La Mettrie’s Man-Machine

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Philosophers’ theories regarding the human soul? Basically there are just two of them: the first and older of the two is materialism; the second is spiritualism.

The metaphysicians who implied that matter might well have the power to think didn’t disgrace themselves as thinkers. Why not? Because they had the advantage (for in this case it is one) of expressing themselves badly. To ask whether unaided matter can think is like asking whether unaided matter can indicate the time. It’s clear already that we aren’t going to hit the rock on which Locke had the bad luck to come to grief in his speculations about whether there could be thinking matter.

The Leibnizians with their ‘monads’ have constructed an unintelligible hypothesis. Rather than materialising the soul, they spiritualised matter. How can we define a being like the so-called ‘monad’ whose nature is absolutely unknown to us?

Descartes and all the Cartesians – among whom Malebranche’s followers have long been included – went wrong in the same way, namely by dogmatising about something of which they knew nothing. They admitted two distinct substances in man, as if they had seen and counted them!

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Dialectics of God: Lautman’s Mathematical Ascent to the Absolute. Paper.

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Figure and Translation, visit Fractal Ontology

The first of Lautman’s two theses (On the unity of the mathematical sciences) takes as its starting point a distinction that Hermann Weyl made on group theory and quantum mechanics. Weyl distinguished between ‘classical’ mathematics, which found its highest flowering in the theory of functions of complex variables, and the ‘new’ mathematics represented by (for example) the theory of groups and abstract algebras, set theory and topology. For Lautman, the ‘classical’ mathematics of Weyl’s distinction is essentially analysis, that is, the mathematics that depends on some variable tending towards zero: convergent series, limits, continuity, differentiation and integration. It is the mathematics of arbitrarily small neighbourhoods, and it reached maturity in the nineteenth century. On the other hand, the ‘new’ mathematics of Weyl’s distinction is ‘global’; it studies the structures of ‘wholes’. Algebraic topology, for example, considers the properties of an entire surface rather than aggregations of neighbourhoods. Lautman re-draws the distinction:

In contrast to the analysis of the continuous and the infinite, algebraic structures clearly have a finite and discontinuous aspect. Though the elements of a group, field or algebra (in the restricted sense of the word) may be infinite, the methods of modern algebra usually consist in dividing these elements into equivalence classes, the number of which is, in most applications, finite.

In his other major thesis, (Essay on the notions of structure and existence in mathematics), Lautman gives his dialectical thought a more philosophical and polemical expression. His thesis is composed of ‘structural schemas’ and ‘origination schemas’ The three structural schemas are: local/global, intrinsic properties/induced properties and the ‘ascent to the absolute’. The first two of these three schemas close to Lautman’s ‘unity’ thesis. The ‘ascent to the absolute’ is a different sort of pattern; it involves a progress from mathematical objects that are in some sense ‘imperfect’, towards an object that is ‘perfect’ or ‘absolute’. His two mathematical examples of this ‘ascent’ are: class field theory, which ‘ascends’ towards the absolute class field, and the covering surfaces of a given surface, which ‘ascend’ towards a simply-connected universal covering surface. In each case, there is a corresponding sequence of nested subgroups, which induces a ‘stepladder’ structure on the ‘ascent’. This dialectical pattern is rather different to the others. The earlier examples were of pairs of notions (finite/infinite, local/global, etc.) and neither member of any pair was inferior to the other. Lautman argues that on some occasions, finite mathematics offers insight into infinite mathematics. In mathematics, the finite is not a somehow imperfect version of the infinite. Similarly, the ‘local’ mathematics of analysis may depend for its foundations on ‘global’ topology, but the former is not a botched or somehow inadequate version of the latter. Lautman introduces the section on the ‘ascent to the absolute’ by rehearsing Descartes’s argument that his own imperfections lead him to recognise the existence of a perfect being (God). Man (for Descartes) is not the dialectical opposite of or alternative to God; rather, man is an imperfect image of his creator. In a similar movement of thought, according to Lautman, reflection on ‘imperfect’ class fields and covering surfaces leads mathematicians up to ‘perfect’, ‘absolute’ class fields and covering surfaces respectively.

