http://www.carleton.ca/~claughli/fuzzy.htm

                         THE FUZZY BRAIN
                                by
                         Charles Laughlin
                       Carleton University
 
In an article published in the Journal of Anthropological Research (Laughlin 1993), I made the suggestion that fuzzy set theory (FST)
offers many useful insights into the nature of natural categories of human thought, culture and communication, especially those
related to transpersonal experiences.  I noted that ethnologists have been slow to avail themselves of these insights, and that they
have upon occasion committed the fallacy of over-crispifying native categories in describing cosmologies and religious systems.
 
I would like to briefly share some thoughts on the possible neurophysiological substrate for fuzziness in natural categories
and reason.  But first, for those of you not familiar with FST, I will briefly summarize the central insights at the core of the
theory.
 
Fuzzy Set Theory
 
FST posits and models the essential fuzziness of natural categories.  The central concept in FST is, of course, fuzziness. 
Fuzziness refers to an easing of the restrictions upon membership in a category -- categories being cognitive classes of objects
"which are considered equivalent" (Rosch et al. 1976:383).  Fuzzy sets are categories with graded membership, and constraints upon
membership in a category are elastic.  Instead of an object either belonging to, or not belonging to a category, as is the case in
classical set theory, the object may be more or less a member of a category.  Examples of fuzzy categories often cited in the
literature include "young child," "good person," "tall man" and "beautiful woman."  Fuzzy numbers include notions like "lots,"
"several," "occasional," "a bunch," "a few," etc.  
By the same token, fuzzy reasoning or logic (Giles 1979, Lakoff 1973) "relates to what might be called approximate reasoning, that
is, a type of reasoning which is neither very exact nor very inexact" (Zadeh 1975:1).  While propositions in classical logic are
considered to be either true or false, in fuzzy logic they may be more or less ("sort of," "maybe," "fairly," "pretty much") true, or
more or less false.  For example, a witness to a crime may make a formal statement of what they saw, and this statement may be
considered more or less accurate, more or less true, relative to the statements of other witnesses.
 
Fuzziness is often signalled in natural language by the use of hedges (Lakoff 1973; see also Sapir 1944).  Such terms as "sort
of," "very," "unlikely," "kind of like," "more or less," "almost," "practically," etc., are commonly used in English to signal fuzzy
intent and to modify the meaning of the focal term (see Lakoff 1973:472 for a lengthy list of English hedges).
 
Fuzziness is contrasted with crispness.  Crispness refers to precise membership within a category.  The crispness of categories
we require in scientific theory, description, measurement and discourse is actually abnormal when considered cross-culturally,
especially as regards religious knowledge.  Indeed, that kind of crispness is rare in each of our own everyday lifeworlds.  FST is,
by the way, a theoretical approach to crispifying fuzziness. 
 
Fuzziness and the Brain
 A real problem we face is that most theories of human categorization used in ethnology are grounded in cognitive
psychology and few give other than passing reference to FST (see Laughlin 1993 for some interesting exceptions).  This is
unfortunate, because ethnologists utilizing cognitive psychological theories are often uncritical of inherent weaknesses in the
cognitive psychological approach.  One severe weaknesses is that few cognitive psychology models are grounded in the neurosciences. 
That means that it is often hard or impossible to map the inferred structures of perception and cognition defined in these theories
onto what we have come to know about how the human brain works. 
This is pretty much the same problem many of us found with Levi-Straus' semiotic structuralism in the 1970s.
 
Moreover, there is considerable controversy in psychology over how categories are learned and whether categorical recognition involves
holistic or analytical strategies (see Medin 1984 for a review). Despite these considerations, there does exist a relevant
literature on the psychophysiology of natural category construction in the nervous system; see Herrnstein (1982), Kay and McDaniel
(1978:617-621), Bieterman (1987).  The work of Irving Bieterman (1987) at SUNY-Buffalo is particularly promising.  Recognizing that
the nervous system is working on abstractions all the way from receptor sites into the central cortical processing structures, he
has hypothesized that there exists a finite set of generalized component detectors in the retina -- structures he calls "geons" --
that produce information about the properties of objects.  Recognition occurs because of a free combination of geons -- the
principle he calls "componential recovery."  He has shown in experimental research that if two or three geons can be recovered
by a subject, they are sufficient for recognition.  Additional research confirming the importance of geons in both recognition and
access to memory may be found in Biederman and Ju (1988).  
 
