It seems intuitive to me, to replace, "is X reducible to Y" with "does Y uniquely characterise ^{[1]} X?".

(I think this adequately captures what is being debated over when we argue that consciousness is "reducible".)

(This may not be how to dissolve "reducible" in general, but it feels appropriate for this abstraction hierarchy.)

Answering the Questions

#2

Does physical state "uniquely characterise" informational/computational state?

We can reinterpret this as the inverse of:

Can the same physical state represent distinct informational/computational states?

I think the answer to the above is "yes". Consider the simple computation "2 + 2 = 4".

We might represent this computation by placing a group of two peas and another group of two peas together to form a new group.

If each pea represents:

The number "1", then we have computed: 2 + 2 = 4

The string "go", then we have computed: ("go", "go") + ("go", "go") = ("go", "go", "go", "go")

Computation is frame and interpretation dependent. The semantic content of a given computational/informational state is indeterminate. We need to know the referent of the different computational elements in idea space. E.g., in the second computation above, the referents are:

Peas: string "go"

Collection of peas: tuple

Placing a group of peas together: tuple addition

It seems that computation is NOT reducible to physics.

#1

Does computational/informational state "uniquely characterise" semantic content/knowledge state?

As best as I can tell, the answer is also "no".

Given an arbitrary string, there's no unique semantic content it refers to. For example, take a UTF-8 encoding of a string "Paris is the capital of France". It's not at all clear to me that the string points to the same semantic content I have associated with the label "Paris is the capital of France".

I can easily imagine another language where:

"Paris" means "two"

"Is the" means "is the"

"Capital of" means "smallest member of"

"France" means "collection of prime numbers"

And the statement becomes "two is the smallest member of the collection of prime numbers" or "two is the smallest prime".

The process in which I read and dereference the constituent tokens of the string in my own world model could be viewed as another algorithm that consumes the string. So, it doesn't mean that computation cannot be reduced to semantic content, but that a snapshot of computational state cannot be reduced to semantic content.

Interlude: "Stateful" Computation

It may be the case that you need to know what algorithm consumes the string (and perhaps the algorithm that produced the string as well).

(Defined inductively/recursively, what program the string is an input to [and so on ad infinitum], [and perhaps what program it's an output from (and so on ad infinitum)] are needed to determine the semantic content of the string.)

The same string can encode wildly different (potentially any) semantic content if you permit arbitrary algorithms to consume it.

Thus, there's no unique semantic content an arbitrary string refers to.

A More Complete Notion of "State"

But if you didn't condition on only the current state, but the entire lifetime of the computational process (every state prior and subsequent), then perhaps the semantic content is uniquely characterised?

(I don't know.)

Properly speaking, I think "computational/informational state" should be understood as a given snapshot together with the full sequence of snapshots it belongs to (and its location in that sequence).

That is properly speaking, we should speak about a triple (s_i, S, i) where:

s_i is a particular snapshot of the computational process

S is a (potentially infinite) sequence of states that s_i belongs to

i is the "index"/"position" of s_i in S

And not merely the snapshot s_i.

A snapshot alone doesn't fully characterise the notion of "state" that we may be interested in.

Confusion on Computation

I don't fully understand computation yet, but:

The definition of

#3

Given that computational state is not reducible to physical state, mental state is not reducible to physical state.

Intermediate Conclusions

I do not know if semantic content is reducible to computational/informational state.

Specifically, I do not know if a (s_i, S, i) triple have a unique "meaning"?

A snapshot s_i does not have a unique "meaning".

Computational/informational state is NOT reducible to physics.

Would this be true even if you condition on the entire light cone that contains a given physical state?

I do not know

Semantic content is not reducible to a given snapshot of physical state

This is implied by #2

It is again unknown if semantic content is reducible to a snapshot of physical state if you condition on the lightcone that produced that snapshot.

And you know the location of the snapshot within said lightcone.

Speculation on Computation and Physics

Mathematics resides in idea space and seems completely independent of the laws of physics of any particular universe.

The theorems of a given collection of axioms hold across all physical universes regardless of their laws.

"2 + 2 = 4" in every model of Peano arithmetic.

Computation seems to implement mathematics within particular physics.

And what computation is possible in a universe depends on the laws of physics of that universe?

So, computation being partially(?) reducible to physics seems plausible, but I don't fully understand it.

The definition of:

The grammar/language

The "model of computation"

Etc.

Seem to exist entirely in idea space and not cash out to physics space?

But I don't intuit theory of computation (I haven't yet learned it}, so maybe I'll come back to this after grokking computation.

Closing Remarks

Interlude

If SQA is true, then a given computational/informational snapshot may represent different qualia.

On the levels of abstraction again:

Subjective experience: semantic content (?)

"State" of the conscious algorithm

"State" of the physical system implementing the algorithm

("State" being a triple of which the snapshot is the first component and not just a snapshot.)

This is a more detailed description of how I think reducing subjective experience should be analysed.

(I think I've been able to decompose: "is consciousness reducible" to a bunch of disambiguated empirical questions.)

Tentative Conclusion

Given SQA:

"Is consciousness reducible to physics?" becomes "is '2 + 2 = 4' reducible to physics?"

(I think the answer to that is "no".)

But "is '2 + 2 = 4' reducible to computation?", has an answer of "yes".

[Computation itself may not be fully reducible to physics.])

^{^}

"Uniquely characterised" is deliberately left vague.

It may characterise/specify/describe the phenomenon under consideration up to some notion of "equivalence" (e.g. within a given mathematical world, we might adopt a definition that is unique up to isomorphism).