Albert Lautman Dialectics in mathematics

Organic and the Orgiastic. Cartography of Ground and Groundlessness in Deleuze and Heidegger. Thought of the Day 43.0

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In his last hermeneutical Erörterung of Leibniz, The Principle of Ground, Heidegger traces back metaphysics to its epochal destiny in the twofold or duplicity (Zwiefalt) of Being and Thought and thus follows the ground in its self-ungrounding (zugrundegehen). Since the foundation of thought is also the foundation of Being, reason and ground are not equal but belong together (zusammenhören) in the Same as the ungrounded yet historical horizon of the metaphysical destiny of Being: On the one hand we say: Being and ground: the Same. On the other hand we say: Being: the abyss (Ab-Grund). What is important is to think the univocity (Einsinnigkeit) of both Sätze, those Sätze that are no longer Sätze. In Difference and Repetition, similarly, Deleuze tells us that sufficient reason is twisted into the groundless. He confirms that the Fold (Pli) is the differenciator of difference engulfed in groundlessness, always folding, unfolding, refolding: to ground is always to bend, to curve and recurve. He thus concludes:

Sufficient reason or ground is strangely bent: on the one hand, it leans towards what it grounds, towards the forms of representation; on the other hand, it turns and plunges into a groundless beyond the ground which resists all forms and cannot be represented.

Despite the fundamental similarity of their conclusions, however, our short overview of Deleuze’s transformation of the Principle of Sufficient Reason has already indicated that his argumentation is very different from Heideggerian hermeneutics. To ground, Deleuze agrees, is always to ground representation. But we should distinguish between two kinds of representation: organic or finite representation and orgiastic or infinite representation. What unites the classicisms of Kant, Descartes and Aristotle is that representation retains organic form as its principle and the finite as its element. Here the logical principle of identity always precedes ontology, such that the ground as element of difference remains undetermined and in itself. It is only with Hegel and Leibniz that representation discovers the ground as its principle and the infinite as its element. It is precisely the Principle of Sufficient Reason that enables thought to determine difference in itself. The ground is like a single and unique total moment, simultaneously the moment of the evanescence and production of difference, of disappearance and appearance. What the attempts at rendering representation infinite reveal, therefore, is that the ground has not only an Apollinian, orderly side, but also a hidden Dionysian, orgiastic side. Representation discovers within itself the limits of the organized; tumult, restlessness and passion underneath apparent calm. It rediscovers monstrosity.

The question then is how to evaluate this ambiguity that is essential to the ground. For Heidegger, the Zwiefalt is either naively interpreted from the perspective of its concave side, following the path of the history of Western thought as the belonging together of Being and thought in a common ground; or it is meditated from its convex side, excavating it from the history of the forgetting of Being the decline of the Fold (Wegfall der Zwiefalt, Vorenthalt der Zwiefalt) as the pivotal point of the Open in its unfolding and following the path that leads from the ground to the abyss. Instead of this all or nothing approach, Deleuze takes up the question in a Nietzschean, i.e. genealogical fashion. The attempt to represent difference in itself cannot be disconnected from its malediction, i.e. the moral representation of groundlessness as a completely undifferentiated abyss. As Bergson already observed, representational reason poses the problem of the ground in terms of the alternative between order and chaos. This goes in particular for the kind of representational reason that seeks to represent the irrepresentable: Representation, especially when it becomes infinite, is imbued with a presentiment of groundlessness. Because it has become infinite in order to include difference within itself, however, it represents groundlessness as a completely undifferentiated abyss, a universal lack of difference, an indifferent black nothingness. Indeed, if Deleuze is so hostile to Hegel, it is because the latter embodies like no other the ultimate illusion inseparable from the Principle of Sufficient Reason insofar as it grounds representation, namely that groundlessness should lack differences, when in fact it swarms with them.

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Theories of Fields: Gravitational Field as “the More Equal Among Equals”

