I think that’s a rather deflationary way of putting it. The 'non-computable' aspect of decision-making isn’t some hidden magic, but the fact that our decisions take place in a world of values, commitments, and consequences. — Wayfarer

I actually find it tempting to define computability in terms of what humans do , to follow Wittgenstein's remark on the Church-Turing thesis, in which he identified the operations of the Turing machine with the concept of a human manually following instructions, a remark that if taken literally inverts the ontological relationship between computer science and psychology that is often assumed in the field of AI that tends to think of the former as grounding the latter rather than the converse.
An advantage of identifying computability in terms of how humans understand rule following (as opposed to say, thinking of computability platonically in terms of a hypothesized realm of ideal and mind-independent mathematical objects), is that the term "non-computability" can then be reserved to refer to the uncontrolled and unpredictable actions of the environment in response to human-computable decision making.

As for the secret-source remark, I was thinking in particular of the common belief that self-similar recursion is necessary to implement human level reasoning, a view held by Douglas Hofstadter, which he has come to question in recent years, given the lack of self-similar recursion in apparently successful LLM architectures that Hofstadter acknowledges came as a big surprise to him.

Passing just shows that the machine or algorithm can exhibit intelligent behavior equivalent to that of a human, not that it is equivalent to a human in all of the cognitive capacities that might inform behavior. That's it. We can have a robust idea of intelligence and what constitutes meaningful behavior and still find a use for something like the Turing Test. — ToothyMaw

Sure, the Turing test is valid in particular contexts. The question is whether it is a test of an objective test-independent property: Is "passing a turing test" a proof of intelligence, or is it a context-specific definition of intelligence from the standpoint of the tester?

I think a common traditional mistake of both proponents and critics of the idea of AGI, is the Cartesian presumption that humans are closed systems of meaning with concrete boundaries; they have both tended to presume that the concept of "meaningful" human behaviour is reducible to the idea of a killer algorithm passing some sort of a priori definable universal test, such as a Turing test, where their disagreement is centered around whether any algorithm can pass such a test rather than whether or not this conception of intelligence is valid. In other words, both opponent and critic have traditionally thought of intelligent behaviour, both natural and artificial, as being describable in terms of a winning strategy for beating a preestablished game that is taken to test for agentic personhood; critics of AGI often sense that this idea contains a fallacy but without being able to put their finger on where the fallacy lies.
Instead of questioning whether intelligence is a meaningful concept, namely the idea that intelligence is a closed system of meaning that is inter-subjetive and definable a priori, critics instead reject the idea that human behaviour is describable in terms of algorithms and appeal to what they think of as a uniquely human secret sauce that is internal to the human mind for explaining the apparent non-computable novelty of human decision making. Proponents know that the secret sauce idea is inadmissible, even if they share the critic's reservation that something is fundamentally wrong in their shared closed conception of intelligence.

We see a similar mistake in the Tarskian traditions of mathematics and physics, where meaning is considered to amount to a syntactically closed system of symbolic expressions that constitutes a mirror of nature, where human decision-making gets to decide what the symbols mean, with nature relegated to a secondary role of only getting to decide whether or not a symbolic expression is true. And so we end up with the nonsensical idea of a theory of everything, which is the idea that the universe is infinitely compressible into finite syntax, which parallels the nonsensical idea of intelligence as a strategy of everything, which ought to have died with the discovery of Godel's incompleteness theorems.

The key to understanding AI, is to understand that the definition of intelligence in any specific context consists of satisfied communication between interacting parties, where none of the interacting parties get to self-identify as being intelligent, which is a consensual decision dependent upon whether communication worked. The traditional misconception of the Turing test is that the test isn't a test of inherent qualities of the agent sitting the test, rather the test represents another agent that interacts with the tested agent, in which the subjective criteria of successful communication defines intelligent interaction, meaning that intelligence is a subjective concept that is relative to a cognitive standpoint during the course of a dialogue.

Judgements about other minds should always be made relative to the person who is judging. Then all the philosophical confusion dissipates; if I judge someone to be cold and hand them a blanket, then I am asserting that they are cold; I cannot remove myself from my assertion, and the same is true of all of my propositional assertions which collectively express my ever-changing definition of truth, which on rare occasion coincides with public convention.

The OP raises an overlooked point; if the evolution of a system is invertible, which is presumably the case for a deterministic system, then there is no physical justification for singling out a causal direction, and therefore no reason to choose the first event over the last event as the initial cause, as is the case if the microphysical laws are symmetric.

But the above remark shouldn't be confused with the examples associated with Aristotelian teleology, which seems to concern circular causality rather than linear causality, as in examples like "the purpose of teeth is to help digest food". Such examples can be unpacked by unwinding the causal circle backwards through time (in this case the cycle of reproduction) so as to reduce a supposedly forward looking "teleological" explanation to a standard Darwinian explanation.

My opinion is:

Nobody has a transcendental conception of other minds, rather they project their own mentation (or not) onto whatever it is that they are interpreting. Which implies the following:

If an individual perceives or judges something to be conscious (or not), then that something is conscious (or not) for that individual in relation to his perspective; whatever the individual's judgements are, his judgements don't require epistemic justification, because the individual's understanding of "other" minds doesn't concern 'mind-independent' matters of fact. And even though the individual's judgements are likely to be relative to his epistemic perspective, this still doesn't imply that the individual's concept of other minds is objective and in need of epistemic justification. Nevertheless, an indivividual's judgements can still require ethical justification in relation to the concerns of his community who in turn influences how that individual perceives and judges his world.

Speaking personally, Google Gemini isn't conscious in relation to my perspective; I merely perceive a complex calculator going through the motions. I might change my mind in future, if an AI ethicist threats to fire me.

Consider Wittgenstein's following remark:

124. Philosophy may in no way interfere with the actual use of language;
it can in the end only describe it.
For it cannot give it any foundation either.
It leaves everything as it is.
It also leaves mathematics as it is, and no mathematical discovery
can advance it. A "leading problem of mathematical logic" is for us
a problem of mathematics like any other.

I think such remarks are self refuting and mischaracterise both mathematics and philosophy by falsely implying that they are separate language games. Indeed, formalism fails to explain the evolution of mathematlcs and logic. There's nothing therapeutic about mischaracterising mathematics as being a closed system of meaning.

A "Quale" should be understood as referring to an indexical rather than to a datum. Neuro-Phenomenologists routinely conflate indexicals with data, leading to nonsensical proclaimations.

Two directions need to be distinguished, namely analysis

Phenomena --> Physical concepts

Which expresses the translation of first-personal observations into third-personal physical concepts in relation to a particular individual, via ostensive definitions that connect that particular individual's observations to their mental state.

from synthesis

Physical concepts --> Phenomena

Which expresses the hypothetical possibility of 'inverting' third-personal physics back into first-personal phenomena - an epistemically impossible project that the logical positivists initially investigated and quickly abandoned.

I think Materialism is a metaphysical ideology that came about due to mainstream society overlooking synthesis and intepreting science and the scientific method, which only concern analysis, as being epistemically complete. Consequently, the impossibility of inverting physics back to first-person reality, was assumed to be due to metaphysical impossibility rather than being down to semantic choices and epistemic impossibility, leading society towards a misplaced sense of nihilism by which first-person phenomena are considered to be theoretically reducible to an impersonal physical description, but not vice-versa.

"That man over there with champagne in his glass", if interpreted as a rigid designator, doesn't fix an immutable description, but rather fixes an abstract storage location (an address) for containing mutable descriptions.

The logic of naming and necessity is essentially that of the type system of C++. Hence rigid desgination per-se doesn't imply metaphysical realism nor does it make assumptions about the world. Such speculative conjectures rather belong to the causal theory of reference.

