Commutative property

From Wikipedia, the free encyclopedia

An operation ${\displaystyle \circ }$ is commutative iff ${\displaystyle x\circ y=y\circ x}$ for each ${\displaystyle x}$ and ${\displaystyle y}$. This image illustrates this property with the concept of an operation as a "calculation machine". It doesn't matter for the output ${\displaystyle x\circ y}$ or ${\displaystyle y\circ x}$ respectively which order the arguments ${\displaystyle x}$ and ${\displaystyle y}$ have – the final outcome is the same.

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations such as the multiplication and addition of numbers are commutative, was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A corresponding property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

Common uses

The commutative property (or commutative law) is a property generally associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation.

Mathematical definitions

The term "commutative" is used in several related senses.^[4][5]

1. A binary operation ${\displaystyle *}$ on a set S is called commutative if:

   ${\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S}$

   An operation that does not satisfy the above property is called non-commutative.
2. One says that x commutes with y under ${\displaystyle *}$ if:

   ${\displaystyle x*y=y*x}$

3. A binary function ${\displaystyle f\colon A\times A\to B}$ is called commutative if:
   ${\displaystyle f(x,y)=f(y,x)\qquad {\mbox{for all }}x,y\in A}$

Examples

Commutative operations in everyday life

The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.

• Putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same. In contrast, putting on underwear and trousers is not commutative.
• The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills are handed over in, they always give the same total.

Commutative operations in mathematics

The addition of vectors is commutative, because ${\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}$.

Two well-known examples of commutative binary operations:^[4]

• The addition of real numbers is commutative, since

${\displaystyle y+z=z+y\qquad {\mbox{for all }}y,z\in \mathbb {R} }$

For example 4 + 5 = 5 + 4, since both expressions equal 9.

• The multiplication of real numbers is commutative, since

${\displaystyle yz=zy\qquad {\mbox{for all }}y,z\in \mathbb {R} }$

For example, 3 × 5 = 5 × 3, since both expressions equal 15.

• Some binary truth functions are also commutative, since the truth tables for the functions are the same when one changes the order of the operands.

For example, the logical biconditional function p ↔ q is equivalent to q ↔ p. This function is also written as p IFF q, or as p ≡ q, or as Epq.

The last form is an example of the most concise notation in the article on truth functions, which lists the sixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp; Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

• Further examples of commutative binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors, and intersection and union of sets.

Noncommutative operations in daily life

• Concatenation, the act of joining character strings together, is a noncommutative operation. For example,

EA + T = EAT ≠ TEA = T + EA

• Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
• Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
• The twists of the Rubik's Cube are noncommutative. This can be studied using group theory.
• Thought processes are noncommutative: A person asked a question (A) and then a question (B) may give different answers to each question than a person asked first (B) and then (A), because asking a question may change the person's state of mind.
• The act of dressing is either commutative or non-commutative, depending on the items. Putting on underwear and normal clothing is noncommutative. Putting on left and right socks is commutative.

Noncommutative operations in mathematics

Some noncommutative binary operations:^[6]

Subtraction and division

Subtraction is noncommutative, since ${\displaystyle 0-1\neq 1-0}$.
Division is noncommutative, since ${\displaystyle 1\div 2\neq 2\div 1}$.

Truth functions

Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for f (A, B) = A Λ ¬B (A AND NOT B) and f (B, A) = B Λ ¬A are

A	B	f (A, B)	f (B, A)
F	F	F	F
F	T	F	T
T	F	T	F
T	T	F	F

For the eight noncommutative functions, Bqp = Cpq; Mqp = Lpq; Cqp = Bpq; Lqp = Mpq; Fqp = Gpq; Iqp = Hpq; Gqp = Fpq; Hqp = Ipq.^[7]

Matrix multiplication

Matrix multiplication is almost always noncommutative, for example:

${\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}$

Vector product

The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., b × a = −(a × b).

History and etymology

The first known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products.^[8][9] Euclid is known to have assumed the commutative property of multiplication in his book Elements.^[10] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.

The first recorded use of the term commutative was in a memoir by François Servois in 1814,^[1][11] which used the word commutatives when describing functions that have what is now called the commutative property. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch." The term then appeared in English in 1838^[2] in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.^[12]

Propositional logic

Rule of replacement

In truth-functional propositional logic, commutation,^[13][14] or commutativity^[15] refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

${\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}$

and

${\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}$

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with."

Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies.

