Class? is an enjoyable card game aiming at grouping colourful cards into meaningful classes. It illustrates facets of reasoning with classifications. In order to introduce Class? progressively, this small gamebook provides a sequence of games before getting to the Class? game itself and beyond. The games are presented in increasing order of difficulty so that a game will benefit from mastering of previous ones.

Introduction

Class? is a card game that we introduced in order to illustrate our work in knowledge representation and evolution. It is based on a set of 81 cards, all different, representing a pattern. This pattern is determined by four characteristics that each takes exactly one value among three. Those are: -the number: 1, 2 or 3 elements, -the color: red, green or blue, -the shape: square, triangle or round, -the filling: empty, striped or filled.

For example, one striped green square , three filled red triangles , or two blue empty circles .

Although Class? is itself fun to play it requires to master some simple and useful notions with which not everybody is familiar. These are the notion of classes of cards defined by shared characteristics, order relations between such classes, etc. Such notions are worth introduced independently from the Class? game and this may be done using the Class? material. This can be done in an entertaining and intriguing way.

Moreover, we noted that the material available to us to teach important ideas of computing were very often based on set theory and arithmetics, but were not very well suited to draw the interest on partial orders, yet a very useful tool.

Hence we designed this small Class? gamebook as a collection of games which use the Class? material and introduce progressively one of these concepts at a time. Some of these games are real games that may provide a lot of fun and others are mere exercises illustrating important techniques. They also can be used as a demonstration to open a discussion. This small book is addressed at those who would like to initiate others to these simple games and concepts. It tries to provide information useful for doing so. All games are played with the Class? deck of cards and sometimes with classification boards. They may require some pens, papers and the template presented in Appendix A.

The games are (totally) ordered so that each one relies on mastering of the previous one. Each game is systematically described along the following outline: Audience the targeted audience; Duration the durations of the exercise; Resources the material support needed; Learning outcomes the goal of the current exercise; Prerequisites the skills that are required to do the exercise; Skills which skills are learned through this exercise; Game play the rules of a game; Example an example of how the game is played;

Learning outcomes

-Learn what a class is and how to describe it; -Learn that cards (objects) may be grouped together into classes, defined by a membership relation; -Learn that there are different classes to which the same card may belong, there is no unique class.

Prerequisites

-Know how to describe the characteristics of the cards: shape (circle, square and triangle), number (1, 2 and 3), color (green, blue and red) and texture (filled, striped, empty).

Skills

-Analyzing differences and similarities.

Game play

1. Pick two Class? cards that are different on up to three characteristics (color, shape, texture or number).

2. Ask what the cards have in common and what is different between them.

3. Describe the smallest class to which they belong: it is defined by what they have in common. 4. Pick two other cards, one which shares the common characteristics and one which does not, and ask whether they belong to it or not. 5. For the card that does not belong: can we describe the class to which it belongs, together with the other cards? How? 6. Repeat the operation.

Example

1. Pick two cards and .

2. They have their shape (circle), number (2) and texture (filled) in common, but their color is different (blue and green). 3. The smallest class to which these belong is the class described by all the cards with two filled circles. This can be indicated as follows: . 4. Pick two other cards and . The second card belongs to that class but the first card does not. 5. The class to which all the cards belong together is the class described by cards with two filled figures. This can be indicated by: . 6. Repeat.

Science

Cards may be grouped in classes, representing a set of cards with specific characteristic values. For example, represents the class of all cards having two blue elements. To denote that the shape and the filling are not specified for this class neutral shapes and patterns, that cannot be found on any of the cards (here star and dotted), are used. These can be replaced by any value (square, triangle or round; empty, striped or filled).

For any set of cards, there is a class to which they belong together. Appendix A provides another way to conveniently represent classes.

The example teaches the following membership equations (noting by ∈ that cards belong to a class):

, ∈ , , ∈ , , , ∈
but also (with the same cards):

, , ∈ , ∈ , ∈
We learn that there are different classes to which a card belongs. For example, belongs to but also to and .

1.10 Questions 1.11 Thoughts to be shared -Cards may be grouped in different ways to form classes, there is no unique way.

-But the smallest class is unique, it is the class described by the characteristics that the cards have in common. -Even cards with no similar characteristic belong to a common class: the class of all cards.

-The order is irrelevant to determine the class of the cards.

1.12 Knowing more/doing more -Try Card splitting classes.

Game 2

Card splitting classes

Audience

Ages 8 and up.

Duration

15 minutes.

Resources

-Cards.

Learning outcomes

-Learn how to create a discriminating class.

Prerequisites

-Know what a class is (What is a class?).

Skills

-Analyzing differences and similarities.

Game play

1. 6 cards are laid out in front of the players. 2. They each choose a class such that three of their cards belong to it and the three others do not belong to it. If it happens that no such class can be found, two further cards from the deck. 3. The first player to provide such a class wins.

Players can play in turn.

Example

Science

With respect to the previous game, 'Card splitting classes' introduces the capability to exclude cards from a class. So, instead of defining a class through the common characteristics, it is also possible to define them through the non common ones.

Trying to find a class discriminating between two sets of cards requires to isolate those characteristic value combinations which applies to only three of the cards. The task is difficult because there are already a large combination of sets of three cards.

Questions

1. ⋆ How many combinations of sets of three cards are there? 2. ⋆⋆ Is it always possible to find such a class? 3. ⋆⋆ If it is possible, is the class necessary unique?

Thoughts to be shared

-A class determines the set of cards belonging to it as well as the set of those non belonging to it (the others).

2.12 Knowing more/doing more -Try Guessing classes.

-You could have a look at the SET! game1 , in which the goal is to find a class of three cards out of 9.

Game 3

Guessing classes

Audience

Ages 12 and up.

Duration

15 minutes.

Resources

-Cards; -Support for noting classes (see Appendix A).

Learning outcomes

Prerequisites

-Knowing how to create a splitting class.

Skills

-Analyzing differences and similarities.

Game play

1. Each player receives 6 cards.

2. They each choose a class such that three of their cards belong to it and the three other ones do not belong to it (splitting class). They keep their class secret (see Appendix A). If it happens that no such class can be found, they can draw two further cards from the deck. 3. One by one, the players ask the other player for a positive (belonging to the class) or negative (not belonging to the class) card for the chosen class of the other player. 4. If the player has no card for the asked category (positive or negative), the player takes cards from the deck and assigns them to the piles until the player finds a card that is asked for (positive or negative). 5. The first player to guess the other's class wins.

Example

Variations

-It is possible to play with the cards visible and the player asks for each card if they are positive or negative. -The card to be disclosed may also be taken randomly, i.e. not asking for positive or negative.

These variations should allow for less strategy. On the contrary, if the players can choose any card to give as example, this can be used strategically (and may last longer).