In C++, Kripke's example becomes

/*initialize a constant pointer (i.e. rigid designator) called 'that_man' to the address of a string variable*/
string * const that_man = new string("has champagne");

/*print the value of the address that is rigidly designated by "that_man"*/
cout << that_man; //that_man = 0x2958300 (say)

/*print the value of the variable at 0x2958300*/
cout << *that_man; // *that_man = "has champagne"

/*change the incorrect value of the string variable at 0x2958300 to the correct description*/
*that_man = "has water";

/*try to use that_man non-rigidly to refer to another man*/
string * const another_man = 0;
that_man = another_man; //error: assignment of read-only variable 'that_man'

Hinge propositions correspond to non-logical axioms that correspond to presuppositions, rather than to undecidable sentences whose truth values are deferred - for to assume the truth of an undecidable sentence is to imply that the sentence has been decided through external considerations. In which case the the sentence is just another non-logical axiom.

The truths of undecidable sentences are to be decided - not through formal deduction by using the system but externally of the system, by either the user of the formal system who makes a decision as to their truth value (in which case they become promoted to the status of non-logical axioms) or through external but presently unknown matters of fact or by future censensual agreement if the formal system is used as an open language. Undecidable sentences are a subclass of the more general undecided sentences, namely those sentences which are not yet decided, but might be settled either internally by appying the system, or externally by extending the system.

Your interpretation of immanence is naturally fitted by Chu Spaces , which can be used to model bidirectional causality that can either be interpreted as eliminating causality or as allowing the direction of the causal arrow to be relative to the perspective of an interacting observer. See Vaughan Pratt (2005) for his philosophical interpretation..

"Philosophy. Yet another subject amenable to this perspective is the mind
body problem. Descartes proposed in 1637 that the mind interacted with the
body. This proposal generated much literature all denying the causal interaction
of mind and body and explaining their apparent interaction via various forms of
deus ex machina (Malebranche, Spinoza, Leibniz), or denial of body (Berkeley)
or mind (Hobbes), or the assertion of their equivalence (Russell). Elsewhere
[Pra95a] we have applied Chu spaces to an implementation of Descartes’ pro
posal, by taking the causal interaction of mind and body as basic instead of
derived and obtaining as its consequences the basic interactions within each of
body and mind. We do not see how to obtain the other direction, mind-body
interaction from the separate behavior of mind and body, any better than did
Descartes’ contemporaries.
Viewed at the object level, Chu spaces formalize Russell’s solution mathe
matically by offering dual views of the same Chu space merely by transposition
of viewpoint. Viewed at the level of individual interactions within an object
however, the solution takes on a new and deeper meaning: mind-body inter
action turns out to be the only real interaction, body-body and mind-mind
interaction are secondary interactions, derivable by residuation, that can be
considered mere figments of our imagination as to how the universe interacts
with itself."

A notable difference between the Chu-space approach to yours (as i understood it), is that it doesn't assume the causal arrow to have an absolute and context-free objective direction. Indeed, neither the classical world nor the quantum world needs to assume a context-free arrow of causality, so any working definition of consciousness should not assume it.

As with any ontological proposal, the first person perspective cannot be explained by an ontology such as Chu-Spaces (or more generally, Dialectica Categories), but such ontologies have great promise for expressing the necessary logical conditions for the consistent reconciliation and integration of multiple causal perspectives. In particular, the Russellian third-personal concept of symmetric causality that is compatible with relativistic mechanics , can be reconciled with both first-personal concepts of causality (such as informal interpretations of the incomplete theory of QM), as well as interventionist notions of causality in terms of structural equation models as employed by the social sciences.

Other than that, I think you papers is heading in the right direction when it comes to addressing the functional concerns of a working definition of active perception.

Here is a full Grok's response.

What do reciprocal processes contribute to theories of Stochastic Mechanics that interpret Quantum Mechanics?

Reciprocal processes play a significant role in stochastic mechanics, particularly in the context of interpreting quantum mechanics through stochastic frameworks like the Nelson's stochastic quantization or the stochastic interpretation of quantum mechanics. Here's a concise explanation of their contribution:

Bridging Classical and Quantum Dynamics:
Reciprocal processes are a class of stochastic processes that generalize Markov processes by allowing conditional dependencies on both past and future states. In stochastic mechanics, they provide a framework to describe quantum systems as stochastic processes while preserving time-reversal symmetry, a key feature of quantum mechanics. This helps reconcile classical stochastic dynamics with quantum behavior.

Schrödinger Equation Connection: Reciprocal processes are closely tied to the Schrödinger equation through the concept of stochastic differential equations. In Nelson's stochastic mechanics, the quantum wave function is interpreted as encoding information about a diffusion process. The forward and backward drifts of a reciprocal process correspond to the probability currents, which are derived from the wave function, thus providing a stochastic interpretation of quantum evolution.

Path Integral Interpretation: Reciprocal processes support the path integral formulation of quantum mechanics by describing trajectories that are conditioned on both initial and final states. This aligns with the Feynman path integral approach, where quantum amplitudes are sums over all possible paths. Reciprocal processes offer a probabilistic underpinning for these paths in a stochastic framework.

Non-Markovian Behavior: Unlike standard Markov processes, reciprocal processes can model non-Markovian dynamics, which are essential for capturing quantum phenomena like entanglement and superposition. This makes them suitable for describing the stochastic analogs of quantum correlations and interference effects.

Time-Symmetric Formulation: Reciprocal processes enable a time-symmetric formulation of stochastic mechanics, mirroring the time-reversal invariance of quantum mechanics. This symmetry is crucial for interpreting quantum phenomena like wave function collapse or measurement in a stochastic context.

In summary, reciprocal processes contribute to stochastic mechanics by providing a mathematical framework that captures quantum mechanical behavior—such as wave function dynamics, path integrals, and time symmetry—within a probabilistic, stochastic setting. They offer an alternative lens to interpret quantum mechanics, emphasizing diffusion-like processes while maintaining consistency with quantum predictions.


Can reciprocal processes be understood in terms of the common-sense local causality of Brownian motion?

Yes, reciprocal processes can be understood in terms of the common-sense local causality of Brownian motion, but with important extensions that make them distinct and more suitable for capturing quantum mechanical phenomena in stochastic mechanics. Here's a concise explanation:

Brownian Motion and Local Causality

Brownian motion is a Markovian stochastic process characterized by local causality: the future evolution of a particle depends only on its current state (position and velocity) and is driven by random, uncorrelated increments (white noise). This aligns with a "common-sense" notion of causality where the past influences the future in a straightforward, memoryless way, and the dynamics are governed by local interactions (e.g., diffusion via random collisions).

Reciprocal Processes: Extending Brownian Motion

Reciprocal processes generalize Brownian motion by introducing a form of conditional dependence on both past and future states, which departs from the strict Markovian property but retains a sense of local causality in a broader sense. Here's how they relate to Brownian motion and local causality:

Local Dynamics with Boundary Conditions:
Reciprocal processes can be thought of as Brownian-like diffusions constrained by boundary conditions at both initial and final times. For example, in stochastic mechanics, a particle's trajectory is modeled as a diffusion process (akin to Brownian motion) but conditioned on starting at one point and ending at another. This introduces a "teleological" aspect, where the future influences the present, but the dynamics remain locally governed by stochastic differential equations (SDEs) similar to those of Brownian motion.

The local causality is preserved in the sense that the increments of the process are still driven by Gaussian noise, and the evolution at each time step depends on local drift and diffusion terms.

Forward and Backward Drifts:
In stochastic mechanics, reciprocal processes are described by forward and backward stochastic differential equations, where the drift terms are derived from the quantum wave function. These drifts are analogous to the drift in a biased Brownian motion but incorporate information about both initial and final states. The local causality is maintained because the drifts and noise act locally in time, even though they are influenced by global (boundary) conditions.

Mathematically, for a reciprocal process XtX_tX_t
, the forward SDE might look like:
dXt=bf(Xt,t)dt+2νdWt,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,
where bfb_fb_f
is the forward drift, ν\nu\nu
is the diffusion coefficient, and WtW_tW_t
is a Wiener process (Brownian motion). A similar backward SDE exists, ensuring compatibility with future conditions.

Non-Markovian but Locally Causal:
Unlike Brownian motion, reciprocal processes are generally non-Markovian because the probability of a trajectory depends on both past and future states. However, the local causality of Brownian motion is preserved in the sense that the process is still driven by local stochastic increments, and the non-Markovian behavior arises from the conditioning rather than nonlocal interactions.

