If the One is not: Beyond the Arithmetization of Being
The Parmenides is a dialogue chiefly introductive. Its collocation within the Platonic corpus leads us to think it prepares the following dialogues, which refer each other in a really unique manner. We must therefore think that these dialogues do anything but develop what is deployed in the Parmenides. The hypothesis I would like to propose is that there is a clear theoretical line from the Parmenides to the Statesman, and it concerns the crisis introduced in the ancient thought by the discovery of the incommensurable magnitudes. I thus consider all the dialogues, which follow the Parmenides (above all the Theaetetus and the Sophist), as seismographs of the conceptual earthquake produced by this discovery: they move in a disrupted situation, which obscured what once was clear and makes again necessary to inquiry what is science, what is Being, what is the logos. I will work up this hypothesis in three parts, and then propose a theoretical conclusion.
- The Mereological Divisibility in the Parmenides
The structure of the Parmenides, beyond its analytical complexity, can be schematized in two principal hypothesis: if the One is, if the One is not. It is a question then of the relation between the One and the Being, on one side, and the One and the Not-Being, on the other, that is of the relation between the One and the two ways the goddess points the finger at in the Preface to Parmenides’ Poem: those of Being and of Not-Being. This means that it does not concern simply the problem of Being, but the problem of its arithmetization, that is of its conceivability in numerical terms (through arithmoi), as One or Many. This “arithmetization of Being” leads to paradoxes, which are systematically analysed in the course of the dialogue, and summed up in the conclusion: “whether one is or is not, it and the others (talla – the word allo alludes to a mere quantitative, not qualitatively differentiated, multiplicity) both are and are not, and both appear and not appear all things in all ways, both in relation to themselves and in relation to each other.” (166c) In fact, to say that “the One is (not)” means to put an equivalence between Being and One, which generates an infinite diairesis. By taking on their own the One and the Being, we can say namely that “the One is” and “the Being is one”; but in its turn each of these claims cleaves (through a recursive process) in these same claims, “the One is” and “the Being in one”, on one side, and “the Being is one” and “the One is”, on the other side, and so forth.
“So again, each of the two parts possesses oneness and being; and the part, in its turn, is composed of at least two parts; and in this way always, for the same reason, whatever part turns up always possesses these two parts, since oneness always possesses being and being always possesses oneness. So, since it always proves to be two, it must never be one.” (142e) This cleavage fragments the One and the Being: it is even due, as Plato writes, to the Being itself, which is said of the One (144e), since, as the first thesis states, the One as such, without saying that it is, is completely unknowable. (142a)
The setting of the problem of the One and the Many through the equivalence between One and Being (or Not-being) leads to a fragmentation, that is to an infinite countability, since “if there is number, there would be many, and an unlimited multitude of beings” (144a): the One is in fact reiterated at any step. However, this situation is nothing but the consequence of that initial identification, that is of the arithmetization of Being, of its conceivability in terms of natural numbers.
- The Introduction of a Not-number (which is also a Relative Not-Being)
The theoretical path the following dialogues outline concludes in the disarticulation of the Pythagorean-Eleatic equivalence between One and Being. This is the issue of the discovery of the incommensurable magnitudes, which affirm the divisibility in infinitum, but no more as mereologic discretization, namely, as division into parts (as it happened in Zeno’s paradoxes). The mereology is actually internal to the arithmetization of Being, since it expresses the division into fractions. The divisibility in infinitum expressed by the incommensurable magnitudes, on the contrary, does not produce any mereology: between the diagonal and the side of the square there is no common fraction. The divisibility, then, assumes another meaning: that of the fundamentality of the relation (of the division) as such over the entities that are in this relation. It can be expressed also as a passage from the conceptuality of the metechein (participation, to be a part of) to that of synechein (to tie together, to be in relation). Namely: from a digital to an analogue conception of Being. In Theaetetus this emerges in the function of the soul (which is called, just as the incommensurable magnitudes, dynamis – see 147e and 185c), which “puts in relation”, compares (analogizomene, 186a) the beings, finding identities and differences among them. In a nutshell, what the Theaetetus lets laboriously emerge is that Being cannot be identified with the One and therefore discloses a different dimension than that of the One: a further dimension, that of the dynamis, of what is not actual (since it does not express a natural number, an arithmos), but can almost become actual (for instance, if raised to the square). The incommensurable is not allo with respect to the One, it does not introduce a multiplication or a division of the One into infinite other Ones, but is heteron, of a different genus, just like the dynamis with respect to the actuality. A not-number that is also a relative Not-being.