Commutativity of conjunction

${\displaystyle (P\land Q)\leftrightarrow (Q\land P)}$

 

Commutativity of disjunction

${\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}$

 

Commutativity of implication (also called the law of permutation)

${\displaystyle (P\to (Q\to R))\leftrightarrow (Q\to (P\to R))}$

 

Commutativity of equivalence (also called the complete commutative law of equivalence)

${\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}$

Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.^[16][17][18]

Mathematical structures and commutativity

• A commutative semigroup is a set endowed with a total, associative and commutative operation.
• If the operation additionally has an identity element, we have a commutative monoid
• An abelian group, or commutative group is a group whose group operation is commutative.^[17]
• A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.)^[19]
• In a field both addition and multiplication are commutative.^[20]

Related properties

Associativity

The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms doesn't change. In contrast, the commutative property states that the order of the terms does not affect the final result.
Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function

${\displaystyle f(x,y)={\frac {x+y}{2}},}$

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, ${\displaystyle f(-4,f(0,+4))=-1}$ but ${\displaystyle f(f(-4,0),+4)=+1}$). More such examples may be found in commutative non-associative magmas.

Symmetry

Graph showing the symmetry of the addition function

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seen in the image on the right.

For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then ${\displaystyle aRb\Leftrightarrow bRa}$.

Non-commuting operators in quantum mechanics

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and ${\displaystyle {\frac {d}{dx}}}$. These two operators do not commute as may be seen by considering the effect of their compositions ${\displaystyle x{\frac {d}{dx}}}$ and ${\displaystyle {\frac {d}{dx}}x}$ (also called products of operators) on a one-dimensional wave function ${\displaystyle \psi (x)}$:

${\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}$

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the x-direction of a particle are represented by the operators ${\displaystyle x}$ and ${\displaystyle -i\hbar {\frac {\partial }{\partial x}}}$, respectively (where ${\displaystyle \hbar }$ is the reduced Planck constant). This is the same example except for the constant ${\displaystyle -i\hbar }$, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

de Sitter invariant special relativity

From Wikipedia, the free encyclopedia

In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution, which requires a cosmological constant or the stress–energy of a constant scalar field to sustain.

The idea of de Sitter invariant relativity is to require that the laws of physics are not fundamentally invariant under the Poincaré group of special relativity, but under the symmetry group of de Sitter space instead. With this assumption, empty space automatically has de Sitter symmetry, and what would normally be called the cosmological constant in general relativity becomes a fundamental dimensional parameter describing the symmetry structure of spacetime.

First proposed by Luigi Fantappiè in 1954, the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean-Marc Lévy-Leblond. In 1972, Freeman Dyson popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered.^[1] The discovery of the accelerating expansion of the universe has led to a revival of interest in de Sitter invariant theories, in conjunction with other speculative proposals for new physics, like doubly special relativity.

Introduction

De Sitter suggested that spacetime curvature might not be due solely to gravity^[2] but he did not give any mathematical details of how this could be accomplished. In 1968 Henri Bacry and Jean-Marc Lévy-Leblond showed that the de Sitter group was the most general group compatible with isotropy, homogeneity and boost invariance.^[3] Later, Freeman Dyson^[1] advocated this as an approach to making the mathematical structure of general relativity more self-evident.
Minkowski's unification of space and time within special relativity replaces the Galilean group of Newtonian mechanics with the Lorentz group. This is called a unification of space and time because the Lorentz group is simple, while the Galilean group is a semi-direct product of rotations and Galilean boosts. This means that the Lorentz group mixes up space and time such that they cannot be disentangled, while the Galilean group treats time as a parameter with different units of measurement than space.

An analogous thing can be made to happen with the ordinary rotation group in three dimensions. If you imagine a nearly flat world, one in which pancake-like creatures wander around on a pancake flat world, their conventional unit of height might be the micrometre (μm), since that is how high typical structures are in their world, while their unit of distance could be the metre, because that is their body's horizontal extent. Such creatures would describe the basic symmetry of their world as SO(2), being the known rotations in the horizontal (x–y) plane. Later on, they might discover rotations around the x- and y-axes—and in their everyday experience such rotations might always be by an infinitesimal angle, so that these rotations would effectively commute with each other.

The rotations around the horizontal axes would tilt objects by an infinitesimal amount. The tilt in the x–z plane (the "x-tilt") would be one parameter, and the tilt in the y–z plane (the "y-tilt") another. The symmetry group of this pancake world is then SO(2) semidirect product with R², meaning that a two-dimensional rotation plus two extra parameters, the x-tilt and the y-tilt. The reason it is a semidirect product is that, when you rotate, the x-tilt and the y-tilt rotate into each other, since they form a vector and not two scalars. In this world, the difference in height between two objects at the same x, y would be a rotationally invariant quantity unrelated to length and width. The z-coordinate is effectively separate from x and y.

Eventually, experiments at large angles would convince the creatures that the symmetry of the world is SO(3). Then they would understand that z is really the same as x and y, since they can be mixed up by rotations. The SO(2) semidirect product R² limit would be understood as the limit that the free parameter μ, the ratio of the height range μm to the length range m, approaches 0. The Lorentz group is analogous—it is a simple group that turns into the Galilean group when the time range is made long compared to the space range, or where velocities may be regarded as infinitesimal, or equivalently, may be regarded as the limit c → ∞, where relativistic effects become observable "as good as at infinite velocity".