Science

With respect to the previous game, 'Guessing classes' requires to keep track of the examples in order to establish the possible classes covering the positive examples and excluding the negative ones.

For instance, , belongs to many different classes. Adding (as another positive example) would determine several common classes such as , , , and . Said differently, each example separates the set of all possible classes into two sets: those classes to which it belongs and those to which it does not belong. The former is selected if the example is positive; the latter if it is negative. Hence, a set of positive and negative examples selects the set of classes to which all positive example belong and none of the negative example belong. 3.13 Knowing more/doing more -This can be understood as a variation of the Mastermind game (in which this is not the player that proposes the result). Can you provide a variation closer to Mastermind? -Try Subclasses.

Questions

Resources

Learning outcomes

-Learn how classes relate to each other and may be minimally extended.

Prerequisites

-Know what a class is (What is a class?).

Skills

-Relations between classes.

-Transitivity of the relations. -Minimality.

Game play

1. Each player is given five cards which are not shown to the others; 2. Each player in turns puts one card in front of her so that the class corresponding to all the cards on the table is larger than what it was before; 3. The game stops when no player can put a card on the table; 4. The player with less cards in her hands win. During this game it is worthwhile that players tell the class they define aloud.

Example

The picture displays the four cards that each player has and, progressively, how they put them on the table. On the exterior, is displayed the corresponding class. 5. The right-hand player wins: she has no card anymore. The subtlety of the game consists of extending classes in a minimal way so that more cards can be played. The cards must be played in a relevant order.

Variations

-It is possible to start with more cards than 5. This will introduce more strategy.

Science

This game introduces an important relation between classes: subclass.

We say that is a subclass of (or a more specific class than) and note it as follows:

⊑ . This is very general in fact, ⊑ ⊑ ⊑ ⊑ .
When a class is a subclass of another, it freezes additional characteristic values over the other, and its elements (cards) are elements of the other. 4.12 Thoughts to be shared -Classes may be related by a subclass relation, where the superclass is the generalization of the subclass, and the subclass a specialization of the superclass. -These relations reflect both the definition of classes (they have increasingly selective) or the inclusion of the set of cards they describe.

Questions

4.13 Knowing more/doing more -Consider Disjoint classes.

Game 5 Disjointness

Audience

Ages 12 and up. This game is more challenging to understand than the previous ones and even the two next ones.

Duration

30 minutes.

Resources

-Cards; -(Possibly) support for noting classes (see Appendix A).

Learning outcomes

-Learn how classes may be specified positively and negatively.

Prerequisites

-Know what a subclass is (Subclasses).

Skills

-Relations between classes.

-Computing intersection.

Game play

1. Each player is given four cards which are not shown to the others;

2. Each player has two cards on the table: one corresponds to the positive (+) class; the other to the negative (-) class; 3. Each player, taking turns:

-either, adds one card to his or her positive class; -or, adds one card to another player's negative class; -or, draws one card from the deck and either plays one of the above actions, or adds it to his or her hand. A card is added to a class only if (a) it belongs to the class, or (b) it extends the class by one single characteristic and there is no card belonging to both the negative and the positive class. 4. The game ends at the end of the first round in which one of the players cannot expand his or her class or in which the deck is empty. This happens when one class has three indifferent characteristics, sometimes before. It is not possible to have four because this would necessarily cover cards in the negative class. 5. The player with the larger class, in terms of cards belonging to the class wins. In case of ex-aequo, the one with the least cards in her hand wins.

Example

The picture displays the initial situation (0) and for each turn the added card and the class corresponding to each sets of cards.

-+ + - Now it is clear that the first player's negative class cannot be extended anymore: it covers all square cards (one third of the cards). Its positive class is the class of all cards containing two circles. It still can be extended, to all circles (but not to all cards with two elements which would cover cards of the negative class).

On the second player's side the positive class contains all cards with one single elements and empty filling and the negative all blue stripped cards. They both can be extended (for the positive by having arbitrary numbers and for the negative side by having arbitrary colors).

Variations

-It is possible to start with empty positive and negative classes: that would probably be more fun and more strategic. -It may be possible to play several cards at once at each run, still with the same constraints.

-The winner may also be the first to finish.

Science

Game 6

Identifying with classifications 6.1 Audience Ages 8 and up.

Duration

15 minutes.

Resources

-Cards; -Classification boards (either printed or generated on computer).

Learning outcomes

-Learn how to find the class of a card in a classification correctly, and if possible efficiently.

Prerequisites

-Know what a class is (What is a class?).

-Know what a classification is (Subclasses).

Skills

-Applying a procedure.

-Matching an instance to a class.

Game play

1. One classification is shown openly; 2. One player draws a card from the deck; 3. The other player asks, in turn, questions among 'which colour is the card', 'what shape is the card?', 'which filling is the card?', 'how many elements are there in the card?' 4. The first player able to tell to which leave of the classification the card belongs wins.

Example

Here is the classification to be used:

A player draws the card .

1. One asks 'what colour is the card?', the player answers 'green'. 2. Another one asks 'what shape is the card?', the player answers 'square'.

3. Yet another one asks 'how many elements are there on this card?', he answers 'two'. Then this player says: it belongs to class .

Variations

-It is possible to change the game so that the questions become 'is the card red?', 'does the card has two elements?', 'are the elements of the card filled?', etc. This variant makes the game less predictable, but it does not really teaches the identification procedure. -Each player may have a hidden classification. These players will ask questions in turns. The first one to have reached a leaf class in her classification wins. Of course, they can take advantage of the answers to the questions of the others.

Science

Classes may be organized into a classification: a tree of classes (growing from top to bottom) like the one below, in which ⊑ is expressed by having the subclass under its superclass.

Each fork divides all the cards belonging to the class into three subclasses according to a single characteristic: the number, the color, the shape or the filling. The top class contains all the cards. And each class contains fewer cards than the class that precedes it and all the class cards that follow it (from top to bottom). On the same level of the tree, the criteria may be different.

In such a classification, each card belongs to one and only one leaf class (class without subclass) also called its most specific class.

The game is teaching how to proceed step by step to find the most specific class of a card in a given classification, using the subclass relation. In the example above that is:

∈ ∈ ∈ ⊒ ⊒
The number of decisions to take (here 2) is far smaller than the number of classes in the classification.

Questions

1. ⋆ What is the maximum number of steps to make to reach a leave? 2. ⋆ Can there be two cards classified in a leaf class? 3. ⋆⋆ Does the card always belong to the same class in all classifications? 6.12 Thoughts to be shared -Classes can be organized into a classifications via subset relations.

-In different classification, cards can belong to different classes. But in one classification, there is a unique class to which the card belongs. -There is a procedure for finding the most specific class of a card (start from the top and choose the correct branch each time to find the most specific class). -It is very similar to games that one would play on a holiday trip (Is it red? Is it an animal? Does Lucy have one?).

