An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.^[1] The word comes from the Greek ἀξίωμα 'that which is thought worthy or fit,' or 'that which commends itself as evident.'^[2][3] As used in modern logic, an axiom is simply a premise or starting point for reasoning,^[4] and equivalent to what Aristotle calls a definition.^[5] Axioms define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true. Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other (theory and domain dependent) truths. An axiom is defined as a mathematical statement that is accepted as being true without a mathematical proof.^[6]

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

Etymology[link]

The word "axiom" comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

The root meaning of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that some things can be done, e.g. any two points can be joined by a straight line, etc.^[7]

Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid's books Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property".^[8] Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.

Historical development[link]

Early Greeks[link]

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics^{[citation needed]}. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of several sciences^{[citation needed]}, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.

An “axiom”, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that

When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.^[9]

The classical approach is well illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

Postulates

1. It is possible to draw a straight line from any point to any other point.
2. It is possible to extend a line segment continuously in a straight line.
3. It is possible to describe a circle with any center and any radius.
4. It is true that all right angles are equal to one another.
5. ("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

Common notions

1. Things which are equal to the same thing are also equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.

Modern development[link]

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.

Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an “axiom” and a “postulate” disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate we get theories that have meaning in wider contexts, hyperbolic geometry for example. We must simply be prepared to use labels like “line” and “parallel” with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.

When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are “propositions that are regarded as true without proof.” Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and logic itself can be regarded as a branch of mathematics. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics on Cantor's set theory. Here the emergence of Russell's paradox, and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent.

The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. The axiom of choice, a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Other sciences[link]

Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics. In particular, the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.

In 1905, Newton's axioms were replaced by those of Albert Einstein's special relativity, and later on by those of general relativity.

Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr, concerned the interpretation of quantum mechanics. This was in 1935. According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics "hidden variables" to enforce determinism. However, thirty years later, in 1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yielded measurably different results using Einstein's axioms compared to using Bohr's axioms. And it took roughly another twenty years until an experiment of Alain Aspect got results in favour of Bohr's axioms, not Einstein's. (Bohr's axioms are simply: The theory should be probabilistic in the sense of the Copenhagen interpretation.)

As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments.

Regardless, the role of axioms in mathematics and in the above-mentioned sciences is different. In mathematics one neither "proves" nor "disproves" an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments always makes sense, since a falsified physical theory needs modification.

Mathematical logic[link]

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).

Logical axioms[link]

These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.

Examples[link]

Propositional logic[link]

In propositional logic it is common to take as logical axioms all formulae of the following forms, where Failed to parse (Missing texvc executable; please see math/README to configure.): \phi , Failed to parse (Missing texvc executable; please see math/README to configure.): \chi , and Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

can be any formulae of the language and where the included primitive connectives are only "Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

" for negation of the immediately following proposition and "Failed to parse (Missing texvc executable; please see math/README to configure.): \to\, " for implication from antecedent to consequent propositions:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to (\psi \to \phi)

1. Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))

1. Failed to parse (Missing texvc executable; please see math/README to configure.): (\lnot \phi \to \lnot \psi) \to (\psi \to \phi).

Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): A , Failed to parse (Missing texvc executable; please see math/README to configure.): B , and Failed to parse (Missing texvc executable; please see math/README to configure.): C

are propositional variables, then Failed to parse (Missing texvc executable; please see math/README to configure.): A \to (B \to A)
and Failed to parse (Missing texvc executable; please see math/README to configure.): (A \to \lnot B) \to (C \to (A \to \lnot B))
are both instances of axiom schema 1, and hence are axioms.  It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus.  It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens.

Other axiom schemas involving the same or different sets of primitive connectives can be alternatively constructed.^[10]

These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus.^[11]

Mathematical logic[link]

be a first-order language. For each variable Failed to parse (Missing texvc executable; please see math/README to configure.): x\,

This means that, for any variable symbol Failed to parse (Missing texvc executable; please see math/README to configure.): x\,,

the formula Failed to parse (Missing texvc executable; please see math/README to configure.): x = x\,
can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by Failed to parse (Missing texvc executable; please see math/README to configure.): x = x\,
 (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): =\,
 has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.

