On September 26, 2022, the documentary “A Trip to Infinity” was released on Netflix. This film introduces a group of mathematicians, physicists and cosmologists who talk about infinity (the mathematical symbol of which ( ∞ ) is called ‘a lemniscate’ or ‘lazy eight’). Along the way, the viewers are confronted with seemingly simple questions concerning the concept of infinity like “can you go beyond infinity?”, “is infinity real or a human invention?” and others. The documentary opens with several paradoxes vis-à-vis infinity, one of them stems from the intuitively correct mathematical equation ∞ + ∞ = ∞. Subtracting ∞ from each side of this equation, which is mathematically permitted, leads to a strange result, wherein ∞ = 0, which understandably is logically inconceivable.
This paradoxical end result is left to the viewers’ puzzlement without any feasible explanation. Yet, had the documentary’s editors choose to, they could have turned to Baruch Spinoza’s Ethics  for a possible elucidation. In propositions one through fifteen of Part One, Spinoza presents the basic features of his depiction of God. “In nature, there cannot be two or more substances of the same nature or attribute”, asserts Spinoza. And later on: “A substance which is absolutely infinite is indivisible”. Spinoza presents God in the sense of the infinite total existence, the one that includes all of existence. In his Latin phrase “Deus sive Natura”, ‘God or Nature’, the word ‘or’ should be taken as ‘is’. That is, Spinoza’s God is nature, it is all there is, it is not separate from the entire universe including us humans and everything we do and create. God according to Spinoza is infinite in the sense that there is nothing that is not part of it. From here arises that, if God is infinite and indivisible, if God is nature, and if there cannot be two or more substances of the same nature or attribute, then two infinites cannot exist side by side. The infinite includes all there is, so there can be only one absolute all-encompassing infinity. Therefore, the mathematical equation ∞ + ∞ is illogical to begin with; ∞ cannot stand alongside ∞, and for this preliminary basic error, the equation ∞ + ∞ ends up as a paradox: ∞ = 0.
With Lacan, we can conceive of the infinite, as concerning feminine jouissance, which is considered objectless, infinite and not limited by the signifier. In his Seminar XX, “Encore”, Lacan stresses that this is a jouissance that is in the realm of the infinite; “jouissance of the Other, of the body of the Other, is promoted only on the basis of infinity”. With Zeno’s paradox, Lacan proclaims that “It is quite clear that Achilles can only pass the tortoise – he cannot catch up with it. He only catches up with it at infinity. Here then is the statement of the status of jouissance insofar as it is sexual. For one pole, jouissance is marked by the hole that leaves it no other path than that of phallic jouissance.”
However, the infinity of feminine jouissance is not Spinoza’s infinity. In Spinoza’s infinity no finitude can exist, infinity is everything. On the other hand, the infinity of feminine jouissance is an infinity of the not-all. It differs from Spinoza’s all-embracing absolute infinity because it seems to both continue and perforate the finite. As such, it is not totally all-encompassing, but it is rather the other jouissance which rests alongside the phallic one, and goes beyond it up to the non-reachable vanishing point. The prime infinite cardinal, Aleph null (ℵ0), is inaccessible precisely in this sense, because any union of finite numbers raised to whatever power of finite numbers still falls within the finite and ℵ0 cannot be reached or accessed from the finite. The case of Achilles and the tortoise bears the same logic in that the tortoise is inaccessible to Achilles because he can never catch up with it. When Lacan puts Briseis in the place of the tortoise , the mathematical notion of inaccessibility (i.e. infinity) concerns the woman, and it becomes illustrative of the absence of sexual rapport, because Achilles can overtake her but he will never reach her. Thus this leads to the definition of the impossible and so to the proposition by which the above mentioned equation (∞ + ∞ = ∞), which ends up in an impossibility (∞=0), corresponds with the impossibility of writing the sexual rapport.
- Jonathan Halperin and Drew Takahashi (Directors), Jonathan Halperin and Alex Ricciardi (writers), A Trip to Infinity. Netflix, 2022; https://www.youtube.com/watch?v=CNFm_DzHDaE
- Spinoza, B. (1996). Ethics. Translated by E.M. Curley. London: Penguin Books.
- Ben-Hagai, K. Pain and jouissance. Lacanian Review Online, 259, November 21st, 2020. https://www.thelacanianreviews.com/pain-and-jouissance/
- Lacan J., The Seminar of Jacques Lacan, Book XX, Encore, edited by J.-A. Miller, trans. B. Fink, New York & London: Norton, 1998, 7-8.
- Ibid, 8.
- Focchi, M. Number in science and in psychoanalysis. Psychoanalytical Notebooks, Issue 27, September 2013. London Society of the New Lacanian School.
- Lacan J., The Seminar of Jacques Lacan, Book XX, Encore, edited by J.-A. Miller, trans. B. Fink, New York & London: Norton, 1998, 8.