Knowing more/doing more

Duration

20 minutes.

Resources

-Cards; -Support for drawing classifications (paper and pen). This game requires the ability to unambiguously draw a classification.

Learning outcomes

-Learn how relations between classes can be used to form full classifications.

-Learn that simple procedures may be used for developing full classifications from data.

Prerequisites

-Know how classes may be related (Subclasses).

-Identify the class in a classification to which a card belongs (Identifying with classifications).

Skills

-Applying a procedure.

-Relations between classes.

-Transitivity of relations.

Game play

1. Each player has 4 cards, visible to all players.

2. Initially, the classification is reduced to its top:

Each taking turns:

-can assign a card to a leaf class in the classification, only if no card is assigned to it, or -can expand one leaf class of the classification to which a card is assigned, by deciding the characteristic along which this class is further split. The assigned card is reassigned to the new leaf class it belongs to. 4. The first to assign all his or her cards wins.

Example

The initial situation: each player has four cards; the classification only has its top level. Player 1 has won. This is the final classification.

Variations

-It is possible to play the same game with hands hidden from the other players;

-It is possible to play the same game with hands as decks from which players draw cards one by one, when they do not want to expand a leaf class. -There could be a penalty (draw another card) when expanding.

-Or, on the contrary, a player may continue to play, as long as she creates subclasses.

Science

The game teaches how to create classifications that distinguish between all cards on the table and only these. This can allow later to identify the objects when they are encountered again.

In this game, the classification is partly determined by these cards: there is no need to make distinction that do not discriminate two classes. It is also determined by the way players play: from the same starting point, depending on their actions, there may be different classifications.

This shows that there is not one unique classification: several such classifications may be equally useful for discriminating objects. This is the way many machine learning algorithm work: learning to discriminate objects by using their different features.

This also shows that these classifications are not universal, they depend on the cards to discriminate and, if new cards are drawn, the classification may be extended.

Questions

1. ⋆ What is the maximum depth (number of successive forks) of a classification? 2. ⋆⋆ What is the smallest number of leaf classes in a classification of maximum height? 3. ⋆⋆ And what is the maximal number of leaf classes? 4. ⋆⋆ What is the number of classes in the largest classification? 5. ⋆⋆⋆ How many largest classification are there? 6. ⋆⋆⋆ How many classifications are there? 7. ⋆⋆⋆ Is the resulting classification dependent from the card played and their order or will this produce a different classification?

7.12 Thoughts to be shared -It is easy to build a classification from observation. By splitting classes with one separating characteristic, and starting from the top-class with no characteristic set, it is possible to create classifications in which each card belong to exactly one most specific class. -There does not seem to be a single classification.

-Classifications do not need to be 'complete' to be used.

Duration

45 minutes-1 hour (depending on the amount of rounds played).

Resources

-Cards; -Classification boards (either printed or generated on computer).

Learning outcomes

-Learn that without knowing the classification, it is possible to properly assign cards to classes.

Prerequisite

-Know what a class is (What is a class?).

-Know what a classification is (Subclasses).

-Identify the class in a classification to which a card belongs (Identifying with classifications).

Skills

-Logical reasoning.

Game play

This is the first part of the Class? game. It should now be easy to play. 1. Each player has ten cards, one player has a classification (that is kept secret and must be respected during the game). 2. One card from the deck is turned, this is the first class. 3. The player with the classification starts. 4. Players play turn by turn placing one or more cards from their hand on the table that must belong to the same leaf class:

-either by adding them to a class already on the table, -or by creating a new class. The player with the classification can create a new class only if he cannot add to an existing class. When creating a new class, players must then draw a new card from the deck that they add to their hand. 5. At each turn, the player knowing the classification tells if this is correct or not with respect to the classification. If not, the player takes his cards back and takes one additional card from the deck. 6. The first player with no card wins. 7. At the end of the game, each player scores as many points as there are cards left. After several rounds (players may exchange roles), the player with the least points wins.

Note

-All the cards that are played in one turn must be played to the same leaf class, -Players are not required to play all their cards belonging to a leaf class at the same time.

Example

We take inspiration from the example given in the rules for the game. At the beginning (the guesser is left), the knower (right) has the classification of Section 7.10, p.25).

One card is drawn from the deck ( ) to make the first class:

The knower (right) is obliged to add the unique card from this class that he has ( ):

The guesser (left) adds one card ( ) to the class, it is correct:

The knower (right) cannot complete the current class, which is already complete. So she creates another class with three cards featuring green circles ( , , ) and draws one new card from the deck ( ):

The guesser (left) plays three cards (all its circles: , and ):

This is incorrect: he takes his cards back and draws a new card from the deck. And so on until one of the players has exhausted her cards.

Variations

-The knower may announce the number of leaf classes at the beginning.

-The game may be played with multiple guessers.

-It is possible to divide the set of cards in half and play two games in parallel, since in general we do not exhaust all the cards. -It is possible to play twice with the same classification but with two different guessers.

Science

There may be many different classifications over the same set of cards. However, it is possible through trial-and-error to find out if cards are classified together or not.

It is even possible to develop strategies in order to find this more surely. For instance, once establishing that some characteristic values are possible in a class (for instance because it contains it can contain any of its characteristics values: one green empty circle) and that other are indifferent (for instance color and shape because it also contains ), then it is possible to attach .

This kind of reasoning can also be played by inducing the most specific class to which the two former cards belong: . It is then clear that the latter card belongs to this class as well. It does not mean that the class is described by , for instance, it still may be or .

Similarly 8.12 Thoughts to be shared -Everyone can organize their knowledge in the way that seems most useful for them, the most appropriate: there is no "good" representation. -This does not prevent from learning very quickly how cards must be classified.

8.13 Knowing more/doing more -Try Guessing the classification.

Game 9

Guessing classifications

This is the second part of Class?.

Audience

Ages 10 and up.

Duration

Resources

-Cards; -Classification boards (either printed or generated on computer).

Learning outcomes

-Learn that from a few examples it is possible to guess a hidden classification.

Prerequisite

-Know what a class is (What is a class?).

-Know what a classification is (Subclasses).

-Identify the class in a classification to which a card belongs (Identifying with classifications).

Skills

-Logical reasoning.

A first remark is that the cards in the blue class vary in number, shape and filling. Hence, the blue class is not further subdivided.

Then, on the green side, the three classes have homogeneous shapes and at least one of them differs from filling and numbers. Hence, the dividing characteristic can only be the shape.

Finally, on the red side, there are four classes, hence subdivision. One of these classes contain heterogeneous shapes ( , and ), hence the shape does not divide the classes. Remains the number and the filling. However, two classes ( and ) differ only on the filling of their cards.

