by Corry Shores
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[Note: All boldface and underlining is my own. It is intended for skimming purposes. Bracketed comments are also my own explanations or interpretations.]
Hegel, Deleuze, and the Critique of Representation.
Dialectics of Negation and Difference
Part 2: Responses to Representation
Chapter 3: Bergsonism
Subdivision 2: Bergson’s Account of Kant and Classical Logic
Very brief summary:
Many theories of our understanding of the world regard there being an isomorphism between the structures of our world and our representations of it. Spencer thinks we evolved to see the world spatially as an adaptation to the way it really is. Kant assumed our representation of space is an a priori representation that conditions our external perceptions so that they are always spatial. Kant also saw an isomorphism between the understanding and the empirical world. Homogeneous space for Bergson is a medium by which atomic things are externally related. Under this analysis, Kant’s transcendental apperception, because it is what allows our understanding to externally relate terms of our judgments, is like homogeneous space. We can trace the concept of homogenous space to Euclid and see it persist throughout the development of physics, to mechanism and to Spencer. But as the Pythagoreans learned when discovering incommensurability, the world does not always match our representations of it. Uncovering this mismatch is a part of Bergson’s method of intuition.
To explain our knowledge of the world, many theories throughout the history of philosophy have regarded there being an isomorphism between the structures of our understanding and the world we are trying to understand. We saw this isomorphism with Kant’s transcendental idealism. Here the subject-predicate structure of synthesized concepts is isomorphic to the subject-property structure of synthesized objects of our intuition. But for Kant, our representation of space is isomorphic with the space we perceive, because our representation is a priori and conditions how we perceive objects. We can only perceive them as being external one to the other in a homogenous space. For Spencer’s evolutionary theory, we begin with these a priori structures because we evolved to accurately represent the real spatial world around us for the sake of survival. Bergson’s criticism of Spencer is that he presupposes the space that evolution is supposed to have produced. Bergson thinks that the mind and matter have progressively adapted to each other so to produce our notion of space. His criticism of Kant is that space and the understanding are identical for Kant, because the understanding provides a sort of space for atomic terms to be related externally (in subject-predicate relations). Bergson, then, erroneously thinks that Kant’s notion of space is simply homogenous. However, Bergson is right to note that that each moment of inner awareness [or judgment] is accompanied but an ‘I think’ that represents the same unified ego each time. So Kant’s transcendental ego is like the homogenous space that allows atomic things to take on external relations [especially the parts of judgments]. Russell’s set theory also has such a homogenous space-like medium that allows atomic parts (and sets considered individually) to take on external relations and be examined apart from those relations. Euclid describes a homogenous geometrical space, where the metric is the same everywhere. This homogenized space is kept throughout many of the subsequent major developments of geometry and physics, and it is reflected in philosophy as well. We can think of time as a homogenous space. With time and space both containing parts with determinate external relations, we have Laplace’s dream of determining all past and future states of the world on the basis of knowledge of everything about one moment and a great intellect to make the necessary calculations. Like Spencer, the Pythagoreans thought that our understanding corresponds with the world. In this case, because the mind understands integers and their relations, it can know the parts of the world that are expressible with number. But the discovery of incommensurability shattered that belief, because here there is something numerical that is not understandable with integers and their relations. So our minds and the world did not coincide isomorphically. Bergson’s method of intution, we will see, involves uncovering the mismatch between our representations of the world and the structure of the world.
Bergson and Deleuze tried to both come to terms with Herbert Spencer’s evolutionary thinking and also show its problems. We will also look at the relation between the structure of knowledge and the structure of the object. We saw how Kant’s system solves this with his transcendental categories. There is an isomorphism between the understanding and the object. Russell also thinks that the object and the system studying it must resemble one another. Russell has an atomistic system, so he solves Zeno’s paradox by saying time is an infinite discrete series. “The resolution of a paradox of thought therefore leads to an alteration of the conception of the object.” (70) In this chapter we will examine this isomorphism in evolutionary theory by looking at a particular concept of homogenous space.
