full - that is, I do
not believe that it can be wholly and exactly represented as mathematical
functions. Despite this, the theory I am about to put forward is one of the
most important sets of theoretical propositions in design and design
thinking. You have used it yourself, so I'd argue that it has to count as a
theory of some kind.

These propositions meet my criteria for a theory: a model that illustrates
or describes how something works by showing its elements in their dynamic
relationship to one another. This is the dynamic demonstration of working
elements in action as part of a structure.

The theory I put forward with the challenge that you wholly and exactly
represent these statements as mathematical functions is Rittel and Weber's
(1973: -166) theory of the wicked problem:

"1. There is no definitive formulation of a wicked problem.

2. Wicked problems have no stopping rule.

3. Solutions to wicked problems are not true-or-false, but good-or-bad.

4. There is no immediate and no ultimate test of a solution to a wicked
problem.

5. Every solution to a wicked problem is a 'one-shot operation'; because
there is no opportunity to learn by trial-and-error, every attempt counts
significantly.

6. Wicked problems do not have an enumerable (or an exhaustively
describable) set of potential solutions, nor is there a well-described set
of permissible operations that may be incorporated into the plan.

7. Every wicked problem is essentially unique.

8. Every wicked problem can be considered to be a symptom of another
problem.

9. The existence of a discrepancy representing a wicked problem can be
explained in numerous ways. The choice of explanation determines the nature
of the problem's resolution.

10. The planner has no right to be wrong."

Later, I hope to add a few comments on your note. In several debates, you
have used a tactic in which you decline to answer a challenge directly.
Rather, you answer a question with a question, or you propose trading
questions and answers.

Since you have asked for a design theory that I feel cannot be represented
mathematically, I am starting with this.

There are other problems, however, with the notion that "all theories (any
discipline) can be wholly and exactly represented as mathematical
functions." This is the shadow of the logical positivist program. Kurt
Goedel (1931, 1962 [1931]) put an end to that program, and even to the idea
that one can build all mathematical propositions that way.

For now, I'll be curious to see Rittel and Webber's theory of wicked
problems - a model, as I've stated - "wholly and exactly represented as
mathematical functions."

Best regards,

Ken

Reference

Goedel, K., 1931, "Ueber formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme I." Monatshefte fuer Mathematik Physik.
Vol. 38: 173-198.

Goedel, Kurt. 1962 (1931). On Formally Undecidable Propositions of Principia
Mathematica and Related Systems. Translated bv B. Meltzer. New York: Basic
Books.

Rittel, Horst W J, and Melvin M. Webber. 1973. Policy Sciences, Vol. 4,
(1973), 155- 169


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