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Learning Intransitivity: from Intransitive Geometrical Objects to "Rhizomatic" Intransitivity

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A new class of intransitive objects – geometrical and mathematical constructions forming intransitive cycles A>B>C>A – are presented. In contrast to the famous intransitive dice, lotteries, etc., they show deterministic (not probabilistic) intransitive relations. The simplest ones visualize intransitivity that can be understood at a qualitative level and does not require quantitative reasoning. They can be used as manipulatives for learning intransitivity. Experimental studies of people’s understanding which relationships are transitive, and which are not are described. Classification of the types of situations in which the transitivity axiom does and does not work is presented. Four levels of complexity of intransitivity are introduced, from simple combinatorial intransitivity to a "rhizomatic" one. A possible version of the main educational message for students in teaching and learning transitivity-intransitivity is presented.
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Learning Intransitivity:
from Intransitive Geometrical
Objects to "Rhizomatic"
Intransitivity
Alexander Poddiakov
National Research University
Higher School of Economics
A B C
18-21 March 2019, Moscow
All of us know that if A>B and B>C than A>C.
This is true but not for all types of situations.
Alongside such relations, which are called transitive,
exist other types of relations, intransitive ones.
In my presentation, I will
- explore the nature of intransitivity and introduce a
new class of intransitive objects geometrical ones
-describe some experimental research on people’s
understanding of intransitive relations
- list 4 types of complexity of intransitive relations
- argue that the message for students should be more
multi-dimensional than “if you have violated the
transitivity axiom, you are not rational”
The transitivity axiom: if AB and BC then AC
where ” means “dominates over”, “is better than”,
“is preferable to” etc.
Many people believe that if “you have violated the
transitivity axiom, … you are not instrumentally
rational. The content of A, B, and C do not matter to
the axiom”.
(Five Minutes with K.E. Stanovich, R.F. West, and M.E.
Toplak, 2016
https://mitpress.mit.edu/blog/five-minutes-keith-e-stanovich-richard-f-west-and-maggie-e-toplak)
In contrast we have a lot of math research on
objects in intransitive relations
(“rock-paper-scissorsrelations)
2 4 9 1 6 8 3 5 7
Lets us consider 3 sets of 3 pencils of different lengths. We
compare the length of each pencil with the length of all the
other pencils.
Numbers are taken
from the magic
square presented by
Gardner (1974)
2 4 9 1 6 8 3 5 7
Red pencils beat green ones 5 out of 9 times
2 4 9 1 6 8 3 5 7
Green pencils beat blues ones 5 out of 9 times
2 4 9 1 6 8 3 5 7
Blue pencils beat red ones 5 times out of 9 times
Thus, transitivity works during
the comparison of 3 pencils for
the relation “to be longer”…
but it does not work in more
complex situations: during
the comparison of 3 sets of 3
pencils for the relation
“to often be longer”
Thus, transitivity works during
the comparison of 3 pencils for
the relation “to be longer”…
but it does not work in more
complex situations: during
the comparison of 3 sets of 3
pencils for the relation
“to often be longer”
Only arrows from winning to loosing sets
are shown
Intransitive dice
National Museum of Mathematics, USA
Purple (A): 4, 4, 4, 4, 0, 0
Yellow (B): 3, 3, 3, 3, 3, 3
Red (C): 6, 6, 2, 2, 2, 2
Green (D): 5, 5, 5, 1, 1, 1
AB, BC, CD, DA
Intransitive competition (“rock–paper–scissors”
competition) is also studied in biology and sociology
Let’s return to math - some references…
Conrey, B. et al. (2016). Intransitive dice. Mathematics Magazine, 89(2).
Gardner, M. (1974). On the paradoxical situations that arise from non-transitive
relations. Scientific American, 231(4).
Gower, T. (2017). A potential new Polymath project: intransitive dice.
https://gowers.wordpress.com/2017/04/28/a-potential-new-polymath-project-
intransitive-dice.
Grime, J. (2017). The bizarre world of non-transitive dice: Games for two or more
players. The College Mathematics Journal, 48(1).
Lebedev, A. V. (2018). [Intransitive triplets of continuous random variables]
http://www.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=21644
Pegg Jr., E. (2005). Tournament dice. http://www.mathpuzzle.com/MAA/39-
Tournament%20Dice/mathgames_07_11_05.html.
Trybuła, S. (1961). On the paradox of three random variables. Applicationes
Mathematicae, 5(4).
Yet all this research deals with numbers. I will present
geometrical and mechanical constructions in
intransitive relations. From a mathematical point of
view, it is a new class of intransitive objects. They
show deterministic (not probabilistic) intransitive
relations.