## Introduction

I recently made an attempt to dissolve the question: "is consciousness reducible"?

(This was borne out of a discussion with Rob Bensinger on Twitter.)

I've repackaged the thread for LessWrong.

## Concepts Hierarchy

I see three relevant/particularly pertinent levels of abstraction:

A. Mental state

B. Computational/Informational state

C. Physical state

I see three relevant questions we can then ask:

## Idea Space

I would like to posit another abstraction, a domain where ideas and abstractions live.

It's in this domain that the axioms of set theory and Peano arithmetic lie.

(Where "2 + 2 = 4" is defined and true.)

I think it's in this domain, that knowledge/semantic content lives. Where the sentence: "Paris is the capital of France" is defined and true.

## Some Assumptions

Potential assumptions that one might make.

## Semantic Qualia Assumption (SQA)

"Mental state" refers to the "knowledge state"/"semantic content" of a mind at a particular location in spacetime.

The "qualia" that a mind experiences, resides in idea space.

I do not claim that this assumption is true, I merely explicate it so that we may consider the implications if the assumption were true.

## Taboo "Reducible"

I propose we taboo the word "reducible".

It seems intuitive to me, to replace, "is X reducible to Y" with "does Y uniquely characterise

^{[1]}X?".(I think this adequately captures what is being debated over when we argue that consciousness is "reducible".)

(This may not be how to dissolve "reducible" in general, but it feels appropriate for this abstraction hierarchy.)

## Answering the Questions

## #2

Does physical state "uniquely characterise" informational/computational state?

We can reinterpret this as the inverse of:

I think the answer to the above is "yes". Consider the simple computation "2 + 2 = 4".

We might represent this computation by placing a group of two peas and another group of two peas together to form a new group.

If each pea represents:

Computation is frame and interpretation dependent. The semantic content of a given computational/informational state is indeterminate. We need to know the referent of the different computational elements in idea space. E.g., in the second computation above, the referents are:

It seems that computation is NOT reducible to physics.

## #1

Does computational/informational state "uniquely characterise" semantic content/knowledge state?

As best as I can tell, the answer is also "no".

Given an arbitrary string, there's no unique semantic content it refers to. For example, take a UTF-8 encoding of a string "Paris is the capital of France". It's not at all clear to me that the string points to the same semantic content I have associated with the label "Paris is the capital of France".

I can easily imagine another language where:

And the statement becomes "two is the smallest member of the collection of prime numbers" or "two is the smallest prime".

The process in which I read and dereference the constituent tokens of the string in my own world model could be viewed as another algorithm that consumes the string. So, it doesn't mean that computation cannot be reduced to semantic content, but that a snapshot of computational state cannot be reduced to semantic content.

## Interlude: "Stateful" Computation

It may be the case that you need to know what algorithm consumes the string (and perhaps the algorithm that produced the string as well).

(Defined inductively/recursively, what program the string is an input to [and so on ad infinitum], [and perhaps what program it's an output from (and so on ad infinitum)] are needed to determine the semantic content of the string.)

The same string can encode wildly different (potentially any) semantic content if you permit arbitrary algorithms to consume it.

Thus, there's no unique semantic content an arbitrary string refers to.

## A More Complete Notion of "State"

But if you didn't condition on only the current state, but the entire lifetime of the computational process (every state prior and subsequent), then perhaps the semantic content is uniquely characterised?

(I don't know.)

Properly speaking, I think "computational/informational state" should be understood as a given snapshot together with the full sequence of snapshots it belongs to (and its location in that sequence).

That is properly speaking, we should speak about a triple (s_i, S, i) where:

And not merely the snapshot s_i.

A snapshot alone doesn't fully characterise the notion of "state" that we may be interested in.

## Confusion on Computation

I don't fully understand computation yet, but:

## #3

Given that computational state is not reducible to physical state, mental state is not reducible to physical state.

## Intermediate Conclusions

## Speculation on Computation and Physics

Mathematics resides in idea space and seems completely independent of the laws of physics of any particular universe.

The theorems of a given collection of axioms hold across all physical universes regardless of their laws.

"2 + 2 = 4" in every model of Peano arithmetic.

Computation seems to implement mathematics within particular physics.

And what computation is possible in a universe depends on the laws of physics of that universe?

So, computation being partially(?) reducible to physics seems plausible, but I don't fully understand it.

The definition of:

Seem to exist entirely in idea space and not cash out to physics space?

But I don't intuit theory of computation (I haven't yet learned it}, so maybe I'll come back to this after grokking computation.

## Closing Remarks

## Interlude

If SQA is true, then a given computational/informational snapshot may represent different qualia.

On the levels of abstraction again:

("State" being a triple of which the snapshot is the first component and not just a snapshot.)

This is a more detailed description of how I think reducing subjective experience should be analysed.

(I think I've been able to decompose: "is consciousness reducible" to a bunch of disambiguated empirical questions.)

## Tentative Conclusion

Given SQA:

"Is consciousness reducible to physics?" becomes "is '2 + 2 = 4' reducible to physics?"

(I think the answer to that is "no".)

But "is '2 + 2 = 4' reducible to computation?", has an answer of "yes".

[Computation itself may not be fully reducible to physics.])

^{^}"Uniquely characterised" is deliberately left vague.

It may characterise/specify/describe the phenomenon under consideration up to some notion of "equivalence" (e.g. within a given mathematical world, we might adopt a definition that is unique up to isomorphism).