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Mieli dalyviai! Visa mano kūryba ir kartu visi šie puslapiai yra visuomenės turtas, kuriuo visi kviečiami laisvai naudotis, dalintis, visaip perkurti. - Andrius



Helmut Leitner pastabos.

Discussion with Helmut Leitner on "everyday language"

GOS redefines a number of words, which may create problems for people to understand GOS. To talk about this, we need a notation. To solve this, we either need a good understanding of all these languageel issues or otherwise it might make sense to change GOS terminology by using other terms or even by inventing new terms.

Notation in test:

  • topology. A word without classification. Default meaning of GOS (local language).
  • "topology". This means: beware of the meaning, there are more than one interpretations possible.
  • topologyel (pronounce 'topology in everyday language'). The word in the meaning of "everyday language". Whatever this may mean (look into Wikipedia).
  • topologygos. The word, explicitely in the meaning of GOS.
  • math. Language of mathematics, probably needed.
  • phil. Language of philosophy, might be needed.

Considered alternatives:

  • GOS.topology (collides with full stop at end of sentence)
  • GOS:topology (collides with wiki namespace syntax)
  • topology:GOS (counterintuitive to namespace syntax)
  • topology/GOS (possible but "less beautiful")
  • topology.GOS (collides with full stop at end of sentence)


topologygos unequal topologyel.

To do.

[ MathWorld's definition of Topology]: Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed... The "objects" of topology are often formally defined as topological spaces... There is also a formal definition for a topology defined in terms of set operations. A set X along with a collection T of subsets of it is said to be a topology if the subsets in T obey the following properties: 1. The (trivial) subsets X and the empty set are in T. 2. Whenever sets A and B are in T, then so is A intersection B. 3. Whenever two or more sets are in T, then so is their union


perspectivegos unequal perspectiveel.

perspectiveel: Typically perspectiveel depend on a certain viewpoint and is unique. A perspectiveel can't contain other perspectivesel. You have to add a dimension to contain perspectives. A coordinate system can be used to map, understand, calculate and contain other perspectivesel.

To do.


languagegos unequal languageel.

languageel: A languageel is a mapping of ideal things (words, symbols) to perceived real things (objects, properties, ideas).

languagegos: I do not yet understand what languagegos means, but it is quite different.

To do.

[ MathWorld's definition of Formal Language]: In mathematics, a formal language is normally defined by an alphabet and formation rules. The alphabet of a formal language is a set of symbols on which this language is built. Some of the symbols in an alphabet may have a special meaning. The formation rules specify which strings of symbols count as well-formed. The well-formed strings of symbols are also called words, expressions, formulas, or terms. The formation rules are usually recursive. Some rules postulate that such and such expressions belong to the language in question. Some other rules establish how to build well-formed expressions from other expressions belonging to the language. It is assumed that nothing else is a well-formed expression. For example, the language of propositional calculus could be defined as follows....

Some associations come to my mind. Some people talk about "the ability to make/see a difference" which means to "compare". Are two things equal or not.

This seems to need interpretation. Old saying: "one can't sum up apples and pears". Others might say: "5 apples plus 3 pears are 8 pieces of fruit". So it depends on the el[EverydayLanguage]!perspective.

Related seems the philosophical problem of "being and nothingness" which - far from Hegels "logic" - seems a matter of interpretation: for example when does human life start (when to we see abortion as murder, avoiding pregnancy as sin)?

On the other hand the operation of "comparison" might be considered fundamental to generate "language". How could we keep a "cat" and a "dog" separate, if we were not able to observe the quality of things, compare and see differences, and give them names (as mapping symbols and perceptions).

GOS uses "language" in a different way, far from everyday language. I think such redefinitions are a problem and should be avoided.