The symmetry group of special relativity is not entirely simple, due to translations. The Lorentz group is the set of the transformations that keep the origin fixed, but translations are not included. The full Poincaré group is the semi-direct product of translations with the Lorentz group. If translations are to be similar to elements of the Lorentz group, then as boosts are non-commutative, translations would also be non-commutative.

In the pancake world, this would manifest if the creatures were living on an enormous sphere rather than on a plane. In this case, when they wander around their sphere, they would eventually come to realize that translations are not entirely separate from rotations, because if they move around on the surface of a sphere, when they come back to where they started, they find that they have been rotated by the holonomy of parallel transport on the sphere. If the universe is the same everywhere (homogeneous) and there are no preferred directions (isotropic), then there are not many options for the symmetry group: they either live on a flat plane, or on a sphere with a constant positive curvature, or on a Lobachevski plane with constant negative curvature. If they are not living on the plane, they can describe positions using dimensionless angles, the same parameters that describe rotations, so that translations and rotations are nominally unified.

In relativity, if translations mix up nontrivially with rotations, but the universe is still homogeneous and isotropic, the only option is that spacetime has a uniform scalar curvature. If the curvature is positive, the analog of the sphere case for the two-dimensional creatures, the spacetime is de Sitter space and its symmetry group is the de Sitter group rather than the Poincaré group.

De Sitter special relativity postulates that the empty space has de Sitter symmetry as a fundamental law of nature. This means that spacetime is slightly curved even in the absence of matter or energy. This residual curvature implies a positive cosmological constant Λ to be determined by observation. Due to the small magnitude of the constant, special relativity with its Poincaré group is indistinguishable from de Sitter space for most practical purposes.

Modern proponents of this idea, such as S. Cacciatori, V. Gorini and A. Kamenshchik,^[4] have reinterpreted this theory as physics, not just mathematics. They postulate that the acceleration of the expansion of the universe is not entirely due to vacuum energy, but at least partly due to the kinematics of the de Sitter group, which would replace the Poincaré group.

A modification of this idea allows ${\displaystyle \Lambda }$ to change with time, so that inflation may come from the cosmological constant being larger near the big bang than nowadays. It can also be viewed as a different approach to the problem of quantum gravity.^[5]

High energy

The Poincaré group contracts to the Galilean group for low-velocity kinematics, meaning that when all velocities are small the Poincaré group "morphs" into the Galilean group. (This can be made precise with İnönü and Wigner's concept of group contraction.^[6])

Similarly, the de Sitter group contracts to the Poincaré group for short-distance kinematics, when the magnitudes of all translations considered are very small compared to the de Sitter radius.^[5] In quantum mechanics, short distances are probed by high energies, so that for energies above a very small value related to the cosmological constant, the Poincaré group is a good approximation to the de Sitter group.

In de Sitter relativity, the cosmological constant is no longer a free parameter of the same type; it is determined by the de Sitter radius, a fundamental quantity that determines the commutation relation of translation with rotations/boosts. This means that the theory of de Sitter relativity might be able to provide insight on the value of the cosmological constant, perhaps explaining the cosmic coincidence. Unfortunately, the de Sitter radius, which determines the cosmological constant, is an adjustable parameter in de Sitter relativity, so the theory requires a separate condition to determine its value in relation to the measurement scale.

When a cosmological constant is viewed as a kinematic parameter, the definitions of energy and momentum must be changed from those of special relativity. These changes could significantly modify the physics of the early universe if the cosmological constant was greater back then. Some speculate that a high energy experiment could modify the local structure of spacetime from Minkowski space to de Sitter space with a large cosmological constant for a short period of time, and this might eventually be tested in the existing or planned particle collider.^[7]

Doubly special relativity

Since the de Sitter group naturally incorporates an invariant length parameter, de Sitter relativity can be interpreted as an example of the so-called doubly special relativity. There is a fundamental difference, though: whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry.^[8][9] A drawback of the usual doubly special relativity models is that they are valid only at the energy scales where ordinary special relativity is supposed to break down, giving rise to a patchwork relativity. On the other hand, de Sitter relativity is found to be invariant under a simultaneous re-scaling of mass, energy and momentum,^[10] and is consequently valid at all energy scales. A relationship between doubly special relativity, de Sitter space and general relativity is described by Derek Wise.^[11] See also MacDowell–Mansouri action.

Newton–Hooke: de Sitter special relativity in the limit v ≪ c

In the limit as v ≪ c, the de Sitter group contracts to the Newton–Hooke group.^[12] This has the effect that in the nonrelativistic limit, objects in de Sitter space have an extra "repulsion" from the origin: objects have a tendency to move away from the center with an outward pointing fictitious force proportional to their distance from the origin.

