Structuralism and Qualia
Clark, 2015-11-13
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I. Thesis: that the spectrum inversion thought experiments do
not demonstrate that there exist qualitative properties of
experience that are distinct from the functional (or more
broadly, structural) properties of experience.
"Quality of appearance" is ambiguous; it can be applied
either to the stimuli that appear that way (the thing that looks
blue) or to the properties of the sensory state in virtue of
which those stimuli appear that way.
The pesky problem: spectrum inversion.
"Qualitative character" will be used only in the latter sense:
they are properties of mental states.
"...imagine your spectrum becomes inverted at a particular
time in your life and you remember what it was like before
that. There is no epistemological problem about
"verification". You wake up one morning and the sky
looks ...[yellow], and your red sweater appears to have
turned ...[green], and all the faces are an awful color, as on a
color negative." (Putnam 1981, p. 80)
Putnam notes that there was a state that yesterday had the
functional role of "signaling the presence of 'objective blue'
in the environment" (that is, of the stimulus class that
yesterday included the color of the clear sunlit sky). He goes
on to say:
"If this functional role [had by the sensation of the color of
the sky] were identical with the qualitative character, then
one couldn't say that the quality of the sensation has
changed....But the quality has changed. The quality doesn't
seem to be a functional state in this case." (Putnam 1981, p.
81).
The qualitative character of a particular sensation will
include all the properties of that mental state that determine
how the stimulus that is being sensed appears to the creature
sensing it.
III. Invertible quality spaces
For the mind of a creature today to be functionally
isomorphic to its mind yesterday, the structure of relations
between stimuli and mental states, among its mental states,
and between mental states and behavior, must be the same.
This requires a color quality space that has a certain kind of
symmetry. Even if it is not found in the actual world, a
creature that has such a space seems readily conceivable:
A reconstruction. One can conceive of a world in which
1. A creature wakes up some morning and finds that
everything with a visible color seems to have changed in
appearance. Today it seems to be the complement of the
color that it was yesterday, as in a color negative.
2. The mind of the creature today is functionally
isomorphic to its mind yesterday.
It would follow that in that world
3. The creature's sensations of a clear sunlit sky today have
the same functional role as they did yesterday.
But then:
4. The qualitative character of that creature's sensations of a
clear sunlit sky has changed.
5. The functional role of that creature's sensations of a clear
sunlit sky has not changed.
Therefore
6. The qualitative character of a creature's sensations of a
clear sunlit sky is not identical to the functional role of
those sensations.
II. Some terminology
That something "looks blue" (and not green) I will call a
"quality of appearance" or "phenomenal property".
The verb of appearance ("looks", "seems", "appears", etc) is
here used to emphasize that the qualities sensed depend upon
the organization of the sensory system of the creature.
8. Showing all of the relative similarities:
Structuralism and Qualia
Clark, 2015-11-13
9. Adding the stimulus classes for these experiences:
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to stimuli, and to behavior. We get a "Ramsey correlate" for
each term to be defined. For example:
z has the appearance as of blue if and only if:
There are B,P,R,O,Y,G such that (for any x)(for any y)
[(Lxy iff ((Gx & By) or (Gx & Yy))) &
(Mxy iff ((Bx & Py) or (Yx & Oy))) &
(Nxy iff ((Px & Ry) or (Ox & Ry))) &
x is B if its stimulus is in Sb &
x is P if its stimulus is in Sp &
x is R if its stimulus is in Sr &
x is O if its stimulus is in So &
x is Y if its stimulus is in Sy &
x is G if its stimulus is in Sg ]
& z is B
10. After inversion:
Functionalism is just one variety of structuralism in
philosophy of science. To get to the broader variety: include
relations other than causal relations. Include any empirically
useful relations among theorized processes and between
theorized processes and observables found in any
empirically successful model. Drop the "function" from
functionalism.
A Ramsey sentence for any such theory will replace each
theoretical term with a description of the relations it bears to
all the other theoretical terms and to observables; those
relations now include any that might be empirically useful.
But structuralism still provides a relational account of
qualitative character. So it too seems threatened by
symmetric structures and spectrum inversion.