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Descartes, in Le Monde, gave a fully relational definition of localization (space) and motion. According to Descartes, there is no “empty space”. There are only objects, and it makes sense to say that an object A is contiguous to an object B. The “location” of an object A is the set of the objects to which A is contiguous. “Motion” is change in location. That is, when we say that A moves we mean that A goes from the contiguity of an object B to the contiguity of an object C3. A consequence of this relationalism is that there is no meaning in saying “A moves”, except if we specify with respect to which other objects (B, C,. . . ) it is moving. Thus, there is no “absolute” motion. This is the same definition of space, location, and motion, that we find in Aristotle. Aristotle insists on this point, using the example of the river that moves with respect to the ground, in which there is a boat that moves with respect to the water, on which there is a man that walks with respect to the boat . . . . Aristotle’s relationalism is tempered by the fact that there is, after all, a preferred set of objects that we can use as universal reference: the Earth at the center of the universe, the celestial spheres, the fixed stars. Thus, we can say, if we desire so, that something is moving “in absolute terms”, if it moves with respect to the Earth. Of course, there are two preferred frames in ancient cosmology: the one of the Earth and the one of the fixed stars; the two rotates with respect to each other. It is interesting to notice that the thinkers of the middle ages did not miss this point, and discussed whether we can say that the stars rotate around the Earth, rather than being the Earth that rotates under the fixed stars. Buridan concluded that, on ground of reason, in no way one view is more defensible than the other. For Descartes, who writes, of course, after the great Copernican divide, the Earth is not anymore the center of the Universe and cannot offer a naturally preferred definition of stillness. According to malignants, Descartes, fearing the Church and scared by what happened to Galileo’s stubborn defense of the idea that “the Earth moves”, resorted to relationalism, in Le Monde, precisely to be able to hold Copernicanism without having to commit himself to the absolute motion of the Earth!

Relationalism, namely the idea that motion can be defined only in relation to other objects, should not be confused with Galilean relativity. Galilean relativity is the statement that “rectilinear uniform motion” is a priori indistinguishable from stasis. Namely that velocity (but just velocity!), is relative to other bodies. Relationalism holds that any motion (however zigzagging) is a priori indistinguishable from stasis. The very formulation of Galilean relativity requires a nonrelational definition of motion (“rectilinear and uniform” with respect to what?).

Newton took a fully different course. He devotes much energy to criticise Descartes’ relationalism, and to introduce a different view. According to him, space exists. It exists even if there are no bodies in it. Location of an object is the part of space that the object occupies. Motion is change of location. Thus, we can say whether an object moves or not, irrespectively from surrounding objects. Newton argues that the notion of absolute motion is necessary for constructing mechanics. His famous discussion of the experiment of the rotating bucket in the Principia is one of the arguments to prove that motion is absolute.

This point has often raised confusion because one of the corollaries of Newtonian mechanics is that there is no detectable preferred referential frame. Therefore the notion of absolute velocity is, actually, meaningless, in Newtonian mechanics. The important point, however, is that in Newtonian mechanics velocity is relative, but any other feature of motion is not relative: it is absolute. In particular, acceleration is absolute. It is acceleration that Newton needs to construct his mechanics; it is acceleration that the bucket experiment is supposed to prove to be absolute, against Descartes. In a sense, Newton overdid a bit, introducing the notion of absolute position and velocity (perhaps even just for explanatory purposes?). Many people have later criticised Newton for his unnecessary use of absolute position. But this is irrelevant for the present discussion. The important point here is that Newtonian mechanics requires absolute acceleration, against Aristotle and against Descartes. Precisely the same does special relativistic mechanics.

Similarly, Newton introduced absolute time. Newtonian space and time or, in modern terms, spacetime, are like a stage over which the action of physics takes place, the various dynamical entities being the actors. The key feature of this stage, Newtonian spacetime, is its metrical structure. Curves have length, surfaces have area, regions of spacetime have volume. Spacetime points are at fixed distance the one from the other. Revealing, or measuring, this distance, is very simple. It is sufficient to take a rod and put it between two points. Any two points which are one rod apart are at the same distance. Using modern terminology, physical space is a linear three-dimensional (3d) space, with a preferred metric. On this space there exist preferred coordinates xi, i = 1,2,3, in terms of which the metric is just δij. Time is described by a single variable t. The metric δij determines lengths, areas and volumes and defines what we mean by straight lines in space. If a particle deviates with respect to this straight line, it is, according to Newton, accelerating. It is not accelerating with respect to this or that dynamical object: it is accelerating in absolute terms.

Special relativity changes this picture only marginally, loosing up the strict distinction between the “space” and the “time” components of spacetime. In Newtonian spacetime, space is given by fixed 3d planes. In special relativistic spacetime, which 3d plane you call space depends on your state of motion. Spacetime is now a 4d manifold M with a flat Lorentzian metric ημν. Again, there are preferred coordinates xμ, μ = 0, 1, 2, 3, in terms of which ημν = diag[1, −1, −1, −1]. This tensor, ημν , enters all physical equations, representing the determinant influence of the stage and of its metrical properties on the motion of anything. Absolute acceleration is deviation of the world line of a particle from the straight lines defined by ημν. The only essential novelty with special relativity is that the “dynamical objects”, or “bodies” moving over spacetime now include the fields as well. Example: a violent burst of electromagnetic waves coming from a distant supernova has traveled across space and has reached our instruments. For the rest, the Newtonian construct of a fixed background stage over which physics happen is not altered by special relativity.

