30 Dec 2008

Genetic Definition in Spinoza's Improvement of the Understanding §§91-97, with Deleuze's Commentary

[The following is summary. I quote Deleuze after the summary. My own commentary is in brackets. The complete quotation of the passages follows at the end.]

Spinoza, On the Improvement of the Understanding (Treatise on the Emendation of the Intellect):

[91] [91e] (1) (2)

Spinoza will discuss clear and distinct ideas. When we are affected physically, as for example when the sun hits our eyes, we receive an impression and we form an idea, in this case, of the sun. These impressions are inadequate, because we do not see the whole chain of causation leading up to the event of that impression. We know something by its cause, so because we do not have the full picture of causation, we do not fully understand the sun as it is in that particular event.

So we will seek clear and distinct ideas not in the imagination but in the pure intellect.

[91] (3)
There are an infinity of ideas. Hence Spinoza will discuss them generally, so that we have a general understanding of all ideas, which will better allow us to understand any given particular idea in its individuality.

[92] (1)
As Spinoza noted earlier, something is understood in either of two ways

a) through its essence
b) through its proximate cause (what immediately causes it)

[92] (2)
When something is not self-existent, it is caused by something else. Such things must be understood by the proximate cause that brought them about. However, things that are self-existent cause themselves, (because their nature or definition necessitates that they exist). Because such things are not caused by anything else, and because their existence follows from their nature (or definition), we must understand them by means of their essence alone.

[92] (3)
For, we better know an effect when we have a "more perfect knowledge of its cause."

[93] (1)
And we must keep clearly separated

a) the thing, and
b) the idea of the thing.

[Spinoza says specifically: "we shall be extremely careful not to confound that which is only in the understanding with that which is in the thing itself." see the entry on Deleuze's explanation of the difference between the objective and formal reality of ideas.]

[93] (1)

In his Ethics, Spinoza offers as his first axiom:

I. Everything which exists, exists either in itself or in something else.

This should be self-evident, in absence of a third option. The axiom applies universally to everything no matter what, so Spinoza calls it infinite. But because it is so general, we cannot begin from this axiom and then deduce particular properties about particular things. So from this axiom, we know that a circle exists either in itself or in something else. But we do not know yet what makes a circle a circle, thus we do not yet have a clear and distinct idea of it. Hence, our knowledge of the circle is not adequate when understood solely through axioms.

So we need a way to distinguish something from everything else. We need something, unlike the infinite axiom, that regards the circle in its finitude and that delimits it from everything else. In other words, we need its definition.

Also, if we knew everything that was essential to the thing, we also would have adequate knowledge of it; and from that essence we should be able to determine the other properties that follow from this essence. So for example, we know from what is essential to triangles and right angles (or from their definitions) that a triangle's interior angles equal two right angles. So a triangle is not defined as being a plane figure whose interior angles equal two right angles, nor is this essential to the triangle; however, this property of triangles follows necessarily from its essence or definition.

[94] (1)
Thus our method for discovering something's true nature is to grasp its definition, and to conceive the things that follow from the definition.

[94] (2)
And the better the definition, the easier it is to discover what follows from it.

[94] (3)
So our method must create definitions that are the most fruitful for producing consequential properties.

[94] (4)
Hence Spinoza will now explain the conditions for good definitions.

[95] (1)
A definition is perfect if it explains what is most essential to the defined thing. Also, the definition is not to list all the properties of the thing; rather, we should construct the definition so that all these properties follow from the definition. [It is in this way that a definition "produces" or generates the properties of the thing. These other properties were no less expressed in the original definition, only they were "involved" or "implied" or "implicit" in the thing's nature, essence, or definition. These properties then "evolve" or "explicate" or become "explicit" when we think rationally about the essence or definition. It is for this reason that Spinoza says that something which is self-caused has an essence that "involves" its existence. What that means is, the property of its existence follows from its essence or definition, that is, it is produced by that definition. For, it is "self-caused," that is, its essence causes its existence because its essence follows from its existence, which is to say, the thing's existence is involved in its essence. See the entry on implication and explication for more on expression.]

[95] (2) (3)
Spinoza considers the circle for an example.

[95] (4)
One of the properties of a circle is that all straight lines drawn from the center to the circumference are equally long. But this is just a property of the circle. It is not what is essential to it. Our definition should be crafted so that this property follows from the definition, and not that this property be explicit in the definition. In other words, our definition must imply the defined thing's properties.