IV. From functionalism to structuralism
Functionalism claimed that theoretical terms in folk
psychology and experimental psychology could be
"functionally" defined. One forms the Ramsey sentence for
a theory by conjoining all its sentences, replacing all the
theoretical terms with variables, then prefxing the thing with
quantifiers. For the structure above the Ramsey sentence
would be:
There are B,P,R,O,Y,G such that (for any x)(for any y)
[(Lxy iff ((Gx & By) or (Gx & Yy))) &
(Mxy iff ((Bx & Py) or (Yx & Oy))) &
(Nxy iff ((Px & Ry) or (Ox & Ry))) &
x is B if its stimulus is in Sb &
x is P if its stimulus is in Sp &
x is R if its stimulus is in Sr &
x is O if its stimulus is in So &
x is Y if its stimulus is in Sy &
x is G if its stimulus is in Sg ]
Then one defines a particular theoretical term by describing
its "functional role": its relations to other theoretical terms,
V. Structuralism in philosophy of mathematics. It has run
into a problem with the same logical structure, which is there
called "non-trivial automorphism".
A group G is a set and binary operation •, which when
applied to any two elements a, b is denoted a • b. The set and
binary operation must satisfy four group axioms:
Closure: For all a, b in G the result of a • b is also in G.
Associativity: For all a, b and c in G, (a • b) • c = a • (b • c).
Identity element: There exists an element e in G such that for
every a in G, a • e = a.
Inverse element: For each a in G there exists an element b in
G such that a • b = e.
If the group operation is also commutative (For all a, b in G
a • b = b • a ) then the group is "abelian".
Examples: the integers under addition are an abelian group.
The natural numbers under addition are not, nor are the
integers under multiplication. The non-zero rational numbers
under multiplication are an abelian group.
11. A bijection (mapping) from set S to set T is a function σ
that is one-to-one onto T. Philosophers call this a "one-toone correspondence". Mathematically: (i) for every element
Structuralism and Qualia
Clark, 2015-11-13
a in S, σ(a) is an element of T; (ii) if a ≠ b then σ(a) ≠ σ(b);
and (iii) for every element b in T, there is an element a in S
such that σ(a) = b. σ(a) can be called "the image of a" or the
"inverse" of a.
Group G with operation • is isomorphic to group H with
operation * if and only if there exists a bijection σ from G to
H such that σ(a • b) = σ(a) * σ(b).
Example: group G is the set of all positive real numbers
under multiplication. Group H is the set of all real numbers
under addition. A one-to-one mapping from G to H is
provided by taking the logarithm: σ(a) = log(a) .
Multiplying positive real numbers is isomorphic to adding
their logarithms. Because log(a x b) = log(a) + log(b), we
have σ(a • b) = σ(a) * σ(b)
The groups G and H have "identical structural properties".
Calculations can be performed in whichever is more
convenient.
12. Any group G is isomorphic to itself. σ just maps every
element to itself, and the group operation • remains •. Such
an isomorphism is called an "automorphism". A non-trivial
automorphism is one in which σ is not the identity mapping:
there is at least one a in G such that σ(a) ≠ a.
Example: the abelian group G of integers under addition has
a non-trivial automorphism. σ(a) = the additive inverse of a;
this maps positive integers to negative ones, and vice-versa.
The additive inverse of (a + b) = the additive inverse of a +
the additive inverse of b, so in H the group operation is +.
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integers, "is the additive inverse of" is irreflexive. Hence
σ(a) ≠ a; their non-identity is assured. Since "additive
inverse" is one of the relations in the structure, a and σ(a)
can be distinguished from one another using nothing but
resources found within the structure.
Weak discernibility is pretty weak. It implies that we can
confirm the non-identity of two elements, but in a certain
sense we still cannot tell "which is which".
VII. Breaking the symmetry: an aside
As soon as one adds some additional expressive resources to
the abelian group of integers under addition, one can "break
the symmetry". Add "is greater than" and we can produce
formuli that are true of just positive integers or of just
negative ones.
These prospects disappear when one considers the most
widely discussed example of a non-trivial automorphism: the
"indiscernibility of i and -i." More precisely it is a nontrivial automorphism of the complex number field that maps
each complex number (a + bi) to its conjugate (a - bi) and
vice-versa.
Burgess (1999) says that i and -i "are not distinguished from
each other by any algebraic properties". Any expression,
formula, or polynomial in the language of complex analysis
satisfied by i will also be satisfied by -i, and vice-versa.
15. But every real number has two distinct square roots, so i
and -i are still weakly discernible.
13. There are mathematical examples that have all the
features of spectrum inversion. Jack may be not only
spectrum inverted relative to Jill, but also addition inverted.