The profound change comes with general relativity (GTR). The central discovery of GR, can be enunciated in three points. One of these is conceptually simple, the other two are tremendous. First, the gravitational force is mediated by a field, very much like the electromagnetic field: the gravitational field. Second, Newton’s spacetime, the background stage that Newton introduced introduced, against most of the earlier European tradition, and the gravitational field, are the same thing. Third, the dynamics of the gravitational field, of the other fields such as the electromagnetic field, and any other dynamical object, is fully relational, in the Aristotelian-Cartesian sense. Let me illustrate these three points.

First, the gravitational field is represented by a field on spacetime, gμν(x), just like the electromagnetic field Aμ(x). They are both very concrete entities: a strong electromagnetic wave can hit you and knock you down; and so can a strong gravitational wave. The gravitational field has independent degrees of freedom, and is governed by dynamical equations, the Einstein equations.

Second, the spacetime metric ημν disappears from all equations of physics (recall it was ubiquitous). At its place – we are instructed by GTR – we must insert the gravitational field gμν(x). This is a spectacular step: Newton’s background spacetime was nothing but the gravitational field! The stage is promoted to be one of the actors. Thus, in all physical equations one now sees the direct influence of the gravitational field. How can the gravitational field determine the metrical properties of things, which are revealed, say, by rods and clocks? Simply, the inter-atomic separation of the rods’ atoms, and the frequency of the clock’s pendulum are determined by explicit couplings of the rod’s and clock’s variables with the gravitational field gμν(x), which enters the equations of motion of these variables. Thus, any measurement of length, area or volume is, in reality, a measurement of features of the gravitational field.

But what is really formidable in GTR, the truly momentous novelty, is the third point: the Einstein equations, as well as all other equations of physics appropriately modified according to GTR instructions, are fully relational in the Aristotelian-Cartesian sense. This point is independent from the previous one. Let me give first a conceptual, then a technical account of it.

The point is that the only physically meaningful definition of location that makes physical sense within GTR is relational. GTR describes the world as a set of interacting fields and, possibly, other objects. One of these interacting fields is gμν(x). Motion can be defined only as positioning and displacements of these dynamical objects relative to each other.

To describe the motion of a dynamical object, Newton had to assume that acceleration is absolute, namely it is not relative to this or that other dynamical object. Rather, it is relative to a background space. Faraday, Maxwell and Einstein extended the notion of “dynamical object”: the stuff of the world is fields, not just bodies. Finally, GTR tells us that the background space is itself one of these fields. Thus, the circle is closed, and we are back to relationalism: Newton’s motion with respect to space is indeed motion with respect to a dynamical object: the gravitational field.

All this is coded in the active diffeomorphism invariance (diff invariance) of GR. Active diff invariance should not be confused with passive diff invariance, or invariance under change of coordinates. GTR can be formulated in a coordinate free manner, where there are no coordinates, and no changes of coordinates. In this formulation, there field equations are still invariant under active diffs. Passive diff invariance is a property of a formulation of a dynamical theory, while active diff invariance is a property of the dynamical theory itself. A field theory is formulated in manner invariant under passive diffs (or change of coordinates), if we can change the coordinates of the manifold, re-express all the geometric quantities (dynamical and non-dynamical) in the new coordinates, and the form of the equations of motion does not change. A theory is invariant under active diffs, when a smooth displacement of the dynamical fields (the dynamical fields alone) over the manifold, sends solutions of the equations of motion into solutions of the equations of motion. Distinguishing a truly dynamical field, namely a field with independent degrees of freedom, from a nondynamical filed disguised as dynamical (such as a metric field g with the equations of motion Riemann[g]=0) might require a detailed analysis (for instance, Hamiltonian) of the theory. Because active diff invariance is a gauge, the physical content of GTR is expressed only by those quantities, derived from the basic dynamical variables, which are fully independent from the points of the manifold.

In introducing the background stage, Newton introduced two structures: a spacetime manifold, and its non-dynamical metric structure. GTR gets rid of the non-dynamical metric, by replacing it with the gravitational field. More importantly, it gets rid of the manifold, by means of active diff invariance. In GTR, the objects of which the world is made do not live over a stage and do not live on spacetime: they live, so to say, over each other’s shoulders.

