**Reading time: 17 minutes**

Translation by AB – January 17, 2021

### Definitions

The term “body” will be used here in the approximate sense of “*lived body*”^{1}, condition of our perceptions . It can, however, be heard in its more common meaning of “material body” without distorting the text. As for “language games”, we are obviously referring to Ludwig Wittgenstein’s philosophy. But it is enough to admit that a “language game” is simply a “game” with words, that one “moves” according to “grammar” rules, without referring to their meaning. A language game is therefore a kind of board game whose pawns would be words or symbols.

### Mathematicians

We redeploy here this thesis: “intelligence”, “consciousness”… are not phenomena that can be understood only by a cognitivist tooling dedicated exclusively to the exploration of the brain and its logical and biological manifestations. It is also necessary to include and understand the *body*, which is eventually seen by cognitivism only as an appendage to be controlled (robotics) or as a set of sensors. This unthought body is however at the deep source of intelligence, this ability to elaborate representations of the world through “language games”. Indeed, everyone knows that language alone is folded on itself and tautological: each word is defined by other words. This infinite regression can only be “silenced” by a gesture, a finger pointed towards what we want to say, a movement of the body: we constantly interrupt our words by gestures of anchoring. The “move” in a linguistic game is always substantiated in the animation of the body. Conversely, the body calls for language games and it is by following the mathematician that we can better understand this. Indeed, the mathematician is this embodied human who has this peculiarity that he exposes his language games (mathematics). It is therefore possible to observe how it “works” for him between his body and his language games. If we admit that intelligence is, as for any human, the result of this interaction, we can then have a serious model to face the question of artificial intelligence. It is in any case, it seems to us, a possible way.

Let’s leave for a short exploration of the connivance between the embodied mathematician and mathematics, between his body and his language games. We will conclude with some words about machines.

### Mathematics and Will

Mathematics is a source of innumerable and complex philosophical rubik’s cubes. These games totally elude the majority of us and, even worse, seem to be useless from every point of view (scientifically, politically, morally, etc.). But they do exist: what is this “vital impulse” that animates some brilliant minds? Can we approach and better observe, thanks to mathematics, this “will to know”?

Mathematicians have had great difficulty in integrating “infinity” into their language games. This problem was identified by Greek mathematicians and more or less solved by German mathematician Georg Cantor, at the end of the 19th century. This two-thousand-year stubbornness and ultimate success ultimately appear to be the pure product of willpower, or effort, to discover despite a body that knows only the present moment. We represent this as follows:

We are not all mathematicians but we all have, in our language games, indications of this Will. They are only less obvious than the word **infinity** or the symbol **∞** within mathematics.

The feedbacks of language games to the body seem even more mysterious. They seem to correspond to “interruptions in the game” or *silences*. We will propose a term to designate this silent call later:

This diagram makes it possible to insist on this essential point. The mathematician’s intelligence has logically regulated the use of the **infinity** and then asks this question to the body itself: is there a point of view in our phenomenological experience corresponding to this **infinity** that we settle in our language game? In the same way that after having “learned”, for example, the logical use of the word **intelligence**, we wonder about its validity in our own experience of the world and its applicability to machines.

We will now explore this back and forth movement between body and language games using a simple and well-known and simple mathematical example.

### Will of “straight line”

We have already started to use the following typographical rule: this **word** is a “pawn” of a language game which obeys only the logical and syntactic rules of this game. Therefore, to what corresponds a **straight line** in the mathematics of today? Let’s show one (in silence):

What do we *see*? A segment, a *finite* piece of a **straight line**. In our everyday language, there are more or less “straight” things, but we cannot show anything like a mathematical **straight line**, infinitely infinite: to the right, to the left, infinitely thin, infinitely straight. How is it possible that this thing which *cannot be shown in silence* appeared in mathematics?

So, we see this segment. Then there is a Will, an effort, a vital impulse seized by thought. Note that “Will” is not a thing in itself but is “will to” or “will of”. Formally, in the case of mathematics, it is the “will of” something in the language game, temporarily detached from the body but coming from it. It is a “will of tool” which, we believe, is part of a vital impetus peculiar to all species. The line segment is thus extirpated from the body, grasped in the language game (**segment**), and subjected to this “will of tool” in the thought consisting in indefinitely prolonging what we have grasped, in improving our tools by making them always bigger, stronger, more beautiful.