While it looks as though this might pick out a preferred point in space—the center of repulsion, it is more subtly isotropic. Moving to the uniformly accelerated frame of reference of an observer at another point, all accelerations appear to have a repulsion center at the new point.

What this means is that in a spacetime with non-vanishing curvature, gravity is modified from Newtonian gravity.^[13] At distances comparable to the radius of the space, objects feel an additional linear repulsion from the center of coordinates.

History of de Sitter invariant special relativity

• "de Sitter relativity" is the same as the theory of "projective relativity" of Luigi Fantappiè and Giuseppe Arcidiacono first published in 1954 by Fantappiè^[14] and the same as another independent discovery in 1976.^[15]
• In 1968 Henri Bacry and Jean-Marc Lévy-Leblond published a paper on possible kinematics^[3]
• In 1972 Freeman Dyson^[1] further explored this.
• In 1973 Eliano Pessa described how Fantappié–Arcidiacono projective relativity relates to earlier conceptions of projective relativity and to Kaluza Klein theory.^[16]
• Han-Ying Guo, Chao-Guang Huang, Zhan Xu, Bin Zhou have used the term "de Sitter special relativity" from 2004 onwards.^{[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]}
• R. Aldrovandi, J.P. Beltrán Almeida and J.G. Pereira have used the terms "de Sitter special relativity" and "de Sitter relativity" starting from their 2007 paper "de Sitter special relativity".^[10][32] This paper was based on previous work on amongst other things: the consequences of a non-vanishing cosmological constant,^[33] on doubly special relativity^[34] and on the Newton–Hooke group^[3][35][36] and early work formulating special relativity with a de Sitter space^[37][38][39]

• From 2006 onwards Ignazio Licata and Leonardo Chiatti have published papers on Fantappié–Arcidiacono theory of relativity pointing out that it is the same thing as de Sitter relativity^{[14][40][41][42][43]}
• In 2008 S. Cacciatori, V. Gorini and A. Kamenshchik^[4] published a paper about the kinematics of de Sitter relativity.
• Papers by other authors include: dSR and the fine structure constant;^[44] dSR and dark energy;^[45] dSR Hamiltonian Formalism;^[46] and De Sitter Thermodynamics from Diamonds's Temperature,^[47] Triply special relativity from six dimensions,^[48] Deformed General Relativity and Torsion.^[49]

Quantum de Sitter special relativity

There are quantized or quantum versions of de Sitter special relativity.

On the Search for the Neural Correlate of Consciousness

June 26, 2002 by David Chalmers
Original link:  http://www.kurzweilai.net/on-the-search-for-the-neural-correlate-of-consciousness
Originally published March 1998 in Toward a Science of Consciousness II: The Second Tucson Discussions and Debates (MIT Press). Published on KurzweilAI.net on June 26, 2002.

There’s a variety of proposed neural systems associated with conscious experience, but no way to directly observe or measure consciousness. Chalmers suggests though that there may be a “consciousness module” — a functional area responsible for the integration of information in the brain, with high-bandwidth communication between its parts.

From the author: This paper appears in Toward a Science of Consciousness II: The Second Tucson Discussions and Debates (S. Hameroff, A. Kaszniak, and A.Scott, eds), published with MIT Press in 1998. It is a transcript of my talk at the second Tucson conference in April 1996, lightly edited to include the contents of overheads and to exclude some diversions with a consciousness meter. A more in-depth argument for some of the claims in this paper can be found in Chapter 6 of my book The Conscious Mind (Chalmers, 1996).

I’m going to talk about one aspect of the role that neuroscience plays in the search for a theory of consciousness. Whether or not neuroscience can solve all the problems of consciousness single handedly, there is no question that it has a major role to play. We’ve seen at this conference that there’s a vast amount of progress in neurobiological research, and that much of it is clearly bearing on the problems of consciousness. But the conceptual foundations of this sort of research are only beginning to be laid. So I will look at some of the things that are going on from a philosopher’s perspective and will see if there’s anything helpful to say about these foundations.

We’ve all been hearing a lot about the "neural correlate of consciousness". This phrase is intended to refer to the neural system or systems primarily associated with conscious experience. I gather that the catchword of the day is "NCC". We all have an NCC inside our head, we just have to find out what it is. In recent years there have been quite a few proposals about the identity of the NCC. One of the most famous proposals is Crick and Koch’s suggestion concerning 40-hertz oscillations. That proposal has since faded away a little but there are all sorts of other suggestions out there. It’s almost got to a point where it’s reminiscent of particle physics, where they have something like 236 particles and people talk about the "particle zoo". In the study of consciousness, one might talk about the "neural correlate zoo". There have also been a number of related proposals about what we might call the "cognitive correlate of consciousness" (CCC?).

