constructive to read the two philosophers as, on the contrary, complementary to each other. There is the brilliant, characteristic, and generally accepted denial by Aristotle of the existence of species and genera – ‘the animal in general’ exists only in the imagination, the species of donkeys perhaps a little more but still not really – and all that really exists is this particular donkey, and another one over there. He places a taboo, in other words, when inspecting this particular donkey, with those particular ears and those big eyes, on hypostasizing donkeyness; a taboo on extending it any further than this particular donkey. Or, if you like, we can eliminate donkeys altogether by counting them as one, two, and three but remembering while doing so that we are counting something that does not exist, and take full responsibility for the fact that we are doing, operating, something that does not exist. That makes all applied mathematics a risky business when, if you forget when counting ‘one thing, two things, three things’, no matter what they may be, it has already slipped into the realm of things that are not there.
And what if applied mathematics is calculating without including what is being counted after the number? The number will have reality if we see it as counting ‘this thing here’: that is, if we are viewing the donkey, not abstracting the number back to a general species or genus. At this point, Aristotle seems unable to help us further and it looks as though (although there can always be a surprise in store) we need to go back to Plato, to see the number itself no less specifically, as ‘thisness’ (haeccaeitas), than the donkey. Can we see the number, can we look straight at it in the same way we can look a donkey straight in the eyes?
Yes, we can, and, oddly enough, the experience will affect us more directly and strongly than our encounter with the donkey.
This experience is not possible with a denumerable number, an element of mathematical operations, because any number there as an element of a set of integers must be abstract and generalized, totally indistinguishable from any other number, since otherwise the set will fall apart. But actually, to our relief, we find we are not being called upon to distinguish between a number with which a close encounter is possible and a mathematical number, because modern mathematics gets by perfectly well without the concept of number and deals instead with structures and processes. In other words, just as geometry repudiates the indefinable and undefined concept of the point and is entirely willing, according to Hilbert, to talk instead about a beer mug, so modern mathematical terminology does not include the concept of number. Philosophy was all for donating number to mathematics (at least, general and philosophical reference books say that number is ‘one of the basic concepts of mathematics’), but now mathematics is returning the gift and philosophy will have to return to it, which is something else for us to look forward to.
Aristotle has experience of encounters with ‘this thing here’, Plato of encounters with forms or, later, with numbers. Just as ‘this thing here’ cannot be part of a calculation, since otherwise it will cease to exist, so Plato’s forms cannot be counted in terms of one, two. This is what I call moving out of metric space. Let us tentatively, without becoming prematurely confident, say that by moving out of the space of counting, auditing, and structuring, we are returning to the early geometric number. I am setting myself that as a task for the future. Just as Plato has long been urging us to think more thoroughly about number, so he is also calling upon us to review geometry. I am merely accepting the challenge which, as an old man, he left at the very end of The Epinomis.
In order to train minds capable of mastering this knowledge,
one must teach the pupil many things beforehand, and continually strive hard to habituate him in childhood and youth. And therefore there will be need of studies: the most important and first is of numbers in themselves; not of those which are corporeal, but of the whole origin of the odd and the even, and the greatness of their influence on the nature of reality. (990c)
I understand this to mean the invariable symmetry of all that exists, which is called real number.
When he has learnt these things, there comes next after these what they call by the very ridiculous name of geometry … and this will be clearly seen by him who is able to understand it to be a marvel not of human, but of divine origin. (990d)
Numbers themselves, like geometry itself rather than the ridiculous way in which it may be understood, are an invitation and a task for us. Let us initially try to approach this task from a fairly easy and uncontroversial direction. A confident counting of things, ‘one window, two windows’, or, which is essentially the same, of numbers, ‘one number, two numbers’, is possible because the generalizations are, as it were, ready and, indeed, waiting for content. Let us imagine a pit that we want to fill up. Who dug the pit, and how, into which the set of natural numbers is thrown? The series can be very long, but the pit is invariably bigger, as if anticipating and inviting it in. Counting things such as the stars in the sky will surely run into difficulties, not because of any lack of space, but only because it is difficult to count them.
Now at this point there is something to which I want to draw your attention, namely what might be called the ‘lure of the pit’, or ‘the lure of the heap’,17 which seems to have a will of its own to grow bigger, to accumulate, and which, if we did not keep our eyes open, did not protest and resist, would level things down even more. A random example: 300 years ago, a census of the population, conducted as it is nowadays, would have been impossible. The difference between a working man and a woman seemed just too great, and this was even more true of an old man or a child, so the count was based on households or smallholdings. In a modern census, all classes of the population are, as it were, lumped together, levelled out. What is causing this virtually limitless trend towards generalization? Obviously, the fact that ultimately everything belongs to one primal category, the totality of all that exists, is, as it were, sucking everything into itself, demanding enumeration, and avidly seeking to be filled up. Moreover, we know in advance that there is no danger of the categories running out of space, that there might not be a receptacle to count everything into. Everything – and that is a lot of stuff – belongs ultimately to the One, to one world, one universe. We have always enumerated too little, ascribing things to the universe. Everything always fits into it. It has room. Everything ultimately can be generalized by the very fact of belonging to one and the same universe.
This power of unity, of wholeness, encompassing, drawing in, clearly can never be captured by counting. The universe is of a size that, no matter how much you put into it, there will still be room for more. Trying to put more and more into One is clearly an inadequate, negative, or deluded response to the challenge of unity. We can see that the numerical series gets its power from its infinity, from the negative knowledge that the power of unity cannot be exhausted by endless listing. The origin of number is thus negative. A numerical series, infinite counting, are based on a tacit assumption that we can safely engage in endless enumeration because unity is strong enough to accommodate that. We enumerate all the constituent parts and abilities of a human being, but then realize that they can be further subdivided, and that there are others yet to be discovered, not yet investigated.
We encounter something analogous with time: its infinity stems from our own irremediably late arrival on the world scene, from our being late for the event of the world. No matter how many years we add to our own lifespan or to that of humankind, we can be reassured by our intuitive certainty that the event of the world has infinite room to draw on. A supply of unified, official time is reliably provided by our lateness; it can, and should, never end.
Just as, apart from official, levelled-down time, there is being-time, so, apart from the negatively defined unit of arithmetic, there is a unit or unity in the experience of the whole, but as a unit of experience we are much more directly affected by it. More directly, I would say, than the live donkey in front of us: the latter affects us on a living and organic level, but the experience of unity is, by definition, everything, integral. The experience of unity is the same as of the point as concentration, of mindfulness, of prayer as the submission of everything to the one who is no part of being, no part of anything, and yet has the most intimate and direct relationship to everything of anything.
For the time being, we do not see how this all-encompassing unit can move, create order, a set or calculation,