An Experience of Infinity

An Experience of Infinity

by Ohad Nachtomy, Professor of Sonia T. Marschak Academic Chair in Humanities and Arts, Israel

An Experience of Infinity in Leibniz’s Notes from Paris 1676

In 1672, an ambitious young man arrived in Paris on a diplomatic mission. The young German arrived with the idea to divert the French army from invading the Netherlands by convincing Louis XIV to invade Egypt instead.[1] The so-called Egyptian plan was schemed by Gottfried Wilhelm Leibniz and his mentor, Baron Boineburg. Leibniz was sent Paris to promote it. While he could not meet the French king, the mission did enable the young Leibniz to move from the provincial town of Mainz to Paris, and to spend four years at the intellectual center of Europe, where much of the activity forming modern science and philosophy was taking place. Leibniz did not waste his time. By the end of his sojourn in Paris he managed to establish an important intellectual network and to come up with some incredible achievements such as the rudiments of his differential calculus, a calculating machine, important philosophical writings such as the Confession of a philosopher and a great variety of notes on metaphysics of remarkable interest and inventiveness. Some of these notes were collected and edited by the Akademie Ausgabe under the title De summa rerum (DSR), which means both “on the highest of things” and “on all things.”[2] Leibniz is well known for his major contributions to the development of the notion of infinity in mathematical, philosophical, and theological contexts. It would not be exaggerated to say that Leibniz had immense influence on the introduction of infinity into the language of science as a description of nature. Indeed, in a famous letter to Foucher (quoted later by Georg Cantor) he wrote: “I am so much in favor of actual infinity that, instead of admitting that nature rejects it, as it is vulgarly said, I hold that it affects it everywhere, for better marking the perfections of its author” (GP I:416). If Leibniz is well known for endorsing actual infinity, arguing that it affects nature everywhere, it escaped our attention that he was also preoccupied with experiencing infinity.

A curious note from 1676 that evokes this theme (but has gone almost unnoticed)[3] reads as follows:

The following operation of the mind seems to me most wonderful: namely, when I think that I am thinking, and in the middle of my thinking I note that I am thinking about my thinking, and a little later I wonder at this tripling reflection. Next, I also notice that I am wondering and in some way wonder at this wonder, and fixed in one contemplation I return more and more into myself, […] So, it comes about that we are finally wearied by this effort and feel a headache or, even (if we persist) become insane (DSR 73).

Leibniz is recounting here a strange personal experience, attending to reflections of his own reflections. The experience has a reiterative structure and could go on ad infinitum. Leibniz finds the experience not only possible but also wonderful. He is experimenting, it seems, with our ability to sense infinity through reiterative reflection. Given that much of his mathematical work during these years is dedicated to dealing with infinity and that much of his philosophical efforts are dedicated to defending the possibility of an infinite Being while rejecting the possibility of an infinite number, it is perhaps less surprising that infinity comes up in his sleepless nights.[4] 

In the sequel to the text cited above, Leibniz even provides some instructions as to how one can have such an experience (the perception of perceptions to infinity):

If anyone wants to experience this, all he has to do is this: at some point in the darkness of the night, when it happens that he cannot sleep, let him begin to think of himself and of his thinking and of the perception of perceptions. (DSR 73).

The perception of one’s perceptions could certainly go on for a long time. Hopefully, one would end up falling asleep rather than being wearied by a headache or, if persisting, becoming altogether insane.

The passage ends with the following conclusion: “and so the perception of perception to infinity is perpetually in the mind and in that there consists its existence per se and the necessity of its continuation” (DSR 75). Leibniz is seeking to support his initial point that there is some memory after death, as there is when we fall asleep (DSR 73). This provides support to his view that an active soul would remain active, unless annihilated by God. It relates to his later notion of “petits perceptions,” perceptions that are always taking place in the background and that we are usually not aware of (such as background noise we notice only when it disappears).

Leibniz’s extraordinary account of his own experience turns out to be important for the other philosophical seeds it sows. While the passage captures a strange nocturnal experience, the ability to have perception of perceptions (later to be called apperception) turns out to mark the difference between the human mind and that of other animals. For Leibniz, the distinctive feature of the human mind is that it can reflect on its own thoughts. This in turn will serve him as one of the necessary requirements for human freedom. This mark of the human mind can be gleaned from other passages in his Paris notes as well. For example, Leibniz writes that, “we do not act as a simple machine but out of reflection, i.e. out of action on ourselves” (DSR 37). He also notes that when I think and reflect on myself for a long time, with continuous reflections on a reflection, there is, as it were, a kind of amazement; and a wondering at this reciprocation. […] If this is the nature of the mind, and it consists in the sense of itself, then I do not see how that sense can be impeded or destroyed (DSR 61, my italics).