A small list of suggestions that have been put forward might include:

• 40-hertz oscillations in the cerebral cortex (Crick and Koch 1990)
• Intralaminar nucleus in the thalamus (Bogen 1995)
• Re-entrant loops in thalamocortical systems (Edelman 1989)
• 40-hertz rhythmic activity in thalamocortical systems (Llinas et al 1994)
• Nucleus reticularis (Taylor and Alavi 1995)
• Extended reticular-thalamic activation system (Newman and Baars 1993)
• Anterior cingulate system (Cotterill 1994)
• Neural assemblies bound by NMDA (Flohr 1995)
• Temporally-extended neural activity (Libet 1994)
• Backprojections to lower cortical areas (Cauller and Kulics 1991)
• Neurons in extrastriate visual cortex projecting to prefrontal areas (Crick and Koch 1995)
• Neural activity in area V5/MT (Tootell et al 1995)
• Certain neurons in the superior temporal sulcus (Logothetis and Schall 1989)
• Neuronal gestalts in an epicenter (Greenfield 1995)
• Outputs of a comparator system in the hippocampus (Gray 1995)
• Quantum coherence in microtubules (Hameroff 1994)
• Global workspace (Baars 1988)
• Activated semantic memories (Hardcastle 1995)
• High-quality representations (Farah 1994)
• Selector inputs to action systems (Shallice 1988)

There are a few intriguing commonalities among the proposals on this list. A number of them give a central role to interactions between the thalamus and the cortex, for example. All the same, the sheer number and diversity of the proposals can be a little overwhelming. I propose to step back a little and try to make sense of all this activity by asking some foundational questions.

A central question is this: how is it, in fact, that one can search for the neural correlate of consciousness? As we all know, there are problems in measuring consciousness. It’s not a directly and straightforwardly observable phenomenon. It would be a lot easier if we had a way of getting at consciousness directly; if we had, for example, a consciousness meter.

If we had a consciousness meter, searching for the NCC would be straightforward. We’d wave the consciousness meter and measure a subject’s consciousness directly. At the same time, we’d monitor the underlying brain processes. After a number of trials, we’d say OK, such-and-such brain processes are correlated with experiences of various kinds, so that’s the neural correlate of consciousness.

Alas, we don’t have a consciousness meter, and there seem to be principled reasons why we can’t have one. Consciousness just isn’t the sort of thing that can be measured directly. So: What do we do without a consciousness meter? How can the search go forward? How does all this experimental research proceed?

I think the answer is this: we get there through principles of interpretation. These are principles by which we interpret physical systems to judge whether or not they have consciousness. We might call these pre-experimental bridging principles. These are the criteria that we bring to bear in looking at systems to say (a) whether or not they are conscious now, and (b) what information they are conscious of, and what information they are not. We can’t reach in directly and grab those experiences and "transpersonalize" them into our own, so we rely on external criteria instead.

That’s a perfectly reasonable thing to do. But in doing this we have to realize that something interesting is going on. These principles of interpretation are not themselves experimentally determined or experimentally tested. In a sense they are pre-experimental assumptions. Experimental research gives us a lot of information about processing; then we bring in the bridging principles to interpret the experimental results, whatever those results may be. They are the principles by which we make inferences from facts about processing to facts about consciousness, so they are conceptually prior to the experiments themselves. We can’t actually refine them experimentally (except perhaps through first-person experimentation!), because we don’t have any independent access to the independent variable. Instead, these principles will be based on some combination of (a) conceptual judgments about what counts as a conscious process and (b) information gleaned from our first-person perspective on our own consciousness.

I think we are all stuck in this boat. The point applies whether one is a reductionist or an anti-reductionist about consciousness. A hard-line reductionist might put some of these points slightly differently, but either way, the experimental work is going to require pre-experimental reasoning to determine the criteria for ascription of consciousness. Of course such principles are usually left implicit in empirical research. We don’t usually see papers saying "Here is the bridging principle, here are the data, and here is what follows." But it’s useful to make them explicit. The very presence of these principles has some strong and interesting consequences in the search for the NCC.

In a sense, in relying on these principles we are taking a leap into the epistemological unknown. Because we don’t measure consciousness directly, we have to make something of a leap of faith. It may not be a big leap, but nevertheless it suggests that everyone doing this sort of work is engaged in philosophical reasoning. Of course one can always choose to stay on solid ground, talking about the empirical results in a neutral way; but the price of doing so is that one gains no particular insight into consciousness. Conversely, as soon as we draw any conclusions about consciousness, we have gone beyond the information given, so we need to pay careful attention to the reasoning involved.

So what are these principles of interpretation? The first and by far the most prevalent such principle is a very straightforward one: it’s a principle of verbal report. When someone says "Yes, I see that table now", we infer that they are conscious of the table. When someone says "Yes, I see red now", we infer that they are having an experience of red. Of course one might always say "How do you know?" — a philosopher might suggest that we may be faced with a fully functioning zombie – but in fact most of us don’t believe that the people around us are zombies, and in practice we are quite prepared to rely on this principle. As pre-experimental assumptions go, this is a relatively "safe" one — it doesn’t require a huge leap of faith — and it is very widely used.

