La Mettrie’s Man-Machine

tumblr_mbxz7eiQ0w1qzb6vvo1_1280

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!

man machine

Advertisement

Weyl and Automorphism of Nature. Drunken Risibility.

MTH6105spider

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

Leibniz’s Compossibility and Compatibility

1200px-NaveGreca

Leibniz believed in discovering a suitable logical calculus of concepts enabling its user to solve any rational question. Assuming that it is done he was in power to sketch the full ontological system – from monads and qualities to the real world.

Thus let some logical calculus of concepts (names?, predicates?) be given. Cn is its connected consequence operator, whereas – for any x – Th(x) is the Cn-theory generated by x.

Leibniz defined modal concepts by the following metalogical conditions:

M(x) :↔ ⊥ ∉ Th(x)

x is possible (its theory is consistent)

L(x) :↔ ⊥ ∈ Th(¬x)

x is necessary (its negation is impossible)

C(x,y) :↔ ⊥ ∉ Cn(Th(x) ∪ Th(y))

x and y are compossible (their common theory is consistent).

Immediately we obtain Leibnizian ”soundness” conditions:

C(x, y) ↔ C(y, x) Compossibility relation is symmetric.

M(x) ↔ C(x, x) Possibility means self-compossibility.

C(x, y) → M(x)∧M(y) Compossibility implies possibility.

When can the above implication be reversed?

Onto\logical construction

Observe that in the framework of combination ontology we have already defined M(x) in a way respecting M(x) ↔ C(x, x).

On the other hand, between MP( , ) and C( , ) there is another relation, more fundamental than compossibility. It is so-called compatibility relation. Indeed, putting

CP(x, y) :↔ MP(x, y) ∧ MP(y, x) – for compatibility, and C(x,y) :↔ M(x) ∧ M(y) ∧ CP(x,y) – for compossibility

we obtain a manageable compossibility relation obeying the above Leibniz’s ”soundness” conditions.

Wholes are combinations of compossible collections, whereas possible worlds are obtained by maximalization of wholes.

Observe that we start with one basic ontological making: MP(x, y) – modality more fundamental than Leibnizian compossibility, for it is definable in two steps. Observe also that the above construction can be done for making impossible and to both basic ontological modalities as well (producing quite Hegelian output in this case!).