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Science and Technology - Pietro Chistolini

Euclides and the amoeba: Improbable dialogue about the foundations of geometry and spatiality in the Western culture.

Pietro Chistolini

On the inner road you may walk darkened or luminous.
Attend to the two paths that open before you.


The Elements by Euclides is said to be the best well-known and most diffused book in the Western world, only second to the Bible. As a point of fact, who hasn’t gone through things like dot, line, triangles, squares, Pythagoras theorem, etc, etc.?

Geometry is the scaffolding, the model physical-mathematical sciences build upon. It is an axiomatic deductive model that uses primitive, intuitive concepts, axioms and theorems, everything rigorously constructed with precision and rationality. It is a model of rationality. And the history of geometry has also been in some way the history of Western society, a point of reference for more than 24 centuries.

In other terms, geometry belongs in the historical and social landscape in which we have been brought up, belongs to the landscape which gave form to science, which gave form to Western society as a whole.

But in a hypothetical new civilization, which could be the role of science? How do the ideas of this New Humanism reconcile with science?

In this sense, I’ve been working for years to put into evidence the centrality of the human being in science, to trace science back to human action, to a tale, I’ve tried to show the mythical background of science, that which links science to the history of humankind. Today, again, we will start from a great myth.

We are in Athens, in the 4th century before our age, at the Platonic Academy. Just a few tens of men belonging in a philosophical school of philosophy that characterised in a most decisive way the fate of the Western world.
Speaking of Plato, let’s bring to mind the myth of the cave, which illustrates his thought so well.
In an underground cave some shackled men are constrained to watch shadows. The men see and reason over shadows which are the projection of shapes that other men keep carrying behind a wall. These shapes are lighted by a fire. The fire illuminates the forms and their shadows are projected on the wall of the cave.

The men see just shadows and know nothing of the forms that generate them; Plato calls these forms ‘ideas’, these are the Platonic ideas. The Greek term ‘eidos’ has been translated as idea but the original meaning was quite different. ‘Form’ would have been a much more appropriate translation.
Men may not see these forms but may have intuitions about them and somehow manage to have a glimpse of that which is the cause of shadows: fire. And they have the capacity to set off on this path of elevation, of ascesis, until they exit the cave and know the world of trees, mountains, birds, and proceed up to what Plato considers the highest good, the One, the sun. In other words, these ideas –and among the most pure of them, the idea of geometry and numbers- these forms are a sort of guide, a hoist, a cause that determines elevation, first towards the fire, then towards the sun.

Aristotle pointed out that the ideas intended as forms were not enough to illustrate all the possible causes which determine this path. So he added the material causes, the final causes and the efficient causes. Besides, he started developing logics, and on his logics –core and foundation of that which would become Western logics- he also formulated his physics and metaphysics, trying to deduce the first principle, the One, following a logical path, through the use of logics. Little by little logics replaces the journey itself. Logics, which should have complemented the journey, now takes its place. The forms, the ideas will no longer show the way, but have become mere entities of our thinking and must be given a certain order. It is like remaining inside the cave and try to put in order, to give meaning to, to find a logical justification to these moving shadows. This is the direction that most of the Western culture has followed.

Putting ideas in order, putting order to thinking, is like our wardrobe at home. We know how difficult it is to find common family rules to keep our clothes in order. How to order our and somebody else’s clothes, however, is just remotely associated with life itself, with experience, with the fact that we need to put on some clothes to go out and act in the world. But what matters is not what I’m wearing, but the experience of living, of living in the world.

This is, in a highly simplified way, the observation that Husserl -the father of phenomenology- made about 1900. What counts is the immediate experience, Husserl calls it erlebnis, Silo calls it registro: that which manifests and is registered by the consciousness, the experience we go through as a result of an act launched by the consciousness, as a result of an intentional act.
But let’s go back to Athens. Plato and Aristotle are still having a dialogue. According to tradition, at the top of the entrance door to Plato’s Academy there was an inscription that read “Let no one enter who does not know geometry”.

Maybe now we will manage to understand why geometry was so important to Plato. It was the reference, the cause for elevation, for the path of ascesis so beautifully described in the myth of the cave.
But something happened. In fact, there are numerous criticisms to geometry in the Republic by Plato. Something happened within that School. It is commonly accepted today that it was Eudoxus, a member of the Academy, who set the basis of what we call today the axiomatic deductive method.