Of course, nothing prevents us, if we wish to do so, from singling out the gravitational field as “the more equal among equals”, and declaring that location is absolute in GTR, because it can be defined with respect to it. But this can be done within any relationalism: we can always single out a set of objects, and declare them as not-moving by definition. The problem with this attitude is that it fully misses the great Einsteinian insight: that Newtonian spacetime is just one field among the others. More seriously, this attitude sends us into a nightmare when we have to deal with the motion of the gravitational field itself (which certainly “moves”: we are spending millions for constructing gravity wave detectors to detect its tiny vibrations). There is no absolute referent of motion in GTR: the dynamical fields “move” with respect to each other.

Notice that the third step was not easy for Einstein, and came later than the previous two. Having well understood the first two, but still missing the third, Einstein actively searched for non-generally covariant equations of motion for the gravitational field between 1912 and 1915. With his famous “hole argument” he had convinced himself that generally covariant equations of motion (and therefore, in this context, active diffeomorphism invariance) would imply a truly dramatic revolution with respect to the Newtonian notions of space and time. In 1912 he was not able to take this profoundly revolutionary step, but in 1915 he took this step, and found what Landau calls “the most beautiful of the physical theories”.

Automorphisms. Note Quote.

GraphAutormophismGroupExamples-theilmbh

A group automorphism is an isomorphism from a group to itself. If G is a finite multiplicative group, an automorphism of G can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity G={1,-1,i,-i} can be written as shown above, which means that the map defined by

 1|->1,    -1|->-1,    i|->-i,    -i|->i

is an automorphism of G.

Looking at classical geometry and mechanics, Weyl followed Newton and Helmholtz in considering congruence as the basic relation which lay at the heart of the “art of measuring” by the handling of that “sort of bodies we call rigid”. He explained how the local congruence relations established by the comparison of rigid bodies can be generalized and abstracted to congruences of the whole space. In this respect Weyl followed an empiricist approach to classical physical geometry, based on a theoretical extension of the material practice with rigid bodies and their motions. Even the mathematical abstraction to mappings of the whole space carried the mark of their empirical origin and was restricted to the group of proper congruences (orientation preserving isometries of Euclidean space, generated by the translations and rotations) denoted by him as ∆+. This group seems to express “an intrinsic structure of space itself; a structure stamped by space upon all the inhabitants of space”.

But already on the earlier level of physical knowledge, so Weyl argued, the mathematical automorphisms of space were larger than ∆. Even if one sees “with Newton, in congruence the one and only basic concept of geometry from which all others derive”, the group Γ of automorphisms in the mathematical sense turns out to be constituted by the similarities.

The structural condition for an automorphism C ∈ Γ of classical congruence geometry is that any pair (v1,v2) of congruent geometric configurations is transformed into another pair (v1*,v2*) of congruent configurations (vj* = C(vj), j = 1,2). For evaluating this property Weyl introduced the following diagram:

IMG_20170320_040116_HDR

Because of the condition for automorphisms just mentioned the maps C T C-1 and C-1TC belong to ∆+ whenever T does. By this argument he showed that the mathematical automorphism group Γ is the normalizer of the congruences ∆+ in the group of bijective mappings of Euclidean space.

More generally, it also explains the reason for his characterization of generalized similarities in his analysis of the problem of space in the early 1920s. In 1918 he translated the relationship between physical equivalences as congruences to the mathematical automorphisms as the similarities/normalizer of the congruences from classical geometry to special relativity (Minkowski space) and “localized” them (in the sense of physics), i.e., he transferred the structural relationship to the infinitesimal neighbourhoods of the differentiable manifold characterizing spacetime (in more recent language, to the tangent spaces) and developed what later would be called Weylian manifolds, a generalization of Riemannian geometry. In his discussion of the problem of space he generalized the same relationship even further by allowing any (closed) sub-group of the general linear group as a candidate for characterizing generalized congruences at every point.