What we end up wanting to logically settle in the language game is the ability to extend any line segment in its natural direction. The Will therefore adds a line segment at the end of the one that the body has shown, then another … until it gets exhausted (or bored). The language then breaks off with a lazy “etc.”. There remains a trace of this desire to repeat the same act in the language game, it is this new “pawn”, this new concept: the **straight line**.

The “line segment” is perceived passively (phenomenologically); the **straight line** is an actively (logically) perceived concept:

### Will to repeat (cetera desunt)

According to our intuition, one can consider that the will to repeat is linked to the feeling of time, to the permanent unknown that constitutes this “moment after” which puts the body “at risk”, to the projection of this moment which does not yet exist on our internal screen for its illumination in thought. We have to imagine this Will which illuminates the way in front of the body as an ultra-rapid phenomenon which is perhaps what we call “consciousness”, and which evaporates in inattention and the ebb of the Will.

We can thus imagine the desire to establish a “front light” for the body which initially leads to an attempt at repetition in the language game. It is then necessary that this repetition draws a stable thing in the thought, or more exactly that the repetition stabilizes something^{2}. The **straight line** is the name of that thing stabilized in thought by the “forward light” of the body observing a line segment. It never ends, not in the sense that it is actually infinite, but in the sense that the will of repeating the extension of the segment can always be exercised until exhaustion (of the Will).

It is obvious that no mathematician has found himself repeating any operation whatsoever on purpose, without stopping, without eating, without sleeping, thus caught in a loop of the language game. It is necessary to interrupt the repetition (the time to “see” a stable form emerging), and there are then two possible attitudes. The mental posture “of everyday life”, operating an active and useful return to the body by presenting it with the perceived stable shape, a straight segment longer than the initial segment for example (in the act of driving, we always anticipate the next extension of the road). Or else the “mathematical” mental posture consists in naming the stable form perceived by a desire for repetition and in keeping this new “pawn” in the language game. The mathematician has the time to will but ends up closing the subject by adding “etc.”. We can imagine the situation as follows:

The repetition gradually reveals a shape (here a circle). We usually settle for the portions actively seen in the repetition (the black portions). In the arithmetic language game this could be, for example, a truncated decimal expansion (**3,14**). But the mathematical game gives a name (**pi**) to the whole of the glimpsed shape:

**π = 3,14 + etc.**

**etc.** is the name of the stable and repeated “algorithm” by which the form is revealed. The mathematician is therefore drawn into a language game “shaken” by the permanent irruption of new “**etc.**-algorithms” capable of defying the only real constraint of any language game: its *consistency*.

### Wills

Even if we do not do here the philosophical exegesis of the “Will”, we are nevertheless browsing a conceptual cloud identified for a long time. This is Spinoza’s “*conatus*“, the effort to exist, the exercise of the power to exist, that is to say, to persevere in one’s being ; it is Nietzsche’s “*will to power*“, according to him the “*innermost essence of being*“, an internal necessity of increase of power; it is the “*willing will*” of the French philosopher Maurice Blondel, who proposes a “*series of activity projects*” (such as the extension of mathematical games by **etc.**-algorithms) captured by the “*willed will*” and that propels the intelligence; or even the Will according to Descartes, which characterizes the practical dimension of infinity (infinitude of the Will); etc.

### Will of “point”

The Greek mathematician Euclid is the author, circa 300 BC. J.-C., from one of the most fundamental mathematical texts in history (“*Elements*“), in which he wrote the very first mathematical language game.

Euclid probably had the intuition of the **straight line** by this effort of Will:

**straight line = segment + etc.**^{3}

But it was not in this form that he “resolved” his intuition. Like all Greeks, he made no room for the actual infinity in the language game but cautiously contented himself with a language congruent on the human everyday scale^{4}. So, he couldn’t use **etc.**-algorithms. Fortunately, on a human scale, two intersecting lines show Euclid a “point of intersection”:

This point of intersection appears to the body as something that seems to belong to each of these straight lines, an “atom” of line in a way. If we make the effort to drag the red line along the blue line (etc.), the point of intersection moves and the blue line appears to consist of all these points of intersection. The **straight line** can therefore be defined by the local concept of **point**, the definition of which by Euclid does not seem to call for any **etc.**-algorithm:

**A point is anything that has no part**

This strange formula arises from a logical necessity internal to the language game. Indeed, if the point were “thick”, no matter how small, it would be possible to consider two other points inside that we could then connect by a segment. A **segment** could therefore not be defined as an alignment of **points**. The coherence of mathematical language therefore requires the idealization of the point, not to an infinitely small object, which a Greek cannot do, but to something that contains nothing. But what then is a **point** for the body, which must be shown when the mathematical game is interrupted? It’s nothing more than a pure place, a *location*^{5}, rather than an “object”, and that’s how children should be taught.