It is easy to see from this type of psychophysiological research that "fuzziness" of natural categories in both perception and
cognition may be due in part from the additive effect of components (geons).  A robin will trigger more geons characteristic of
"birdliness" than will a bat.  I suspect, however, that we will find the picture much more complex than can be modelled by geon
theory.  I have always argued that adult perceptual and cognitive structures develop from nascent structures ("neurognosis" in
biogenetic structural theory; see Laughlin 1991) that are "already there" in the newborn and infant.  It may well be that the origins
of developed geons are to be found in retinal neurognosis, as well as the deeper neurocognitive structures mediating perceptual and
cognitive categories.  There is considerable evidence in favor of this view with respect to, say, facial and hand recognition in
humans and non-human primates.
 
Experiential Proximity Hypothesis
 
In the 1993 article, I proposed what I call the "Experiential Proximity Hypothesis."  The hypothesis states: The more a state of
consciousness is oriented on direct experience, the more fuzzy will be the categories informing experience.  What was unstated, but
implicit in the article was the fact that we now know that virtually all interactions within the nervous system are
reciprocal.  That is, for every afferent pathway between two areas of nervous tissue, there exists a reciprocal efferent pathway.  It
used to be thought for example that the thalamus is a passive gateway into higher cortical processes.  We now know that the
relationship between thalamic nuclei and the cortical areas they innervate is a reciprocal one.  The innervation is two-way.  Both
the nuclei and the cortical areas interact -- they communicate with each other.  The same is the case between sensory receptor sites
and tissues "higher-up" in the nervous system. 
 
The "experiential proximity hypothesis" tacitly recognizes that a state of consciousness may be initiated from the exteroceptor,
interoceptor or sensorial structures (a butterfly suddenly appears before my eyes, or I imagine a chocolate milkshake) and propagate
toward intentional processes, or it may be initiated from intentional structures and propagate outward (I reason out a
problem, and then orient my senses and body to implement the solution).  I suspect that fuzziness of natural categories
manifests most obviously where the intention is from peripheral (exteroceptive or interoceptive) toward higher cortical processes,
and least operative where intention derives from higher cortical processes.  Overly crisp categories only naturally operate where
reasonable precision is required, as is the case for example in traditional pharmacological systems.
 
Notes
 
References
 
     Bieterman, I., 1987, Recognition-by-Components: A Theory of Human Image Understanding. Psychological Review 94(2):115-147.
     Biederman, I. and G. Ju, 1988, Surface versus Edge-Based Determinants of Visual Recognition. Cognitive Psychology 20: 38-64.
     Giles, R., 1979, A Formal System for Fuzzy Reasoning. Fuzzy Sets and Systems 2:233-257.
     Herrnstein, R.J., 1982, Stimuli and the Texture of Experience. Neuroscience and Biobehavioral Reviews 6:  105-117.
     Kay, P. and C.K. McDaniel, 1978, The Linguistic Significance of the Meanings of Basic Color Terms. Language 54(3):610-646.
     Klir, G.J. and T.A. Folger, 1988, Fuzzy Sets, Uncertainty, and Information. Englewood Cliffs, NJ: Prentice-Hall.
     Lakoff, G., 1973, Hedges: A Study in Meaning Criteria and the logic of Fuzzy Concepts. Journal of Philosophical Logic 2:458-508.
     Laughlin, C.D., 1991, Pre- and Perinatal Brain Development and Enculturation: A Biogenetic Structural Approach." Human Nature
2(3):171-213.
     Laughlin, C.D., 1993, "Fuzziness and Phenomenology in Ethnological Research: Insights from Fuzzy Set Theory. Journal of
Anthropological Research 49(1).
     Medin, D.L., 1984, Concepts and Concept Formation. Annual Review of Psychology 35:113-138.
     Rosch, E., C.B. Mervis, W.D. Gray, D.M. Johnson, and P. Boyes- Braem, 1976, Basic Objects in Natural Categories. Cognitive
Psychology 8:382-439.
     Sapir, E., 1944, Grading: A Study in Semantics. Philosophy of Science 11:93-116 [also pp. 122-149 in Selected Writings of Edward
Sapir in Lauguage, Culture and Personality (ed. by D.G. Mandelbaum). Berkeley: University of California Press, 1968.]
     Zadeh, L.A., 1965, Fuzzy Sets. Information and Control 8:338- 353 [reprinted in Yager et al. 1987].
     Zadeh, L.A., 1975, Calculus of Fuzzy Restrictions. Pp. 1-39 in Fuzzy Sets and Their Applications to Cognitive and Decision
Processes (ed. by L.A. Zadeh, K.-S. Fu, K. Tanaka & M. Shimura). New York: Academic Press.
     Zimmermann, H.-J., 1990, Fuzzy Set Theory and Its Applications. Boston: Kluwer.