So the principle here is that when information is verbally reported, it is conscious. One can extend this slightly, as no one believes that an actual verbal report is required for consciousness; we are conscious of much more than we report on any given occasion. So an extended principle might say that when information is directly available for verbal report, it is conscious.

Experimental researchers don’t rely only on these principles of verbal report and reportability. These principles can be somewhat limiting when we want to do broader experiments. In particular, we don’t want to just restrict our studies of consciousness to subjects that have language. In fact just this morning we saw a beautiful example of research on consciousness in language-free creatures. I’m referring to the work of Nikos Logothetis and his colleagues (e.g. Logothetis & Schall 1989; Leopold & Logothetis 1996). This work uses experiments on binocular rivalry in monkeys to draw conclusions about the neural processes associated with consciousness. How do Logothetis et al manage to draw conclusions about a monkey’s consciousness without getting any verbal reports? What they do is rely on a monkey’s pressing bars: if a monkey can be made to press a bar in an appropriate way in response to a stimulus, we’ll say that that stimulus was consciously perceived.

The criterion at play seems to require that the information be available for an arbitrary response. If it turned out that the monkey could press a bar in response to a red light but couldn’t do anything else, we would be tempted to say that it wasn’t a case of consciousness at all, but some sort of subconscious connection. If on the other hand we find information that is available for response in all sorts of different ways, then we’ll say that it is conscious. Actually Logothetis and his colleagues also use some subtler reasoning about similarities with binocular rivalry in humans to buttress the claim that the monkey is having the relevant conscious experience, but it is clearly the response that carries the most weight.

The underlying general principle is something like this: When information is directly available for global control in a cognitive system, then it is conscious. If information is available for response in many different motor modalities, we will say that it is conscious, at least in a range of relatively familiar systems such as humans and primates and so on. This principle squares well with the previous principle in cases where the capacity for verbal report is present: availability for verbal report and availability for global control seem to go together in such cases (report is one of the key aspects of control, after all, and it is rare to find information that is reportable but not available more widely). But this principle is also applicable more widely.

A correlation between consciousness and global availability (for short) seems to fit the first-person evidence — the evidence gleaned from our own conscious experience — quite well. When information is present in my consciousness, it is generally reportable, and it can generally be brought to bear in the control of behavior in all sorts of different ways. I can talk about it, I can point in the general direction of a stimulus, I can press bars, and so on. Conversely, when we find information that is directly available in this way for report and other aspects of control, it is generally conscious information. I think one can bear this out by consideration of cases.

There are some interesting puzzle cases to consider, such as the case of blindsight, where one has some kind of availability for control but arguably no conscious experience. Those cases might best be handled by invoking the directness criterion: insofar as the information here is available for report and other control processes at all, it is available only indirectly, by comparison to the direct and automatic availability in standard cases. One might also stipulate that it is availability for voluntary control that is relevant, to deal with certain cases of involuntary unconscious response, although that is a complex issue. I discuss a number of puzzle cases in more detail elsewhere (Chalmers 1996, forthcoming), where I also give a much more detailed defense of the idea that something like global availability is the key pre-empirical criterion for the ascription of consciousness.

But this remains at best a first-order approximation of the functional criteria that come into play. I’m less concerned today to get all the fine details right than to work with the idea that some such functional criterion is required and indeed is implicit in all the empirical research on the neural correlate of consciousness. If you disagree with the criterion I’ve suggested here – presumably because you can think of counterexamples — you may want to use those counterexamples to refine it or to come up with a better criterion of your own. But the point I want to focus on here is that in the very act of experimentally distinguishing conscious from unconscious processes, some such criterion is always at play.

So the question I want to ask is: if something like this is right, then what follows? That is, if some such bridging principles are implicit in the methodology of the search for the NCC, then what are the consequences? I will use global availability as my central functional criterion in the discussion that follows, but many of the points should generalize.

The first thing one can do is produce what philosophers might call a rational reconstruction of the search for the neural correlate of consciousness. With a rational reconstruction we can say, maybe things don’t work exactly like this in practice, but the rational underpinnings of the process have something like this form. That is, if one were to try to justify the conclusions one has reached as well as one can, one’s justification would follow the shape of the rational reconstruction. In this case, a rational reconstruction might look something like this:

(1) Consciousness <-> global availability (bridging principle)
(2) Global availability <-> neural process N (empirical work)

so

(3) Consciousness <-> neural process N (conclusion).

According to this reconstruction, one implicitly embraces some sort of pre-experimental bridging principle that one finds plausible on independent grounds, such as conceptual or phenomenological grounds. Then one does the empirical research. Instead of measuring consciousness directly, we detect the functional property. One sees that when this functional property (e.g. global availability) is present, it is correlated with a certain neural process (e.g. 40-hertz oscillations). Combining the pre-empirical premise and the empirical result, we arrive at the conclusion that this neural process is a candidate for the NCC.