===Not Hegel===

I see no parallels to Hegel, who encyclopedically tries to reach acceptable interpretations of the reality, if an interpretation turns out wrong - no problem - this is part of his dialectic system. Andrius, as a mathematician is "not really interested in reality" (his own words) and would be shocked by any part of GOS that can be proven wrong. Mathematics is always an essence, not an encyclopedia. {{HelmutLeitner}}

{{Andrius}}: I add that I might be even more shocked if I was proven right. I'm not proving anything. Instead, I'm documenting what I observe. And I'm observing what's real to me - the constraints on my own mind - as I have no way to constrain the physical world, and it ultimately isn't directly relevant to who I am. This world can come and go, but I supposedly continue, and may very well continue. I think the validation of my observations is given, and will be given, by their usefulness. I also know that there are many checks on my observations, places where it becomes apparent that an observation is murky, or incomplete, or needs to be corrected, or is simply wrong. But I try to focus on simply writing down my observations as best I can, so that I could make progress. Likewise, it's really only possible to communicate this with other people who are similarly choosing to map out what constrains their imagination. Because people who aren't interested in the thought that their minds are constrained, aren't going to allow for that, but may typically instead focus on constraints in the physical world, and so there is nothing for them to talk about with me, or me with them. That's a reason why it means a lot to me that people focus on their MinciuSodas/keyConcept, as it makes it possible to converse with somebody regarding their own inner life, their own accountability, their own self-imposed, self-chosen, self-accepted constraints, rather than some distant physical reality (or even more distant theories such as "the earth is round", "the universe is expanding", etc. which we could live perfectly complete lives without knowing, and which we could turn out to be partially or completely wrong and not really change anything). Also, I note that many parts of this Glossary of Structure (defining the {{Topologies}}, EmotionalResponses, {{Counterquestions}}, GoodWillExercises,DirectionsFromTheGood etc.) are driven by empirical data (of subjective experiences) which I sorted through looking for how various things sort out. Such empirical data will be the touchstone to confirm whether the three {{Languages}} are properly defined. And I do expect them to be encyclopedic, covering all concepts, including the ones that we find in physics, etc. But that is still to come. So I'm focusing on getting right the mechanics that will generate those languages.

I don't see why I should be considered a mathematician. Why should we categorize people as if that made them different? I studied math (but also physics and the humanities) as tools for this work.

{{HelmutLeitner}}: Andrius, when you talk here about "category theorie" and "morphism" and "topology" then you are using an extremely mathematical language and way of thinking. You are building GOS on axioms, like a mathematician. For example you assume that the constraints that you experience in your thinking are universal in the sense that they are constraints of "the thinking" that everyone shares. If you expect encyclopedic usability you assume that there can be no fundamental problem between "theory=language=thinking" and "reality=nature=universe". In a spiritual or biblical sense I agree that it wouldn't change our life if the earth were flat but disregarding such aspects of reality doesn't make you exactly a natural scientist. I experience you as a philosopher and a "empirist of the mind" but also as a 90% mathematician drawing strongly from this abstract and symbolic background.

See also: Welcome

AndriusKulikauskas: I am working to uncover the structural language by which we live our lives. My current work is at Summary and I am also writing an Introduction. Earlier I hope to work with others to create a GlossaryOfStructure and perhaps that might be possible some day.

Here is HelmutLeitner's understanding of this work. See also:

  • [SomeParallels Some Comparisons and Parallels] by FranzNahrada

{{HelmutLeitner}}: When I look at Andrius's GlossaryOfStructure I feel the desire to add an outside view to this system, in a way translate it into everyday language, so that it becomes easier to understand. I hope that the necessary simplifications will not create misunderstandings but an overall picture that motivates the readers to dig deeper.

===Why is GlossaryOfStructure difficult to understand?===

GlossaryOfStructure develops a model for the structure of our thinking. This is not what we usually do. Maybe we reflect on our doing or even our thinking, but we rarely go deeper. GlossaryOfStructure goes deeper and in doing so it must lack the words to describe this. So Andrius has to do what many mathematicians and philosophers have to do: reuse common words and give them a highly specialized meaning. So he creates his own language of words that look familiar but aren't. Look at the pages of words like {{Everything}}, {{Structure}}, {{View}} or even {{Truth}} and you will see the problem.