[95] (5)
Although in this example it is not so important whether or not the properties follow from the essence, it is however very important that the properties of physical beings follow from their essence. For, we have no other way of understanding the properties of physical things except by understanding their essences.

[95] (6)
Spinoza claims that according to the order of Nature, properties follow from their essences. So if we do not begin with the essence before obtaining the properties, we are "perverting" the succession of ideas which reflects the order of Nature, and we thereby fall astray from the object we are trying to understand.

A. Rules for any definition:

So that we do not begin first by understanding something's properties before understanding its essence, we must observe the following rules:

A. Rule I:
[96] (1) The definition must contain the proximate cause of the thing. [96] (2) So for example, we should define the circle as the figure traced by the end of a moving line rotating around its other endpoint, which is fixed in place, like how we draw a circle with a compass. (Spinoza writes specifically that a circle is definied as "the figure described by any line whereof one end is fixed and the other free").

A. Rule II:
[96] (4) We must be able to deduce all the thing's unique properties from its definition. We see that given our above definition of the circle, it follows that all straight lines drawn from the center to the circumference are of equal length. [96] (5) So because we must deduce all the thing's properties from its definition, the definition then must be affirmative. [96] (6) Spinoza here is speaking of intellectual affirmation, and not verbal affirmation.

B. Rules for the Definition of self-caused, uncreated things:

B. Rule I:
Only the definition is needed to explain the thing. We do not need anything outside the thing to account for it, which is the same as saying that it has no exterior cause.

B. Rule II:
The thing's existence cannot be doubted given the definition. So, the definition must make it absolutely necessary that the thing exists. In other words, the thing's existence must follow from the definition the same way the circle's properties follow from its definition.

[We might define something by genus and species, for example, that a circle is a plane figure whose points are equidistant from a center point. Here we subsume the notion of circle under something taken as more abstract, namely, the figure. This is different from Spinoza's definition of the circle as "the figure described by any line whereof one end is fixed and the other free," because "figure" here is not taken as some abstraction that includes circles, but rather it is the result when we produce the circle. But in the case of the genus-species definition, we could have also turned the substantive (noun) "figure" into an adjective by saying that a circle is something figural whose points are equidistant from a centerpoint. We can turn this substantive into an adjective, because the substantive "figure" is abstract, and includes the circle as being something more specific. But in this way, our definition has turned to something presupposed and extrinsic in order to explain the defined thing. Hence,]

B. Rule III:
The definition cannot contain substantives (nouns) which can be turned into adjectives; or in other words, the definition may not explain the thing abstractly.

B. Rule IV:
[which is not absolutely necessary], We should be able to deduce all the thing's properties from its definition.

Deleuze, Commentary, Expressionism in Philosophy: Spinoza:

We have an adequate idea to the extent that, from a thing, some of whose properties we conceive clearly, we give a genetic definition, from which follow all of its known properties (and still others that we do not know). It has often been noted that the only role of mathematics in Spinoza is to provide such a genetic process. The cause as sufficient reason is what, being given, means that all the thing's properties are also given, and, being withdrawn, means that all the properties are withdrawn with it. We define the plane by the movement of a line, the circle by the movement of a line with one endpoint fixed, the sphere by the movement of a semicircle. To the extent that a thing's definition expresses its efficient cause or the genesis of what it defines, the thing's idea itself expresses its own cause, and we have rendered the idea adequate. Thus Spinoza says that the second part of Method is primarily a theory of definition: "the chief point of this second part of the Method is concerned solely with this: knowing the conditions of a good definition . . . ."


Deleuze, Spinoza et le problème de l'expression:
Nous avons une idée adéquate dans la mesure où, de la chose dont nous concevons clairement certaines propriétés, nous donnons une définition génétique, d'où découlent au moins toutes les propriétés connues (et même d'autres que nous ne connaissions pas). On a souvent remarqué que les mathématiques chez Spinoza avaient exclusivement le rôle d'un tel processus génétique. La cause comme raison suffisante est ce qui, étant donné, fait que toutes les propriétés de la chose le sont aussi, et, étant supprimé, fait que les propriétés le sont toutes également. Nous définissons le plan par le mouvement de la ligne, le cercle par le mouvement d'une ligne dont une extrémité est fixe, la sphère par le mouvement d'un demi-cercle. Dans la mesure où la définition de la chose exprime la cause efficiente ou la genèse du défini, l'idée même de la chose exprime sa propre cause: nous avons fait de l'idée quelque chose d'adéquate. C'est en ce sens que Spinoza dit que la seconde partie de la méthode est d'abord une théorie de la définition: « Le point principal de toute cette seconde partie de al méthode se rapporte exclusivement à la connaissance des conditions d'une bonne définition ... »


