all, as I say, agreeable clichés that only seem to be important. All the positive attitudes of thoughtful, humane, artistic people towards the world we live in will remain no more than hot air, because what will decide not only the fate of things, not only what people do with them, but their very emergence will be mindfulness. ‘You look at the forest from the sidelines,’ a certain knowledgeable and sympathetic person told me. In fact, however, we are looking at it from the standpoint of mindfulness, which is crucial.
There are different kinds of mindfulness. A hack is attentive: he pays attention to where and how there is dead money to be made. Investigative, probing attention will see what needs to be seen in the forest by someone who is assured of three good meals a day in comfortable surroundings and whose priority is to ensure that situation continues.
Mindfulness is not a tool to be used by humans but a state into which they can and should fall. It is not targeted at the forest. Let us instead attempt a provisional conjecture about how it is that the forest captivates us and lifts us out of metric space. It is in fact possible for mindfulness and the forest, two polar opposites, to come together. Nicholas of Cusa’s coincidentia oppositorum develops something that Aristotle all but explicitly spells out.5 We are moving towards our reading of Aristotle with a working hypothesis that for him matter, like that of Plato, will prove to be eidos.
Before we read them, let me dictate this to you: we expect to find in Plato and Aristotle conclusions that we have arrived at ourselves. What arrogance, to attribute our own concepts to distant classical authors! For anyone inclined to that view, let us add insult to injury by saying that not only do we expect to find our own viewpoint in Aristotle but that, if he does not confirm our hypotheses, we will consider there has been no point in reading him.
To anyone complaining that this is a departure from the proprieties of the history of philosophy and philology, we can and will argue that, on the contrary, it is the only proper way to read Aristotle. The history of thought, and history in general, will crumble and I will find myself a New Russian in the most deplorable sense of the word if we do not insist that the most important occupation of humans has never changed: it is working on themselves, digging down to their true self. All our talk about the classical world will prove empty (the ‘Axial Age’, the ‘beginnings of our civilization’) if we fail to notice that these were different times, when the air was purer and visibility was better. The millennia act as a filter, so that in our modern, murky, unfiltered thinking we can presently only have conjectures as to what we will find there, in the classical world, taken as if it lay in the future. We shall be capable of rising to the challenge of the future, of modernizing, only if we lose no time in rising to the challenge of the classical world, and before we can do that we must once and for all repudiate the mentality that avers ‘they had not yet …’. If there is something we talk about now that we do not find in them, that means only that we for some reason are still talking about something they long ago gave up talking about. Their silence is not because they have nothing to say: it conveys a more important message.
One such thunderous silence is Plato’s doctrine of physical bodies. They are, he tells us, composed of geometrical shapes, so that, for example, fire is composed of tetrahedra.6 Did Plato really not know that a point, a line, a triangle, a plane surface, have no volume? That they are abstractions and that the ideal tetrahedron will never be found? That we can spend the rest of time trying to establish the exact line of its perfect edge, just as Achilles will never catch up with the tortoise? Not only the edge but the ideal tetrahedron in its entirety is wholly elusive. What has happened to matter? Where is it? Well, that’s just philosophical idealism for you! It needs to be fixed with materialism!
Let’s try to fix it. Let’s point out to Plato: you have failed to understand, you have failed to notice that there is matter in this tetrahedron we have drawn. There is the wood of the blackboard, the chalk, even the energy used in drawing the lines: these are all material. We find your tetrahedron bewildering, but we do understand chalk on a blackboard (materialism deals with tangible things). But these soon melt away, along with the Platonic solid! The blackboard, the line, the classroom, the ‘solid’, are all clear and easy to understand until we come to mindfulness. When we come to mindfulness, we find, to our amazement, that we must take our leave of metric space, and these material things become no more familiar and comprehensible to us than they would be to someone who was drunk out of their mind. We can have absolutely no doubt that Plato managed to do what we can do: namely, he crossed the threshold of unceasing mindfulness which grinds up the material objects of traditional perception.
Plato’s silence in response to our bewilderment, our indignant ‘Where has matter gone?’ and ‘How can they burn, these ideal tetrahedra nobody has ever seen?’, firstly results from his reluctance to chop logic with us in the delusive space of traditional thinking and, secondly, delivers a message we are not yet ready, not yet mature enough, to hear. In the forest, there is not only indefinability, a superseding of images, a silencing of thought, wonder and horror. There is also geometry. More than that, there is actually nothing there but geometry. There is a sign at the entrance that reads:
‘Anyone Who Has Never Studied Geometry: Keep Out!’7 Geometry is our introduction to philo-sophy, the love of wisdom.
This is all the more unexpected because the first thing we noticed about the forest, and it is the reason we chose it as the topic for these two semesters, is what it inspires and instils by propelling us out of metric space. Of course, in the forest there is none of the geometry of lines and projections such as get drawn on a blackboard. The ungeometrical nature of the organic was noted some time ago, if by geometry we understand nothing beyond diagrams. So, is there another geometry, and is the forest pushing us out of metric space in order to make us finally understand geo-metry? Geometry as taking the measure of the earth? What would that be? It would be pre-Euclidean geometry, in the sense of not giving Euclid what he is asking for (or demanding) in his postulates and, first and foremost, not accepting his points as something that can be determined or found. Let us again follow the path of our heuristics, of expecting something we already surmise. The point in this early geometry will be Parmenidean and Zenonian, an unattainable focus on precision extending to everything. There is only one point and it coincides with everything. The small difficulty that my work on points in various courses about porosity and Wittgenstein has not been published8 will be put right shortly, so I will not repeat myself, and anybody interested can look up the coincidence of opposites, of the absolute minimum and absolute maximum in a point, in my index to the two-volume collection of the works of Nicholas of Cusa.9
Given that our concern is not with the problems of geometry, we need go no further than the ‘point’. The more so because the fundamental issue, which we should never have forgotten, is that Euclid asked to be conceded his point, that it was a convention. We conceded it and promptly forgot all about it. The result is that now that point seems to have as much right to exist in reality as a cup of coffee. We ought to have remembered, as Euclid himself always did, that the point on which all his geometry is based is only a convention. That, thank God, was finally recalled in the twentieth century.
Toward the end of the nineteenth century the keenest thinkers in the field of geometry became increasingly concerned about the lack of true rigour in Euclid’s presentation. Undoubtedly, the invention of non-Euclidean geometries did much to spur the search for a correct and complete treatment of classical geometry. The most notable work of the new type was Hilbert’s Grundlagen der Geometrie [Principles of Geometry], published in 1899.10 David Hilbert (1862–1943) began by stating 21 axioms involving six primitive or undefined terms. [Chief among these was the point, Bibikhin.] He once made a famous comment (not actually published until 1935) to emphasize the importance of keeping the undefined terms totally abstract, that is, devoid of preconceived meaning: ‘One must be able to say at all times – instead of “point, line, and plane” – “tables, chairs, and beer mugs.”’ Such a viewpoint was not widely accepted until well into the twentieth century and, of course, had never occurred to Euclid or his followers.11
That is a good rule for keeping the debate on the ground. We should take a beer mug and place it on a table that extends as far as another beer mug.