know what a point is, and that becomes less clear every time we encounter one. Up until the present day, the ‘point’, as well as other geometrical terms, has existed only thanks to our ability to grasp and hold an object. It was not the point we could see and hold; all we had was our ability to grasp and hold, and the thing we did that to most often, and with great gusto, we called a point or something derived from a point.
Everything about geometrical constructs is beautiful and indisputable, but the elements of geometry are floating in thin air. The fact that they are mere conventions is blindingly obvious, and they hold true only because people have agreed to believe that they should. In Euclid’s original definition, a point ‘is that which has no part’. The ungraspability of a point is established with a deliberate paradox. I say it was deliberate, because in the classical world it was a commonplace that something that has no parts has nothing with which to have contact with anything else. If it merged with something else, it would add nothing to the whole of that other, or, if it entered as a constituent part of that other, it would rend it apart, because nothing in the other would be able to be in contact with a thing that had no part.
Whether it is easy, difficult, or impossible to obtain a point, the only way it can be done is by concentration. The still unresolved paradoxes of Zeno and Parmenides show that concentration of the mind alone is insufficient. I have no time to repeat what has been said in earlier lectures, and would ask those who are thinking about this for the first time to take on trust, without ifs or buts, that in the most literal manner, in the problem of the point, the basis of geo-metry, we come back to mindfulness. All the observations on the relationship of the point to time, about the point as the present, according to Aristotle and Hegel, which has also been previously discussed in detail, provide the context of what we need now to engage with. Our topic is Plato’s unexpected geometry where we would have expected to find matter, the geo-metry of the forest, the forest as geo-metry without Euclidean metric space. Here is what a modern historian of thought has to say about Plato’s ‘eidetic atomism’, which means that in place of the expected atoms of matter we encounter pure eide: ‘Diese kühne und in ihrer Weise großzügige Theorie der Materie ohne Prinzip der Materie hat weder im Altertum noch später Nachahmung, ja auch nur Verständnis gefunden.’ ‘This bold and, in its way, ambitious theory of matter without the actual principle of matter was not taken up, or even understood, either in classical times or subsequently.’12
Whether or not it was understood, whether it is easy or difficult, we need to buckle down. As Plato says in The Epinomis (992a), ‘[T]his is the way, this the nurture, these the studies, whether difficult or easy, this the path to pursue.’13 The Epinomis here is a legacy, bequeathing the law of bliss, the law in the sense we have been speaking of, the law of humankind, of its nature; happiness in the sense that it was experienced by the Pilgrim when he discovered unceasing prayer, constant mindfulness. The whole history of thought speaks with striking unanimity of the happiness of humanity when it returns to the law. We read the final pages of The Epinomis, and this is so similar to the joyous pages of The Pilgrim. When he comes back to the law, to religion and faith, Plato tells us, a man will be freed from his distress (which we referred to, following the Gospels, as ‘the issue of blood’):
And the man who has acquired all these things in this manner is he whom I account the most truly wise: of him I also assert, both in jest and in earnest, that when one of his like completes his allotted span at death, I would say if he still be dead, he will not partake any more of the various sensations then as he does now, but having alone partaken of a single lot [i.e. he will be vouchsafed wholeness, unity, monasticism, Bibikhin] and having become one out of many, will be happy and at the same time most wise and blessed. (992b)
We shall need to return to The Epinomis, because our main themes of the law, religion, constant mindfulness, and the forest in the sense of matter come together there really very clearly. But we shall remain firmly focused on what has just intrigued us most: matter as number; the forest whose very breeze, we cannot doubt, despatches metric space; and geography, which, according to Plato, has pure geometry as its law. We are not ready for this, to proceed by way of clues and glimpses, such as one that has been suggested to us, of seeing the pillars of a mosque, symbolizing the pillars of the universe, as a forest, as trees, and in some way subordinate to the sacred number of ‘17’. We can, if so inclined, link anything to anything else, any idea to any other. We, however, will prefer to admit failure, inability, rather than rush to associate the forest with a number, or agree with the majority of historians of thought who believe Plato was getting a bit carried away here, coming up against the limitations of idealism, or even, as Alexey Losev manages to claim in line with his preoccupations, that Plato was a man of his time who ‘lived and worked’ in slave ownership, with the result that he projects the callousness of a slave owner on to the world of ideas, and that his geometry is based on heartlessness. ‘Because number, devoid of qualities or indifferent to them, is precisely his basic principle, lacking any personal or “spiritual” dimension. Accordingly it is entirely predictable that Plato’s philosophy, having developed as far as its limitations allowed, ends up with the doctrine that his eternal and divine forms are numbers.’14
This is sad. I do not think Marxism was the cause of this blunder. More probable is what Nicolai Hartmann noted: that matter as number is such an amazing leap in Plato’s thinking that it has not been understood to this day.15 If Losev effectively passed on this, preferring to say nothing, that is because in his curious formulation he only hinted at a strict, harsh discipline, also to be found in the structure of the ancient polis, which succeeded in binding in slavery anybody unwilling to accept the risk of being free and taking responsibility for themselves. It was a degree of disciplining of thought that we have forgotten, but which is no less unyielding than mathematics. We have worked our own way round to the discipline and school of constant mindfulness. There is, of course, plenty about that in Plato. It is clear that without schooling, without strict discipline, it will be impossible for us to understand the riddle and mission of Plato, namely matter as number. Discipline is undoubtedly a necessary condition here, but is it a sufficient condition? We may not have enough vision. Does anybody know why the forest is digital?
The forest is wood, and wood is fuel. We take heat and light from fire, the burning of matter. Fire, according to Plato, is a tetrahedron. Not something in the form of a tetrahedron, not a tetrahedron filled up with something, just a tetrahedron. It is not that somewhere baffling processes are taking place with the primary elements in attendance and the tetrahedron is a kind of assembly or abstract function signalling or symbolizing them.
To our difficulties, deadlock actually, must be added the fact that if Plato understands the element of number, the one, in what is called a substantialist way, as a simple concentration, a happy totality, as a blessed fulfilment of everything, how is he to get anything out of such a one, which is clearly singular and clearly equivalent to the maximum integer of the universe? How is he to construct anything with it when a tetrahedron requires at least four different points? We land ourselves in the problem of the difference between the substantialist so-called Pythagorean number and the arithmetical number, which are completely different things. A lot of care has been needed to avoid getting burned by this distinction.
Again the admirable Nicolai Hartmann warns us:
The theory of Timaeus represents, in terms of its content, a synthesis of atomism and the doctrine of forms, which should be considered impossible in view of the natural antithesis of these two doctrines. There has to this day been no thorough investigation of this historical topic. This is one of the numerous gaps in classical historiography of philosophy in the last hundred years that result from its deficient understanding of the problem.16
That is, the problem is not in the method and apparatus, but in too much understanding: everything is immediately abundantly clear to the researcher, just as the slave-owning undertow of Plato’s idea was, unfortunately, only too clear to Losev. Hartmann sees no problem where Plato did, and is perhaps even a little smug at having been able to sort it out as Plato, unable to resolve the contradictions between materialism and idealism, failed to.
Owing to the fact that the ‘contradictions’ between Plato