- The Paradoxies of the Arithmetization of Being
This new conception of Being is enshrined in the Sophist, in the famous definition according to which Being is dynamis (247e), and as such anterior, or different, with respect to the One. This definition comes after an examination of the monist and pluralist ontologies preceding Plato, which led to the same aporias highlighted in the Parmenides.
In 237d it is firstly stated that who says something says always also one thing. This claim puts a strict correlation between the something (tì) and the number (arithmós), so that the number itself is counted among the things that are (tôn ónton, 238a): everything that is, is number, and vice versa. The difficulties arising from this claim concern a certain confusion, due to a rather easygoing way of talking (242c). All those who tried to reflect about being, Plato says, tried to solve the question in a mythic manner, by talking us stories, “as we were children”: they reduced the problem of the tò ón to a problem of elementary numbering, that is, by making reference to one, two, three or many entities. This childish way of thinking unites the monists’ and the pluralists’ ontologies. In both, it is a question of counting as we make with fingers: one, two, three, etc.
The Stranger observes, however, that, if we identify the Being with only one of two principles, the other one will be a Not-being, so that there will be by fact only one principle; if, on the contrary, we pose two or even more principles, and all are understood as Being, the result is paradoxically the same: since all are “being”, they turn once again into the one. It follows “that the two are one.” (244a)
It seems then that, if we start from a numeric consideration of the problem of being, there is no alternative to the Parmenidean monism and its paradoxes. Either if the being is one or if we introduce a multiplicity of principles, the result is always the same: “two is one” or “one is two”, that is, “one is also many.” (244a; 245b)
This conclusion is entirely analogous to that one, by which the existence of the incommensurable magnitudes is proved. The negation of their existence – that is: the claim that all is commensurable – involves actually that the one is two, or, more in general, that the odd numbers are identical with the even numbers. So Aristotle says in the Prior Analytics: if the diagonal of the square were commensurable to the side, then the odd numbers were equal to the even numbers (An. Pr. I, 23, 41a 23-31), and then the one would be equal to the two. An absurdity that, how Socrates says in the Theaetetus, one could not believe even by sleeping (190b). The acceptation of their existence saves then from this ontological collapse and represents the definitive overcoming of Eleatism.
- Conclusion
By developping the aporias of the Parmenides, resulting from the arithmetizing identification of Being and One, the Theaetetus and the Sophist come to a new ontology: an ontology that abandons this identification and thinks of Being as the incommensurable itself (Joly, Le renversement platonicien, 1994; Toth, Platon et l’irrationnel mathématique, 2011). As such, Being is located, however, in a different dimension with respect to that of countable entities: the dimension of the dynamis. This result involves some theoretical consequences that goes beyond the Parmenides and are not fully grasped neither by Plato nor by the Greek thought in general, because of the persistence of the henologic principle, which is also a principle of actuality and positivity. To my opinion, they can be summed up under the following titles:
1) Modality. Being is no more conceivable in quantitative terms, but in modal terms: it is different from the actual, it is a principle of the transformation of reality, not of its computability. This resolves also the aporia of the divisibility in infinitum: Being as dynamis is a potentiality, as Aristotle will say, and not an actuality. The divisibility in infinitum does not mean the copresence of infinite beings, but the possibility of the process of division, which is a relational and generative process.
2) Relationality. Moreover, as Hegel will state in the Science of Logic, the incommensurable magnitudes does not represent entities, they are not quanta (finite magnitudes) which result from a division of the Being into parts: they rather signify in a clear manner that Being is a relation and nothing exists out of a relation. The meaning of the division in infinitum is not therefore the fragmentation in infinitesimal parts, in many Ones, which replicate in infinitum this same situation; “division in infinitum” means that the division, that is the relation, the ratio, is the very ontological prius.
3) Positionality. The idea that the One is the principle is overcome by the positional conception of numbers, which involves the introduction of the zero, unknown by the Greeks. From a positional, and not cardinal, point of view, the zero does not mean the nothing: it means the principle, the arché as starting point of something. In the Indian thought, where it seems to have been introduced, the zero is understood as a power, something that is not yet positive or actual, but can arise everything positive and actual.
It is probably in this direction that the second hypothesis of the Parmenides has to be assumed. Although not pursued by the following philosophy, basically henologic, which privileged rather the first hypothesis, it is instead more interesting from a mathematical point of view: “the One is not” leads to the hypothesis of the continuous and to the positional and not cardinal conception of number.