Moreover, Weyl argued that the enlargement of the physico-geometrical automorphisms of classical geometry (proper congruences) by the mathematical automorphisms (similarities) sheds light on Kant’s riddle of the “incongruous counterparts”. Weyl presented it as the question: Why are “incongruous counterparts” like the left and right hands intrinsically indiscernible, although they cannot be transformed into another by a proper motion? From his point of view the intrinsic indiscernibility could be characterized by the mathematical automorphisms Γ. Of course, the congruences ∆ including the reflections are part of the latter, ∆ ⊂ Γ; this implies indiscernibility between “left and right” as a special case. In this way Kant’s riddle was solved by a Leibnizian type of argument. Weyl very cautiously indicated a philosophical implication of this observation:

And he (Kant) is inclined to think that only transcendental idealism is able to solve this riddle. No doubt, the meaning of congruence and similarity is founded in spatial intuition. Kant seems to aim at some subtler point. But just this point is one which can be completely clarified by general concepts, namely by subsuming it under the general and typical group-theoretic situation explained before . . . .

Weyl stopped here without discussing the relationship between group theoretical methods and the “subtler point” Kant aimed at more explicitly. But we may read this remark as an indication that he considered his reflections on automorphism groups as a contribution to the transcendental analysis of the conceptual constitution of modern science. In his book on Symmetry, he went a tiny step further. Still with the Weylian restraint regarding the discussion of philosophical principles he stated: “As far as I see all a priori statements in physics have their origin in symmetry” (126).

To prepare for the following, Weyl specified the subgroup ∆o ⊂ ∆ with all those transformations that fix one point (∆o = O(3, R), the orthogonal group in 3 dimensions, R the field of real numbers). In passing he remarked:

In the four-dimensional world the Lorentz group takes the place of the orthogonal group. But here I shall restrict myself to the three-dimensional space, only occasionally pointing to the modifications, the inclusion of time into the four-dimensional world brings about.

Keeping this caveat in mind (restriction to three-dimensional space) Weyl characterized the “group of automorphisms of the physical world”, in the sense of classical physics (including quantum mechanics) by the combination (more technically, the semidirect product ̧) of translations and rotations, while the mathematical automorphisms arise from a normal extension:

– physical automorphisms ∆ ≅ R3 X| ∆o with ∆o ≅ O(3), respectively ∆ ≅ R4 X| ∆o for the Lorentz group ∆o ≅ O(1, 3),

– mathematical automorphisms Γ = R+ X ∆
(R+ the positive real numbers with multiplication).

In Weyl’s view the difference between mathematical and physical automorphisms established a fundamental distinction between mathematical geometry and physics.

Congruence, or physical equivalence, is a geometric concept, the meaning of which refers to the laws of physical phenomena; the congruence group ∆ is essentially the group of physical automorphisms. If we interpret geometry as an abstract science dealing with such relations and such relations only as can be logically defined in terms of the one concept of congruence, then the group of geometric automorphisms is the normalizer of ∆ and hence wider than ∆.

He considered this as a striking argument against what he considered to be the Cartesian program of a reductionist geometrization of physics (physics as the science of res extensa):

According to this conception, Descartes’s program of reducing physics to geometry would involve a vicious circle, and the fact that the group of geometric automorphisms is wider than that of physical automorphisms would show that such a reduction is actually impossible.” 

In this Weyl alluded to an illusion he himself had shared for a short time as a young scientist. After the creation of his gauge geometry in 1918 and the proposal of a geometrically unified field theory of electromagnetism and gravity he believed, for a short while, to have achieved a complete geometrization of physics.

He gave up this illusion in the middle of the 1920s under the impression of the rising quantum mechanics. In his own contribution to the new quantum mechanics groups and their linear representations played a crucial role. In this respect the mathematical automorphisms of geometry and the physical automorphisms “of Nature”, or more precisely the automorphisms of physical systems, moved even further apart, because now the physical automorphism started to take non-geometrical material degrees of freedom into account (phase symmetry of wave functions and, already earlier, the permutation symmetries of n-particle systems).

But already during the 19th century the physical automorphism group had acquired a far deeper aspect than that of the mobility of rigid bodies:

In physics we have to consider not only points but many types of physical quantities such as velocity, force, electromagnetic field strength, etc. . . .

All these quantities can be represented, relative to a Cartesian frame, by sets of numbers such that any orthogonal transformation T performed on the coordinates keeps the basic physical relations, the physical laws, invariant. Weyl accordingly stated:

All the laws of nature are invariant under the transformations thus induced by the group ∆. Thus physical relativity can be completely described by means of a group of transformations of space-points.