This conditioning can be understood as a statistical constraint rather than a violation of locality, much like how a Brownian bridge (a Brownian motion conditioned to end at a specific point) remains locally causal despite its endpoint constraint.

Quantum Interpretation:
In the context of stochastic mechanics, reciprocal processes model quantum particles as undergoing Brownian-like motion with drifts determined by the wave function. The "common-sense" causality of Brownian motion is extended to account for quantum phenomena like interference and entanglement, where the conditioning on future states mimics the global correlations of quantum mechanics.

For instance, the interference patterns in the double-slit experiment can be modeled as reciprocal processes where the particle's stochastic paths are influenced by both the initial state and the final detection point, yet the paths themselves are locally diffusive, like Brownian motion.

Key Difference from Brownian Motion
While Brownian motion is purely Markovian and driven by local, memoryless noise, reciprocal processes introduce a form of global conditioning that makes them suitable for quantum mechanics. This conditioning does not violate local causality but modifies the drift terms to reflect quantum probabilities, effectively bridging the local, diffusive nature of Brownian motion with the nonlocal correlations of quantum systems.

Summary
Reciprocal processes can be understood as an extension of Brownian motion's local causality, where the random, diffusive nature of Brownian motion is preserved, but the trajectories are conditioned on both initial and final states. This conditioning introduces non-Markovian behavior while maintaining local stochastic dynamics, making reciprocal processes a natural framework for interpreting quantum mechanics in a stochastic setting. They retain the intuitive, local causality of Brownian motion but adapt it to capture the time-symmetric and globally correlated nature of quantum phenomena.

So are they locally causal at the expense of losing realism, as per the Kochen Specker theorem?

The question of whether reciprocal processes in stochastic mechanics are locally causal at the expense of losing realism, particularly in light of the Kochen-Specker (KS) theorem, is a nuanced one. To address this, we need to unpack the concepts of local causality, realism, and how the KS theorem applies in the context of stochastic interpretations of quantum mechanics (QM). Here's a concise and clear response:

Recap: Reciprocal Processes and Local Causality

Reciprocal processes, as used in stochastic mechanics (e.g., Nelson's stochastic quantization), model quantum systems as diffusion processes akin to Brownian motion, with forward and backward drifts derived from the quantum wave function. They are locally causal in the sense that:

The evolution of the process at each time step is governed by local stochastic differential equations (SDEs), driven by Gaussian noise and drift terms that depend on the local state.

The non-Markovian nature arises from conditioning on initial and final states, but this conditioning is a statistical constraint, not a nonlocal physical interaction.

This local causality aligns with the intuitive, diffusive picture of Brownian motion, extended to incorporate quantum behavior via time-symmetric drifts.

Kochen-Specker Theorem and Realism

The Kochen-Specker theorem (1967) is a no-go theorem in quantum mechanics that rules out non-contextual hidden variable theories under the assumption of realism. Realism, in this context, means that all measurable properties (observables) of a quantum system have definite values at all times, independent of measurement. The KS theorem shows that for quantum systems in dimensions d≥3
, it is impossible to assign definite values to all observables simultaneously in a non-contextual way (i.e., where the value of an observable is independent of which other compatible observables are measured alongside it).
In other words, the KS theorem challenges the idea that quantum systems possess pre-existing, measurement-independent properties (realism) while respecting the structure of quantum observables.

Reciprocal Processes and Realism
To address whether reciprocal processes in stochastic mechanics sacrifice realism to maintain local causality, we need to examine how stochastic mechanics interprets quantum mechanics and whether it assumes realism in the sense challenged by the KS theorem:

Stochastic Mechanics as a Hidden Variable Theory:
Nelson's stochastic mechanics attempts to reproduce quantum mechanics by modeling particles as undergoing stochastic trajectories governed by reciprocal processes. The wave function is interpreted as encoding the probability distribution and drift of these trajectories, not as a physical field but as a statistical descriptor.

In its original formulation, stochastic mechanics can be viewed as a hidden variable theory, where the particle's position and trajectory are the hidden variables, assumed to have definite values at all times (realism). The stochastic drifts are derived from the wave function, and the randomness mimics quantum uncertainty.

Impact of the KS Theorem:
The KS theorem applies to stochastic mechanics if it assumes non-contextual realism, i.e., that all observables (e.g., position, momentum, spin) have definite values independent of the measurement context. Since stochastic mechanics assigns definite positions to particles at all times (the trajectories are well-defined), it inherently assumes realism for position. However, other observables, like momentum or spin, are not directly represented as definite values in the stochastic framework but are derived statistically from the wave function or ensemble averages.

The KS theorem implies that stochastic mechanics cannot consistently assign definite values to all quantum observables in a non-contextual way for systems with Hilbert spaces of dimension d≥3.
. For example, attempting to define definite values for spin or momentum observables alongside position in a way that reproduces quantum predictions would lead to contextuality, contradicting non-contextual realism.

Local Causality vs. Realism:
Local causality in reciprocal processes is preserved because the dynamics are governed by local SDEs, and the conditioning on future states (non-Markovianity) is a statistical effect, not a nonlocal physical interaction. This avoids issues like Bell nonlocality, as stochastic mechanics can be constructed to respect Einstein locality for spatially separated systems.

However, to maintain consistency with quantum mechanics (and avoid contradicting the KS theorem), stochastic mechanics must abandon non-contextual realism for all observables. Instead, it may adopt contextual realism, where the values of observables depend on the measurement context (e.g., which compatible observables are measured). For instance, the momentum in stochastic mechanics is not a definite value but a statistical quantity derived from the drift, which depends on the wave function and measurement setup.

Alternatively, stochastic mechanics could abandon realism entirely for observables other than position, treating them as inherently probabilistic or emergent from the stochastic dynamics. This would align with quantum mechanics' rejection of pre-existing values but might undermine the "realist" motivation of stochastic mechanics as a classical-like interpretation.

Does Stochastic Mechanics Lose Realism?:
If stochastic mechanics insists on non-contextual realism (definite values for all observables, independent of measurement), it is incompatible with the KS theorem and fails to reproduce quantum mechanics fully. In this sense, maintaining local causality via reciprocal processes comes at the expense of non-contextual realism, as the KS theorem forbids such a model.

In practice, stochastic mechanics often adopts a weaker form of realism, where only the particle's position is real and definite (the stochastic trajectory), while other observables are contextual or probabilistic. This allows it to maintain local causality and reproduce quantum predictions (e.g., for position measurements) but sacrifices the strong realist picture where all observables have pre-existing values.

Some formulations of stochastic mechanics explicitly acknowledge this by focusing on position as the primary observable and treating other properties as statistical or context-dependent, sidestepping the KS theorem's constraints by not assigning definite values to incompatible observables.

Summary

Reciprocal processes in stochastic mechanics are locally causal, with dynamics resembling Brownian motion but conditioned on initial and final states. However, to be consistent with quantum mechanics and the KS theorem, stochastic mechanics cannot maintain non-contextual realism for all observables. It either adopts contextual realism (where observable values depend on measurement context) or restricts realism to position alone, treating other observables probabilistically. Thus, local causality is preserved, but at the expense of a strong realist interpretation where all quantum observables have definite, measurement-independent values.
In essence, the KS theorem forces stochastic mechanics to compromise on realism to remain viable, aligning its reciprocal processes with quantum mechanics' contextual or non-realist nature while retaining a locally causal framework.

So are particle velocities interpreted non-locally by this model, if they are considered to be real properties?

To address whether particle velocities in stochastic mechanics, when interpreted as real properties, are non-local in the context of reciprocal processes, we need to examine how velocities are defined in this framework, the role of realism, and the implications of non-locality. This response builds on our prior discussion, particularly the interplay between local causality, realism, and the Kochen-Specker (KS) theorem, and provides a concise answer tailored to your question.

