Pluralist Mathematics, Minimalist Philosophy: Hans Reichenbach. Drunken Risibility.

H_Reichenbach

Hans Reichenbach relativized the notion of the constitutive a priori. The key observation concerns the fundamental difference between definitions in pure geometry and definitions in physical geometry. In pure geometry there are two kinds of definition: first, there are the familiar explicit definitions; second, there are implicit definitions, that is the kind of definition whereby such fundamental terms as ‘point’, ‘line’, and ‘surface’ are to derive their meaning from the fundamental axioms governing them. But in physical geometry a new kind of definition emerges – that of a physical (or coordinative) definition:

The physical definition takes the meaning of the concept for granted and coordinates to it a physical thing; it is a coordinative definition. Physical definitions, therefore, consist in the coordination of a mathematical definition to a “piece of reality”; one might call them real definitions. (Reichenbach, 8)

Now there are two important points about physical definitions. First, some such correlation between a piece of mathematics and “a piece of physical reality” is necessary if one is to articulate the laws of physics (e.g. consider “force-free moving bodies travel in straight lines”). Second, given a piece of pure mathematics there is a great deal of freedom in choosing the coordinative definitions linking it to “a piece of physical reality”, since… coordinative definitions are arbitrary, and “truth” and “falsehood” are not applicable to them. So we have here a conception of the a priori which (by the first point) is constitutive (of the empirical significance of the laws of physics) and (by the second point) is relative. Moreover, on Reichenbach’s view, in choosing between two empirically equivalent theories that involve different coordinative definitions, there is no issue of “truth” – there is only the issue of simplicity. In his discussion of Einstein’s particular definition of simultaneity, after noting its simplicity, Reichenbach writes: “This simplicity has nothing to do with the truth of the theory. The truth of the axioms decides the empirical truth, and every theory compatible with them which does not add new empirical assumptions is equally true.” (p 11)

Now, Reichenbach went beyond this and he held a more radical thesis – in addition to advocating pluralism with respect to physical geometry (something made possible by the free element in coordinative definitions), he advocated pluralism with respect to pure mathematics (such as arithmetic and set theory). According to Reichenbach, this view is made possible by the axiomatic conception of Hilbert, wherein axioms are treated as “implicit definitions” of the fundamental terms:

The problem of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations which are neither true nor false, and that only the logical properties of a system – its consistency, independence, uniqueness, and completeness – can be subjects of critical investigation. (p 3)

It needs to be stressed here that Reichenbach is extending the Hilbertian thesis concerning implicit definitions since although Hilbert held this thesis with regard to formal geometry he did not hold it with regard to arithmetic.

On this view there is a plurality of consistent formal systems and the notions of “truth” and “falsehood” do not apply to these systems; the only issue in choosing one system over another is one of convenience for the purpose at hand and this is brought out by investigating their metamathematical properties, something that falls within the provenance of “critical investigation”, where there is a question of truth and falsehood. This radical form of pluralism came to be challenged by Gödel’s discovery of the incompleteness theorems. To begin with, through the arithmetization of syntax, the metamathematical notions that Reichenbach takes to fall within the provenance of “critical investigation” were themselves seen to be a part of arithmetic. Thus, one cannot, on pain of inconsistency, say that there is a question of truth and falsehood with regard to the former but not the latter. More importantly, the incompleteness theorems buttressed the view that truth outstrips consistency. This is most clearly seen using Rosser’s strengthening of the first incompleteness theorem as follows: Let T be an axiom system of arithmetic that (a) falls within the provenance of “critical investigation” and (b) is sufficiently strong to prove the incompleteness theorem. A natural choice for such an axiom system is Primitive Recursive Arithmetic (PRA) but much weaker systems suffice, for example, IΔ0 + exp. Either of these systems can be taken as T. Assuming that T is consistent (something which falls within the provenance of “critical investigation”), by Rosser’s strengthening of the first incompleteness theorem, there is a Π01-sentence φ such that (provably within T + Con(T )) both T + φ and T + ¬φ are consistent. However, not both systems are equally legitimate. For it is easily seen that if a Π01-sentence φ is independent from such a theory, then it must be true. The point being that T is ∑10-complete (provably so in T). So, although T + ¬φ is consistent, it proves a false arithmetical statement.