Stylized Tractors with Intransitive Towing Couplers
Which tractor should the driver choose as the leading one to
fit into to bring two tractors to a destination point?
The driver should choose A in pair A-B, B in pair B-C, and C in
pair A-C. That is, AB, BC, CA.
This model does not require quantitative comparisons. The
comparison of geometrical shapes is all that is necessary.
A B C
Intransitive
Double
Levers
With the same rotation force applied to the shaft,
Lever A will overpower lever B (as lever arm A is shorter
than lever arm B).
Lever B will overpower lever C.
Lever C will overpower lever A.
A A
B B C C
Levers and gears are “close relatives” in the family
of simple machines, so…
Intransitive Double Gears with intransitive speeds
of rotation
A more paradoxical and complicated version:
Oskar van Deventers
Non-Transitive Gears-and-Ratchets
https://i.materialise.com/forum/t/non-transitive-gears-by-oskar/1167
Whatever element the first player chooses (a knob or a
gear), the second player can always choose an element
rotating faster than the element chosen by the first
player.
I have shown that Oskar van Deventer’s Non-Transitive
Gears-and-Ratchets can be used for a three player
game
If the first two players choose an
element (a knob or a gear) each,
the third player always chooses
an element and the direction of
its rotation so that it will rotate
faster than the elements chosen
by other players.
Moreover, in 75% of cases the
third player can not only win but
can also (!) control the
distribution of the second and
third places of the other players.
“Transitivity and intransitivity are fascinating
concepts that relate both to mathematics and to the
real world we live in”.
Teachers can “interest and engage students as
they seek to discover which relationships are
transitive, and which are not.
Roberts, T. S. (2004). A ham sandwich is better than
nothing: Some thoughts about transitivity. Australian
Senior Mathematics Journal, 18.
Experimental studies of people’s understanding
“which relationships are transitive,
and which are not”
Beliefs about intransitive relations are domain-
specific. The participants believe that intransitive
relations exist in some domains but not in others
(though really they are possible in the latter domains
as well).
Poddiakov, 2010, 2011; Bykova, 2018 (under
Poddiakov’s supervision)
Participants: 169 people (17-28 yrs, 121 females, 48 males)
Method
Participants were asked questions about the possibility of the
domination of A over B, B over C, and C over A for various A, B,
and C.
“Is it possible that
A beats/dominates over B,
B beats/dominates over C,
C beats/dominates over A?” in various areas.
1. There are 3 straight rigid sticks where the 1st stick is longer
than the 2nd stick, and the 2nd stick is longer than the 3rd
stick. Is it possible that the 3rd stick can be longer than the 1st
stick?
98% of answers “say no” (right answer)
2. There are 3 objects of different mass where the 1st object is
more massive than the 2nd object, and the 2nd object is more
massive than the 3rd object. Is it possible that the 3rd object
can be more massive than the 1st object?
93% of answers “say no” (right answer)
3. Is it possible that chess computer A regularly prevails over
chess computer B, B regularly prevails over C, but C regularly
prevails over A?
79% of answers “say it’s possible” (right answer)
4. There are 3 boxes with 6 pencils of different lengths in each
box. We compare the length of each pencil with the length of
all the other pencils. We learn that the pencils from the 1st
box are often longer than the pencils from the 2nd box, and
the pencils from the 2nd box are often longer than the pencils
from the 3rd box.
Is it possible that the pencils from the 3rd box are often
longer than the pencils from the 1st box?
29% of answers “say it’s possible”
(right answer)
the simpler example of intransitive
pencil sets is on the right) 2 4 9 1 6 8 3 5 7
5. There are 3 teams with 6 wrestlers in each team. During
their tournament, each wrestler of one team meets and
wrestles with each wrestler from two other teams. It is known
that the wrestlers of the 1st team beat the wrestlers of the 2nd
team more often than are beaten by them, and the wrestlers of
the 2nd team beat the wrestlers of the 3rd team more often
than are beaten by them.
Is it possible that the wrestlers of the 3rd team beat the
wrestlers of the 1st team more often than are beaten by them?
81% of answers “it’s possible” (right answer)
!!!
Problems “3 boxes with 6 pencils in each box” and “3 teams
with 6 wrestlers in each team” are of the same logical
structure.
But, paradoxically, the most participants (81% ) believe in
intransitivity of wrestlers’ teams, and only 29% think that
intransitivity of length in pencils’ boxes is possible.
Why? We have not studied it yet. Only some hypotheses can be
formulated.
6. Is it possible that microorganisms A force out B, B force out
C, and C force out A?
91% of answers “it’s possible” (right answer)
(Concept “intransitive competition” is widely accepted in
biology.)