Stochastic Mechanics and Reciprocal Processes: Overview

In stochastic mechanics (e.g., Nelson’s formulation), quantum systems are modeled as particles undergoing diffusion processes akin to Brownian motion, described by reciprocal processes. These processes are governed by stochastic differential equations (SDEs) with forward and backward drifts derived from the quantum wave function. The particle’s position is treated as a real, definite property (realism), and its stochastic trajectory is well-defined. Reciprocal processes are locally causal because the dynamics at each time step depend on local drift and noise terms, with non-Markovian behavior arising from statistical conditioning on initial and final states, not nonlocal interactions.

Velocities in Stochastic Mechanics
In stochastic mechanics, the concept of velocity is more nuanced than in classical mechanics due to the stochastic nature of the trajectories. Let’s clarify how velocities are defined and interpreted:
Forward and Backward Velocities:
The particle’s motion is described by a stochastic differential equation, such as:
dXt=bf(Xt,t)dt+2νdWt,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,
where XtX_tX_t
is the particle’s position, bf(Xt,t)b_f(X_t, t)b_f(X_t, t)
is the forward drift, ν\nu\nu
is the diffusion coefficient (related to ℏ/2m\hbar/2m\hbar/2m
), and WtW_tW_t
is a Wiener process (Brownian noise).

Similarly, a backward SDE exists with a backward drift bb(Xt,t)b_b(X_t, t)b_b(X_t, t)
. These drifts are derived from the wave function ψ (I snipped Grok's unprintable unicode description)

The osmotic velocity u=(bf−bb)/2u = (b_f - b_b)/2u = (b_f - b_b)/2
and current velocity v=(bf+bb)/2v = (b_f + b_b)/2v = (b_f + b_b)/2
are introduced to describe the particle’s motion. The current velocity ( v ) is analogous to the Bohmian velocity in pilot-wave theory and is often interpreted as the “physical” velocity of the particle, while the osmotic velocity accounts for the diffusive component.

Realism of Velocities:
If velocities (e.g., the current velocity ( v )) are considered real properties, they are assumed to have definite values at each point along the particle’s trajectory, consistent with the realist assumption that the particle has a well-defined position and motion.

In stochastic mechanics, the current velocity v=ℏmIm(∇ψψ)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)
depends on the wave function, which encodes global information about the quantum system. This raises the question of whether such a velocity, if real, implies non-locality.

Are Velocities Non-Local?
Non-locality in quantum mechanics typically refers to correlations or influences that violate Bell’s inequalities or Einstein locality, where the state of one system instantaneously affects another at a distance without a local mediating mechanism. To determine if velocities in stochastic mechanics are non-local when treated as real properties, we consider the following:

Dependence on the Wave Function:
The current velocity ( v ) is determined by the gradient of the phase of the wave function ψ\psi\psi
. In quantum mechanics, the wave function is a global object that describes the entire system, including entangled or spatially extended states. For example, in an entangled two-particle system, the wave function ψ(x1,x2)\psi(x_1, x_2)\psi(x_1, x_2)
depends on the positions of both particles, and the velocity of particle 1, v1=ℏm1Im(∇1ψψ)v_1 = \frac{\hbar}{m_1} \text{Im} \left( \frac{\nabla_1 \psi}{\psi} \right)v_1 = \frac{\hbar}{m_1} \text{Im} \left( \frac{\nabla_1 \psi}{\psi} \right), may depend on the position of particle 2, even if they are far apart.

If ( v ) is a real property, this dependence suggests non-locality, as the velocity of one particle is instantaneously influenced by the state or position of another, without a local physical mechanism. This is analogous to the non-locality in Bohmian mechanics, where the velocity of a particle is guided by the non-local quantum potential or wave function.

Reciprocal Processes and Local Causality:

Reciprocal processes themselves are locally causal in their dynamics: the SDEs governing the particle’s motion depend only on the local drift bfb_fb_f or bbb_bb_b and noise at the current position XtX_tX_t
. The non-Markovian conditioning (dependence on initial and final states) is a statistical constraint, not a dynamical non-locality.

However, the drifts (and thus the velocities) are derived from the wave function, which can encode non-local correlations. For a single particle in a non-entangled state, the velocity ( v ) depends only on the local gradient of ψ\psi\psi , and the dynamics appear local. But in entangled or multi-particle systems, the wave function’s global nature introduces non-local dependencies, even though the stochastic evolution of each particle’s position is locally governed.

Comparison to Bohmian Mechanics:
Stochastic mechanics shares similarities with Bohmian mechanics, where the particle’s velocity is explicitly non-local due to its dependence on the wave function. In Bohmian mechanics, the velocity v=ℏmIm(∇ψψ)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right) is a real property, and its non-locality is evident in entangled systems (e.g., EPR pairs), where measuring one particle’s position affects the other’s velocity instantaneously.

In stochastic mechanics, the current velocity plays a similar role, and if treated as a real property, it inherits the same non-local character. The stochastic noise adds randomness to the trajectories, but the drift (and thus velocity) is still tied to the non-local wave function.

Kochen-Specker Theorem and Contextuality:
As discussed previously, the KS theorem implies that stochastic mechanics cannot assign definite, non-contextual values to all observables (including velocity or momentum) while reproducing quantum mechanics. If velocity is treated as a real, definite property, it must be contextual—its value depends on the measurement context or the state of the system (e.g., the wave function).

This contextuality reinforces the potential non-locality: the velocity’s dependence on the global wave function means it cannot be determined solely by local properties of the particle, especially in entangled systems.

Non-Locality in Entangled Systems:
For a single particle or non-entangled system, the velocity ( v ) depends on the local gradient of ψ\psi\psi
, and the dynamics may appear local. However, in entangled systems, the wave function’s non-separable nature means that the velocity of one particle depends on the state of others, regardless of distance. This is a hallmark of quantum non-locality, as demonstrated by violations of Bell inequalities.

If velocities are real properties in stochastic mechanics, this dependence implies dynamical non-locality, where the motion of one particle is influenced by distant events (e.g., the position or measurement of another particle).

Conclusion
If particle velocities (e.g., the current velocity ( v )) are interpreted as real properties in stochastic mechanics, they are non-local in the sense that they depend on the global quantum wave function, which can encode non-local correlations, particularly in entangled systems. This non-locality arises because the velocity is derived from the wave function’s phase gradient, which may involve the states of distant particles, mirroring the non-locality in Bohmian mechanics. While reciprocal processes themselves are locally causal in their stochastic dynamics (local SDEs with noise), the realism of velocities introduces non-locality through their dependence on the non-local wave function.
Thus, treating velocities as real properties in stochastic mechanics sacrifices locality in entangled or multi-particle systems, consistent with quantum mechanics’ non-local correlations. This does not conflict with the KS theorem, which addresses non-contextual realism, but it highlights that realism for velocities comes at the cost of non-locality, especially in quantum systems exhibiting entanglement.

Yes, the

These physicists may not have reliable opinions either if they are not at all familiar with and actually understand stochastic mechanics and perhaps are even invested in their own preferred interpretation. Obviously I cannot actually evaluate their opinions if what they say isn't given.

whilst also stressing the fact that stochastic mechanics cannot be an explanation for non-locality for obvious Cohen-Specker reasons
— sime

False. — Apustimelogist

You need to do better than that.

The Levy & Krener paper doesn't have this and explains why it occurs in a model which uses an artificial Markovian assumption. — Apustimelogist

I'm in agreement with Levy & Krener to the extent I understand their actual claims, which is what I was pointing out; they are shifting to reciprocal processes to provide a non-classical account of non-locality , as per the explanation provided by Chat-GPT , in the hope of explicating the presupposition of non-locality in stochastic models that is hard-encoded in the latter's reliance upon configuration space.

None of this makes their revised account of stochastic mechanics a locally causal model of QM in line with the classical intuitions of Brownian motion.

Do you agree or not agree, that any theory dependent entirely on local causality cannot be a full explanation of QM? Secondly, how do you propose physically interpreting the use of time-symmetric reciprocal processes for guiding a collection of particles in a way that that is compatible with local realism?