By this argumentation Weyl described a deep shift which ocurred in the late 19th century for the understanding of physics. He described it as an extension of the group of physical automorphisms. The laws of physics (“basic relations” in his more abstract terminology above) could no longer be directly characterized by the motion of rigid bodies because the physics of fields, in particular of electric and magnetic fields, had become central. In this context, the motions of material bodies lost their epistemological primary status and the physical automorphisms acquired a more abstract character, although they were still completely characterizable in geometric terms, by the full group of Euclidean isometries. The indistinguishability of left and right, observed already in clear terms by Kant, acquired the status of a physical symmetry in electromagnetism and in crystallography.

Weyl thus insisted that in classical physics the physical automorphisms could be characterized by the group ∆ of Euclidean isometries, larger than the physical congruences (proper motions) ∆+ but smaller than the mathe- matical automorphisms (similarities) Γ.

This view fitted well to insights which Weyl drew from recent developments in quantum physics. He insisted – differently to what he had thought in 1918 – on the consequence that “length is not relative but absolute” (Hs, p. 15). He argued that physical length measurements were no longer dependent on an arbitrary chosen unit, like in Euclidean geometry. An “absolute standard of length” could be fixed by the quantum mechanical laws of the atomic shell:

The atomic constants of charge and mass of the electron atomic constants and Planck’s quantum of action h, which enter the universal field laws of nature, fix an absolute standard of length, that through the wave lengths of spectral lines is made available for practical measurements.

Schelling, Iain Hamilton Grant and Differential Nature(s) 1.0

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Schelling has often been at the receiving end for his idiosyncrasies or the frequent jumps that he undertook providing a lack of synthetic conflation and therefore missing on a philosophical system. He has most importantly been confined to near total oblivion in the English-speaking fraternity of philosophers and has had to face rebarbative charges against him. Although, there are some sympathetic voices emanating from the continental tradition in trying to revive his importance, like Slavoj Zizek, who has extensively fused the German with Lacanian psychoanalysis, citing Marx’s critique of speculative idealism as derived from Schellingian formulations of post-Hegelian universe of finitude-contingency-temporality. Zizek even goes a step ahead by crediting Schelling over Heidegger as the progenitor of ‘Artificial Earth’. But, it is Grant’s ‘Philosophies of Nature After Schelling’, which takes up the issue of graduating Schelling to escape the accoutrements of Kantian and Fichtean narrow transcendentalism.

Schelling gave a new twist to understanding nature by going past the Kantian nature as subject to necessary laws, as for Kant, nature enjoyed a formal sense. Kant overlooks the phenomenological deficit by arguing for subject’s access to forms of intuition and categories to bear upon what it perceives. Schelling discovers the problematic by raising the issue of subject’s spontaneity to judge in terms of categories. This dynamism of ‘becoming’ is what incites Grant to look into the materialist vitalism in Schelling’s understanding of nature. Grant frees Schelling from the grips of narrow minded inertness and mechanicality in nature that Kant and Fichte had presented nature with. This idea is the Deleuzean influence on Grant. Kant himself pondered over this dilemma, but somehow couldn’t come to terms with subject taking a leap from its determinism in crafting episteme. For, if nature was formal in its adherence to necessary laws, then splitting this boundedness to nature from subject’s autonomous or self-determining cognitivity would arrest the leap from determinism. In a way, Kant falls into the pit that he tries to negotiate, but comes out in conceding to nature the generation of self-determining organisms that possibilizes disinterested aesthetic pleasure in his third critique.  It didn’t take Schelling any Herculean effort to underline the central problems with this position of Kant, but it has taken a path of deliberate neglect of Schelling’s discovery of nature as more subject than object in modern readings of the philosopher.

Grant affirms the cardinality of Schelling’s naturephilosophie as the core, rather than just a phase as against Heidegger’s proclamation of Schelling’s discovery of nature as a fleeting episode, despite Heidegger paying fullest respects to Schelling for his profoundest grasp of spirit because of his commencing from the philosophy of nature. In a remarkable tour de force, Grant takes the accusation of Eschenmayer’s against Schelling head on and helps resurface the identity between nature and history. This identity is derived from Schelling’s insistence on freedom arising from nature, as the latter’s final and most potentiating act, the idea that constantly irritated Eschenmayer. Nature is history also helped Schelling cut the umbilical cord between evolution and teleology, in that he could fix his impressions on Carl Friedrich Kielmeyer’s signaling of a new epoch in natural history, thus getting over with transcendental philosophy’s obsession with fixed forms. That the inertness of nature was already on the way of getting dislodged, was proved by Kielmeyer’s influence on the earliest programme of the German comparative Biology, by which Schelling had himself been mightily influenced. As Kielmeyer had noted in his writings,

“I myself would like to derive all variation in the material of inert nature from a striving for heterogenesis, analogous to that in the organism, in the soul of nature.”