Hence, the first dividing characteristic is necessary the filling. Then, there are two classes with same filling (full), hence the class is necessarily subdivided. Since, one of these has heterogeneous shape, their only differing characteristic is number, thus it is divided by number. This information leads to the following classification:

Two observations can be made here. On the one hand, the classification is not yet complete: the class is further subdivided, but the players cannot know this because they have only one card ( ). This reflects uncertainty with respect to the status of this class on the side of the players. A similar uncertainty occurs for classes , , and . This is the reason why they have been kept smaller, although they are leaves at the moment: there is no way to know if they are further subdivided or not.

On the other hand, the players know that the class is not futher subdivided because the cards belonging to it ( and ) do not share other characteristic values than those of the class. The same observation holds for classes , and for similar reasons.

Having more cards on the table obviously helps to decide if the classes are further divided. But it is not always easy to decide. For instance, consider that there is an additional class based on a single card:

. This would help players to find out that is further subdivided because the two cards belonging to it ( and ) are assigned to different classes. However, because they differ on two characteristics, number and filling, it is not possible to know which one is the dividing characteristic. Hence, the players know that there are subclasses, but they also know that they do not know which ones.

Variations

Science

By not knowing an exact classification, but only some groups of cards belonging to the same leaf class, it is sometimes possible to guess the actual classification. Sometimes, the groups of cards do not provide enough information to do it, or to do it with certainty, but often it is possible. Sometimes it is also possible to be sure of the unique classification leading to such leaf classes. In this case they provide a proof of this.

Questions

1. ⋆ Is it always possible to guess the classification? 2. ⋆ If we exhaust all the cards of the deck, do you think we will know exactly what the classification is? 3. ⋆⋆ Can this capability to guess the classification be useful to play Classifying in an unknown classification? 4. ⋆⋆⋆ Could you design a procedure to build the minimal valid classification? 9.12 Thoughts to be shared -It is possible to guess a classification from a few examples with yes/no answers. -There are situations in which we do not guess the full classification, because some classes may not be played at all. Even though, we can still communicate. We do not need complete knowledge to communicate. -Even if we use all the cards, we might not know what the classification is because we do not know in what way the classes are organized.

9.13 Knowing more/doing more -Check the Class? rules.

-Check out Future developments (beyond Class?).

Game 10

Future developments (beyond Class?)

Various more sophisticated games may be considered from here:

Teaching classifications The goal of a player is that the opponents learn the classification (without words of course). This should help understanding the importance of disclosing information or not. Evolving classifications Players may modify their classifications to better communicate with others... Does this lead to the same classification? This game could be very related to https://classic.csunplugged. org/wp-content/uploads/2014/12/PhylogeneticsUnplugged.pdf. Ordering classifications It is possible to order classifications, because some are more precise than the others. It would also be possible to take advantage of identification in one classification, to identify in another. This could be the basis of yet another game. Matching classifications Both players have a classification and should build an alignment between their classification through guessing the position of cards in each other classifications.

It is also possible to render the games more interactive. For instance, it is possible to introduce the possibility to ask questions to the oracle. These could be questions like: 'does this card belong to this class?' Should it be one question by turn? By game? Should these cost points?

It may also be possible to introduce gossip by allowing players to communicate information, true or false, in order to help each others.

For each of these games, it is possible to ask for an algorithm that plays the game. Algorithms may embed explicitly the strategies developed by players.

On a more 'plugged' side, we have developed a computer program to play Class?. It features automatic players with different strategies. It also provides an application programming interface (API) so that anyone can implement their own strategies and see if they work well. This may be the occasion for 'hackathons', contests and experimenting with learning strategies. This may be taken as an opportunity to introduce different techniques such as constraint programming or general game playing in order to program the game.

-

  Learn how to guess a class from a set of examples and counter-examples (examples to show the opposite).

1. 2 .

 2 ⋆⋆ How many cards are necessary to determine the class of all cards, ? ⋆⋆ Do the examples and counter-examples always correspond to a unique class? 3. ⋆⋆⋆ How many cards do we need to play to be sure to have determined the class?3.12 Thoughts to be shared -Classes can be guessed from examples and counter-examples. -A set of examples and counter-examples may correspond to several classes: the cards of Player 2 in the example that belong to the class may describe the classes , or .

4. 2

 2 Duration 20 minutes.

  ) support for noting classes (see Appendix A).

3 . 4 .

 34 left-hand player puts down her card and the right-hand player ; The corresponding class is the one containing only these cards; 2. The left-hand player puts down her card and the right-hand player ; in the former case, this extends the class by two characteristics (shape and filling); in the latter, only shape has been extended; The left-hand player put down her card leading to the class of all cards: ; the right-hand player plays ; this extends the class by two characteristics (color and number); The left-hand player cannot further extend its class which already covers all cards; the right-hand player can play leading to the class of all characteristics;

4 . 5 . 6 .

 456 first player (left) cannot add a card to either her positive class or the other player's negative class.She drawsfrom the deck and puts it in her positive class. 2. The second player (right) cannot add to his positive class, but can add to the negative class of the other player, which he does. 3. The first player now has the choice to add or to his positive class. He decides to add . The second player (right) still cannot add to his positive class, but can add to the negative class of the other player, which she does. The first player cannot anymore add to his positive class. Indeed, the resulting class would cover the two filled blue squares now belonging to the negative class. He cannot either add anything to the other player negative class. He draws the card or from the deck that he cannot use. Hence, he keeps it in his hand. The second player cannot play anything, he draws from the deck which she may either add to his positive class or the other's negative class, he chooses the former. 7. The first player draws from the deck that he adds to the second player's negative class. 8. The second player draws from the deck that he can add to his positive class, but this does not change the class. 9. Now the first player can play in the negative class of the second player. 10. The second player cannot play to his positive class because the resulting class would cover which belongs to the class! She thus draws from the deck, that he can play to the negative class of the first player.

7. 13

 13 Knowing more/doing more -It is time to play with the Class? game of Classifying in an unknown classifications.

=

  how it is possible to compare two such classes? It is possible to see that class is subsumed by class (or class subsumes class ) because one can go from the template of class to that of class , by only pushing ticks outwards (or inwards for subsumes).This is not possible betweenThe template can be found from the Class? web site as all other material.

  A first set of cards is laid out on the table as:

	Player 1 proposes red cards (	), Player 2 proposes cards with two elements (	), Player 3
	proposes cards with two filled elements (	). Players 1 and 3 are correct, but not Player 2 since there
	are 4 cards with two elements.
	A second set of cards is drawn as:
	Player 1 proposes filled cards (		), Player 2 proposes green circles (	), Player 3 proposes
	filled circles (	). Players 1 and 2 are correct, but not Player 3 since there are only two filled circles.

  1. ⋆ Are each pair of classes in subclass relations? 2. ⋆ Is there a class that is in subclass relation with all other classes? 3. ⋆⋆ What other relations between classes can you think of? Give examples.