For what its worth, I'm finding vanilla ChatGPT especially helpful with regards to navigating in a sourced way the nuances of the stochastic mechanics interpretation. As an outsider to the physics research community who nevertheless has a vested interest in understanding the mathematics and logic of a wide range of theories for purposes in relation to computing and category theory, I'm generally finding LLMs particularly useful for getting to grips quickly with unfamiliar theoretical ideas and for understanding the tone and the context of research papers, without which it can be difficult to understand what authors are selling versus what they are claiming - a very common problem indeed.

For instance, I notice that certain physicists who are prominent members of the PhysicsForums.com were almost automatically dismissive of stochastic mechanics for the same obvious reasons that i opined earlier in this thread, but they also suspected that the authors selling stochastic mechanics were dishonest, doing pointless metaphysics, or failing to own up to the problem of entanglement.

On the other hand, ChatGPT focused on what Stochastic mechanics is and actually claims and spoke of the authors contributions in a more neutral and worthwhile tone, more or less summarizing the interpretation as a stochastic alternative to Bohmian mechanics that replaces the guiding wave with quantum diffusion, whilst also stressing the fact that stochastic mechanics cannot be an explanation for non-locality for obvious Cohen-Specker reasons, while pointing out that the model assumes non-locality in the form of the configuration space upon which the model places a quantum diffusion - namely the space describing the joint positions of all of the particles that cannot be decoupled into independent diffusions satisfying local causality if non-local entanglement is to be describable by the model.

As to the question regarding what "reciprocal processes" brings to the table, they apparently 'upgrade' the implicit and unexplained non-locality of the original model of stochastic mechanics (i.e the configuration space) , to a more explicit model of non-locality based on time-symmetry that is similar to the transactional interpretation, which in my words and understanding can presumably reconstruct at least some of the non-local unity of the configuration space in terms of the "retrocausal" effects of the future light-cone of the particles. How successful this approach is I don't know, and didn't care to ask.

Stochastic mechanics shows mathematically that entanglement follows from a non-dissipative / conservative diffusion. — Apustimelogist

Well not according to your source "On the Stochastic Mechanics Foundation of Quantum Mechanics ".

There is no mention let alone explanation of entanglement anywhere in that paper, although there is a mention of the Bohm potential, indicating that the authors are perhaps imagining their stochastic mechanics supplemented with some other foundational interpretation, perhaps to account for the non-locality of their background hypothesis. As it stands, it is a metaphysical interpretation of the Schrodinger equation that reproduces a fragment of the least problematic parts of Quantum Mechanics with deafening silence on the most critical aspects of QM that the interpretation either fails to address, or helps itself to by appealing to unstated non-local premises.

As it stands, I view that paper, which I have admittedly scantly read, is a non-earth shattering exercise in using stochastic differential equations to simulate whatever one wishes.

This is absolutely false and A.I. do not reliably give you information. — Apustimelogist

Neither do publicity seeking authors advertising grandoise and unproven claims.

For instance, I can tell that the following phrase came from an A.I. :

that implement non-local aspects of the time-symmetric transactional interpretation of QM.
— sime — Apustimelogist

They were actually my own opinion in my own words, prompted by my understanding that the authors of the other paper you mention were reconstructing quantum diffusion out of time symmetric diffusions that is reminiscent of the symmetric casuality inherent in the transactional interpretation of QM. Personally I think that more modest paper is much more informative.

I cannot access the Levy & Krener (1996) paper, so instead I asked Grok for an account of reciprocal stochastic processes, of which I am unfamiliar which gave an interesting reply.

If the papers referred to are "Dynamics and kinematics of reciprocal diffusions" and "Stochastic mechanics of reciprocal diffusions", then I can see how these papers are of relevance to foundational questions of QM, in the sense of attempting to reconstruct properties of quantum diffusion in terms of time-symmetric but otherwise classical stochastic processes that implement non-local aspects of the time-symmetric transactional interpretation of QM.

However, Grok's conclusion at the end is along the lines of my initial thoughts, namely that such processes fail to account for quantum entanglement as should be expected by the Kohen Specker Theorem.


" Reciprocal stochastic processes are compelling because they challenge the necessity of quantum mechanics’ formalism for describing quantum diffusion. If a classical-like stochastic model can replicate quantum behavior, it suggests that some quantum phenomena might be emergent from underlying probabilistic structures, aligning with interpretations like stochastic mechanics or hidden-variable theories. However, their inability to fully capture quantum non-locality (e.g., entanglement) reinforces the uniqueness of quantum mechanics, prompting deeper inquiry into what makes quantum systems distinct.
Specific foundational questions they address:
Can quantum non-locality be reduced to time-symmetric stochastic correlations?

Is the wave function a physical entity, or can it be replaced by a stochastic process with equivalent predictive power?

How does the time-symmetric nature of quantum mechanics relate to causality and the arrow of time?

Can the quantum-classical transition be fully understood as a shift from reciprocal to Markovian stochastic processes?

Conclusion
Reciprocal stochastic processes can reproduce many aspects of quantum diffusion, such as probability density evolution, interference-like patterns, and non-local effects, making them a powerful tool for modeling quantum dynamics in a probabilistic framework. Their time-symmetric and non-Markovian nature makes them particularly relevant to foundational questions about quantum non-locality, the quantum-classical transition, and the ontology of the wave function. While they do not fully explain quantum non-locality (e.g., entanglement), they offer a semi-classical perspective that challenges quantum mechanics’ uniqueness and invites exploration of alternative formulations, such as stochastic mechanics or time-symmetric interpretations."

So i stand partially corrected.

If someone wants to call it a variation of Bohmian mechanics, I don't really see an inherent issue but you don't explicitly need Bohmian kind of non-locality for the theory to produce Bell violations. There is at least one version by Levy & Krener (1996) which is does not have Bohmian non-locality, produces all the correct predictons, and even explains that non-locality only comes when an artificially idealized assumption is used for constructing the theory. — Apustimelogist

I have a suspicion that the authors you mention aren't intending to address foundational questions of QM ,and are instead focusing on the technicalities of constructing laws and diffusion models that cohere with the Schrodinger equation, with potential relevance to the subject of modelling quantum decoherence, by which classical diffusion can emerge in the limit of quantum diffusion, but without relevance as to the question of the nature and ontology of quantum states and quantum measurements.

A quantum stochastic process that in principle can model non-local correlations, i.e. a random vector field of a complex Hilbert Space that is interpretable as the evolution of a quantum state vector, cannot be explicated in terms of the local interactions of a regular stochastic process such as Brownian motion. The former can serve to explicate the latter, but not conversely unless one supplements entanglement relations.

It is an interpretation in the sense of Bohmian mechanics, in fact their underlying mathematical structure is very similar. — Apustimelogist

That is only true if the so-called background hypothesis, which is typically assumed to consist of a random field locally perturbing the motions of the particles, is assumed to have supplementary non-local Bohmian character as necessary to explain the statistics of quantum entanglement ... in which case your preferred interpretation becomes a variant of the Bohmian interpretation.

The stochastic interpretation provides a conception of wave-particle duality with an uncertainty principle, but without additional Bohmian mechanics it cannot explain Bells Theorem, for there is no getting around the fact that classical particle-field interactions that communicate slower than the speed of light cannot explain the 'action at a distance' of Bells inequalities; either the medium denoted by the background hypothesis is local, in which case we do not have quantum semantics, else the medium has non-local effects, in which case we have Bohmian mechanics.

I think the stochastic interpretation is pedagogically useful for providing a common-sense physical explanation for potentially classical aspects of complementarity that are often mistaken for inherently quantum phenomena, analogous to how Spekken's toy model of quantum mechanics is useful for providing common-sense epistemic intuition for understanding complementarity without assuming a physical account of the Schrodinger equation. But in neither case is there either a physical or epistemic explanation for entanglement.