We know that the axiomatic-deductive structure of geometry moves from axioms and definitions from which all other statements that characterize geometry are derived, are deduced, through the so-called theorems. But in order to express an axiom it is necessary, in turn, to resort to some primitive concepts, to self-evident principles. In the case of geometry, dot, line and surface have been the primitive concepts for more than two thousand years. We have all learnt them at school and they seem to us unquestionable, clear-cut, self-evident concepts. In the minutes to come, just as an exercise, we will call into question these statements, these seemingly a priori concepts of dot, line and surface.

The axiomatic-deductive method, again, consists in fixing a list of coherent axioms and, as from these  axioms, building the entire edifice –for instance the entire edifice of geometry- regardless of the origin of the axioms and without caring too much about finding a justification for them.

But for Plato the study of forms and axioms should not only lead to the construction of theorems, to logical rules. It is necessary to go deeper, to bring the origin of axioms to light, to try to go beyond, to the very antipodes of what Eudoxus asserted. For Eudoxus the purpose of the construction was to put order in the ideas of thought, whilst for Plato forms and all other ideas must serve to go beyond that symbolic fire. Plato openly expresses his disappointment.

But what does it entail to start off from axioms rather than from registers, from immediate experience? It ultimately means to ignore the intentional activity of the human being, it means to build an agglomerate of rules that take on a life of their own, almost independently of man himself, and in the end we find a science in which the observer has wholly disappeared, or almost. There are laws that overpower the human being, that cancel man’s presence. But are these laws capable of giving an answer to existential questions? May these laws go beyond the fire Plato has shown? Obviously not: we are still dealing with shadows.

And this externalizing, this alienation of registers may only lead us to a sort of obscuration, to that which many have defined as the nihilist trend of Western society. In this sense, science as well needs to retrieve and dis-cover the role of intentionality.

But, how to do this?

Let’s go back to the exercise. Let’s go back to the primitive concepts of geometry: dot, line, surface. We have said that any primitive concept will do. For instance, about 1879 Clifford said that in order to build the edifice of geometry one could start from intuition, from the intuition of a generic solid objet and from the intuition of space. We will call surface that which separates the object from the rest of space. If we join two surfaces, they will overlap along a line. If we make two lines intersect they will meet at a given point. Thus, briefly, from the primitive concepts of object and space, we manage to derive the old primitive concepts of Euclidean geometry. At this point, mathematicians are pleased and contented: it’s enough for them to know that these new primitive concepts are equivalent to the old ones, and that the entire geometry has remained unchanged.

But, let’s try to understand things better. In the sense of what Plato tried to convey.

Here’s a glass of water. We immediately identify a surface as that which separates water from air. The surface is that thing that we can see separating the water from the air. But what are we actually seeing? What is this surface?

Can we isolate a surface from all the rest? It would be like saying that a doughnut has a hole –not all doughnuts have holes. The doughnut must have a hole. Can we separate and isolate the hole from the rest of the doughnut? Makes no sense, does it? Do surfaces really exist or are they an artefact of our minds? An abstraction. If we get closer with the help of a microscope, we will see what happens between the air and the water; what this surface is. Well, we will discover funny things that we call molecules of air and molecules of water.

What do they look like? At a given magnification level they exhibit globular, pseudo-spherical forms. Again we find a surface. We need a higher level of magnification. Now we find atoms, electrons, nuclei, neutrons, protons, spheroids once again and once again the problem is that we represent them with a surface. We must then go deeper and in so doing we enter the world of quantum mechanics and quantum mechanics does not tell us what a surface is at all!!! We are projecting our abstractions!

The only answer that quantum mechanics can give us is an indeterminate answer: the wave-corpuscle dualism –you have probably heard about this before- means that matter may behave either as a particle or as a wave according to the way in which we set up the experiment, according to the way in which we ask questions to nature: it depends on the act. There is a binding, inescapable structure between the action of observing and the object perceived; a structure that may not be resolved in any way, no matter the magnifying power we may achieve.

The attempt to furnish a description of the world leaving consciousness aside, leaving intentionality aside, does not work. There is always the act-object structure, the consciousness-world structure, beneath.

