12 Jan 2009

Leibniz' Mens Momentanea (Momentary Mind) in Studies in Physics and the Nature of Body

by Corry Shores
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At the beginning of chapter 2 of Difference & Repetition, Deleuze refers to mens momentanea. We will examine Leibniz' use of the term in his "Studies in Physics and the Nature of Body," principles 1 through 17. Skip to 17 for what is specifically relevant to Difference & Repetition.

Leibniz "Studies in Physics and the Nature of Body"

I. The Theory of Abstract Motion: Fundamental Principles, [Praedemonstrabilia]

A continuum contains actual parts.

For Descartes, the indefinite is found in the thinker and not the thing. However, the continuum's parts are actually infinite.

If something has a position, it is surrounded by other things that do not necessarily touch each other. So if something could not be so surrounded, it would not have position.

Something with a magnitude should have a position, hence it should be able to have things surrounding it which do not touch one another. But, if the thing's magnitude were zero, then its surrounding bodies would touch. For, there would be nothing between them. Hence, a magnitude cannot be zero, and therefore there is no minimum in space or in a body.

Also, if there were a minimum value equal to zero, then there would be just as great an infinity of minima in the part as in the whole. [See the entry on Spinoza's 12th letter on the infinite for more on this debate.] But this is absurd, so there cannot be such zero-valued minima.

Any given space, body, motion, or time has a beginning and an end. For, it extends. But, does its beginning itself extend? Leibniz says no, and offers the following proof.

Consider line ab.

We place point c half-way between the endpoints.

Half-way between a and c we place point d.

Similarly for point e.

We can imagine placing halfway-points on-and-on toward a.

If both the beginning of something along with its remainder were removed, nothing would be left. And if we remove line segment dc, something remains, ad. Therefore, segment ac could not be the beginning. For, we may take-away a right-side portion of ac without destroying its left side, so the beginning must be found somewhere between a and d But, we may remove ed without destroying all of ad. Therefore ad is not the beginning, and we must seek it in ae.

So nothing is a beginning from which something on the right side can be removed. But that from which nothing extended can be removed is unextended. Therefore the beginning of body, space, motion or time namely, a point, conatus, or instant is either nothing which is absurd, or unextended, which was to be demonstrated. (140a)

The beginning of something cannot be extended. For, everything extended is divisible; but, if something is divisible, it itself is not the beginning. Rather, the beginning is to be found within it. But we also know that the beginning cannot be nothing, because we already presumed that there is a beginning. Hence the beginning must not have extension. [It has magnitude, so it has intensive magnitude. See Deleuze's derivation of intensive magnitude from such infinitely small divisions in his commentary on Spinoza's Letter on Infinity in Cours Vincennes 20-01-81. Also, compare this argument to Zeno's debate with Eudemos (I.C, Only finitely extending things can exist). Zeno argues that if you add something to something else, without also thereby increasing the second thing's size, then the first one does not exist. Leibniz here makes the contrary argument. It is premised on unextended magnitudes, that is, intensities in the sense of differentials. See Leibniz' helpful visualization of calculus differentials, or the remarkably illuminating introductory calculus lecture by MIT's Prof. David Jerison.]

So because beginnings are indivisible and unextended, there are indivisibles or unextended beings. (139-140)

Cavalieri's method of indivisibles takes as its ground that a surface is composed of an indefinite number of parallel lines, and that a solid is made up of an indefinite number of planes. These parts are the constituent indivisibles of the forms. This theoretical ground is demonstrated in his method, which is able to determine the area of a plane figure by totaling these indivisibles. Leibniz builds from this theoretical groundwork. He says that there are no points whose part equals zero, or whose parts lack distance. We can conceive and designate the magnitude of any point. Even though such points are not extensive, they still have a value. However, this magnitude is still less than any given or givable value. In other words, it is infinitesimal [and intensive rather than extensive.]

A point's extension in space does not equal zero. But rest is zero motion [compare to Leibniz' law of continuity where he seems to make a different claim. In that context he describes rest as an infinitely small amount of motion.]. So the ratio of rest to motion is not the ratio of a point to space. For, because the point is infinitely small, the ratio of a point to space is one to infinity. However, the ratio of rest to motion is equal to the ratio of zero to one; for, their difference is absolute.