===What is GlossaryOfStructure?===

GlossaryOfStructure is a mathematical and philosophical model for the structure of our thinking. Just as mathematics gives us the structures of regular solids like these (images with courtesy of and only these 5 regular bodies:

GlossaryOfStructure says that the most important concepts of thinking are arranged in a finite number of structures. In a simpified way you could imagine the concepts to sit at the edges of such bodies, but the structures have no such nice 3-dimensional pictures going with them. The structures of GlossaryOfStructure have been named {{Nullsome}}, {{Onesome}}, {{Twosome}}, {{Threesome}}, {{Foursome}}, {{Fivesome}}, {{Sixsome}} and {{Sevensome}}. If you look at these pages you will get a rough impression, what types of concepts they model and what logic they contain.

===What is GlossaryOfStructure useful for?===

As these structures model our thinking, the complete structures represent what is thinkable at all. Not in detail but in its abstract form. So this is an extremely powerful tool to create order in all types of interesting systems. For example to analyze a complex problem or to create a complicated project or organization. You can be sure to put ideas into the right order. You can be sure, not to forget an important aspect. GlossaryOfStructure promises to be a system to provide consistency and completeness of thought.

===How does GlossaryOfStructure relate to religion or worldview===

If you'll read on you will notice, that {{Truth}}, {{Good}} and {{God}} are among the fundamental concepts. It's astonishing that a mathematical theory yields moral concepts (see {{Sixsome}}). Also Andrius often cites from the Bible which shows his Christian roots. IMHO this doesn't mean that the GlossaryOfStructure is religiously bound to Christianity although it's certainly compatible. The concept of {{God}} is very abstract and one may safely assume that it is compatible with all religions and worldviews representing {{Good}} intentions.

===How does GlossaryOfStructure relate to the Open Leader network===

The Open Leader network is built using the knowledge of the GlossaryOfStructure. Done right, this should guarantee that the key concepts are correctly identified and put into an order that allows the whole network and its individuals to unfold organically. This means that each network member can have increased trust that this endeavour will succeed. On the other hand, customers of the independent thinkers network can have trust that even the toughest problems can be solved.

===Examples "GOS at work"===

One of the most outstanding, challenging, inspring and influencial works is Christopher Alexander's "The Nature of Order". After decades of work he arrives at 15 principles of living systems, unable to explain why 15 nor being entirely sure of the completeness or formulation of the list. Andrius is able to offer a 5x3 structure of these principles and an interpretation that may lead to a deeper understanding of these principles: see PrinciplesOfLife. Even if one shouldn't agree with his interpretation, it is a big step forward to have an additional viewpoint to reflect on this otherwise incomprehensible collection. {{HelmutLeitner}}

Further examples will be collected at ApplicationExamples.


{{Andrius}}: Helmut, thank you, this is very helpful. I look forward to your help with our {{Booklet}}.

{{HelmutLeitner}}: Andrius, I'm relieved that you like it. At the moment I feel a bit empty, but I'll help if I can. ... For the moment I'm thinking about {{TermsAndTranslations}}.


Andrius, you present an extremely formal way of thinking, symmetrical like mathematics. While this may be understandable for some people (I have big difficulties to understand although I was educated as a natural scientist), for most people it will present an obstacle. Maybe you impress some people, but for most people it will seem like a weird and complicated way to look at things and IMHO it will hinder to achieve your online cooperation goals. -- {{HelmutLeitner}}

Helmut, online cooperation is not my goal! It is just a tool. What are your goals? -- {{AndriusKulikauskas}}

Andrius, as a human my overall goal is to understand other humans. As a natural scientist and philosopher my goal is to create a meta-worldview that allows to build bridges between different views. As a software developer my goal is to do professional work and to help my customers to make good use of new technologies (e. g. wiki) and make a living from that. As a father and husband my goal is to care for my family and have a good time. -- {{HelmutLeitner}}