Of course it doesn’t work nearly so simply in practice. The two stages are very intertwined; our pre-experimental principles may themselves be refined as experimental research goes along. Nevertheless I think one can make a separation, at least at the rational level, into pre-empirical and experimental components, for the sake of analysis. So with this sort of rational reconstruction in hand, what sort of conclusions follow? There are about six consequences that I want to draw out here.

(1) The first conclusion is a characterization of the neural correlates of consciousness. If the NCC is arrived at through this sort of methodology, then whatever it turns out to be, it will be a mechanism of global availability. The presence of the NCC wherever global availability is present suggests that it is a mechanism that subserves the process of global availability in the brain. The only alternative that we have to worry about is that it might be a symptom rather than a mechanism of global availability; but that possibility ought to be addressable in principle by dissociation studies, by lesioning, and so on. If a process is a mere symptom of availability, we ought to be able to empirically dissociate it from the process of global availability while leaving the latter intact. The resulting data would suggest to us that consciousness can be present even when the neural process in question is not, thus indicating that it wasn’t a perfect correlate of consciousness after all.

(A related line of reasoning supports the idea that a true NCC must be a mechanism of direct availability for global control. Mechanisms of indirect availability will in principle be dissociable from the empirical evidence for consciousness, for example by directly stimulating the mechanisms of direct availability. The indirect mechanisms will be "screened off" by the direct mechanisms in much the same way as the retina is screened off as an NCC by the visual cortex.)

In fact, if one looks at the various proposals that are out there, this template seems to fit them pretty well. For example, the 40-hertz oscillations discussed by Crick and Koch were put forward precisely because of the role they might play in binding and integrating information into working memory, and working memory is of course a central mechanism whereby information is made available for global control in a cognitive system. Similarly, it is plausible that Libet’s extended neural activity is relevant precisely because the temporal extendedness of activity is what gives certain information the capacity to dominate later processes that lead to control. Baars’ global workspace is a particularly explicit proposal of a mechanism in this direction; it is put forward explicitly as a mechanism whereby information can be globally disseminated. All of these mechanisms and many of the others seem to be candidates for mechanisms of global availability in the brain.

(2) This reconstruction suggests that a full story about the neural processes associated with consciousness will to do two things. Firstly, it will explain global availability in the brain. Once we know all about the relevant neural processes, we will know precisely how information is made directly available for global control in the brain, and this will be an explanation in the full sense. Global availability is a functional property, and as always the problem of explaining the performance of a function is a problem to which mechanistic explanation is well-suited. So we can be confident that in a century or two, global availability will be straightforwardly explained. Secondly, this explanation of availability will do something else: it will isolate the processes that underlie consciousness itself. If the bridging principle is granted, then mechanisms of availability will automatically be correlates of phenomenology in the full sense.

Now, I don’t think this gives us a full explanation of consciousness. One can always raise the question of why it is that these processes of availability should give rise to consciousness in the first place. As yet we have no explanation of why this is, and it may well be that the full details concerning the processes of availability still won’t answer this question. Certainly, nothing in the standard methodology I have outlined answers the question; that methodology assumes a relation between availability and consciousness, and therefore does nothing to explain it. The relationship between the two is instead taken as something of a primitive. So the hard problem still remains. But who knows: somewhere along the line we may be led to the relevant insights that show why the link is there, and the hard problem may then be solved. In any case, whether or not we have solved the hard problem, we may nevertheless have isolated the basis of consciousness in the brain. We just have to keep in mind the distinction between correlation and explanation.

(3) Given this paradigm, it is likely that there are going to be many different neural correlates of consciousness. I take it that this is not going to surprise many people; but the rational reconstruction gives us a way of seeing just why such a multiplicity of correlates should exist. There will be many neural correlates of consciousness because there may well be many different mechanisms of global availability. There will be mechanisms of availability in different modalities: the mechanisms of visual availability may be quite different from the mechanisms of auditory availability, for example. (Of course they may be the same, in that we could find a later area that integrates and disseminates all this information, but that’s an open question.) There will also be mechanisms at different stages of the processing path whereby information is made globally available: early mechanisms and later ones. So these may all be candidates for the NCC. And there will be mechanisms at many different levels of description: for example, 40-hertz oscillations may well be redescribed as high-quality representations, or as part of a global workspace, at a different level of description. So it may turn out that a number of the animals in the zoo, so to speak, can co-exist, because they are compatible in one of these ways.

I won’t speculate much further on just what the neural correlates of consciousness are. No doubt some of the ideas in the initial list will prove to be entirely off-track, while some of the others will prove closer to the mark. As we philosophers like to say, humbly, that’s an empirical question. But I hope the conceptual issues are becoming clearer.