The next problem was inspired by Assyrian wheeled battering
rams
https://commons.wikimedia.org/wiki/File:Assyrian_Attack_on_a_Town.jpg
7. There are 3 kinds of stylized wheeled battering rams
competing with one another: each ram tries to mark another
ram with its inserted felt-tip pen.
It is known that the 1st ram marks the 2nd ram, but avoids
being marked by the last one; and the 2nd ram marks the 3rd
ram, but avoids being marked by the last one.
Is it possible that the 3rd ram marks the 1st ram, but avoids
being marked by the last one?
86% of answers “it’s possible”
(right answer,
a possible example is
on the right)
8. There are 3 double-gears joined in pairs, the 1st double-gear
rotates faster than the 2nd double-gear, and the 2nd double-
gear rotates faster than the 3rd double-gear.
Is it possible that the 3rd double-gear rotates faster than the
1st double-gear?
26% of answers “say it’s possible” (right answer)
a possible example is below.
Conclusion on the experiment
Beliefs about intransitive relations are domain-
specific. The participants believe that intransitive
relations exist in some domains but not in others
(though really they are possible in the latter domains
as well).
The axiom of transitivity was applied by the
participants very selectively. In many situations it was
the correct solution.
M.S. Permogorskiy (2016, under Poddiakov’s
supervision) proposed the following original hypothesis:
Inclusion of additional information
“it is known that A>B and B>C”
into problems that deal with objectively intransitive
relations reduces the success rate (i.e., prevents
participants from drawing the right conclusion that
C>A).
A problem posed for the experimental group
There are 3 sets of 6 pencils of different lengths in each set.
Set А: 8 5 5 3 3 3
Set B: 7 7 7 2 2 2
Set C: 6 6 6 4 4 1
We have compared the length of each pencil with the length of
all the other pencils. We have learnt that pencils from Set A
are often longer than pencils from Set B, and pencils from Set
B are often longer than pencils from Set C.
Based on the data, which of the following statements is right?
- Pencils from Set A are often longer than pencils from Set C?
OR
- Pencils from Set C are often longer than pencils from Set A?
The problem posed for the control group
There are 3 sets of 6 pencils of different lengths in each set.
Set А: 8 5 5 3 3 3
Set B: 7 7 7 2 2 2
Set C: 6 6 6 4 4 1
We have compared length of each pencil with length of all the
rest pencils.
Based on the data, which of the following statements is right?
- Pencils from Set A are often longer than pencils from Set C?
OR
- Pencils from Set C are often longer than pencils from Set A?
Results
54% of correct answers - for problems with the information that
“pencils from Set A are often longer than pencils from Set B,
and pencils from Set B are often longer than pencils from Set
C”.
71% of correct answers - for problems without this information.
Conclusion on the experiment
Information such as “A>B and B>Ccan lead the participants
to assume that the relationship is transitive. This can
negatively influence solving problems on objectively
intransitive relations.
Discussion
We have a rich tradition of math studies of various
intransitive objects. Yet there is no such tradition of
psychological and educational studies on
understanding-misunderstanding of the objectively
intransitive relations by people.
It seems reasonable to distinguish between four types of
situations (Poddiakov, 2010).
(1) Relations are objectively transitive and problem solvers
make correct conclusions about their transitivity.
(2) Relations are objectively transitive, but problem solvers
wrongly consider them as intransitive. Most studies are
conducted in this paradigm
(see e.g. May, 1954; Tversky, 1969; Regenwetter et al, 2011).
(3) Relations are objectively intransitive and problem solvers
make correct conclusions about their intransitivity
(e.g. intransitivity of intransitive dice, lotteries etc.)
(4) Relations are objectively intransitive, but problem solvers
wrongly consider them as transitive (e.g. because of taking the
transitivity axiom for granted).
Surprisingly, this type has been minimally studied by cognitive
and educational psychology.
Intransitive relations:
4 levels of complexity
The following classification is not exhaustive and serves to
mark some reference points.
1. Simple combinatorial intransitivity between non-interacting
objects (e.g. in intransitive dice sets, intransitive sets of sticks
etc.). Each object can be precisely described by a few
parameters (like numbers on the sides of dice). The parameters
are additive, without interactions.
2 4 9 1 6 8 3 5 7
2. Interactive intransitivity without qualitative transformations
of the objects in the intransitive relations. The information
about the objects and their interactions is complete.
3. Interactive intransitivity
with qualitative
transformations of the
objects in intransitive
relations.
Information about the
objects and their
interactions is complete.
Position A (White)
Position B (Black)
Position B (Black)
Position C (White)
Position C (White)
Position D (Black)
Position D (Black)
Position A (White)
Intransitive chess positions
” means “is better than”
4. Interactive "rhizomatic"
(multiple, intertwining)
intransitivity in real complex
systems.