  This game brings together disjointness and subclass.When in disjointness relation, no element of one class belongs to the other. There are connections between disjointness and subclass: For any classes C, C ′ and C ′′ , if C ⊑ C ′ and C ′ ⊥C ′′ then C⊥C ′′ . For instance, from Are classes containing one single card necessarily disjoint (have no card in common)? 2. ⋆⋆ Could both players have the same positive class in the end?5.12 Thoughts to be shared -Not all classes are related via subclasses, there are pairs of classes that are different but have overlap or that have no card in common (this is disjunction). -It is possible to expand two classes under the constraint that they remain disjoint.

		We say that	is disjoint from	and note it as follows:	⊥	.
				⊥	and	⊑	one can deduce
	⊥	.
	5.11 Questions
	1. ⋆ 5.13 Knowing more/doing more

-Try Identifying with classifications.

  ⋆⋆⋆ What happens if you delete a card category (for example all red cards)? And what happens if we add one? 2. ⋆⋆⋆ Can the game work if instead of the question "What color is the card?" we ask ourselves "Is the card green?". What would be the consequences?

	, if another class contain the card	then not all card containing one element are in the
	same class. It is also sure that	belongs to yet another class.
	8.11 Questions
	1.

  -To play this game independently from Classifying in an unknown classifications, it is possible that one the knower draws a classification and 15 cards that she dispose on the table according to the classification. Then the guessers have to guess it. -Another game would consists of drawing a classification and 16 cards at random, and arranging them on the table according to a classification. Then, each player, in turn, have the right to withdraw one card (alternatively, as many cards she wants) from one single class, so that it is still possible to guess the classification. The last player to withdraw a card wins. -This game suggests another game, that can be played alone: take a classification, put on the table a minimal number of cards, arranged in classes, which correspond to only one classification. Is it always possible?

https://www.csunplugged.org/.

https://members.loria.fr/MDuflot/files/med/index.html

https://setgame.com

Do not hold your breath too long, there is no plan for a big book. You'd better try this one now.

Acknowledgements. The development of Class? has been supported by INRIA.

The Computer science unplugged project 1 (or tangible coding in another style) obviously inspired this game collection. We also took inspiration from the page of Marie Duflot-Kremer 2 for the description of exercises.

Game play

1. The table features several leaf classes as sets of cards. This may be the state of the table at the end of playing Classifying in an unknown classification. 2. Each player must describe the classification corresponding to these classes. 3. The player who describes it correctly wins.

When part of the Class? game, in the event of a tie, the player who guessed the classification correctly most often wins.

Note

-The description of the classification may be given by explaining by which characteristic each class is split starting from the top class (see Example).

Example

Consider the state of the table at the end of the example of the previous game. It features two classes:

What can be stated about the actual classification (Section 7.10, p.25)? By analysing, these two classes, it can be observed that the left-hand class has cards with the same number (1), same colour (red) and same filling (filled), and the right-hand side class has cards with the same colour (green) and the same shape (circle) but different filling and number. A classification starts from the top by dividing classes along one single characteristic. From these two classes it is possible to deduce that this characteristic is not shape, from the left-hand class, and not number nor fillings, from the right-hand class. Hence, it can be deduced that the first characteristic along which the top is divided is colour. It is thus this way: Actually, it is not possible to deduce more from these two classes.

For the sake of the exercise, consider that all cards that the players had in their hands, have been dispatched. This would lead to the following classes on the table :  Appendix A

Class template

In order to avoid cheating, it is possible to write down the chosen class, or mark it with the following template:

It describes the class characteristics in a square divided in four along the diagonals: -top: filling; -bottom: colour; -left: number; -right: shape.

Appendix B

Answers to questions

Here are the (very) short answers to the questions. They are here for the reader to confirm answers.

If you do not agree with some answer or think that you have better questions, think about it and if you are convinced, contact us. Detailed answers to questions may be available elsewhere.

B.1 What is a class?

1. ⋆ Is there a unique class to which the cards belong? No. 2. ⋆⋆ Is there a unique smallest class to which the cards belong? Yes. 3. ⋆⋆ What if we pick cards that do not share any characteristic (color, shape, texture or number), for instance and ? Is there a class to which they both belong? Yes.

4.

⋆⋆⋆ Is the class chosen in step 3 of the game dependent on the order of picking cards? Is there a difference if we first pick and then ? No.


Bernard Guy
email: guy@emse.fr

Does movement precede space and time? The contribution of relational thinking and embodied cognition

Keywords: space, time, movement, embodied cognition, relational thinking, epistemology, World 3.0

Movement precedes space and time that are derived from it. In order to justify this proposal, one must guard against two temptations: a) that of believing that reality imposes on us in a univocal way the words to describe it, each one referring to the proper qualities of the corresponding objects; b) that of believing that the concepts can reach a state of purity that abstracts them completely from human knowledge, rooted in the functioning of the body. In response, two dimensions of thought are particularly adapted: 1) a relational dimension: one can only compare phenomena to other phenomena (movements to other movements) and conventions are necessary. Space and time are the names of these comparisons; 2) an embodiment dimension: the concepts of space and time have a meaning first in the experience of the body inside the world (lived in terms of movements), before the words. Our proposal is understood within the plane described by these two dimensions, and inscribed in the history of philosophy. The way to express the objectivity of embodied knowledge is discussed. The numerous consequences of the proposal to better link concrete space and concrete time are listed; they will be developed in further works.

Introduction

We discuss here some aspects of the relations between the concepts of movement, space and time, and our basic proposal: movement precedes both space and time that are derived from it 1, 2 . Although somewhat paradoxical, this way of understanding works and opens many avenues. In some respects, some elements of the proposal are not new. But we insist that remarkable consequences appear, not only for the humanities and the social sciences, but for physics itself; they are, and will be, developed elsewhere.

In order to do justify our statement, it is necessary to go back to some kind of philosophy, and discuss the way we build our words, our concepts, our knowledge, trying to obtain a correspondence with the world. We must do with an epistemological deepening which brings to light the hidden foundations. On this level, we think it is essential to guard against two temptations: a) that of believing that reality imposes on us in a univocal way the words to describe it, each one referring to their own qualities, for themselves, of the corresponding objects; b) that of believing that the concepts can reach a state of purity that abstracts them completely from human knowledge, which is rooted in the functioning of the body.

In response to these two temptations, two dimensions of thought are particularly adapted to our study: 1) a relational dimension: one can only compare phenomena to other phenomena (movements to other movements) and conventions are necessary. Space and time are the names of these comparisons; 2) an embodiment dimension: the concepts of space and time have a meaning first in the experience of the body inside the world (lived in terms of movements), before the words. These two dimensions provide an indispensable grid of analysis, defining two axes of a plan where various mental maps can be drawn: in this scene we can express our proposal and compare it to a whole series of proposals provided by the history of thought.