(4) This way of thinking about things allows one to make sense of a idea that is sometimes floated: that of a consciousness module. Sometimes this notion is disparaged; sometimes it is embraced. But this picture of the methodology in the search for an NCC suggests that it is at least possible that there could turn out to be such a module. What would it take? It would require that there turns out to be some sort of functionally localizable, internally integrated area, through which all global availability runs. It needn’t be anatomically localizable, but to qualify as a module it would need to be localizable in some broader sense. For example, the parts of the module would have to have high-bandwidth communication among themselves, compared to the relatively low-bandwidth communication that they have with other areas. Such a thing could turn out to exist. It doesn’t strike me as especially likely that things will turn out this way; it seems just as likely that there will be multiple independent mechanisms of global availability in the brain, scattered around without any special degree of mutual integration. If that’s so, we will likely say that there doesn’t turn out to be a consciousness module after all. But that’s another one of those empirical questions.

If something like this does turn out to exist in the brain, it would resemble Baars’ conception of a global workspace: a functional area responsible for the integration of information in the brain and for its dissemination to multiple nonconscious specialized processes. In fact I should acknowledge that many of the ideas I’m putting forward here are compatible with things that Baars has been saying for years about the role of global availability in the study of consciousness. Indeed, this way of looking at things suggests that some of his ideas are almost forced on one by the methodology. The special epistemological role of global availability helps explain why the idea of a global workspace provides a useful way of thinking about almost any empirical proposal about consciousness. If NCC’s are identified as such precisely because of their role in global control, then at least on a first approximation, we should expect the global workspace idea to be a natural fit.

(5) We can also apply this picture to a question that has been discussed frequently at this conference: are the neural correlates of visual consciousness to be found in V1, in the extrastriate visual cortex, or elsewhere? If our picture of the methodology is correct, then the answer will presumably depend on which visual area is most directly implicated in global availability.

Crick and Koch have suggested that the visual NCC is not to be found within V1, as V1 does not contain neurons that project to the prefrontal cortex. This reasoning has been criticized by Ned Block for conflating access consciousness and phenomenal consciousness (see Block, this volume); but interestingly, the picture I have developed suggests that it may be good reasoning. The prefrontal cortex is known to be associated with control processes; so if a given area in the visual cortex projects to prefrontal areas, then it may well be a mechanism of direct availability. And if it does not project in this way, it is less likely to be such a mechanism; at best it might be indirectly associated with global availability. Of course there is still plenty of room to raise questions about the empirical details. But the broader point is that for the sort of reasons discussed in (2) above, it is likely that the neural processes involved in explaining access consciousness will simultaneously be involved in a story about the basis of phenomenal consciousness. If something like this is implicit in their reasoning, Crick and Koch might escape the charge of conflation. Of course the reasoning does depend on these somewhat shaky bridging principles, but then all work on the neural correlates of consciousness must appeal to such principles somewhere, so this can’t be held against Crick and Koch in particular.

(6) Sometimes the neural correlate of consciousness is conceived of as the Holy Grail for a theory of consciousness. It will make everything fall into place. For example, once we discover the NCC, then we’ll have a definitive test for consciousness, enabling us to discover consciousness wherever it arises. That is, we might use the neural correlate itself as a sort of consciousness meter. If a system has 40-hertz oscillations (say), then it is conscious; if it has none, then it is not conscious. Or if a thalamocortical system turns out to be the NCC, then a system without that system is unlikely to be conscious. This sort of reasoning is not usually put quite so baldly as this, but I think one finds some version of it quite frequently.

This reasoning can be tempting, but one should not succumb to the temptation. Given the very methodology that comes into play here, there is no way to definitely establish a given NCC as an independent test for consciousness. The primary criterion for consciousness will always remain the functional property we started with: global availability, or verbal report, or whatever. That’s how we discovered the correlations in the first place. 40-hertz oscillations (or whatever) are relevant only because of the role they play in satisfying this criterion. True, in cases where we know that this association between the NCC and the functional property is present, the NCC might itself function as a sort of "signature" of consciousness; but once we dissociate the NCC from the functional property, all bets are off. To take an extreme example, if we have 40-hertz oscillations in a test tube, that almost certainly won’t yield consciousness. But the point applies equally in less extreme cases. Because it was the bridging principles that gave us all the traction in the search for an NCC in the first place, it’s not clear that anything follows in cases where the functional criterion is thrown it away. So there’s no free lunch here: one can’t get something for nothing.

Once one recognizes the central role that pre-experimental assumptions play in the search for the NCC, one realizes that there are some limitations on just what we can expect this search to tell us. Still, whether or not the NCC is the Holy Grail, I hope that I have said enough to make it clear that the quest for it is likely to enhance our understanding considerably. And I hope to have convinced you that there are important ways in which philosophy and neuroscience can come together to help clarify some of the deep problems involved in the study of consciousness.

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Copyright © 1998 MIT Press, from Toward a Science of Consciousness II: The Second Tucson Discussions and Debates. Used with permission.