The title of this presentation mentions the amoeba. What can the amoeba, a little unicellular animal, a protozoan, tell us? What can it say to us other than what Euclides and his geometry have told us already? To begin with, we can see that, in spite of being unicellular, the amoeba has several primitive functions. It has sensory functions but not senses; it has no sight; it has this sort of primitive touch resulting from its membrane, but no senses strictly speaking. It has no nervous system because it consists of only one cell.

It moves by deforming its membrane; by forming pseudopodia, these finger-like protrusions, but has no muscles, no limbs… eppur si muove (and yet it moves). It features all the functions that we will later find in more complex organisms, but at an absolute primitive level, in only one cell. There is visceral activity, reproductive, motor, sensory activity … Is there mental activity? Does it have consciousness? I would tend to say it doesn’t. However, if by consciousness we intend that something which coordinates vegetative functions, etc. we could say that it has a sort of memory since it can recognise food.

Let’s see how an amoeba moves and eats. It recognises food through this primitive sense. And it must have an automatic perception of spatiality to be able to move and detect the presence of an object, a solid body to feed on. It has therefore what we have called the basic intuitions of Euclidean geometry: a solid body and spatiality. It’s already there, in the amoeba! But we also find this tension, pro-tension, intention towards the object of its feeding; we find a definitely primitive form of intentionality that we may, therefore, associate with an equally primitive form of consciousness.

Let’s add dynamics to the scene. This funny little animal has all the fundamental functions that characterize a living being even though it is only one cell: it has some perception of spatiality and some sort of intentionality as well. It’s quite easy then to imagine that, through an evolutionary process of growing adaptation to the environment, these functions must have perfected to be found later on in more complex, evolved structures such as senses proper, limbs, muscles, eyes, of all possible forms. But what is it that triggers, feeds, gives direction to all this? This force that gives impulse to evolution is intentionality, and it is already present in this unicellular organism!

We are approaching the conclusion.

The theses we have briefly discussed here are the following:

1. Geometry belongs in the landscape of formation of Western society.

2. Geometry is a model of the axiomatic-deductive method that underlies Western science and rationality

3. As the axiomatic-deductive method took root, an attitude consolidated in the West by which the role of the human being has been marginalized, not to say nullified, before the so-called “fundamental laws”.

4. Also science will have to “re-dis-cover” the omnipresence and centrality of human intentionality, of the consciousness-world structure and “re-start” from immediate experience, from internal registers.

5. A philosophical school of a few tens of individuals, with their activity of study and reflection, have been able to strongly affect Western culture, to such an extent that –in order to conceive the foundation of a new civilization- it is necessary to go back in time and try to understand what happened within that school, it is necessary to go back and reconsider the nucleus of reverie of an entire civilization.

I wanted to speak about geometry because it seems something truly objective, something indestructible, but it isn’t. Geometry is not a world outside or above man; it is rather the vehicle of somebody’s intentionality. In any case, it is something that keeps taking us back to the path described in the myth of the cave.

But where today are these subjects discussed, where can one practise, where and how can one set upon this path?

Right here, for instance, in a park like this or in any of the other parks of study and reflection around the world. These parks are witness to Silo’s Message, to Universal Humanism.

I started this chat by quoting Silo, where he indicates the two directions of the inner road, and then discussed it in relation to the path taken by the entire Western civilization.

I would like to finish –now I’m serious- with another of Plato’s allegories: the allegory of the second navigation.

Plato tells us that human beings are like sea vessels with full-blown sails. We need the wind in the sails to go into the world. The winds are our senses, our perceptions, our opinions… but when there is no wind we must lower the sails. When there is no wind, sails are useless. When the sea is dead calm, when there’s the void, we may carry on with the second navigation: we take up the oars and start to row. Navigation becomes more demanding, less automatic. We start probing the depths with the oars. It’s just a glimpse, a soft grazing touch of the profound.  Plato is interested in this second navigation, he is interested in reaching beyond sensations and opinions. With the allegory of the oar cleaving the surface of the sea, Plato sets off along the path of ascesis he had described for us in the myth of the cave.

... and it’s also my wish that we all embark on a deep and intense second navigation.

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