Kant tried to explain how synthetic a priori statements were possible. He applied the subject’s categories to intuition, which conditions experience. “As the same categories play a role in both conditioning experience and conceptualizing it, their application to experience is vouchsafed.” (71) We saw previously how Deleuze wanted to break with this parallelism between empirical and transcendental. “Spencer attempts to provide a genetic interpretation of the development of our categories of thought that would allow us to see exactly why space appears to us in the particular a priori form that it has.” (71) Bergson rejects Spencer’s concept of space, but builds from this analysis. Kant proves the a priori nature of space on the basis of the subject’s ability to represent itself or to conceive the world in certain ways. Kant’s first argument is that in order for us to represent the referents of sensations as being alongside one another (being in different places), we must suppose we already have a representation of space. So we can only have empirical representations on the basis of an a priori representation of space. He second argument is that while we can think of an absence of objects in space, we cannot think of there being an absence of space in the world. This argument “follows a similar structure, placing the weight of the proof on the subject's capacity to conceive of certain relations between space and objects. In other words, what we cannot conceive of, in terms of empirical phenomena, cannot be the case. It is this reliance on the possibilities of representation that at first appears to be the target of evolutionary theory” (71). Evolutionary theory will not argue that we conceive space in accordance to how we condition empirical givenness (as being spatial). Instead, it argues that
our cognitive capacities are based on the fitness of the organism to the environment. Thus, alternative, partial conceptions of space lead to an organism that is not optimally attuned to its environment and therefore has a lower chance of survival than one with a more practical conceptual schema. As natural selection eliminates those organisms with suboptimal | representational schemata, a more optimal scheme becomes sedimented in the organism so that what is a posteriori for the species becomes a priori for the individual. (71-72)
So conceivability seems not to be the support for the argumentation, but rather conceivability becomes a function of fitness. Soon we will see that Spencer’s approach leads to a reversal of this.
What is life for Spencer?
For Spencer, life "in its simplest form is the correspondence of certain inner physico-chemical actions with certain outer physico-chemical actions" so that "each advance to a higher form of life consists in a better preservation of this primary correspondence by the establishment of other correspondences." Evolution is thus characterized by the gradual broadening of the correspondence between the world itself and the set of physico-chemical reactions that lie at the heart of our interactions with the world. (72)
As these correspondences increase, so too does the creature’s milieu. And as the milieu widens, so too does the organism’s subjective representation of the world near coextensity with the world itself. Truth, then, is the accurate correspondence of relations between subject and objects. Error is the absence of this correspondence. More correspondence means more chance to survive, and less means greater chance to die.
Thus, finally, pragmatic truth, when it is applied to a milieu that encompasses the entirety of the real, becomes objective truth. As Spencer considered the evolution of humanity to have reached this endpoint, primarily because of the successes of the scientific enterprise, for him, the validity of the a priori categories is restored, as they now once again correspond to reality (71).
So the reason we cannot imagine their being no space is because our evolution has reached such a point that we are unable to produce an erroneous view of the world’s spatial nature. Thus Spencer’s evolutionary theory also leads to the a priori nature of our representation of space and the categories, but of course for different reasons. (72-73) “The a priori nature of the categories is now grounded in a pragmatic correspondence between the organism and its environment rather than through the conditioning of the object by the subject.” (73)
Although Bergson notes how Spencer avoids the Kantian criticism, he also thinks Spencer failed to provide an adequate alternative to Kant’s theory. Spencer’s theory of evolution presupposes that we are evolving in space. So he begins by presupposes what he will conclude will be the outcome of evolution. “Spencer therefore fails to provide an adequate response to Kant, as his explanation of the genesis of space presupposes an account of a space such as that which Kant gives wherein this genesis takes place.” (73) However, Bergson builds from certain of Spencer’s metaphysical ideas. (a) We need an account of space that does not accept it as ready made. This account would retrace space’s genesis, which he thinks Kant achieves. (b) Bergson will say that Kant’s theory of space does not allow for an account of space’s nature. Bergson looks at the three possible relations between subject and world in Kant. Either  the mind is determined by things,  things are determined by the mind, or  there is some mysterious agreement between the mind and things. Bergson then proposes a fourth possibility:  “ ‘intellect and matter have progressively adapted themselves one to the other in order to attain at last a common form’ ( CE, 206).” (73) But this process of progress adaptation cannot precede space, like with Spencer. So Bergson agrees with Kant that space is an ideal feature of the world and also that we cannot use an empiricist account of our conception of space. Bergson notes that Kant does not consider the possibility of degrees of spatiality. For Kant, space is either given or not given. [In order to say that our understanding is connected to both pure space and to its degrees or indeterminate forms, Bergson will have to argue that the understanding is wider than Kant conceived it to be]. Bergson is trying to explain the genesis of space, which highlights a limitation in Kant’s account. First we examine what for Bergson is the connection between homogenous space and the understanding for Kant and Russell. (74)
For Kant, judgments are made through the relations between fully determined terms. The faculty of understanding (which is external to the terms) is what allows the terms to be thought together. So understanding is a third term underlying the unity of judgment. Bergson notes that because we here are conceiving of the elements of consciousness having the subject-predicate form (their being objectival), we are compelled to regard them as united by an artificial bond, “ ‘a formless ego, indifferent and unchangeable, on which it threads the psychic states which it has set up as separate entities’ (CE, 3).” (74) Bergson thinks that Kant’s conception of the atomistic components of judgment is reflected in his conception of the world’s elements, and so he has an atomistic conception of space. We will explore this because it leads to the two forms of multiplicity.