Schelling and Kielmeyer were fellow travelers in the sense that both recognized the fundamental delusion of the Kantian possibility of using a piori principles in deducing external nature. Grant makes a very affirmative intervention in here, when he elevates Deleuzean admonition to the fact that only contemporary French philosophy offers a scathing attack on the modern philosophy since its inception by Descartes holding the verdict of ‘nature not existing for itself’. This whole notion of becoming over being is wrought about by seemingly imperceptibly small and infinitely many changes. Or as Schelling maintains:

“Nature admittedly makes no leap; but it seems to me that this principle is much misunderstood if we try to bring into a single class of things which nature has not only separated, but has itself opposed to one another. That principle says no more than this, that nothing which comes to be in nature comes to be by a leap; all becoming occurs in a continuous sequence.”

This continuous sequential becoming is what has made Schelling to look at forces more potently rather than at phenomena as the measure of the differentials between the things that are separated by nature, but only as factors pertaining to becoming. This is a direct supplement to Kielmeyerian account of natural history, converting the principles underlying transcendental philosophy from the phenomenal and the somatic nature to making the somatic into the phenomenal products of a priori dynamics,  without making the phenomenal somatic coextensive with nature as such.  Products as such, for Schelling were discontinuities in nature and therefore not in the real sense speculative, as this was based on the principle of an Idea of nature as against nature and as ‘materiality is not yet corporeality’.

 

Lev Semënovič Vygotskij’s anti-dualism

Blow Your Mind
At the beginning there were Cognitive Sciences, which promise to solve once and for all the so-called mind-body problem. Mind, this was the core of the promise, is to software such as body is to hardware. This proportion is the power of an image and an extraordinary research project: human mind is a function of neural circuits. It seemed that the mind-body dualism was overcome. But the result was the creation of new – and at the beginning unnoticed -dualisms: internal vs. external, thought vs. action, mind vs. environment. Perhaps the most serious of all dualisms created by Cognitive Sciences is the individual mind vs. community dualism. If mind coincides with the activity of a single brain, then mind doesn’t need society and social relations to exist and function. This is the reason why, for a long time, Cognitivism embraced Chomsky’s innatist thesis: if mind has an innate structure, it doesn’t need relationship with external environment. Chomsky’s innatism, that spread out in Cognitivism in 60s and 70s, now come up again through the reductionist ideas of contemporary Neurosciences, and it has a long story: as the same Chomsky clarified in Cartesian linguistics (1966) this setting of human mind, which is provided with faculties whose functioning is unconnected with historical and cultural context, dates back to Descartes.

There was no more than one answer to solve these problems, that is to try to widen individual mind’s tight boundary. It was born this way that which seemed a new model of mind: the theory of extended mind. The mind extends up to include external instruments, scaffolds that complete and widen the internal capacities of our mind. This paradigm, genuinely, is not new at all, on the contrary it is but the incomplete and unaware resumption of the theory of one of the most important psychologists of  the last century, the Russian Lev Semënovič Vygotskij (1896-1934). Vygotskij has been the first psychologist to suggest an overall description of human mind as social and historical entity. Vygotskij’s anti-dualism is more radical than the anti-dualism of extended mind, for which human mind is a natural entity, and it cannot be understood if we abstract from the social and historical relationships in which it is necessarily and naturally involved in.

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On the contrary, Vygotskij’s thesis is radical: human nature is not closed into individual body, because in reality it coincides with the set of historical and social relationships that the body keeps up with its environment. This is a complete overturning of Cognitivist paradigm. We think that there isn’t, to this day, a naturalistic description of psychology of human animal more consistent than that of Vygotskij. In effect, cognitive scientists that recognize and appreciate the heritage of Vygotskij (for example Andy Clark and Michael Tomasello) were wholly unable to exceed the dualistic prejudice (between mind and body and between individual mind and society) that is the trademark of Cognitive Sciences. Accordingly, the point is not how much we are disposed to extend the boundary of individual mind, rather than how we intend to put at the first place the naturalistic notion of social relation.