Advertisement

Expressivity of Bodies: The Synesthetic Affinity Between Deleuze and Merleau-Ponty. Thought of the Day 54.0

6

It is in the description of the synesthetic experience that Deleuze finds resources for his own theory of sensation. And it is in this context that Deleuze and Merleau-Ponty are closest. For Deleuze sees each sensation as a dynamic evolution, sensation is that which passes from one ‘order’ to another, from one ‘level’ to another. This means that each sensation is at diverse levels, of different orders, or in several domains….it is characteristic of sensation to encompass a constitutive difference of level and a plurality of constituting domains. What this means for Deleuze is that sensations cannot be isolated in a particular field of sense; these fields interpenetrate, so that sensation jumps from one domain to another, becoming-color in the visual field or becoming-music on the auditory level. For Deleuze (and this goes beyond what Merleau-Ponty explicitly says), sensation can flow from one field to another, because it belongs to a vital rhythm which subtends these fields, or more precisely, which gives rise to the different fields of sense as it contracts and expands, as it moves between different levels of tension and dilation.

If, as Merleau-Ponty says (and Deleuze concurs), synesthetic perception is the rule, then the act of recognition that identifies each sensation with a determinate quality or sense and operates their synthesis within the unity of an object, hides from us the complexity of perception, and the heterogeneity of the perceiving body. Synesthesia shows that the unity of the body is constituted in the transversal communication of the senses. But these senses are not pre given in the body; they correspond to sensations that move between levels of bodily energy – finding different expression in each other. To each of these levels corresponds a particular way of living space and time; hence the simultaneity in depth that is experienced in vision is not the lateral coexistence of touch, and the continuous, sensuous and overlapping extension of touch is lost in the expansion of vision. This heterogenous multiplicity of levels, or senses, is open to communication; each expresses its embodiment in its own way, and each expresses differently the contents of the other senses.

Thus sensation is not the causal process, but the communication and synchronization of senses within my body, and of my body with the sensible world; it is, as Merleau-Ponty says, a communion. And despite frequent appeal in the Phenomenology of Perception to the sameness of the body and to the common world to ground the diversity of experience, the appeal here goes in a different direction. It is the differences of rhythm and of becoming, which characterize the sensible world, that open it up to my experience. For the expressive body is itself such a rhythm, capable of synchronizing and coexisting with the others. And Merleau-Ponty refers to this relationship between the body and the world as one of sympathy. He is close here to identifying the lived body with the temporization of existence, with a particular rhythm of duration; and he is close to perceiving the world as the coexistence of such temporalizations, such rhythms. The expressivity of the lived body implies a singular relation to others, and a different kind of intercorporeity than would be the case for two merely physical bodies. This intercorporeity should be understood as inter-temporality. Merleau-Ponty proposes this at the end of the chapter on perception in his Phenomenology of Perception, when he says,

But two temporalities are not mutually exclusive as are two consciousnesses, because each one knows itself only by projecting itself into the present where they can interweave.

Thus our bodies as different rhythms of duration can coexist and communicate, can synchronize to each other – in the same way that my body vibrated to the colors of the sensible world. But, in the case of two lived bodies, the synchronization occurs on both sides – with the result that I can experience an internal resonance with the other when the experiences harmonize, or the shattering disappointment of a  miscommunication when the attempt fails. The experience of coexistence is hence not a guarantee of communication or understanding, for this communication must ultimately be based on our differences as expressive bodies and singular durations. Our coexistence calls forth an attempt, which is the intuition.

Simultaneity

Untitled

Let us introduce the concept of space using the notion of reflexive action (or reflex action) between two things. Intuitively, a thing x acts on another thing y if the presence of x disturbs the history of y. Events in the real world seem to happen in such a way that it takes some time for the action of x to propagate up to y. This fact can be used to construct a relational theory of space à la Leibniz, that is, by taking space as a set of equitemporal things. It is necessary then to define the relation of simultaneity between states of things.