In Kant’s transcendental aesthetic, there is a difference between the object insofar as it appears to us and the space that possibilizes its appearing. So space, in Bergson’s view here, is logically prior to the objects that occupy it. Space is the medium that allows objects to relate and interact. Analogously, the ego provides the ‘space’ in which terms relate.
In the cases of both the mind and the understanding of space, therefore, we find the model of a medium through which the elements can interact. The ego enables judgment, and space, in coordination with the categories of the understanding, enables perception of the object. (74)
This spatial metaphor is also at work in Russell’s set theory, because it creates domains that relate to one another.
We saw that set theory provides a hierarchical model of relations, sets being related to others by the relative domains of objects over which they range. Cantor's | definition of the set as "a collection of definite, well discernible objects" emphasizes the use of spatial metaphors at the foundation of the discipline of classical logic, which allows the graphical representation of logical results through, for instance, Venn diagrams. (74-75)
In this set theory, we view the elements of propositions as discrete, and they function in a space that is “inert to their interactions.” This allows us to regard the relations between the members of sets as being purely external to the terms themselves. And in fact, even the relations between sets themselves is seen as external to the sets. Because the relations are external, we can encounter paradoxes of self-referentiality that result when a set has an external relation to itself that is in contradiction with its own definition. But this allows Russell to give an extensive definition of the set, because he could enumerate objects that relate to one another externally on the basis of shared properties, and it also had advantages for analysis. Because relations are external, we can put them aside and merely analyze the terms. Bergson sees something similar in Kant. For Kant, we may examine the parts synthesized into manifolds each individually. This presupposes a spatial understanding of the parts, because they are related by external relations.
Thus, the spatial model allows the method of analysis to develop, where a complex phenomenon can be broken down into its component parts in order to understand the whole through a later process of synthesis. (75)
For Bergson, every homogenous medium is a space. He sees Kant’s faculty of understanding as a homogeneous medium, thus for him, Kant’s understanding and space are identical. Somers-Hall recounts that
In chapter 1, we saw that insofar as Kant takes the empirical to have the same form as the transcendental, he is unable to take account of the generation of the empirical. This led us so see that by privileging the structure of judgment, Kant belonged to the tradition of representation. (75)
Deleuze however sees Kant’s theory of space in less simplistic terms than Bergson.
As Deleuze notes in relation to Kant's argument from incongruent counterparts (DR, 13, 26), Kant recognizes an "inner difference" within space that escapes the understanding. We can further note that spatiotemporal objects for Kant cannot be described 'atomically, ' as "all substances, in so far as they can be perceived to coexist in space, are in thoroughgoing reciprocity" (CPR, B256). 
Bergson’s simplification no longer allows us to make his analogy between space and the understanding. However, Bergson’s argument against the limitations of the understanding do still hold, and it will be important for Deleuze’s attack on finite representation. Recall how for Kant, the understanding produces judgments by subsuming representations under other representations. Each term is self-sufficient, so it needs something additionally to relate them, and this is the ‘I think’ whose unity provides the grounds from the representations to come together. So Bergson’s critique applies to the ‘I think’, which can be thought of as like homogeneous space that relates atomic externally related parts.