Why do you need to use the word "God" to say "the simple laws of nature"? "Everything" does not have a wish, it merely appears that Everything tends to wish to SURVIVE, as in the Evolution, the things that had WISH to SURVIVE, were more successful in that. Natural selection selected to exist those who had more of this tendency to survive. But if something has this tendency, that doesn't mean that it has its own wish to do it. Human wants it, cause there are already developed mechanisms that stop the body from making suicides, but that couldn't be other way, cause there is simple laws of nature. -- {{Inyuki}}

This page is an extension of OutsideView. It ia meant to provide a complete list of GOS terms, together with one definition using GOS terminology and another definition using everyday language, if at all possible. Of course, this is no way authoritative, but an experimental try. There will also be many QuestionsToAndrius. -- {{HelmutLeitner}}

Basis for this work is the {{PageIndex}} assuming that each GOS term has or will have a page of his own.

work in progress ... complete when 100+ terms are available.


{{Slack}} is the difference between {{Everything}} and {{Anything}}. It is anti-structure.

{{Slack}} is hard to define in everyday language. It seems near to freedom (not being bound to certain rules) and of individuality (deviating from an ideal norm), randomness and redundancy. The difference between a living being and a dead perfect cristall?


{{Anything}} is the structure of life.

Probable "life" is misleading in the definition above. All things that exist, living or not, are unique and individual. But ... the more a system is a living system, the more this become visible. So {{Anything}} seem to symbolize that all existing things in the cosmos are individual. This is in oppositon to abstract or absolute concepts symbolized by {{Everything}}.


{{Structure}} is currently not directly defined in terms of GOS.

Probably {{Structure}} means something like "fundamental quality". This way "anything is the structure of life" would translate to "individuality is the fundamental quality of all things in the cosmos".


{{Everything}} is currently not directly defined in terms of GOS.

{{Everything}} seems to match with the concept of abstraction, as used in our thought processes. Everything seems to mean "EL: everything but not in individual detail".

When Andrius says "I want to know everything and apply that usefully" I understand "I want to understand the structure of all our thinking and make that usable for the good of people". See also what {{JosephGoguen}} writes about his understanding of this sentence.

Helmut: Andrius, how does the concept of {{Thought}} fit into the GOS and how does it relate to God?