If you wish the long version of the paper on fuzziness and phenomenology, it is in a zipped wp file which you can find on the "Selected Articles" page!

http://www.iit.nrc.ca/IR_public/fuzzy/fuzzyJDocs/overview.html

Overview of Fuzzy Concepts

Fuzziness
FuzzyVariable, FuzzySet and FuzzyValue
FuzzyRule

In the real world there exists much fuzzy knowledge, i.e., knowledge that is vague, imprecise, uncertain, ambiguous, inexact, or probabilistic in nature. Human thinking and reasoning frequently involve fuzzy information, possibly originating from inherently inexact human concepts and matching of similar rather then identical experiences. In systems based upon classical set theory and two-valued logic, it is very difficult to answer some questions because they do not have completely true answers. Humans, however, can give satisfactory answers, which are probably true. Expert systems should not only give such answers but also describe their reality level. This level should be calculated using imprecision and the uncertainty of facts and rules that were applied. Expert systems should also be able to cope with unreliable and incomplete information and with different expert opinions.

Fuzziness

Fuzziness occurs when the boundary of a piece of information is not clear-cut. For example, words such as young, tall, good, or high are fuzzy. There is no single quantitative value which defines the term young when describing a fuzzy concept (or fuzzy variable) such as age. For some people, age 25 is young, and for others, age 35 is young. The concept young has no clean boundary. Age 1 is definitely young and age 100 is definitely not young; however, age 35 has some possibility of being young and usually depends on the context in which it is being considered. In fact an age can have some possibility of being young and also some possibility of being old at the same time (note that these are NOT probabilities and the sum of all the possibilities does not need to sum to 1.0). The representation of this kind of information is based on the concept of fuzzy set theory [Zadeh, Cox, Tsoukalas and Uhrig, Kosko]. Unlike classical set theory where one deals with objects whose membership to a set can be clearly described, in fuzzy set theory, membership of an element in a set can be partial, i.e., an element belongs to a set with a certain grade (possibility) of membership. More formally a fuzzy set A in a universe of discourse U is characterized by a membership function

         A : U [0,1]

which associates a number A(x) in the interval [0,1] with each element x of U. This number represents the grade of membership of x in the fuzzy set A (with 0 meaning that x is definitely not a member of the set and 1 meaning that it definitely is a member of the set). For example, the fuzzy term young might be defined by the fuzzy set in the Table below.
 

 Fuzzy Term young

Age

Grade of Membership

25

1.0

30

0.8

35

0.6

40

0.4

45

0.2

50

0.0

One might also write

  young(25) = 1, young(30) = 0.8, ... , young(50) = 0

Grade of membership values constitute a possibility distribution of the term young as applied to the fuzzy variable age. The table can also be shown graphically (see Figure below).

                  Possibility distribution of young

The possibility distribution of a fuzzy concept like somewhat young or very young can be obtained by applying arithmetic operations to the fuzzy set of the basic fuzzy term young, where the modifiers somewhat and very are associated with specific mathematical functions. For instance, the possibility values of each age in the fuzzy set representing the fuzzy concept somewhat young might be calculated by taking the square root of the corresponding possibility values in the fuzzy set of young (see Figure below). These modifiers are often referred to as hedges. A number of modifiers are supplied with FuzzyJ (see Modifiers).