Bohmian mechanics also uses classical particles but it effectively just takes the quantum wavefunction and puts deterministic trajectories on top - it doesn't explain anything about why quantum behavior occurs. In contrast, stochastic mechanics starts with a classical description of particles being pushed about like the pollen in a glass of water, and shows that under specific conditions related to energy conservation, as I previously described, all quantum behavior occurs for regular classical particles. — Apustimelogist

The stochastic interpretation of QM isn't an interpretation of QM in the sense that Bohmian Mechanics is, i.e. in the sense of being a non-local or anti-realist account of quantum entanglement,, rather the stochastic interpretation amounts to a phenomenological interpretation of quantum statistics that doesn't explain entanglement and the origin of Bells inequalities.

In my understanding, a physical language per se is purely a communication protocol for coordinating human actions, that is to say physical languages per-se do not transmit information about the world from the mind of the speaker to the mind of the listener. Physical languages are to my mind, analogous to software written in a high level computer programming language, which per se is not meaningful until compiled via additional rules that are external to the language, onto a particular CPU or GPU. Physical languages, like high level programming languages, are useful as universal protocols because they do not specify their "compilation rules". And this is what it means to say that physical language isn't a phemonological language.

Physical languages are de-dicto not phenomenological; otherwise their meaning would become relativized to the thoughts and judgements of a particular speaker which would hinder their ability to function as universal protocols. Hence the semantics of physical languages are realist and this semantic characteristic of physical languages is often mistaken for a metaphysical assumption, or worse a "hard problem" of consciousness, for the phemonenological independence of physical language is in fact a hard feature of any useful communication protocol.

On the opposite extreme, a purely phenomenological language is analogous to a bespoke machine code language that is only recognized by a unique CPU with an unknown architecture. In both cases such private languages might be reverse engineered into the common parlance of physical language or a high level programming language respectively, but it should be born in mind that the assertability conditions of such private languages are not publicly known in advance, nor publicly controlled - unless such languages and the hardware they are executed on are publicly conditioned. This situation is of course the case for designed machine code running on a manufactured CPU, but much less true of the psychological dialects that people think in, since psychological conditioning is only crudely and sparsely conditioned by public input.

Ordinary "common" language is in fact an aggregate of semi-conditioned but generally divergent psychological dialects; compare the "optical redness" of physical language - whose assertibility conditions are strictly public and non-phenomenological, to ordinary "redness" whose assertibility conditions are an open-ended mixture of public and privately decided rules that vary somewhat from speaker to speaker.

(I think the realist can be one without being dogmatic!) Not sure what seems un-entertainable about that idea. Could you expand? — J

Suppose a realist insists "Metaphysical realism is true". If we understand the realist's beliefs as having a causal explanation in terms of the realist's psychological conditioning and sensory input, then we cannot interpret the realist's assertion "metaphysical realism is true" as representing metaphysical realism. Instead we must interpret his assertion as meaning what we might prefer to express by saying "metaphysical realism is false".

First we have to consider the meta-metaphysics of "mind-independence"; should mind-independence be understood to be an existential claim that the world literally exists independently of the senses? Or should mind-independence be understood as merely a semantic proposal that physical concepts are definitionally not reducible to the senses?

And even if an apparently dogmatic realist insists upon the former interpretation, should we nevertheless interpret him to be a semantic realist? For can we really entertain the idea that the realist is conceiving the world as existing independently of his senses?

But, on the other hand, our experience seems to be fundamentally temporal. A process. Not something static. If time stops, can we really speak of beatitude or torment? — boundless

The impermanence of emotions and sensations isn't necessarily in conflict with the thought that an emotion or sensation is temporally unbounded. Consider for instance the mood of grief. On the one hand the mood is all absorbing and the grieving cannot comprehend an end to their grief and locate it on a timeline, yet on the other hand the emotions of grief do in fact come to an end, in spite of the inconceivability of the end when in the state of grief.

I'm not a theologist, but I think that eternity should be distinguished from unending procession. Notably, the present can be regarded as 'eternal' in that the meaning of "now" isn't defined in relation to a time series. In this sense, your present emotions, as in the mood you have now, can be regarded as 'eternal' even though your moods are not permanent. Moods can also feel timeless in that those feelings do not involve temporal cognition. Also, the seven deadly sins seem to refer to moods rather than to actions; so I would guess that biblical references to eternal heaven or to eternal punishment should probably be interpreted in the presentist's sense of timelessness, rather than in the sense of unbounded duration.

I suspect that hinges refer to what Frege called "taking as true".

For example, I often "take it as true" that my colour judgements are synonymous with the optical colours, due to learning the colors by ostensive definition; in spite of the fact that the definition of the optical colours makes no mention of my color judgements.

If this is the case, then hinges represent an extension of thought from Wittgenstein's earlier remarks in relation to private language, and possible represent a footnote to, or even an attack on, Frege's anti-psychologism that sought to clearly delineate truth from "taking as true".

Can you explain a bit more thoroughly what you mean by "resource-conscious"? — Metaphysician Undercover

Resource conscious logics such as Linear Logic don't automatically assume that the premise of a conditional can be used more than once. They are extensions or refinements of relevance logic. The best article relating resource-sensitivity to the principles of quantum mechanics is probably nlabs description of linear logic

https://ncatlab.org/nlab/show/linear+logic

As for uncertainty principles:

Recall that classical logic has the propositional distributive law, that for all A, B and C

A ∧ ( B ∨ C) = (A ∧ B) ∨ ( A ∧ C)

Here, the meaning of "and" is modelled as the Set cartesian product, and the meaning of "or" by set disjunction, neither of which are resource conscious - therefore one always has the same cartesian product, even after taking an element from one of its sets. The negation of this principle is more or less a definition of the uncertainty principle and characterizes the most remarkable aspect of quantum logic, which is in fact a common-sense principle that is used extensively in ordinary life.

The connectives of Linear logic cannot be interpreted in terms of the cartesian product and set disjunction. Instead it has the tautology

A ⊗ ( B ⊕ C) ≡ ( A ⊗ B ) ⊕ ( A ⊗ C )

If this formula is interpreted to be a true conclusion that needs to be proven with respect to unknown premises , then it has the interpretation "Assume that we are sent an A i.e. an element (a : A), and that we are also sent either (b : B) or (c : C) at our opponent's discretion, neither of which consume the (a : A) (that is to say B and C are independent of A). Then we end up with either (a : A) and (b : B), or (a : A) and a (c : C)".

Likewise, our opponent's side of this interaction is then described by the tautology

¬A ⅋ ( ¬B & ¬C) ≡ ( ¬A ⅋ ¬B ) & ( ¬A ⅋ ¬C)

"If our sending of (a : A) also implies our sending of either (b : B) or (c : C), where B and C are independent of A , then we either send both (a : A) and (b : B), or we send both (a : A) and (c: C).

But there isn't the theorem

A ⊗ ( B & C) ≡ ( A ⊗ B ) & ( A ⊗ C )

The inability to derive this theorem is the common-sense uncertainty principle of linear logic: getting an A together with a choice of B or C for which this act of choosing is independent of the existence of A, isn't equivalent to the outcome of the choice being independent of the existence of A.

(Imagine winning a bag of sugar together with a choice between winning either ordinary ice cream or diet ice cream. It might be that the awarders of the prizes use the awarded bag of sugar to produce the chosen ice-cream.)

By analogy, by using a resource-conscious logic as the foundation of an alternative calculus, smoothness and pointedness can be reconciled by defining them to be opposite and incompatible extremes of the state of a mutable function that is affected by the operations that are applied to it. This is also computationally realistic.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.

Is this in any way motivated by the uncertainty principle? — Moliere

If you mean the Heisenberg uncertainty principle no - although I'm tempted to think that Zeno was close to discovering a logical precursor to the Heisenberg Uncertainty Principle on the basis of a priori arguments.

The semantic problems of calculus with regards to Zeno's arguments stem from the fact that calculus isn't resource conscious. Sir Isaac Newton and Leibniz had no reason in 17th century to formulate calculus that way, given the use cases of calculus that they had in mind.

A notable feature of resource-conscious logics is how they naturally have "quantum-like" properties, due to the fact their semantic models are state spaces of decisions that are generally irreversible, thereby prohibiting the reuse of resources; indeed, the assumption that resources can be reused, is generally a cause of erroneous counterfactual reasoning, such as when arguing that a moving object must have a position because it might have been stopped.