She counts 1, 2, 3, ... while he counts -1, -2, -3, ... (which he
calls "1", "2", "3"...). But all their sums and differences
match.
VIII. Applied to intrasubjective spectrum inversion
So do we need monadic attributes of positivity and negativity
to distinguish 3 from -3?
16. In fact their relations imply that the two are distinct from
one another. They are caused by distinct and nonoverlapping stimulus classes and occur on different
occasions. Even simpler: Y and σ(Y) are qualitatively
dissimilar from one another--about as dissimilar as any two
qualities in that space can be. A quality of appearance cannot
be dissimilar from itself.
14. The mapping function σ that takes us from G to H maps
3 to -3 and -3 to 3. But it is very clear within either group
that 3 and -3 are distinct integers. They are non-zero, and
one is the additive inverse of the other. Therefore, the
structure itself implies that they are not identical.
Noted by Saunders (2003) and Ladyman (2005, 2007);
source Quine (1961, 230). Two elements are "absolutely
discernible" if there is a formula in the system free in one
variable that is true of exactly one of the two. They are
"relatively discernible" if there is some asymmetric relation
R that applies to the two in one order only.
15. Elements x and y are weakly discernible if they stand in
an irreflexive relation to one another. Among non-zero
Our Traffic Signal Species has a color quality space with a
non-trivial automorphism. It does not follow that the
appearance as of yellow has the same structural properties as
the appearance as of blue.
17. Putnam assumed that the functional role of the sensation
of a color is identical to the functional role of the sensation
of its inverse. This is the crux of the argument. It is a
natural assumption, but it is false.
18. Even in a symmetric quality space the qualities Y and B
do not have "identical functional roles". We do not need to
introduce some new class of monadic properties to
distinguish them. The structure itself implies that they are not
identical to one another.
Structuralism and Qualia
Clark, 2015-11-13
19. A diagnosis: it was very easy to confuse isomorphism and
identity of functional roles.
20. Isomorphism between systems is conventionally taken to
license saying that those two systems have "identical
structural properties". But it does not license saying that two
elements within one system, where one is mapped by σ to the
other, have "identical structural properties".
Given the Ramsey sentence in section IV, one can write
down a second Ramsey correlate for the node "appearance as
of yellow", and compare it to the one for blue. Those two
Ramsey correlates are not identical predicates, and they have
distinct extensions. But for all that they do have the same
logical form. One can be converted to the other merely by
substituting variables for variables. So there may be some
temptation to say they describe "isomorphic relational
properties", whatever those are. But they do not describe
identical relational properties.
IX Intersubjective inversion
Suppose Jack and Jill are members of our Traffic Signal
Species, and Jack is spectrum inverted relative to Jill.
Intrasubjectively there is a vast psychological difference
between sensing something that looks blue and sensing
something that looks yellow.
We can schematically describe a psychological difference
between Jack and Jill: the same stimulus of the clear blue sky
puts Jill into the psychological state of sensing something
that appears to be Q, and Jack into the state of sensing
something that appears to be the inverse of Q. Q is either
quality B or quality Y, and we don't know which. Jack and
Jill are not in identical psychological states, even though
their psychological organizations are isomorphic.
Epistemic costs of living in an invertible world.
X Costs we bear already
Unforeseen by common sense: we can determine that there
are two non-identical entities, but we lack the resources to
frame a description true of only one of them. They are in
that sense "indiscernible"--we cannot tell which is which-but not identical.
Ladyman and Ross (2007) argue that we are there already:
two or more particles in an entangled state may possess
exactly the same monadic and relational properties that are
expressed by the formalism of the theory. Consider a pair of
electrons in the orbital of a helium atom for example. They
have the same energy eigenstate, and the same position state
(which is not localized) ... Clearly, according to this state
description, there is no property of particle 1 that cannot also
be predicated of particle 2. (Ladyman & Ross 2007, 135)
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in an entangled state with respect to some observable each
particle has no state of its own with respect to that
observable but rather enters into a product state. The only
intrinsic properties than an entity in an entangled state has
that are independent of the other entities in that state are its
state-independent properties such as mass, charge, and so
on... (Ladyman & Ross 2007, 150)
The spin state of the two electrons in the helium orbital "is
such that in any given direction in space they must have
opposite spins" (135).
One can determine the aggregate of quantum particles, but
they cannot be specifically "enumerated" (or identified).
How this might be possible.
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