[The following is the text in full for these summarized Spinoza passages]

[91] [91e] (1) Now, in order at length to pass on to the second part of this method, I shall first set forth the object aimed at, and next the means for its attainment. (2) The object aimed at is the acquisition of clear and distinct ideas, such as are produced by the pure intellect, and not by chance physical motions. (3) In order that all ideas may be reduced to unity, we shall endeavor so to associate and arrange them that our mind may, as far as possible, reflect subjectively the reality of nature, both as a whole and as parts.
[92] (1) As for the first point, it is necessary (as we have said) for our purpose that everything should be conceived, either solely through its essence, or through its proximate cause. (2) If the thing be self-existent, or, as is commonly said, the cause of itself, it must be understood through its essence only; if it be not self-existent, but requires a cause for its existence, it must be understood through its proximate cause. (3) For, in reality, the knowledge, [92f] of an effect is nothing else than the acquisition of more perfect knowledge of its cause.
[93] (1) Therefore, we may never, while we are concerned with inquiries into actual things, draw any conclusion from abstractions; we shall be extremely careful not to confound that which is only in the understanding with that which is in the thing itself. (2) The best basis for drawing a conclusion will be either some particular affirmative essence, or a true and legitimate definition. (93:3) For the understanding cannot descend from universal axioms by themselves to particular things, since axioms are of infinite extent, and do not determine the understanding to contemplate one particular thing more than another.
[94] (1) Thus the true method of discovery is to form thoughts from some given definition. (2) This process will be the more fruitful and easy in proportion as the thing given be better defined. (3) Wherefore, the cardinal point of all this second part of method consists in the knowledge of the conditions of good definition, and the means of finding them. (4) I will first treat of the conditions of definition.
[95] (1) A definition, if it is to be called perfect, must explain the inmost essence of a thing, and must take care not to substitute for this any of its properties. (2) In order to illustrate my meaning, without taking an example which would seem to show a desire to expose other people’s errors, I will choose the case of something abstract, the definition of which is of little moment. (95:3) Such is a circle. (4) If a circle be defined as a figure, such that all straight lines drawn from the center to the circumference are equal, every one can see that such a definition does not in the least explain the essence of a circle, but solely one of its properties. (5) Though, as I have said, this is of no importance in the case of figures and other abstractions, it is of great importance in the case of physical beings and realities: for the properties of things are not understood so long as their essences are unknown. (6) If the latter be passed over, there is necessarily a perversion of the succession of ideas which should reflect the succession of nature, and we go far astray from our object.
[96] In order to be free from this fault, the following rules should be observed in definition:— I. (1) If the thing in question be created, the definition
must (as we have said) comprehend the proximate cause.
(2) For instance, a circle should, according to this rule,
be defined as follows: the figure described by any line
whereof one end is fixed and the other free. (3) This
definition clearly comprehends the proximate cause. II. (4) A conception or definition of a thing should be such
that all the properties of that thing, in so far as it is
considered by itself, and not in conjunction with other
things, can be deduced from it, as may be seen in the
definition given of a circle: for from that it clearly follows
that all straight lines drawn from the center to the
circumference are equal. (5) That this is a necessary
characteristic of a definition is so clear to anyone, who reflects on the matter, that there is no need to spend time
in proving it, or in showing that, owing to this second
condition, every definition should be affirmative. (6) I
speak of intellectual affirmation, giving little thought to
verbal affirmations which, owing to the poverty of language,
must sometimes, perhaps, be expressed negatively, though
the idea contained is affirmative.
[97] The rules for the definition of an uncreated thing are as follows:— I. The exclusion of all idea of cause — that is, the thing
must not need explanation by Anything outside itself. II. When the definition of the thing has been given, there must
be no room for doubt as to whether the thing exists or not. III. It must contain, as far as the mind is concerned, no
substantives which could be put into an adjectival form;
in other words, the object defined must not be explained
through abstractions. IV. Lastly, though this is not absolutely necessary, it should
be possible to deduce from the definition all the properties
of the thing defined. All these rules become obvious to anyone giving strict attention to the matter.


Spinoza. On the Improvement of the Understanding (Treatise on the Emendation of the Intellect). Transl. R.H.M. Elwes.
Full text available at:

Deleuze, Gilles. Spinoza et le problème de l'expression. Paris: Les Éditions de Minuit, 1968.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Trans. Martin Joughin. New York: Zone Books, 1990.

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