In the following (second) part of the paper (materials and methods), we will describe our twodimensional approach used to discuss space-time-movement issues. In the third part (results), we will present our proposal, and inscribe it in the history of philosophy and science. In the fourth part (discussion), we will make a stop on some epistemological issues referring to the status of our proposal, and briefly list the consequences that may be foreseen. We will end by some lines of conclusion.

Materials and methods: two dimensions of intellection

In figure 1, the first dimension is marked by the horizontal axis putting in scene the relational mode of rationality, fundamental for our business: we are inside the world, we can only compare the phenomena to each other, without escaping the world nor qualifying its objects in themselves in a substantial way. We will thus oppose mobility to immobility: these words have a concrete character, without prejudging the existence, or the meaning, of concepts such as time and space. We can grasp the pair mobility / immobility from a human experience, extended in different ways by conceptual generalizations. They are qualified as relative, to underline, if need be, the comparative point of view, at a certain scale of speeds: a whole domain extends where to put the limit between the "concrete" times (on the mobility side) and spaces (on the immobility side) considered, variable according to what we are able, or want, to look at. We cannot avoid going in circles and making conventions (such as that between concrete time and space along the horizontal axis) [START_REF] Morin | Introduction to complex thinking[END_REF] . Words have relative meanings to each other, depending on the context, with inevitable uncertainties and fragilities.

The second dimension, according to the vertical axis, opposes two poles, which epistemologists, psychologists, etc., have identified in our appropriation of the world 4 to 15 .

We take this duality here for granted. It is about, on the one hand, downwards, the apprehension by the body, the gestures, the images and the imaginary, before the words; this mode includes, by extension, the experimental devices and their implementation in the measurements. And on the other, upwards, the use of words, concepts and all the discursive formalism. This dimension, contrasting words to images, is as much the one that opposes the universal to the singular, the whole to the part, the common name to the proper name, the philosophical to the physical.

Results: understanding space, time and movement

We can compare two main types of evolution in the representation plane (Fig. 2).

Our proposal

In figure 2A we find, in the lower part, the relational source of the movements compared to each other, which leads to a first step, with two concepts of concrete (experienced, measured) space ("stopped movements") and time ("continued movements"); both are concerned by the movements where they are rooted; they are not independent.

Continuing the evolution of thought up the drawing, we arrive at the independent fictions (à la Vaihinger [START_REF] Vaihinger | The philosophy of as if. Preface and translation by Christophe Bouriau[END_REF] ) of space and time; they get rid of the relational constraints and of the attachment to matter. It is interesting to consider each of them as an endpoint, after a whole path, rather than "a priori forms", or substances different by nature, pre-existing us (this avoids considering them as enigmas!).

Other proposals in the literature

Figure 2B presents the system most often encountered in the history of science, drawn in a very caricatural way: the concepts of space and time are not proposed by paths of the previous type (anchored in the movement of the body) and their fictional character is not clearly affirmed. Their genesis is not always made explicit, it varies according to the authors (a priori form, property of nature given to us and over which we have no control, etc.). Links exist between the concept of time and the movements offered by the world (lower part of the diagram), but they do not appear as constitutive of the genesis of time.

Let us give further details about the way some selected authors deal with these issues.

Aristotle (4th century BC) [START_REF]Aristotle[END_REF] . We can certainly wonder to what extent the projection of this author's system in our diagram is practicable, when for him movement has nothing relational but expresses a distinctive property of an object! In any case, what seems to us to be noteworthy is, on the one hand, the strong link between movement and time; on the other hand, as far as space is concerned, a vision very different from the one common today, opposing the imperfect sublunary world, where each body has its preferred place (where it remains motionless if it gets there), to the world of the celestial sphere whose perfect circular motion serves to measure time; this last motion is to be positioned in the upper part of the diagram. There are in Aristotle's works (the Metaphysics, the Physics) long discussions to define the separation between time and movement [START_REF] Zahar | The eternity of motion in Ibn BaggA (AVEMPACE): from generic to numerical definition. The commentary to chapters 1 and 2 of book VIII of the Physics[END_REF] . They are the premises of a true abstraction of time from movement, which will operate in Galileo 19 and Newton 20, 21 . The fact remains that we do not have in Aristotle anything resembling the equation movement = space / time, i.e. a space interval travelled during a period of time.

Classical China (a few centuries before and after Jesus Christ). It has been abundantly emphasized that in classical China, that is, from the first centuries before Christ until more recent times, there is no need for the concept of time 22 . Nor is there any separate space. On the temporal side (on the side of mobility), we are faced with an assortment of seasons, epochs; these are inseparable from a set of spatial domains. There is no movement in our Western sense, but a composition of processes in endless correlation and interaction (cf. Yin and Yang). Space and time are said to be "the world as it extends" and "the world as it lasts".

Isaac Newton (1687). It is interesting to look at Newton, who clearly shows, almost in spite of himself (he wants to free himself from it), the rooting of the concepts of space and time in concrete experience (in particular human experience). According to him, we must aim at something purer, truer, which he calls absolute time and absolute space, respectively. For us, these are fictions à la Vaihinger [START_REF] Vaihinger | The philosophy of as if. Preface and translation by Christophe Bouriau[END_REF] that have shown great fruitfulness. In the spirit of the present text, we can say: the link between experience and absolute space and time is not to be unlearned, as Newton asks: it is foundational and we must go back upstream, to experience. In Newton, space and time are separated.

Immanuel Kant (1781). Kant's a priori forms are another way of looking at the duality that the human mind needs in order to talk about the world and that it projects onto it 23 . To express oneself in this way (a priori forms) improperly gives to words an anteriority over the experience to be narrated. In Kant, space and time are always separated.

Ernst Mach (1903). This author expresses unequivocally the process of abstraction of time from the comparison of different movements, one of which will be chosen arbitrarily as a standard. The "Mechanics" should be read particularly clearly from this point of view 24 . One can notice that, as generally, space is not discussed; does Mach think it is self-evident? The two concepts of space and time are seen in a dissociated way, despite the fact that space is present in movement. Movement has its modern meaning as the travel of a piece of space in the course of time (whose speed is the ratio between the amplitudes of space and time involved).

Albert Einstein (1905). The original point of A. Einstein is that he does not try to define at length the concepts of space and time [START_REF] Einstein | On the electrodynamics of moving bodies[END_REF] . He takes them for granted and relies from the outset on instruments, rulers and clocks, and their working. The discussion of simultaneity, with a method of synchronization to test it, is also remarkable. This is already a way of linking time and space. With the two postulates, in particular the Second one, we arrive at the theory of relativity. The readings that we make on the rulers and the clocks are linked, but the corresponding concepts of space and time remain distinct. Clocks and rulers are idealized, the first are point-like, the second rigid, which is debatable. Note that, if Einstein had gone to the end of his idea, i.e. to look at how the instruments actually work, he would have opened the clocks and found that they were hiding a movement.