When something is moving, it does so continuously. In other words, in motion there are never tiny pauses during which the body is at rest.

When something is at rest, it will remain at rest until something else causes it to move. If motion had little moments of rest, then the motion would tend to be at rest until acted-on by something else.

And we know also that when something is in motion, it will move at the same velocity and direction unless something else acts on it. So if rest tends to stay resting, and motion tends to stay moving, then there could not be little intervals of rest in motion. For, there would need to be something external to cause it to come to rest and likewise to cause it to return moving.

[While writing this, Leibniz did not yet distinguish force from motion. Also, what Hobbes called "endeavor" is here "conatus." editor's note page 144.] Because a point is infinitely small, and not nothing, the ratio of a point to space is one to infinity. Conatus is endeavor. It is motion at a point, or instantaneous velocity. So the ratio of conatus to motion is analogous to the ratio of a point to space: both are ratios of one to infinity. For, conatus is the beginning and end of motion, just as the infinitely small magnitude was the beginning and the end of the line [as explained in 4].

When something very small hits something much larger than it, the smaller thing seems to stop its motion. But at the moment of contact, the tiny body is still striving to move, even though in the next moment it will not move. Instead, that endeavor to move is transferred to the larger object, which moves ever so slightly, even if infinitely little. So even if the larger body was not tending in any motion, even the smallest body when contacting it will cause it also to tend somewhere, even if that tendency is too small to change its location. So when bodies make contact

1) they transfer their tendencies to each other, and
2) they cause each other to move, even if ever so slightly.

Conatus is the beginning and the end of motion. So in a collision, the end of the motion of the one body is the same as the beginning of the motion of the other body. This, again, is motion at a point.

Body c is moving toward point b on line ab. And body d is moving toward point a.

They continue toward each other

and then collide

At the moment of collision, the two bodies are still
1) striving forward,
2) transferring their conatus, and thus
3) striving backwards
all in the same moment.

Hence there can be many contrary conatuses in the same body at the same time.

But this holds only for a moment, because immediately after their exchanged-conatuses are co-present, the balls bounce back away from each other, carrying the other's conatus with them.

Conatus is an instantaneous motion. There is an infinitely small movement during an infinitely small amount of time. Nonetheless, any point on that body is tending some direction, so in that instantaneity any one point on the body crosses many points in space.

One point of a moving body at the time of conatus, or in a time less than any assignable time, is in many places or points of space.

So, because the moving body's points are simultaneously in many points in space, the body will fill a space greater than its own size. [see the entry on Zeno's paradox of movement in place. Zeno thinks that it is absurd for something to be in two places at once.]

the body will fill a part of space greater than itself, or greater than it would fill at rest or if moving more slowly, or if striving in one direction only. (140d)

But the ratio between the point on the body to the point of space it takes-up is still infinitely small.

Consider a circle with a tangent line and another line intersecting that point of tangency, like this:

And consider this other line pivoting on the point of tangency, and gradually moving closer to the tangent.

The line intersects another point on the circumference, that slowly moves toward the point of tangency. As it continues to get closer, we will need to enlarge the area around the point of tangency to see what is happening.

We see that the lines' angles get closer to each other the more the moving circumference-point nears the point of tangency.

Gradually, the moving line comes closer to being at the same angle as the tangent. But so long as the point moving along the circumference is not precisely the same as the point of tangency, the angle will always be off by at least a little.

[Above, the dark band is the circle's circumference, and the two smaller lines are the tangent and moving line.]
But as the point moving along the circumference reaches the limit, there will no longer be any extensive space between them. Thus the angle separating the lines will be infinitely small. It will not be the same as zero, but also there is no way to quantify its extent. Leibniz likens the ratio between those angles as being similar to the ratio of the point on the moving body to the points of space it takes-up. So in other words, there is a ratio, but it is infinitely small.

So something moving is in more than one spatial point and temporal point in any instant. For, an instant is not a zero amount of time, but rather is an infinitely small amount. And in that infinitely small amount of time, the body moves an infinitely small amount of space.

whatever moves is never in one place when it moves, nor indeed in one instant or least moment of time, because whatever moves in time strives, or begins and stops moving, in that instant, that is, it changes its place.