So in the case of a resource-conscious calculus that avoids mathematical interpretations of Zeno's paradoxes (as in a function having a gradient but also consisting of points), a function must be treated as a mutable object whose topology undergoes a change of state whenever the function is projected onto a basis of functions that "measure" the function's properties -- Thus the uncertainty principle of Fourier analysis has to be part of the foundations of a resource-conscious calculus rather than a theorem derived from real-analysis of the continuum that is the cause of the semantic unsoundness of calculus with respect to the real world.

An obvious candidate for contributing to the foundations of such an alternative calculus is some variant of differential linear logic, which incidentally has many uses in quantum computing applications.

Is your point that Zeno treats motion as a series of steps, while both physics and maths treat it as continuous?

I'll go along with that. — Banno

More or less in the case of Zeno. Mathematics is often said to resolve the paradox in terms of the topological continuity of the continuum, by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, meaning that the paradox resurfaces when the continuum is deconstructed in terms of points.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.



Mathematical limits are proved in two steps using mathematical induction - which obviously does not involve a literal traversal of each and every rational number in order, which leads nowhere. (The proof of a limit is intensional, whereas the empirical concept of motion is extensional).

The mathematical interpretation of Zeno's paradox seems straightforward to me. Evaluating limits makes the so-called paradox disappear. What is illogical about that? And what does this have to do with calculus. Representing a continuum as an infinite series of infinitesimals seems like a good model of how the universe works, simple and intuitive. — T Clark

Zeno's dichotomy paradox corresponds to the mathematical fact that every pair of rational numbers is separated by a countably infinite number of other rational numbers. Because of this, a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p. Hence calculus does not say that f(x) moves towards L as x moves towards p.

Not something with which I am familiar. But in intuitionistic type theory, isn't a theorem synthetic if its truth depends on constructive proof rather than mere definitions? That is, not all synthetic theorems contain existential quantifiers. Consider "Every red bead appears before every blue bead on the string", which is not analytic, which must be determined by inspecting the arrangement of beads, and which uses universal quantification only. I may be misunderstanding your point, but being synthetic is not dependent on existential quantification only. However if your point is just that theorems containing an existential quantification are always synthetic because they require constructive proof, then yep. — Banno

Apologies for any misleading. To clarify, in type theory synthetic judgments can be identified with existential quantification due to the fact all propositions are types: having a proof that proposition A is true is equivalent to constructing a term a of type A, written a : A.

When referring to existential quantification, Lof wasn't referring to an existential quantifier within a proposition, but to an existential quantifier over terms representing a proof of a proposition type. Furthermore, the terms of a proposition type are definitionally equal by fiat, i.e a proposition type is the equivalence class of all proofs of that proposition.

My example referring to the swans was potentially misleading for conflating the two sorts of existential quantification, but nevertheless valid. A term cannot be constructed for the proposition type "All swans are white", indeed for any proposition containing a universal quantifier over an infinite domain, unless the proposition is interpreted intuitionistically such that the proposition can be proved by mathematical induction.

Perhaps a better example is the proposition "Nothing can accelerate beyond the speed of light". In relativity, a proof of that proposition implies contradiction. Hence presumably, the negation of the proposition is analytic in the theory special relativity, meaning that the proposition doesn't imply the physical impossibility of faster than light travel.

So we need necessity in order to do physics; but we must debar it from logic. A difficult path to tread. — Banno

Compare Quine to Martin Lof, the inventor of intuitionistic type theory. According to Lof, analytic sentences, at least in the context of intuitionistic type theory, are de-dicto definitions that are regarded to consist of perfect information, as in a complete table.

So in terms of your beads example, Lof would regard your proposed function mapping numbers to colors as analytic. But it is important to note the utility of calling this function definition "analytic" is only in relation to existentially quantified propositions about the analytic definition, which Lof classifies as "synthetic". E.g the theorem "there exists three red beads" is synthetic for Lof in relation to your bead function definition, because to determine the truth of the theorem requires checking.

In general, Lof regards a theorem in relation to intuitionistic logic to be 'synthetic' if the theorem contains an existential quantifier whose existence requires a proof in relation to the analytic definitions provided. Lof regards a synthetic theorem to be 'a priori' if the theorem can be proved de dicto via a process of deduction using the supplied analytic definitions that makes no recourse to facts about the external world. This is of course the case with intuitionistic logic, since its deductive system is constructive, i.e. de dicto. Hence for Lof, most of the theorems of intuitionistic mathematics are synthetic a priori (with the exception being postulated mathematical axioms). Generally, synthetic a priori propositions are undecidable.

Of course one might question whether the rules of the deductive system are correctly applied or whether one's analytic definitions are correct, in which case one's definitions are treated as being truth apt synthetic propositions in relation to some other underlying analytic definitions. So the analytic-synthetic distinction Lof intended is pragmatic without implying an absolute metaphysical distinction.

I think that Lof's reasoning is very much in line with Quine, whose notion of "physical necessity" I understand to be synthetic a posteriori, being in relation to the external world, but nevertheless also in relation to an analytic definition of physical terminology that undergoes constant revision on the basis of a posteriori evidence.

For example I imagine that Quine would consider the theorem "All swans are white" to be an analytic definition in the sense that Lof referred to, namely that the theorem doesn't contain a non-negated existential quantifier and so cannot be regarded as "true" except in the de dicto sense. This of course doesn't imply that the theorem's negation is analytic, which consists of a non-negated existential quantifier that answers to de re evidence. To me, such examples suggest that when counter-examples cause theory change, the falsified older theory is often not even wrong, in that the older theory cannot express the counter-example that it is wrong about.

It is my understanding that the appropriate mathematics didn’t exist in Zeno’s time. — T Clark

I'm not aware of a mathematical definition of an alternative continuum that resolves all of the logical puzzles posed by Zeno.

Zeno's paradoxes when interpreted mathematically, pose fundamental questions concerning the relationship between mathematics and logic, and in particular the question as to the logical foundation of calculus. The existence and utility of the classical continuum is also called into question.

I consider the most important and radical implication of Wittgenstein’s later work to be his critique of Frege’s theory of sense as reference. — Joshs

A critique of Frege's theory of sense and reference by Wittgenstein isn't possible, because Frege never provided an explicit theory or definition of sense. Frege only demonstrated his semantic category of sense (i.e. modes of presentation) through examples. And he was at pains to point out that sense referred to communicable information that leads from proposition to referent - information that is therefore neither subjective nor psychological. Therefore Fregean sense does not refer to private language - a concept that Frege was first to implicitly refer to and reject - but to sharable linguistic representations that can be used.

The later Wittgenstein's concept of language games, together with his commentary on private language, helps to 'earth' the notion of Fregean sense and to elucidate the mechanics of a generalized version of the concept, as well as to provide hints as to how Frege's conception of sense was unduly limited by the state of logic and formal methods during the time at which Frege wrote.

Frege - the first ordinary language philosopher? ;-)

Frege remained mired in a formalistic metaphysics centered on logic, without ever grasping f Wittgenstein’s distinction between the epistemic and the grammatical. — Joshs

Definitely not, for that makes it sounds like Frege was a dogmatic contrarian as opposed to the innovative and respectable founder of analytic philosophy - apparently the only thinker for whom Wittgenstein expressed admiration. As previously mentioned, Frege had already distinguished the epistemic from the grammatical when he introduced the turnstile. He knew the maxim "garbage in, garbage out".

Yet Frege's perception of propositions having eternal truth suggests that Frege might have been dogmatically wedded to classical logic that has no ability to represent truth dynamics. Indeed, I suspect that the later Wittgenstein's anti-theoretical stance was not a reaction against logic and system-building per-se, but a reaction against the inability of propositional calculus and first-order logic to capture the notion of dynamic truth and intersubjective agreement - a task that requires modern resource sensitive logics such as linear logic, as well as an ability to define intersubjective truth or "winning conditions", as exemplified by Girard's Ludics that breaks free from Tarskian semantics.