Reflections on reflections belong to the nature of the human mind. But it is also a defining feature of the divine mind. As Leibniz write: “it must be demonstrated rigorously that he [God] senses his own actions on himself, for there is nothing more admirable than for the same being to sense and to be affected by itself” (DSR 27). This view echoes Aristotle’s nous (νοῦς) as a mind contemplating itself (hê noêsis noêseôs noesis, 1074b21–35, spec. ll. 34-35).[5] In the Leibnizian context, reflections of reflections acquire additional meaning. Self-reflection is constitutive of God’s mind. It also plays a role in the conceiving of all possibilities, which is central to Leibniz’s metaphysical and modal system. Indeed, as I describe elsewhere, in the De summa rerum, these possibilities are conceived by God as he reflects on his own attributes.[6] In much later texts, Leibniz explicitly refers to a “Divine Mathematics and Metaphysical Mechanism” figuring at the “radical origin of things” (AG 151). This mechanism explicates the way all possible worlds are conceived in God’s understanding. As Leibniz famously notes, “Cum Deus calculat et cogtationem exercet fit mundus”.[7] What often escapes attention is that such a calculating mechanism presupposes divine self-reflection.

But there is no space for the details here. Instead, in conclusion, I want to draw attention only to the fact that the very ability of reiterative reflection upon reflections turns out to be a feature that the human mind shares with God’s mind. It may well be seen as one of the ways in which humans are created in God’s image (imago Dei), and one of the ways in which humans share and manifest the infinity of God. Thus, a rather strange and somewhat obscure nocturnal experience the young Leibniz has in the middle of his sleepless nights in Paris turns out to bear some significant consequences for the development of his philosophy, and probably philosophy more broadly.

The relation between the nature of the mind (in distinction from a body comes up in Leibniz’s early thought. As he notes in a famous line from the Theory of Abstract Motion [winter 1670-1671, “Each body is namely a momentary mind, or one that lacks memory”. Since “it lacks memory, it lacks the sense of its own actions and passions, it lacks thought” (A VI, 2, 266).  

In the Elements of Natural Law [second half of 1671 (?)], Leibniz defines striving as “the beginning of action” and thought as “action on itself”,

Whatever acts on itself, has some memory (for we remember when we sense that we have sensed); and consequently the perception of harmony or disharmony or of lust or pain, through the comparison of an old and a new sensory impression, and also an opinion or an expectation derived from this of a future sensory impression and from this again the conatus to act, i.e., the will. (A VI, 1, 483).

In On the Union of the Soul and the Body, Leibniz points out that “we do not act as a simple machine, but out of reflection, i.e., of action on ourselves” (A VI, 3, 480; DSR 37). According to On Memory and the Reflection of the Mind on Itself, the perception of perceptions is what constitutes the per se existence of a mind and the necessity of its continuation (A VI, 3, 517; DSR 75).  In On Existence, one of the last papers of his Paris years, he puts the point thus: “Thought, or the sensation of oneself, or action on oneself, is necessarily continued” (A VI, 3, 588; DSR 113). A slightly earlier paper, On Forms or Attributes of God, introduces this explicitly as a criterion for substantiality: “Thought is not duration, but that which thinks is something that endures. And this is the difference between substance and forms (A VI, 3, 514; DSR 69) .

 
[1] See M. R. Antognazza, Leibniz. An Intellectual Biography, Cambridge: Cambridge University Press, 2009.

[2] Most of these texts were translated into English by G.H.R. Parkinson in the Yale Leibniz series in 1992. My citations here refer to this edition, abbreviated as DSR.

[3] Parkinson, for example, notes that, for Leibniz, “to think is to reflect” (Introduction to DSR, p. xlvi) but ignores Leibniz’s point about reflections on reflections. For discussion, see S. di Bella, “Memoria e individualità. L'ontologia della temporalità nelle note parigine di Leibniz (1676): un confronto a distanza con Spinoza?”, in Ontologia e temporalità. Spinoza e i suoi lettori moderni, ed. G. D’Anna-V. Morfino, Milan: Mimesis, 2012, pp. 81–107, esp. pp. 85–88. But, as can be seen from the title, the focus here is the much-debated confrontation between Leibniz and Spinoza.

[4] This is clearly exhibited in the two short pieces Quod Ens Perfectissimum Sit Possible (DSR 91–95), followed by Ens Perfectissimum Existit (DSR 96–100). This is Leibniz’s amendment to Descartes’ proof for the existence of God, which Leibniz also shows Spinoza in their single meeting during Leibniz’s trip from Paris back to Germany. According to Leibniz, only if the notion of the most perfect being is shown to be possible, would it follow that he exists. I have argued for these points at length in Living Mirrors: Infinity, Unity, and Life in Leibniz’s Philosophy, Oxford: Oxford University Press, 2019.

[5] Aristotle’s Metaphysics Book Λ/XII, chapter 9 and De anima III 1.

[6] This is related to Leibniz’s early work on the art of combination (1666) and other texts from Leibniz’s early writings. I have written on this extensively in my Possibility, Agency, and Individuality in Leibniz’s Metaphysics, Dordrecht: Springer, 2007.

[7] “As God calculates and exercises his thoughts, the world is made” (A VI 4, Part A N. 8, 22, 1677)

Ohad Nachtomy is a Sonia T. Marschak professor at the Humanities and Arts department at Technion, Israel Institute of Technology. At 2020 I was elected to turn a large teaching operation into an excellent research department with the aim of forging productive links between the humanities and the sciences. His research is focused on early modern philosophy, philosophy and history of biology, mind-body relations, Wittgenstein’s philosophy, and multicultural theory (especially in the Israeli context).

September 5, 2025