  • {{Nothing}}, {{Something}} (empty page), {{Anything}}, {{Everything}}, {{Activity}} - {{Waiting}} {{Recurring}} {{Acting}} ({{Engaging}} ?) {{Unfolding}} (GoingBeyondOnself)
  • {{Slack}}, {{Structure}} - {{Spirit}}
  • {{Operations}}, AddNull, AddOne, AddTwo, AddThree
  • {{Equations}}, NullAddOne, OneAddOne, TwoAddOne, ThreeAddOne, FourAddOne, FiveAddOne, SixAddOne, SevenAddOne, NullAddTwo, OneAddTwo, TwoAddTwo, ThreeAddTwo, FourAddTwo, FiveAddTwo, SixAddTwo, SevenAddTwo, NullAddThree, OneAddThree, TwoAddThree, ThreeAddThree, FourAddThree, FiveAddThree, SixAddThree, SevenAddThree
  • {{View}}, {{Perspective}}, {{Observer}} ({{Viewer}} ?), {{Observed}} ({{Viewed}} ?), ObservationalPlane
  • {{Representations}}
    • {{Nullsome}}: {{Significant}}, {{Constant}}, {{Direct}}, {{True}}, {{God}}
    • {{Onesome}}: {{Order}}, {{Context}}, {{Algorithm}}, {{Structure}}, {{Concept}}, {{Everything}}
    • {{Twosome}} / {{Divisions}}, (FreeWill) {{Random}}? - ({{Fate}}) {{Causal}} ?, {{Inside}} -- {{Outside}}, {{Theory}} - {{Practice}} ({{Reality}} ?), {{Same}} - {{Different}},
    • {{Threesome}} / {{Topologies}}: {{Existing}} - {{Acting}} - {{Thinking}}, {{One}} - {{Many}} - {{All}}, {{Subject}} - {{Process}} - {{Object}}, {{Necessary}} - {{Actual}} - {{Possible}}
    • {{Foursome}}: {{Why}}, {{how}}, {{What}}, {{Whether}},
    • {{Fivesome}}: {{Cause}}, {{Effect}}, {{Time}}, {{Space}},
    • {{Sixsome}}: {{Judgement}}, {{Choice}}, {{Duty}} - {{Justice}} - {{Loyalty}}, {{Caring}} - {{Obeying}} - {{Believing}}, {{Intimacy}} - {{Beauty}} - {{Love}}, {{honesty}} - {{Courage}} - {{hope}}, {{Cognition}}, {{Emotion}}
    • {{Sevensome}}: {{Choosing}}, {{Moving}} - {{Resisting}} - - {{Listening}} ? - {{Observing}} ? - {{Understanding}} ? - {{Waiting}}? - - {{Following}} ? - {{Supporting}} ?
  • {{SecondaryStructures}} / {{Factoring}}:
    • {{Divisions}}
    • {{Representations}}
    • {{Topologies}}
    • {{Languages}}: {{Verbalization}}, {{Narration}}, {{Argumentation}}
  • {{Sign}}, {{Signed}}, {{Symbol}}, {{Index}}


Andrius, my difficulties in understanding are still immense, I still think this is a problem of language. While I start to understand terms like "everything" I still do not understand the meaning of terms like "view", "perspective" or "representation".

Explanations that are based on the local terminology (e. g. a view of a view is a view) do not reach me because:

  • a number multiplied with a number also yields a number
  • a vector added to a vector is a vector

these are structural aspects (a kind of closedness) but do not explain what a number, a vector or a view is.

Examples - however short or simple - do help and are extremely important to help in understanding. With respect to the divsions:

I know of only four ways of conceiving such a division:

  • "free will" and "fate"
  • "outside" and "inside"
  • "theory" and "practice"
  • "same" and "different"

I assume that you mean "types or classes" of divisions. You have provided two dicvisions with examplary explanations:

  • outside/inside: For example, if I am outside a system, then it also has an inside, and so opposites coexist. But if I am sucked inside that system, then there is only inside, as in that we are inside our universe.
  • same/different: Likewise, if items are the same, then they must also be different in that they are distinct. But if they are different, then they are simply different.

So at this point I'd ask you to provide similar explanations for

  • free will/fate:
  • ...
  • theory/practice:
  • ...

You also did not explain how these subtleties in divisions do add to our understanding of our thinking.

I also see (and miss) a certain order:

  • same/different -- the most general "making a difference" (maybe "what")
  • inside/outside -- difference with respect to an arbitrary "system boundary" (maybe "where" in space and time)
  • reality(not practice!)/theory -- difference with respect to "modelling" or "representation" or "reference"
  • free will/fate -- different view of a process with respect to "causality"

As a working hypothesis I would doubt that these four are a complete list and try to challenge you by adding further types of divisions, like:

  • created/grown
  • chaotic(individual)/ordered
  • following/opposing (this is maybe 4-fold: moving/following/opposing/standing-by)
  • form/content
  • causal/stochastic (maybe causal/unrelated/free/random)
  • independent/correlated(resonating)
  • centered/peripheral (near/far, relative to center)
  • continuous/atomic (unstructured/structured)

Of course I know that adding a single additional division would break existing symmetry and you must resist that - but it is not my job to make you happy.

So I would assume that you might be then able to fold them into the existing four types, but this would at least make clearer, what's characteristic about them.


What is a [{{Morphism}} #] in the context of GOS?