I agree. Intuition isn’t really what I was after. Wittgenstein said it better. — Joshs

Many of Wittgenstein's contemporaries said it better than Wittgenstein by formally distinguishing assertions from propositions. In particular, Frege introduced turnstile notation to make the distinction between propositions on the one hand, and assertions about propositions that he called judgements on the other.

If P denotes a proposition, then ⊢ P expresses a judgement that P holds true. Judgements can also be conditioned on the hypothetical existence of other judgements, written Q ⊢ P, where Q expresses a hypothetical judgement.

Notably, turnstile expressions don't denote truth values but rather practical or epistemic commitments, and the logical closure of such implications forms bedrocks of reasoning referred to as syntactic consequence. Of course, this does not preclude the possibility of such a collection of judgements from being treated as an object language, thereby allowing such judgements to be analysed, derived or explicated in terms of the finer-grained meta-judgements of a meta-language.

I presume the later Wittgenstein's remarks were not directed towards Frege or Russell - who essentially robbed the turnstile of philosophical significance by automating it, but at his earlier self who argued in the Tractatus that the turnstyle of logical assertion is redundant, due to thinking of propositions as unambiguous pictures of reality whose sense automatically conveyed their truth. But if this Tractatarian notion of the proposition is rejected, thereby leaving a semantic gap between what a proposition asserts and its truth value, then what does the gap signify and how must it be filled?

Evidently Frege was content to leave the gap unfilled and to signify it with a turnstile, and every logician since Russell has been content to build mathematics upon the turnstile by restricting the role of deduction to mapping judgements to judgements.

Logicians generally aren't bothered by the implication of infinite regress when explicating the judgements of object languages in terms of the meta-judgements of meta-languages, as aren't software engineers who often don't rely upon any meta-logical regression (with occasionally horrific consequences). but it apparently took Wittgenstein more time to feel comfortable with the turnstile and to reach a similar pragmatic conclusion.

A suggested computational analogy:

Non-rigid designators: Reassignable Pointers. Namely, mutable variables that range over the address space of other variables of a particular type. E.g, a pointer implementing the primary key of a relational database.

A rigid designator: A pointer that cannot be reassigned, representing a specific row of a table.

An indexical: A non-rigid designator used as a foreign key, so as to interpret its meaning as context sensitive and subject to change.

I can appreciate the distinction you are pointing out between stipulation and observation. Indeed, classical probability theory explicitly accommodates that distinction, by enabling analytic truths to be identified with an a priori choice of a sample space together with propositions that describe the a priori decided properties of the possible worlds in terms of measurable functions that map worlds to values. By contrast, statistical knowledge referring to observations of the sample space is encoded post-hoc through a choice of probability measure. I think this to be the most natural interpretation of classical probability theory, so I am tempted to think of probability theory as modal logic + statistics.

In particular, we can define a proposition p to be analytically true in relation to a possible world w if p is "True" for every path that includes w (or 'pathlet' if transitivity fails), in an analogous fashion to the definition of modal necessity for a Kripkean frame. (But here, I am suggesting that we say p is analytically true at w rather than necessarily true at w, due to the assumption that the sample space was decided in advance, prior to making observations).

By contrast, we can define p to be necessarily true at w if the set of paths including w for which p is true is assigned a probability equal to one. Thus a proposition can be necessarily true without being analytically true, by there existing a set of paths through w that has probability zero for which p is false.

That's what I thought. "One simple space" - so the step-wise structure disappears? That would presumably be the case if we implemented S5 in this way. — Banno

I'm not quite sure what you meant there, but to clarify, a sample space S can fully and faithfully represent any relation that is defined over a countable number of nodes, in terms of a set of infinite paths over those nodes.

However, speaking of probability theory in the same breath as modal logic seems to be uncommon, in spite of the fact that modal logic and probability theory have practically the same models in terms of Boolean algebras with minor changes or small additional structure that has no bearing with respect to the toy examples that are used to demonstrate the meaning of the theories.

Notably, the logical quantifiers of any decidable theory that has a countable number of formulas can be eliminated from the theory by simply introducing additional n-ary predicate symbols. And since modal logic refers only to fragments of first order logic, then unless the modalities/quantifiers are used with respect to undecidable or uncountable sets of propositions, then they have no theoretical significance and one might as well just stick to propositional logic. To me this raises a philosophical paradox, in that the only propositions that give the quantifiers/modalities philosophical significance are the very propositions that the quantifiers/modalities cannot decide.

In considering this I have been struck by how accessibility in modal logic resembles a Markov process, with states resembling possible worlds and transition probabilities resembling Accessibility relations. A directed graph resembles a Kripke frame... but Markov processes are not binary, unlike modal logic. Would that I had a stronger background in the maths involved. — Banno

Your suggestion is essentially equivalent to what I suggested in my last post, and indeed the likely tool for constructing the sample space i was referring to.

A Markov Kernel on a measurable space (S,B) onto itself, i.e. (S,B) --> (S,B), is a direct way of defining a state-transition probability matrix on a generally infinite set S. But as you indicate, what is needed is a binary valued state-transition matrix rather than a probability matrix. This just means swapping the state-transition probability measure B x S --> [0,1] for an unnormalized binary valued measure B x S ---> {0,1}. By iterating this 'markov process', one obtains a trip on S. The construction I suggested earlier that directly identified trips with events, has one sample space that consists of the product of n copies of S:

S1 x S2 x .... Sn.

in which the sigma algebra of possible trips obeys the accessibility relation.

But surely, ignorance is directly related to probabilities. If an event has a probability of 1, you can predict it perfectly; if all the probabilities are equal, then its like maximal unpredictability. — Apustimelogist

The distribution of an unknown random number generator could equal anything. If an analyst knows that he doesn't know the rng, then why should he represent his credence with a uniform distribution? And why should the ignorance of the analyst be of interest when the important thing is determining the function of the unknown distribution?

The probability that some hypothesis was the cause of your observation; and even if your prior is wrong, probability theory is the only logical way of changing probabilities when you see the evidence if you know the likelihood afaik. — Apustimelogist

Ever heard of imprecise probability?

K. In probability theory possible worlds are elements in a sample space, which consists in all possible outcomes of some experiment. These possible worlds are fixed by the definition of the probability space, they are mutually exclusive in that only one world can be the outcome of any one experiment. They are not hypothetical, but points in a mathematical space.

Wearers possible worlds in modal logic are stipulated, are not mutually exclusive and sit within a structure R which determines what worlds are accessible, one form the other. — Banno

Yes, you're right to challenge my previous post, as I realize that I wasn't quite correct in my interpretation of possible worlds in probability theory. But I still see no fundamental incompatibility.

Ultimately, i think the question we're addressing is "Can a set of possible worlds be adequately modelled in terms of a sigma algebra defined over a sample space?"

I think the key is to think of an element of the sample space as a trip through possible worlds that obeys the accessibility relation. This is essentially how finance uses probability theory when modelling movements of a stock price, where an element of the sample space is a sequence of binary values representing a sequence of price directions. Following this approach,

- An event is a possible trip through possible worlds.
- The sigma algebra defined on the sample space represents the possible history of the trip at each stage.
-A stochastic process represents possible histories of observations as the trip proceeds.
- An additional element can be added to the sample space to represent termination of the trip.

in probability theory the possible worlds are the outcome of a stochastic process, a coin flip or whatever. But in Modal Logic possible worlds are stipulated, hypothetical stats of affairs. They are not the same sort of thing. Care is needed in order to not be misled by the analogy. — Banno

No, it is the same in probability theory. There, the "set of possible worlds" refers to the sample space, where a "possible world" is normally referred to as an event or element of the sample space. A coin flip or stochastic process refers to a random variable, namely a function whose domain is the sample space and whose codomain is another set, usually the reals or the naturals.

So the input to a stochastic process is a particular possible world, of which the output is a set of observations of that possible world.

Any accessibility relation defined on a set of possible worlds can be interpreted as placing restrictions on the probability measure defined on (a sigma algebra of) sets of the possible worlds.

(post recently edited due to a mistake when describing the codomain of random variables)