Leaving aside his argument for this point, given the apparent differences between the structure of the understanding and the structure of space, what will actually be important to Deleuze is the multiplicity that underlies this conception of space and the recognition that it is possible to draw a distinction between extensity and space. (76)
We will now try to understand the importance of space for Bergson. We must first note that we can now formulate different kinds of geometry, Euclidean and non-Euclidean ones. In this chapter we deal with Euclidean, and the next with Riemannian. All of Euclid’s theorems derive from just five axioms, and the fifth interests us here: “through a point not on a given straight line, one and only one line can be drawn that never meets that given line”. (76) [Or, ‘only one line can be drawn through a point parallel to another line.' The basic insight here is that any other line than a parallel one would eventually intersect with the first line.] On the basis of this axiom, we may arrive upon homogenous space. [So if the two lines are parallel, that means the distance we measure at one place will be the same as in any other place. So space has a homogenous metric. We can lift up those parallel lines and move them somewhere else, and still they will not meet, because the metric of space will be the same in that other place as well. The lines are determined, but they are related through a homogenous space. Now consider how Kant’s ego, is homogenous, because it is the same self for every ‘I think’. Euclidean space, as the relational medium between determinate figures, is like Kant’s ego as the homogenous relational medium between instances of inner acts with their accompanying ‘I think’.]
With the fifth axiom, which can be restated as asserting that "through a point not on a given straight line, one and only one line can be drawn that never meets that given line," we arrive at a conception of space as fundamentally homogenous. This means that a particular metric applied at one point within a Euclidean space can equally be applied at any other point. Euclidean space therefore has the fundamental property of measurability, in that we can compare the objects within it by their superposition upon one another. A consequence of this is that an object within a Euclidean space is invariant to transformation by displacement, or in other words, that the space of Euclidean geometry functions as a homogenous medium where position does not affect the constitution of objects within it. Euclidean geometry therefore provides the ideal model of how we are to understand something like the ego as that which allows the relation of already determined concepts. (76)
Descartes invents the algebraic representation of geometry. This, along with his notion of inert matter, further allows for “the conception of physics as the interaction of quantitatively characterized matter within the field of homogenous space defined by this geometry.” (76) “By moving to a purely quantitative definition of matter, Descartes allowed for the application of mathematical concepts to the world, which in tum was to open up the possibility of the mechanics of Newton.” (77) Spencer’s evolution is the closer approximation to this structure of the world.
The final stage of Spencer's phylogenie account is the mirroring of an internal world grounded on the invisible thread of consciousness and the external world grounded in the understanding's relations to the homogenous field of space. For Spencer, Newtonian physics therefore represents the final milieu of the development of the organism, one that allows the complete representation of the world to the organism. (77)
This view of a quantifiable metric space of externally related parts helped form Russell’s discrete view of time, and it would help fulfill Laplace’s dream of an unlimited intelligence and knowledge of the state of everything in the universe at one moment being able to determine all past and future moments. “The mathematico-analytic approach therefore allows the analysis of any closed system in a similar fashion, that is, the quantitative”. (77)
This approach creates difficulties for the Pythagoreans, who believe that all knowable things have number. But with the discovery of incommensurable numbers, it was seen that there are numbers which cannot be reduced to integers and their relations. Our understanding of number then did not match the world we are trying to understand.
When Pythagoras' theorem is applied to the square, we find that the length of the diagonal of the square is √ 2 times the length of the side. As the square root of 2 is irrational, the ratio of the length of the side to the diagonal is irresolvable into an integral ratio. The Greek concept of number, built on the idea of the integral numerical progression, was unable to incorporate the idea of a number which could not be reduced to integers or their relations. With the discovery of incommensurable numbers, Pythagoreanism collapsed, the man who disclosed the difficulty being said to have died in a shipwreck as a result. What therefore defeated the Pythagorean model was the discovery of a mismatch between the world and the subject's ability to conceptualize the world. (78)
Just like the Pythagoreans, “Spencer's model of the gradual adequation of the mind to the world produced the corollary that the structure of the mind was isomorphic with the structure of the world.” (78) Bergson’s method of intuition is “a method of recognition of the mismatch between our representation of the world and the structure of the world itself.” (78) We explore this method of intuition in the next section.
Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.