                Possibility distribution of somewhat young

FuzzyVariable, FuzzySet and FuzzyValue

Fuzzy concepts are represented using fuzzy variables, fuzzy sets and fuzzy values in the FuzzyJ Toolkit. A FuzzyVariable defines the basic components that are used to describe a fuzzy concept. It consists of a name for the variable (for example, age or hot water temperature) the units of the variable if required (for example, years or Degrees C), the universe of discourse (UOD) for the variable (for example a range from 0 to 125), and a set of fuzzy terms that can be used to describe the particular fuzzy concepts for this variable. The fuzzy terms are described using a term name such as old, along with a FuzzySet that represents that term. The fuzzy variable terms along with a set of fuzzy modifiers (such as very or slightly), the operators and and or (fuzzy set intersection and union respectively) and the left and right parentheses provide the basis for a grammar that allows one to write fuzzy linguistic expressions that describe fuzzy concepts in an english-like manner. These linguistic expressions are encoded in a FuzzyValue which holds a specific fuzzy concept such as age is very old.

For example the linguistic expression,

     very old or young

consists of the terms old and young, along with the fuzzy modifier very.

A FuzzyValue is normally created by specifying a FuzzyVariable and a linguistic expression. The example below shows how this is done in Java code.

//definition of fuzzy variable ‘age’ with terms ‘young’ and ‘old’
FuzzyVariable age = new FuzzyVariable(“age”, 0, 120, “years”);
age.addTerm(“young”, new ZFuzzySet(25, 50));
age.addTerm(“old”, new SFuzzySet(50, 65));
// definition of FuzzyValue for concept ‘age is somewhat young’
FuzzyValue ageSomewhatYoung = new FuzzyValue(age, “somewhat young”);

                  FuzzyJ Code to define the concept 'age is somewhat young'

Without getting into too much detail at this point, above we created a FuzzyVariable, age, that provides the basis for building fuzzy concepts about age. The domain for the variable is from 0 to 120 years and we have defined 2 linguistic terms, young and old, that we can now use to represent specific age concepts. The terms were defined using FuzzySets. In this case young is represented by a sub-class of FuzzySet, ZFuzzySet, which defines a Z-shaped fuzzy set like the one shown in Figure 1 above. Once the FuzzyVariable and the terms that are to be used to describe concepts about that variable are ready, we then can create FuzzyValues. In this case we create an object, ageSomewhatYoung. Note here that the definitions of FuzzyValues are actually much more flexible than shown here but this example shows the simplest 'linguistic' (i.e. in english-like phrases) way to create them. Let's also note here that the difference between a FuzzySet and a FuzzyValue may be a bit confusing -- why have both. Ultimately the FuzzyValue is just a FuzzySet (or at least it contains a FuzzySet) but it also has a context, the FuzzyVariable it is associated with. A FuzzySet on its own can be operated on, for example, intersected with another FuzzySet. But this has no real meaning. A FuzzyValue on the other hand can also be manipulated, but only with other FuzzyValues that share the same FuzzyVariable. This is reasonable since doing the intersection of 'hot water' and 'low pressure' makes no sense. This is like adding 5 apples to 6 oranges. Keep this in mind when reading about FuzzySets and FuzzyValues.

FuzzyRule

Now let's continue the example and add a simple FuzzyRule. In general, in rule based systems, rules look something like:

If A1 and
   A2 and
   ...
   An
then
   C1 and
   C2 and
   ...
   Cm

where the Ai are the antecedents on the left hand side (LHS) of the rule and the Cj are the conclusions on the right hand side (RHS) of the rule. In this format, if all of the antecedents (conditions) on the LHS of the rule are true then the rule will fire and the conclusions will be asserted. In the FuzzyJ Toolkit, the fuzzy rules that are supported have FuzzyValue antecedents and FuzzyValue conclusions. (If you are familiar with FuzzyCLIPS then this is equivalent to the so-called fuzzy-fuzzy rules). The rule we are going to build is:

If temperature is hot and
   flow is low
then
   change cold valve positive big amount and
   change hot valve zero

and we will provide 2 input values to the rule, fire the rule and look at the rule outputs. We'll put this simple example together a bit at a time and show the final results of the firing of the rule. There is a normal sequence of events in the execution of a set of fuzzy rules. First the inputs, which are normally crisp values, are fuzzified. This corresponds to representing the uncertainty in the measure of the inputs (in science class this can be equivalent to reading the value of an instrument like a scale and recognizing that the value is actually plus or minus the reading by some factor that represents the accuracy of the instrument or your ability to read the instrument). It is also required for these rules since the FuzzyValue antecedents must be matched against FuzzyValue inputs. Once the rule has its inputs the rule can be fired. It produces a set of FuzzyValue outputs. These outputs can be left as FuzzyValues but in general, and particularly for this example, we go through a process of defuzzification of the outputs to get crisp values. This allows the system to take some real world action, such as moving the hot and cold water valves by some real-valued amount.

Step 1 (define the FuzzyVariables for temperature, flow, cold valve change and hot valve change)

// Temperature Fuzzy Variable has terms cold, OK and hot
FuzzyVariable outTemp = new FuzzyVariable("temperature", 5.0, 65.0, "Degrees C");
outTemp.addTerm("cold", new TrapezoidFuzzySet(5.0, 5.05, 10.0, 35.0));
outTemp.addTerm("OK", new PIFuzzySet(36.0, 3.5));
outTemp.addTerm("hot", new SFuzzySet(37.0, 60.0));
// Flow Fuzzy Variable has terms low, OK, and strong
FuzzyVariable outFlow = new FuzzyVariable("flow", 0, 100.0, "litres/minute");
outFlow.addTerm("low", new TrapezoidFuzzySet(0.0, 0.025, 3.0, 11.5));
outFlow.addTerm("OK", new PIFuzzySet(12.0, 1.8));
outFlow.addTerm("strong", new SFuzzySet(12.5, 25.0));
// hotValveChange has terms NB, NM, NS, Z, PS, PM, and PB which correspond to
// negative big, negative medium, negative small, zero, positive small,
// positive medium and positive big
FuzzyVariable hotValveChange = new FuzzyVariable("hotValveChange", -1.0, 1.0, "");
hotValveChange.addTerm("NB", new ZFuzzySet(-0.5, -.25));
hotValveChange.addTerm("NM", new TriangleFuzzySet(-.35, -.3, -.15));
hotValveChange.addTerm("NS", new TriangleFuzzySet(-.25, -.15, 0.0));
hotValveChange.addTerm("Z", new TriangleFuzzySet(-.05, 0.0, 0.05));
hotValveChange.addTerm("PS", new TriangleFuzzySet(0.0, .15, .25));
hotValveChange.addTerm("PM", new TriangleFuzzySet(.15, .3, .35));
hotValveChange.addTerm("PB", new SFuzzySet(.25, .5));
// coldValveChange has terms NB, NM, NS, Z, PS, PM, and PB which correspond to
// negative big, negative medium, negative small, zero, positive small,
// positive medium and positive big
FuzzyVariable coldValveChange = new FuzzyVariable("coldValveChange", -1.0, 1.0, "");
coldValveChange.addTerm("NB", new ZFuzzySet(-0.5, -.25));
coldValveChange.addTerm("NM", new TriangleFuzzySet(-.35, -.3, -.15));
coldValveChange.addTerm("NS", new TriangleFuzzySet(-.25, -.15, 0.0));
coldValveChange.addTerm("Z", new TriangleFuzzySet(-.05, 0.0, 0.05));
coldValveChange.addTerm("PS", new TriangleFuzzySet(0.0, .15, .25));
coldValveChange.addTerm("PM", new TriangleFuzzySet(.15, .3, .35));
coldValveChange.addTerm("PB", new SFuzzySet(.25, .5));

Step 2 (define our rule)