Henri Poincaré (1905). With Poincaré we always have two different natures for space and time, but this author (similarly to Mach) shows an interesting lucidity concerning the conventional choice of a standard (movement) phenomenon for time: it is the most convenient and not the most true 26 . H. Poincaré also deliberates on space, contrary to the authors already mentioned, and insists on the movements of our body (including those accompanied by muscular sensations) in the construction of space. This does not, however, bring time and space closer together.

Carlo Rovelli (2018). In his writings [START_REF] Rovelli | The order of time[END_REF] , this contemporary physicist does not stop on the concept of space (except to proclaim its quantization, or granularity, at the microscopic level); by contrast, time is more problematic, as it is for most authors, and is the subject of long developments. If it does not exist at the fundamental level, time appears at the macroscopic level, according to a so-called "thermal" emergence [START_REF] Connes | Von Neumann algebra automorphisms and time thermodynamics relation in generally covariant quantum theories[END_REF] ; it comes out on its own, without the question of the associated space being raised, while, on the other hand, the author proclaims relativity as the obligatory passage of any physical theory, time and space forming an inseparable whole. C. Rovelli does not address the idea of a deep link between space and time, nor that of a link between space and movement. Like space, time also has the quality of granularity. It remains to be seen how this is combined with the emergence of macroscopic time. Quantization appears as an a priori property of matter in itself. Following our own guidelines, we may ask whether quantization could rather be considered as relational, as an outcome of the comparison of phenomena at different scales in a probabilistic vision?

Discussion

Epistemological remarks: the « World 3.0 »

If our proposal extends the history of thought and relieves certain difficulties encountered in it, it nevertheless deserves an epistemological consolidation. Does embodied knowledge lead to an objective knowledge? We need find a way to name this knowledge, independently of all the ways to support it (images, gestures, neurons, etc.). To do this, let us take inspiration from the philosopher Karl Popper 29 to 31 . This author proposed to distinguish three "Worlds": World 1 of physico-chemical phenomena and material objects, World 2 of psychic activity and subjective experiences, and World 3 of objective knowledge (reduced mainly for this author to linguistic propositions allowing logical argumentation, demonstration, communication and debate; the facts are reported as propositions on the same level as the other propositions). On the basis of the previous reminders, we suggest dividing the "World 3" of objective knowledge into two worlds: -"World 3.0" refers to all objective knowledge outside words; and -"World 3.1." corresponds to Popper's "World 3" (Fig. 3). World 2 has overlapping domains with World 3, but it has its own parts that escape objectivity. Worlds 3.0 and 3.1 constitute two non-watertight poles, between which specialists (semioticians, psychologists, neurophysiologists, anthropologists, etc.) will find intermediaries (see, for example the epistemology of sign language). The whole question of the passage from gesture to words arises here [START_REF] Virole | Deafness and human sciences[END_REF] , and would deserve a complementary discussion: it is at the basis of the reflections of the present text.

There is a certain porosity between World 1 and World 3.0: but the latter is more than a "raw" sum of phenomena, more than an unconscious passive incarnation. If we speak about knowledge, it is because the phenomena appear there as put to the test, tried, experienced, repeated. The appeal to this substratum is made in a conscious, organized way, with a certain validation by the results (cf. the football player who goes to recover the ball at the very place where it is necessary, without writing mechanical equations; but who can also miss his shot!). This knowledge is "structured", as a form of intuition, or internal adhesion, guarantees it.

Shifts in meaning and misunderstandings

The meanings of the words time, space and movement, shift depending on where they are pronounced in our diagram. The horizontal shift corresponds to the concrete time-space transition according to the choice of a {spatial frame / standard mobile} pair. Vertically, we distinguish, on the one hand, concrete space and time (stopped movements / continued movements), chosen via a convention of stopping by a given community in a relational process, which is the only one practically possible; and, on the other hand, the fictions of space and time that allow us to think and say; the first behave like proper nouns, the second like common nouns. There are possible intermediaries between the ones and the others, varying according to the context, and they cannot be separated once and for all; hence the possible misunderstandings. But we commonly use the same vocabulary, it is the most convenient, it would not be practical to operate otherwise. The shift in meaning is difficult to avoid. We must have a tool with a certain generality (it gathers by abstraction all the singular objects inspected), and at the same time be able to speak of such and such a particular object, with the same word. Or again, to say a concrete relation between A and B, we need the fiction of separate A and B; and we still speak of A and B.

We can compare these language processes to what Vaihinger says about fiction, using the example of the word "tree". A "tree in general" (a common noun) is a fiction, and as such not free of contradictions: it is both green and not green (red or orange in autumn); it has both leaves and no leaves (thorns), branches or not, fruits or not, etc. But we arrived at this tool of thought after having seen and compared multiple plant objects. The tree in general does not exist: no scientific study can be made of it, no analysis. On the other hand, when we are faced with a particular object, the association of the word tree and the designation ("look at this tree") almost transforms the word into a proper name, on which we can effectively make an analysis. It is more convenient to speak of a tree, located at such and such a place (we also detach the tree from space and time) rather than finding a proper name for each tree.

In order to avoid misunderstandings caused by meaning shifts, we propose to use, at least momentarily, specific words, in order to make the status of our proposal clearer (Fig. 4). We can thus substitute the proposition "movement precedes space and time" by: "pre-concept of movement (and of space and time), situated in World 3.0, precedes the pure concepts of space and time (situated in World 3.1)". This formulation is a shortcut that hides several steps detailed in the figure.