{{Andrius}}: Morphism is one of the SixMethodsOfProof. It is proof by matching analogous structures. Morphism is the strongest method of proof in that it is least contingent on the details of a system.

{{HelmutLeitner}}: Andrius, please give examples for [{{Morphism}} #]. Its important that at least one example is outside of GOS and outside of abstract mathematics.

{{Andrius}}: See the example of the [ Road Network Paradox]. The road networks are compared to a system of springs. There is a spring that links two weak springs and thus reduces the load that the system can carry. Cutting that spring improves the system. Similarly, removing a road can improve traffic. This is a morphism because it is mapping a problem from one domain into an analogous domain where it may be easier to understand, at least the relevant aspect. In general, metaphorical thinking is based on morphism, mappings, transformations.

{{HelmutLeitner}}: ok, but how can you consider such a method of transformation a proof?

{{Andrius}}: It's the essence of a proof, the heart of a proof. What is lacking? Perhaps more rigor and care is needed to make the mapping explicit. (Or in the above example, to make clear what exactly carries over from one domain to the other domain.) But what makes it a proof (sloppy or careful) is the mapping. That is where the argument stands. If the mapping is valid, then the argument is proven. Whether or not a mapping (or any mapping) can be validated (and accepted) depends also on the pragmatic context. It's possible to not accept any proof, and it's possible for all proofs to be wrong. But that doesn't mean that they are not proofs. But perhaps I didn't understand you? Or perhaps another word might be better? And yet I'm using proof in the same spirit as mathematicians do in their work. They don't spell out all the details, they make sure the main idea is compelling and when it is, the details clarify themselves and are worked out further. They don't write their papers this way, but this is how they think through their conclusions. So, for example, the above analogy makes clear that adding a road could make traffic worse. It also suggests why traffic is worse (the extra spring moves the weight to the weak springs; the extra road diverts traffic to the less efficient roads). Or vice versa. We've moved the problem from one domain to another one where possibly we may have better intuition. What is actually proven depends on the actual morphism. And note that formalism can make the proof more rigorous (more applicable regarding the details), but it can't make it stronger (clearer as to whether it is inherently valid or invalid). Yes, a proof can be stitched together, but that doesn't convince us that it will hold together. Morphism is the strongest type of proof because it is cut from whole cloth. It is all or nothing - either it applies or it doesn't.

Here are some examples of different ways of proving the same thing: [{{SelfEducation/methods}} of proof]. Morphism is the strongest (and convincing) proof, then induction, and substitution (algebraic manipulation) or examination of cases are much more fragile, much more contingent. A good morphism may be wrong in one case, but then it will be right about another, perhaps more important case. A good morphism can change what questions we choose to ask, it can have us redefine our terms and change our focus.

Here is another example: A right triangle is half of a rectangle. This is a morphism! I mean, this is to say that a right triangle is a concept that bridges the world of rectangles and the world of triangles. Consequently, the shape of the right triangle is given by:

  • the size of an acute angle (the world of triangles)
  • the ratio of any two sides (the world of rectangles)

This defines six maps between these two worlds! and they are the trigonometric functions. We don't need to know any more details - this is the heart of trigonometry, and enough for us to start to apply it practically. In fact, this morphism tells us the kinds of geometries for which we can have a trigonometry.

{{HelmutLeitner}}: I think the [ Road Network Paradox] doesn't hold. While the symmetrical standard situation produces 83x6000=498000 time units, A maybe counterintuitive traffic A->C->B->D actually reduces the total time and so improves average time as expected (e. g. 5600 x 80.8 + 400 x 111.2 = 496960 time units). So either the mechanical example also doesn't hold or the morphism is wrong.