Moreover, there is no portion of space or time that equals zero; for, the infinitely small still has a magnitude, just not an extensive one. Thus there is no sense in saying that the instantaneous motion strives in a minimum space and minimum time (see number 3).

In fact,

at the time of impulsion, impact, or collision, the boundaries or points of two bodies either penetrate each other or are in the same point of space. For when one of two colliding bodies strives into the position of the other, it begins to be in it, that is, it begins to penetrate or to be united.
Conatus is the beginning and end of motion. And when bodies collide, their boundaries coincide in the same points of space. For, their conatus expands them an infinitely small amount into each other. Hence conatus is beginning, end, and penetration.

The bodies are therefore in the beginning of union, or their boundaries are one. (141a)

When bodies collide, they transfer their conatus, and thereby push or impel each other. And, at the moment of impact, their boundaries are one, because the conatus extends them both forward into the same infinitely small amount of space. Leibniz agrees with Aristotle's claim that bodies sharing boundaries are continuous or cohered.

For if two things are in one place, one cannot be put in motion without the other.
So when two bodies impact each other, their boundaries coincide in the same space, but only for a moment. When a body hits our own body, we have a sensation. For example, when the sun's rays hit our eyes, we have a sensation of light. But the two contrary conatuses only coincide for a moment. So the body only retains such an instantaneous motion for merely an instant. The mind however, may remember the sensation; we may recall easily the painful sensation of the sun light hitting our eyes. So only in the memory of our minds may two contrary tendencies last for longer than an instant:

No conatus without motion lasts longer than a moment except in minds. (141b)
Leibniz calls the body a mens momentanea, that is, a momentary mind, or momentaneous mind. [We might also call it an instantaneous mind.]

[Deleuze implicitly refers to this passage in the second chapter of Difference & Repetition, so we reproduce below two translations of the passage, along with the Latin.]

No conatus without motion lasts longer than a moment except in minds. For what is conatus in a moment is the motion of a body in time. This opens the door to the true distinction between body and mind, which no one has explained heretofore. For every body is a momentary mind (mens momentanea), or one lacking recollection [recordatio], because it does not retain its own conatus and the other contrary one together for longer than a moment. For two things are necessary for sensing pleasure or pain action and reaction, opposition and then harmony and there is no sensation without them. Hence body lacks memory; it lacks the perception of its own actions and passions; it lacks thought.
(Leibniz, Philosophical Papers and Letters, 141b.c)

,,No motive momentum (conatus) without motion exceeds the moment except with the minds. For what is the motive momentum at one moment is the motion of a body in a time series. At this point, every person who wants to proceed has the door of the true distinction which no one has explained so far between body and mind open to him. For every body is a momentaneous mind (mens momentanea), that is to say, a mind without memory. For it does not hold on to its own, and at the same time to its alien, contrary motive momentum longer than the moment (as both action and counter-action, that is to say, equation and thus harmony are necessary for any sort of sensory sensation, and without this no sensation exists, neither pleasure nor pain); [the mere body] therefore has no memory; it has no sensation of its actions and passions; it has no consideration."
(Leibniz, qtd in Busche 151b)

,,Nullus conatus sine motu durat ultra momentum praeterquam in mentibus. Nam quod in momento est conatus, id in tempore motus corporis: hic aperitur porta prosecuturo ad veram corporis mentisque discriminationem, hactenus a nemine explicatam. Omne enim corpus est mens momentanea, seu carens recordatione, quia conatum simul suum et alienum contrarium (duobus enim, actione et reactione, seu comparatione ac prionde harmonia, ad sensum, et sine quibus sensus nullus est, voluptatem vel dolorem opus est) non retinet ultra momentum: ergo caret memoria, caret sensu actionum passionum-que suarum, caret cogitatione"; A VI 2, 266, 13-20.

Leibniz. Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956

Leibniz. qt in Busche, H. "Mind and Body in the Early Leibniz." in Individuals, Minds and Bodies: Themes from Leibniz. Eds. Massimiliano Carrara, Antonio-Maria Nunziante, Gabriele Tomasi. Franz Steiner Verlag, 2004.
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  1. thanks for this post, helps so much

  2. Thank you too; I am glad I can help. Corry