Consequences foreseen in humanities and physics

If we believe there is some "efficiency" to the two dimensions of thought presented, it seems important to outline their numerous consequences, as an encouragement to their appropriation (remember Bergson: "an idea is enlightened by what it enlightens" [START_REF] Bergson | Thought and motion[END_REF] ). In all fields of thought, when it comes to deepening this or that situation, we consider that there are serious drawbacks to separating concrete space and time. Physicists have every interest in not letting themselves be intimidated by the mystery of fictional time, making the mistake of taking it for a reality. We must try as much as possible to go back to the source: spatial amplitude and temporal process are the two faces of the relation between movements 2 . It is fruitful to consider a velocity as a ratio of movements, a lapse of time as a piece of movement: it provides, for example, a key to discuss the twins paradox, to alleviate the problems of Wigner-Thomas rotations in the composition of non-collinear Lorentz transformations (time has a direction that projects itself like the other spatial directions), to take up the expression of the gamma factor of the Lorentz transformation (after opening the clocks and taking into account the movements that are hidden there [START_REF] Guy | The Lorentz transformation, time and space. Generalization of the gamma factor as a function of the hidden movement in the clocks[END_REF] ), or to discuss the abnormal velocities involved in black matter and dark energy [START_REF] Guy | Revision of the status of "light velocity" and examination of some cosmological problems[END_REF] . It is fruitful to consider an amplitude of space as a piece of movement, it allows in particular to understand the definition of the meter. It is fertile to put time and space on the same level as the different pairs of quantities of physics; this allows, among other things, a new look at the relations between quantum mechanics and general relativity 36 , at the second law of thermodynamics [START_REF] Guy | Arrows of time and space. An understanding of the second principle of thermodynamics[END_REF] , with its spatial and temporal gradients or at the unexpected familiarity between the second law and the second postulate (relativity) [START_REF] Guy | On the cross-fertilization of the foundations of thermodynamics, quantum mechanics and relativity theory[END_REF] . It is profitable to strongly match time, space and movement in the human and social sciences [START_REF] Guy | Human and social sciences and the trilogy space / time / movement[END_REF] : it brings new insights on the aporias of time, on the relations between history and geography, on the role of space on the identity of migrant peoples, on the convergence of the words of time and space in linguistics, on the understanding of rhythms [START_REF] Guy | Rhythm at the crossroads of the movements of man and the movements of nature[END_REF] , on the genesis of spatio-temporal notions in anthropology [START_REF] Guy | Urban ruptures, a spatio-temporal pragmatics[END_REF] , psychology and neuroscience.

And so on. We would find it difficult to concede that all the elements of this ensemble, which only asks to grow, are only a product of the Coué method?

Conclusion

In conclusion, we promote a strong distinction between, -on the one hand, space and time as separate fictional objects (seen according to a philosophical viewpoint) indispensable for thinking and saying, the endpoint of a whole path, and not as founding principles; and -on the other hand, space and time that count (real, actual, singular, lived, experienced, measured, ...), linked like two sides of the same coin (the comparison of movements : stopped movements / continued movements); it is them that physics studies, upstream of the fictions they will generate. Between the two extremes, there is a shift of meaning or polysemy, and, much more, a change of status of words. The return to concrete space and time, linked in movement, opens up a great many avenues, both in the human and social sciences and in the "hard" sciences (even in the way equations are played and calculations are made): as we have already shown in our works, it is very fruitful to put movement upstream of space and time that may thus be compared / considered together in the same thought experiments. Galileo (1632). Dialogue concerning the two greatest systems in the world. Points Sciences, Paris, 2000. Vertical axis: dimension of incarnation, seeing a continuum between knowledge in the body (downwards) and more abstract concepts (upwards).

Shifts in meaning and color codes: depending on the position in the plane, the same words may have different meanings. We put dissimilar colors to the words according to the levels of altitude where they are located: in red, downwards, for the incarnation in the body; in blue after first comparisons and conventions, in a median position; in green upwards, concepts which tend to separate from their material sources, up to fictions. Thus, the word "movement" will have different meanings in the figures according to its altitude.

In the text, we insist on the "ascent" of meaning from the knowledge of the body to the concepts used by the mind; but there is another movement, a descending one, which allows us to attach a word to what is played out at the bottom thanks to the linguistic tools created at the top. This descending line completes the complex circularity (à la Morin) between the different levels. Both lines (ascending / descending) are inseparable. 2B on the right is a kind of summary of the current standard understanding. In A space and time are first understood in a concrete way as comparisons of movements (this is the source, red and blue) before "becoming", as the result of a thought process, pure fictions in the sense of Vaihinger.

In contrast, in B, time and space appear rather as starting points, whether or not they present a relationship, to be specified, with movements. The stage of extraction, choice and convention from the movements is absent, the said and experienced movements not being seen as different from each other. The absence of the blue stage in 2B especially manifests the absence of the use of relational rationality.

The paths in A are presented ascending; the descending paths are implicit: they are the ones that allow us to draw the blue and red words with the green vocabulary (which is itself the top of the ascending paths). Proposal for the development of World 3 (objective knowledge) into two Worlds 3.0 and 3.1. One does not go directly from World 1 to a formulation in World 3.1 (formulated knowledge), a "step" is necessary in World 3.0 (embodied knowledge). This is the name of the path marked here by the two arrows connecting the three points of the drawing. The limits of such a representation lie in the difficulty of isolating pieces of the world from each other and representing them by the contours of set theory, and in the porosity between the different "Worlds".

Figure 4

Movement, space, time: from pre-concepts to pure concepts Association of different expressions to the words time, space and movement, in order to make understand the shifts of meanings and of epistemological status. The numbers follow the progression of thought. We will thus speak at the beginning (step 1) of the pre-concept of movement, to designate the sensory experience (vision, proprioception, ...) of movement, before the words. The sensory experience of space (one bumps into something, one cannot continue the movement) and of time (something can continue elsewhere) follow together (this is why they are assigned the same step 2 / 2'). One can then name and formulate, respectively for space and for time, and independently now of each other, true discursive concepts in relation to what one will have shown (steps 3, 4); one speaks about proto-concepts because it is still associated with a particular world; one can deduce from them a proto-concept of movement (5). In a final step, through a process of inductive generalization, space and time are completely freed from their relational aspect and from their material support, and we build concepts qualified as "pure" for space (6), time (7), from which we derive a concept of pure movement (8). The steps 1, 2 and 2' situate us in the world 3.0; the steps 3, 4, 5 are situated in the world 3.1, but at the border with the world 3.0. Steps 6, 7, 8 are clearly in world 3.1.

Figure 1 :

 1 Figure 1: Two dimensions useful for our discussion Two dimensions: {concepts / body} along the vertical axis, {mobility / immobility} along the horizontal axis. Horizontal axis: relational dimension, opposing mobility to immobility; or better, what goes slower to what goes faster (cf. the qualifier of relative); these are the germs of space and time, qualified as concrete or physical; after a separation at a border subject to convention.

Figure 2 .

 2 Figure 2. Intellection of space and time, and the possible role of movement

Figure 3 .

 3 Figure 3. The three "Worlds" of Karl Popper

Britannica, 1952.

Jullien F. (2001). About " time ". Elements of a philosophy of living. Grasset, Paris.

Kant E. (1781). Critique of Pure Reason. Garnier Flammarion, Paris, 1976.

Acknowledgements I thank all the people with whom I discussed the foregoing issues: the thermodynamics communities: JETC (Joint European Thermodynamics Conference), Thermodynamics 2.0 (where I obtained Arthur Iberall Gold Medal, 2020); the multidisciplinary seminars of EHESS (Ecole des Hautes Etudes en Sciences Sociales) and Collège International de Philosophie (Paris); of Groupe d'Etude sur la Contradiction (Saint-Etienne and Lyon), SFP (Société Française de Physique) Lyon, UMR CNRS EVS (Unité Mixte de Recherche du Centre National de la Recherche Scientifique, Environnement, Ville, Société).