{{Andrius}}: I think the point about the Road Network Paradox (and the reason that it holds) is that if you link the two bad roads with a shortcut, now it is possible for people to use two bad roads (to get from A to D) rather than just one bad road. And, in fact, you are rewarding the people who take both bad roads! increasing the number of cars on those bad roads. More and more people will take both bad roads (A to B and C to D) until things get so bad that there's no advantage to doing that (and it's better to take the longer routes B to D or A to C). Of course, morphism is a helpful approach precisely because it can make it easier for us to check whether an idea is right or wrong.

{{HelmutLeitner}}: The paradox is artifically constructed because people are assumed to have to go in the wrong direction. It doesn't hold, when some people go the long way on the good roads. Then average travel time decreases. Check the numbers I've given.

{{Andrius}}: Helmut, I think your alternate route is clever. (And they may not have taken that into account.) I can't take the time to think through the numbers. But is your formula suggesting that some people will travel 111.2 minutes and others will travel 80.8 minutes? They are specifically ruling that out by assuming that people will switch routes until everybody's travel time is the same. Also, from a common sense point of view, is the introduction of this clever long route an "improvement" over the original solution. That would be hard to believe (though, perhaps possible). Finally, the point that got me to believe the paradox is that the shortcut is much shorter than the long roads. Taking the shortcut makes for a 30km trip rather than a 50km trip. So people have a reason to sit through two traffic jams instead of just one. And that makes the traffic jams much worse.

{{HelmutLeitner}}: Arguments about that will take longer than checking the numbers. If there are multiple routes from some A to some B then it is improbable that the times of drivers will ever be equal, this is only because the sample model is symmetrical, so the condition - if it exists, doesn't make sense. In real life not even on a single route drivers will experience equal times. Drivers can't compare and can't optimize. Another example: I have four routes to my work, the shortest is 8 min but sometimes jammed to 30 min, so I often take other routes, the longest route guarantees me about 12-15 min. So if I have to be in time and I have 15+ minutes, I'll take the longest route. If I have enough time I take the shortest route, because it is statistically fastest. But I also take other routes regularly - because otherwise I have no experience about their qualities. -- But all such considerations are outside of the abstract model of the paradox. There is no guarantee that the model corresponds to reality, a model is a simplification. -- In effect they introduce a one-way-road and its quite clear that a one-way-road in the wrong direction will not be helpful - but this wouldn't be entertaining. If they talk about a street I have the right to assume that people may use it in either direction. I have the right to assume that they talk about the real world because they strive for a - very doubtful - real world explanation for "they add streets and traffic doesn't get better" which may have quite different reasons (e. g. that good streets attract traffic and that traffic increases). -- Also the mathematical model is doubtful: If we have a street E->G with length L and capacity C (T=L+F/C) and reinterpret it cut in to pieces as E->F->G with the same length (L/2 + L/2) and capacity, the calculated time increases (T=L+2*F/C). This has no influence onto all the aspects we discussed but it does not correspond to traffic or flow reality. Of course changes in the mathematic calculations can change the situations but can't be reflected in the morphism to the physical spring model!

{{Andrius}}: Helmut, you're right that reality is complex, as you say. But here are their assumptions: We can see that if traffic is split unevenly, one route will be faster than the other. People using the slower route will realize that they could do better by changing to the other, and no doubt will do so. The only stable situation is where half the traffic goes each way, because then anyone changing to the other route will find they are on a slower route, and thus will change back. For modeling purposes, those may or may not be good assumptions, and in your case they would at least require averaging. For example, in economics often such assumptions are made (perfect knowledge), occasionally they are not (imperfect knowledge), and there are cases where either model is appropriate. I think it's fair to allow that they are not trying to model all road systems, just some road systems. Here is the website of [ Dietrich Braess] who wrote the original paper in 1968, and which has been generated a lot of research, including observations of this behavior in computer networks. And [ here's another explanation].

{{HelmutLeitner}}: Andrius, I do not doubt that there are assumptions that are consistent with their findings. I do doubt that they say something about real traffic or physical systems. Morphism - the way they use it - just seems to provide insight. Can you provide a better example of a morphism that actually proofs something?


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