// the fuzzy rule ...
// if temperature is hot and flow is low
// then change the hot valve zero and the cold valve positive big
FuzzyRule hotLow = new FuzzyRule();
hotLow.addAntecedent(new FuzzyValue(outTemp,"hot"));
hotLow.addAntecedent(new FuzzyValue(outFlow,"low"));
hotLow.addConclusion(new FuzzyValue(hotValveChange,"Z"));
hotLow.addConclusion(new FuzzyValue(coldValveChange,"PB"));

Step 3 (provide the fuzzified inputs for the rule)

// temperature reading is 45.0 degrees C and the flow is 8.0 litres/minute
double showerTemp = 45.0;
double showerFlow = 8.0;
// create fuzzy values from the crisp values
inputTemp =  new FuzzyValue(outTemp, new TriangleFuzzySet(showerTemp-0.05, showerTemp, showerTemp+0.05));
inputFlow =  new FuzzyValue(outFlow, new TriangleFuzzySet(showerFlow-0.05, showerFlow, showerFlow+0.05));

Step 4 (execute the rule with these inputs)

// remove any inputs associated with the rule, then add the new inputs to the rule
// Note: the order inputs are added is important since the inputs must correspond to the antecedents
hotLow.removeAllInputs();
hotLow.addInput(inputTemp);
hotLow.addInput(inputFlow);
// fire the rule, the result of firing is a special vector of FuzzyValues that represent the outputs
FuzzyValueVector fvv = hotLow.execute();

Step 5 (defuzzify the outputs to get crisp values)

// get the output FuzzyValues from the vector
FuzzyValue hotValveChangeFval = fvv.fuzzyValueAt(0);
FuzzyValue coldValveChangeFval = fvv.fuzzyValueAt(1);
// calculate the deffuzified value
double crispHotValveChange = hotValveChangeFval.momentDefuzzify();
double crispColdValveChange = coldValveChangeFval.momentDefuzzify();

Step 6 (let's look at the fuzzy sets of the outputs and the corresponding crisp values)

System.out.println(hotValveChangeFval.plotFuzzyValue("+"));
System.out.println(coldValveChangeFval.plotFuzzyValue("+"));
System.out.println("Defuzzified hot valve change is: " + crispHotValveChange);
System.out.println("Defuzzified cold valve change is: " + crispColdValveChange);

The results will appear as follows:

Fuzzy Value: hotValveChange
Linguistic Value: ??? (+)

 1.00
 0.95
 0.90
 0.85
 0.80
 0.75
 0.70
 0.65
 0.60
 0.55
 0.50
 0.45
 0.40
 0.35
 0.30
 0.25                         +
 0.20                        + +
 0.15
 0.10
 0.05
 0.00++++++++++++++++++++++++   ++++++++++++++++++++++++
     |----|----|----|----|----|----|----|----|----|----|
   -1.00     -0.60     -0.20      0.20      0.60      1.00

Fuzzy Value: coldValveChange
Linguistic Value: ??? (+)

 1.00
 0.95
 0.90
 0.85
 0.80
 0.75
 0.70
 0.65
 0.60
 0.55
 0.50
 0.45
 0.40
 0.35
 0.30
 0.25                                  +++++++++++++++++
 0.20
 0.15                                 +
 0.10
 0.05                                +
 0.00++++++++++++++++++++++++++++++++
     |----|----|----|----|----|----|----|----|----|----|
   -1.00     -0.60     -0.20      0.20      0.60      1.00

Defuzzified hot valve change is: 0.0
Defuzzified cold valve change is: 0.6531280609133145


Of course this example is simple and there is much more to be mastered before one can produce useful fuzzy systems with the NRC FuzzyJ Toolkit. Also the types of rules and complexity of rules can be enhanced by integration with the Jess  rule-based system.
 

http://members.ams.chello.nl/jsteenis/mathematicalE.htm

MATHEMATICAL CHIP Mathematics and philosophy

HOME
F. Fractals and the mathematics of chaos

Chapter E.  FUZZY LOGIC

    The uncertainty relation of Heisenberg undermines the mechanistic worldview of Descartes that continues to dominate mathematics and physics. Heisenberg noticed that something was wrong with the physical laws and improved the laws by introducing uncertainty in the old laws. He did not make new laws based on uncertainties. He determined that when you knew exactly where a very small particle was, it was impossible to know anything about the speed of that particle. Inside the deterministic doctrine not everything could be measured in a discrete manner. There do however exist non-discrete ways as in the fuzzy logic but that was something Heisenberg could not know yet.