A body of biological studies is
devoted to the complex
intransitive competitions in
ecological niches. Such
competition transforms
participants. Information about
the participants, their features
and interactions is incomplete.
Pic. from: Three Minute Theory:
What is the Rhizome?
https://www.youtube.com/watch?v=gnteiRO-XfU
Conclusion
The important message for students should be more multi-
dimensional than “if you have violated the transitivity axiom,
you are not instrumentally rational, and the content of A, B,
and C do not matter”.
In complex and multi-variable situations, intransitive choices
are perfectly rational because the choice options are in
intransitive relations. That is, transitive choices in intransitive
options are a fallacy.
Various educational tools can be used to support students'
understanding of intransitive relations, and geometrical
intransitive manipulatives can be included here.
Bykova, A. (2018). [Understanding of intransitivity of superiority in various
contexts]. Diploma work. (in Russian)
Permogorskiy, M. S. (2016). [Psychological predictors of the use of the transitivity
heuristic]. PhD thesis. (in Russian) https://www.hse.ru/sci/diss/175064641
Poddiakov, A. (2006). [Intransitive character of superiority relations, and decision
making]. https://psy-journal.hse.ru/en/2006-3-3/27139364.html (in Russian)
Poddiakov, A. (2010). Intransitivity cycles, and complex problem solving.
https://www.researchgate.net/publication/237088961.
Poddiakov, A. (2017). [Non-transitivity is a fount for inventors]. English Google
translation:
https://translate.google.com/translate?sl=ru&tl=en&u=https://trv-
science.ru/2017/11/21/netranzitivnost-kladez-dlya-izobretatelej/
Poddiakov, A. (2018). Intransitive machines. https://arxiv.org/abs/1809.03869
Poddiakov, A., & Valsiner, J. (2013). Intransitivity cycles and their transformations:
How dynamically adapting systems function.
https://www.researchgate.net/publication/281288415
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An experimental study of changes of opinions about intransitivity caused by observation of deterministic intransitive objects was conducted. Aim of the experiment was to investigate influence of adults’ observation of triads of objects (The Intransitive Mobile Assault Towers and The Intransitive Gears) being in intransitive relations of superiority, on changes of their opinions about possibility/impossibility of existence of other "intransitive" objects in different domains. It was shown that opinions about intransitive relations of superiority were domain-specific: the participants thought that objects, being in intransitive relations, are possible in some domains and impossible in other domains (though in reality they are possible in the other domains as well). The transitivity axiom was applied by the participants very selectively, if at all. In many situations it was the right solution. Demonstration of different "intransitive" objects caused different effects. Some of the models provide more opportunities for right generalizations (have more "heuristic power") than others. Both types of models can be used for investigation of understanding of transitivity/intransitivity.
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Full-text available
A most interesting area in transitivity / intransitivity relations consists of relations like dominance / subordination, superiority / inferiority, preferences, etc. If A dominates B and B dominates C, must it be so that A dominates C? If A is superior to B, and B is superior to C, must it be so that A is superior to C? What happens if superiority/inferiority (dominance /subordination, etc.) relations form a cycle, an intransitive loop? Human rationality is often assumed to be based on the logical relation of transitivity. Yet, although transitivity fits relationships between physical objects or human decisions about targets that are independent of one another, it fails to fit the phenomena of systemic and developmental organization. Intransitivity has been shown to be present in various kinds of systems, ranging from the brain to society. In cyclical systems transitivity constitutes a special case of intransitivity. In this chapter, we examine different forms of emergence of intransitivity cycles, fixation of transitive parts in these cycles, and the organization of different levels of reflexivity within the systems. We conclude that reflexivity of cognitive processes— rather than transitivity in specific forms of thought—is the defining criterion of rationality.
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With nontransitive dice, you can always pick a dice with a better chance of winning than your opponent. There are well-known sets of three or sets of four nontransitive dice. Here, we explore designing a set of nontransitive dice that allows the player to beat two opponents at the same time. Three-player games have been designed before using seven dice. We introduce an improved three-player game using five dice, exploiting a reversing property of some nontransitive dice.
Intransitive dice. Mathematics Magazine
  • B Conrey
Conrey, B. et al. (2016). Intransitive dice. Mathematics Magazine, 89(2).
A potential new Polymath project: intransitive dice
  • T Gower
Gower, T. (2017). A potential new Polymath project: intransitive dice. https://gowers.wordpress.com/2017/04/28/a-potential-new-polymath-projectintransitive-dice.
Intransitive machines
  • A Poddiakov
Poddiakov, A. (2018). Intransitive machines. https://arxiv.org/abs/1809.03869