### Geometric language game preview

To logically adjust the formal dialogue between the **point** and the **straight line**, Euclid had proposed, among other things, this kind of rule:

**(1) – For every two distinct points there is a single straight line passing through them**

**(2) – Straight lines that are non-parallel intersect at a single point**

Euclid did not propose exactly these rules but the spirit of the mathematical game is respected: we are able to “shut up” after each one and see what it is about. These rules seem in any case valid on our human scale (a building, a road, a natural landscape…) without our needing to resist.

But what happens in a pure language game that ignores this scale? There seems to be an internal drive towards *homogeneity* and *symmetry*, perhaps for reasons of mental handling capacity (only a machine would be technically capable of playing a complex game for a long time, its only limit being physical: time, energy …). We thus seem to have a sort of internal, purely logical Ockham razor that applies exclusively to the language game to “symmetrize” it. There thus appears a quasi-symmetry between rules **(1)** and **(2)**, with the exception of this detail, this logical hypertelia: why specify **non parallel** in rule **(2)**? Why not propose, symmetrically to rule **(1)**:

**(2) – Two distinct straight lines intersect at a single point**

But once said this, a return to the body is immediately required because where can be the location, this **point of intersection**, for two **parallel straight lines**?

### Return of Will

No matter how hard we try to browse these two parallel lines, whether to the left or to the right, we never seem to meet their intersection. Besides, isn’t this the intuition of the definition of parallelism: never to cross each other? But the symmetrization of rule **(2)** of the language game seems to summon the body to find this intersection. There is a “monster” to be found and tamed, as Liu Hui did with irrational numbers. It was not until the 15th century that the language game and the body began to synchronize, at the cost of actualizing things to the “infinite”, a domain until then reserved for God. Indeed, this point of intersection can only be located infinitely far away (or rather: this point can only refer to an infinite distant location). Mathematicians have therefore called it a **point at infinity**.

For this **point at infinity**, there is no etc.-algorithm that progressively draws its appearance, as for the **straight line** (**etc.(segment)**) or the number **π** (**etc.(3,14)**). It is a new effort of the Will in the form of a jump or a clap that makes it appear “suddenly” to infinity. The **point** required by the logic of rule **(2)** is thus shown by the body of the mathematician at this impossible and, above all, practically irrelevant place.

But this “answer” is not yet satisfactory since only one direction is explored while there are two (says the body). This same **point at infinity** must therefore be shown (possibly encountered if the Will is as great as Faith) in both directions:

The symmetrization reflex of the language game thus produces logical concepts (“pawns”) which, in turn, must be adjusted by the mathematician’s body. Like the yogist’s body, the mathematician’s body is trained to be flexible and generally responds to the “inventions” of logic. But this flexibility has limits: sometimes the language game produces authentic fictions that will remain syntactic pawns for a long time; sometimes it is opposed to the culture of the time and to semantics already made by other language games (the divine infinite and the religious language game for example).

Actively showing this **point at infinity** is an authentic disruption that undermines, so to speak, the entire cultural edifice. Let us recall, in another scientific field, the fate of Giordano Bruno, delivered to the torment of the flames in 1600 after an extraordinary “semantic” process of eight years, for having proposed this emanation of his language game:

**There is no star in the middle of the universe, because it extends identically in all its directions**

If Bruno had the body of a yogist, his executioners did not! Sometimes, mathematicians have to exercise their bodies in the shadows for a long time, and show absolute boldness, willpower and freedom at the price, sometimes, of decay or madness.

### End of the exercise

We are just beginning to consider, with this mathematical example, what a language game is and why its “silence / stop” in the direction of the body is not neutral and engages, even modifies, it. The emergence of the **point at infinity**, refering to this *location* at infinity, is not the only consequence of the initial minor symmetrization of the game. We can indeed go on and get hold of the rule **(1)**:

**(1) – For every two distinct points there is a single straight line passing through them**

Therefore, through any **point** “**A**” and one of these new **points at infinity**, “**Ω**”, passes a single straight line “**AΩ**”:

Without going into the details of the logical reasoning, all the **straight lines** passing through “**Ω**” are parallel to “**AΩ**“. Thus, another line “**BΩ**” line looks like this:

The return to the body then shows us how a **point** can appear, phenomenologically, at infinity: as a direction in space (a point is thus interpretable as a location or as a direction). The body thus concretely seizes the **point at infinity**, which is this point “on the horizon” that we show by *saying* “it’s that way”. The pictorial language game has called him **vanishing point** and its embodied observation is no longer strictly from above (“Euclidianly” ) but from the side, in perspective:

These lines are not parallel on the paper (on which the play of pictorial language unfolds) but for the view (the body). There is always this sensation that is difficult to grasp: the logical modification of a language game, for example as here by symmetrization, modifies the relationship of the body to the world. This is how the mathematician’s body, which validates language games much more complex than this one, differs significantly from ours.

We propose to end this exercise with a final effort consisting in turning one’s gaze in another direction, thus towards another **point at infinity** “**Ω’**“. Let us recall one last time the rule **(1)**:

**(1) – For every two distinct points there is a single straight line passing through them**

So there is this new deduction (move) in the language game: through the two **points** “**Ω**” and “**Ω’**” passes a single straight line “**ΩΩ’**“. This is what mathematicians have called the **line to infinity** and which calls us to recognize it as this horizon line that encircles us:

### Coupling

Mathematical practice thus reveals a kind of “structural coupling” at work (The Mirrors of the “I”) between body and language. A Will sets the thought in motion: what can happen next? The question “etc.?” is therefore addressed to the language game which either already has an **etc.**-algorithm and answers by deduction, or regulates the appearance of new concepts (like **point to infinity**). Once the language game is regulated and symmetrized, the body adjusts, new “horizon lines” appear which change the perspective and our relationship to the world.

We will (temporarily) call the interruption of the language game (the intervention of silence) a “somatic” return or a “somatization”. The coupling scheme therefore looks like this:

Symmetrization can sometimes be so daring and “violent” that it causes intense somatizations: astonishment, illumination, revelation (which can sometimes border on schizophrenia).

If we do not consciously perceive the effects of this coupling in our daily life, it is because it takes place at high speed and with very little impact on the body (except perhaps in childhood and in rare moments of intense somatization). On the other hand, few of us know or want to modify our language games through the effort of the open question “etc.?”. These games are generally transmitted to us through education and the cultural environment. We do not “symmetrize” them, like voluntary mathematicians (who are genuine sportsmen), but we are satisfied most of the time to play with them, to produce deductions and tautologies which, instead of proposing new horizons, tighten the grip of our “convictions”, “ready-made thoughts”, “doxas” and other “creeds”.

### Conclusion – Back to the machines

Contrary to what cognitivists claim, intelligence is therefore not only an activity of manipulating symbols, which is a tautological manifestation of a game whose rules are established, the essentially deductive principle of a “ready-made thought” which acts on us. Intelligence is rather, as these mathematical games show, the manifestation of a *structural coupling* by which the game and the body are constantly adjusting (rapidly or on the contrary very gradually). Intelligence would therefore rather be relative to an “activity of producing language games to be somatized” and therefore largely dependent on the body. Therefore, not only are we a long way from having created a real artificial intelligence, but we don’t even have the slightest idea of what it might look like, which is the body in which its language game could somatize and from which a Will of “etc.” Could unfold.

Finally, can a simple algorithm (set of instructions) produce or modify language sets? Maybe … But it’s about knowing a reason to do it! We suggested above to modify rule ** (2)** for the sake of symmetry and simplification. But in the case of machines with much higher computing power than humans, what reasons would they have to symmetrize? Why produce “simple” language games (the ZFC axiomatic of mathematics, for example) rather than complicated?

Isn’t there a requirement here which, in addition to the Will, characterizes the living?

1. ↑ Wikipedia – *Lived Body*

2. ↑ Moreover, no concept should essentialize (like “will”, “infinity”, “data” …) but should always be the name of a process in a language game (the “will to”, the “infinitization”, the “reflection”…). This note is saved for later

3. ↑ That we should write more rigorously: **straight line = etc.(segment)**

4. ↑ What exceeds this scale, whether towards the horizon or towards the microscopic, this always possible extension of the gaze, cannot be grasped as a mathematical object but is the very condition or the “background” of this object. Likewise, the canvas for the painter, which obviously always exceeds his work, or the silence for the musician.

5. ↑ Continuing the idea of note 1, the “desessentialization” of the concept of “point” in language thus leads to the concept of “location” for the body.