    Everybody uses fuzzy concepts. How does someone know that something is a chair when he sees only a very small part of the object? Why is a chair called chair when you can hardly sit on it – which is by the way often the case when so-called artists design chairs. And many chairs resemble tables – and some tables can be used as chairs. We could say that some chairs are chair for the fully hundred percent. Other objects are chairs for 90 % and again others are maybe 60 % chair, 20 % decorative and for the rest a table. The concept long is also fuzzy. We say that someone is long even when he has a much longer stick in his hand. Women are on average smaller than men are and a woman can be long while a man of the same size is not called long. We use the same word long for two different measurements and the human brain is not troubled. Discrete you could say that someone is long when he is longer then one point ninety meter. But is someone who is one meter and eighty-eight centimetres not long? In the collection of humans you can say that someone with a length of two meter is long for about ninety percent while someone with a length of one meter and ninety centimetres is only long for seventy percent. We can live with fuzzy concepts but it is very difficult to make calculations with fuzzy numbers.

    Chess players often have a different judgement of the same position on the board. One prefers the white position (but in how far?) and the other the black one. They look at the different parts of the position (pawn structure, activity of the pieces, possibility of an attack and so on). But judgement is different because the first prefers the pawn structure while the second gives more weight to the activity of the pieces. It is very difficult to quantify these vague concepts. Therefore it is difficult to use such concepts in computers, which uses discrete numbers (ones and zeros). You need a different method of calculation to connect vague concepts. It is not sufficient to translate vague values into discrete ones after which you can use normal algorithms and then at the end you translate the found discrete numbers again in fuzzy concepts.

    In Japan they use a kind of fuzzy logic based on discrete numbers. Washing machines determine with sensors a value that tells something of the amount of dirt. The weight is also measured and third sensor looks at the colours. Then an algorithm determines how much water and washing powder is needed and what the temperature of the water has to be. These variables are independent from each other and they can be quantified. So you get a washing machine with thousands of programs. This pseudo fuzzy logic is also used in lifts, which on their own accord travel to those floors where most people are waiting. And in cameras to compensate for tremors in the human hand. But this is not real fuzzy logic. The fuzzy situation is split up into very small steps after which discrete values are allotted to the parts. These values can be used in calculations according to the Boolean logic. Boolean logic is by the way a special part of fuzzy logic. It looks fuzzy but it is still discrete. This soft computing is still computing. Vague concepts are being converted into numbers that can be used in a computer. But our brain works with real fuzzy computing.

    In these industrial applications quantification is possible. Quantification of fuzzy chess values is nearly impossible. The variables in chess do not have a discrete value. The variables are dependent on the judgement and the attitude (more aggressive, more defensive and so on) of the observer. Besides the fuzzy values are interdependent. A change in the pawn structure has a vague influence on the activity of the pieces, a decrease of the number of pieces increases the importance of the pawn structure.

    Fuzzy numbers are vague. The value of a certain chess position could be established by the strength of the position of the pawns and the activity of the pieces. In a certain position the strength of the pawns is between 0.4 and 0.7 on a scale that goes from 0.0 to 1.0. The value includes the expectation about how in future – when all pieces have disappeared – the value of the position of the pawns is in the ending. Fuzzy means also that the value 0.4 is not less probable than the mean value 0.55. The distribution of values does not follow the normal distribution of Gauss. The diagram of the lengths of the Dutch population resembles a bell in which the values in the middle are more frequent than the values on the edge. Fuzzy numbers can not be represented in diagrams.