Let x and y be two things with histories h(xτ) and h(yτ), respectively, and let us suppose that the action of x on y starts at τx0. The history of y will be modified starting from τy0. The proper times are still not related but we can introduce the reflex action to define the notion of simultaneity. The action of y on x, started at τy0, will modify x from τx1 on. The relation “the action of x on y is reflected to x” is the reflex action. Historically, Galileo introduced the reflection of a light pulse on a mirror to measure the speed of light. With this relation we will define the concept of simultaneity of events that happen on different basic things.

Untitled

Besides we have a second important fact: observation and experiment suggest that gravitation, whose source is energy, is a universal interaction, carried by the gravitational field.

Let us now state the above hypothesis axiomatically.

Axiom 1 (Universal interaction): Any pair of basic things interact. This extremely strong axiom states not only that there exist no completely isolated things but that all things are interconnected.

This universal interconnection of things should not be confused with “universal interconnection” claimed by several mystical schools. The present interconnection is possible only through physical agents, with no mystical content. It is possible to model two noninteracting things in Minkowski space assuming they are accelerated during an infinite proper time. It is easy to see that an infinite energy is necessary to keep a constant acceleration, so the model does not represent real things, with limited energy supply.

Now consider the time interval (τx1 − τx0). Special Relativity suggests that it is nonzero, since any action propagates with a finite speed. We then state

Axiom 2 (Finite speed axiom): Given two different and separated basic things x and y, such as in the above figure, there exists a minimum positive bound for the interval (τx1 − τx0) defined by the reflex action.

Now we can define Simultaneity as τy0 is simultaneous with τx1/2 =Df (1/2)(τx1 + τx0)

The local times on x and y can be synchronized by the simultaneity relation. However, as we know from General Relativity, the simultaneity relation is transitive only in special reference frames called synchronous, thus prompting us to include the following axiom:

Axiom 3 (Synchronizability): Given a set of separated basic things {xi} there is an assignment of proper times τi such that the relation of simultaneity is transitive.

With this axiom, the simultaneity relation is an equivalence relation. Now we can define a first approximation to physical space, which is the ontic space as the equivalence class of states defined by the relation of simultaneity on the set of things is the ontic space EO.

The notion of simultaneity allows the analysis of the notion of clock. A thing y ∈ Θ is a clock for the thing x if there exists an injective function ψ : SL(y) → SL(x), such that τ < τ′ ⇒ ψ(τ) < ψ(τ′). i.e.: the proper time of the clock grows in the same way as the time of things. The name Universal time applies to the proper time of a reference thing that is also a clock. From this we see that “universal time” is frame dependent in agreement with the results of Special Relativity.

Quantum Entanglement, Post-Selection and Time Travel

If Copenhagen interpretation of quantum mechanics is to be believed, nothing exists in reality until a measurement is carried out. In the double slit experiment carried out by John Wheeler, post-selection can be made to work, after the experiment is finished, and that by delaying the observation after the photon has purportedly passed through the slits. Now, if post-selection is to work, there must be a change in the properties in the past. This has been experimentally proved by physicists like Jean-François Roch at the Ecole Normale Supérieure in Cachan, France. This is weird, but invoking the quantum entanglement and throwing it up for grabs against the philosophic principle of causality surprises. If the experimental set up impacts the future course of outcome, quantum particles in a most whimsical manner are susceptible to negate it. This happens due to the mathematics governing these particle, which enable or rather disable them to differentiate between the course of sense they are supposed to undertake. In short, what happens in the future could determine the past….

….If particles are caught up in quantum entanglement, the measurement of one immediately affects the other, some kind of a Einsteinian spooky action at a distance.

 A weird connection was what sprang up in my mind this morning, and the vestibule comes from French theoretical consideration. Without any kind of specificity, the knower and the known are crafted together by a meditation that rides on instability populated by discursive and linguistic norms and forms that is derided as secondary in the analytical tradition. The autonomy of the knower as against the known is questionable, and derives significance only when its trajectory is mapped by a simultaneity put forth by the known.
Does this not imply French theory getting close to interpreting quantum mechanics? just shocking weird….
Anyways, adieu to this and still firmed up in this post from last week.