    The adding of two vague numbers is very difficult. In the above mentioned chess position the activity of the pieces lies somewhere between 0.2 and 0.6. What is the value of the whole position? It is not correct to calculate the mean of the sum of 0.3 (0.2 + 0.6 divided by 2) and 0.55 (0.4 + 0.7 divided by 2). You have also to take into account that the position will change after the next move. When for example a pawn advances the position of the pawns is weakened while the activity of the pieces can grow. Some calculation techniques have been developed but they are mostly based on the converting of fuzzy numbers in discrete ones. In our brain other processes take place in which the unconscious brain adds fuzzy numbers by means of an unknown technique after which the conscious part – by regarding the proposed result from the outside – looks at the result. When the result does not satisfy our conscious brain orders the unconscious brain to apply another fuzzy technique to the fuzzy numbers in order to arrive at a different answer. It is probable that the brain uses fuzzy techniques, which can not be used by computers.

    So you start with a fuzzy input that leads via fuzzy dependency, fuzzy thinking, fuzzy judgement and fuzzy logic to a fuzzy output. On the basis of a fuzzy complex the brain decides which piece or pawn has to move. And the brain includes in her judgement also fuzzy ideas about such fuzzy facts as the aggressiveness of the opponent. The discrete position on the board gives rise to a fuzzy process that results in a discrete decision: only one move can be executed on the board. For this kind of fuzzy process techniques we do not have any theory. We do not know how we can obtain a fuzzy output from a fuzzy input. About a century ago science stopped to contemplate about such problems.

    To use fuzzy logic thinking one must not be dominated by the idea that was already brought forward by Galilee when he remarked that the book of nature was written in a mathematical language. He pointed to a discrete and not to a fuzzy ordering. People think sometimes fuzzy for example when they are parking their car in the neighbourhood (a vague concept, thus fuzzy) of the sidewalk. But they have to learn to use fuzzy thinking in a more conscious way and leave discrete calculations to the computer. Humans mostly fall back on discrete values but it is better to think in probabilities and uncertainties. Though the thinking of chess players has many fuzzy elements, they regard their ratings mostly as an absolute value of their strength. The strength of chess players is given by an ELO-number. Someone with an ELO of 2000 is stronger than someone with an ELO of 1800. Now the weaker player is often afraid of the stronger one and he regards a win as a surprise. Nonsense! ELO-numbers are not discrete, chance but also fuzzy has much to do with ELO-ratings. But they give only an indication of the strength. When a match is played between two players with ELO’s of 2000 and 1800 the strongest player is expected to win by 7 ½ against 2 ½ points. So it is quite normal when the weakest player wins two or three times in every ten games the players play. The ELO-number is of course also built up from many fuzzy elements. There is a difference in strength when someone plays with the white or the black pieces, the strength varies when someone slept well, does not have troubles outside the game, plays his own trusted opening. The ELO-number is made by a fuzzy addition of many fuzzy factors. Of course someone with an ELO of 2000 does not always make moves that belong to a strength of ELO 2000. Sometimes he makes beautiful moves, sometimes he blunders – it is all included in the number 2000. But someone with an ELO of 1800 makes less often beautiful moves than someone with an ELO of 2000. By the way, ELO-numbers are not real fuzzy numbers but probabilities in which fuzzy elements are included. The example of the ELO-number makes clear that humans have to change their way of thinking in the direction of vague concepts.

    It is already fairly difficult to get accustomed to fuzzy numbers and probabilities in case of the aforementioned relatively easy cases. When the problems get complicated thinking becomes more difficult and it is even more important to change the way of thinking. It is striking that in conversations most people understand fuzzy concepts fairly well. Words as about, maybe, long, short, nice and agreeable are all fuzzy. In a conversation these words are never described exactly. Even the question what is life is fuzzy. Viruses grow but they can not reproduce. Are they alive? People are alive but when does human life begins? With the first two cells from which later a human will grow? Or is it needed that there are 4, 8, 16, 32, 64, 1024, 8192 or even still more cells before we can call a living entity human? The whole abortion discussion turns around such fuzzy concepts. Fuzzy exists, the consequences are everywhere and it is strange that scientists mostly avoid this reality. In some simple cases they use fuzziness but when problems get more complex fuzzy disappears and all is expressed in absolute values. But exactly in complicated problems precise descriptions become meaningless and meaningful descriptions are not precise. Let humans become a little more chaotic.

F. Fractals and the mathematics of chaos
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