Another, more interesting example axiom scheme, is that which provides us with what is known as Universal Instantiation:

in a first-order language Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{L}\,

and a term Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!
that is substitutable for Failed to parse (Missing texvc executable; please see math/README to configure.): x\,
in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

Where the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): \phi^x_t

stands for the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,
with the term Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!
substituted for Failed to parse (Missing texvc executable; please see math/README to configure.): x\,

. (See Substitution of variables.) In informal terms, this example allows us to state that, if we know that a certain property Failed to parse (Missing texvc executable; please see math/README to configure.): P\,

holds for every Failed to parse (Missing texvc executable; please see math/README to configure.): x\,
and that Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!
stands for a particular object in our structure, then we should be able to claim Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)\,

. Again, we are claiming that the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x \phi \to \phi^x_t

is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. Actually, these examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization:

in a first-order language Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{L}\,

and a term Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!
that is substitutable for Failed to parse (Missing texvc executable; please see math/README to configure.): x\,
in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

Non-logical axioms[link]

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.^[12]

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought^{[citation needed]} that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.

Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

Examples[link]

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method.

Arithmetic[link]

The Peano axioms are the most widely used axiomatization of first-order arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.^[13]

We have a language Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{L}_{NT} = \{0, S\}\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): 0\,
is a constant symbol and Failed to parse (Missing texvc executable; please see math/README to configure.): S\,
is a unary function and the following axioms:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x. \lnot (Sx = 0)

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x. \forall y. (Sx = Sy \to x = y)

1. Failed to parse (Missing texvc executable; please see math/README to configure.): ((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x)

for any Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{L}_{NT}\,
formula Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\ 
with one free variable.

The standard structure is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{N} = \langle\N, 0, S\rangle\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \N\,
is the set of natural numbers, Failed to parse (Missing texvc executable; please see math/README to configure.): S\,
is the successor function and Failed to parse (Missing texvc executable; please see math/README to configure.): 0\,
is naturally interpreted as the number 0.

Euclidean geometry[link]

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries. If one also removes the second postulate ("a line can be extended indefinitely") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

Real analysis[link]

The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires use of second-order logic. The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

[edit] Role in mathematical logic

Deductive systems and completeness[link]

A deductive system consists, of a set Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda\,

of logical axioms, a set Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma\,
of non-logical axioms, and a set Failed to parse (Missing texvc executable; please see math/README to configure.): \{(\Gamma, \phi)\}\,
of rules of inference.  A desirable property of a deductive system is that it be complete.  A system is said to be complete if, for all formulas Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

,

that is, for any statement that is a logical consequence of Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma\,

there actually exists a deduction of the statement from Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma\,

. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma\,

of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,
such that neither Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,
nor Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot\phi\,
can be proved from the given set of axioms.

There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion[link]

Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view axioms may be any set of formulas, as long as they are not known to be inconsistent.

See also[link]

	Mathematics portal
	Logic portal

• Axiomatic system
• List of axioms
• Model theory
• Regulæ Juris
• Theorem

References[link]

1. ^ "A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, Posterior Analytics I.2.72a18-b4.
2. ^ Cf. axiom, n., etymology. Oxford English Dictionary, accessed 2012-04-28.
3. ^ Oxford American College Dictionary: "n. a statement or proposition that is regarded as being established, accepted, or self-evidently true. ORIGIN: late 15th cent.: ultimately from Greek axiōma ‘what is thought fitting,’ from axios ‘worthy.’ http://www.highbeam.com/doc/1O997-axiom.html - via HighBeam Research, subscription needed. Please see WP:HighBeam.
4. ^ "A proposition (whether true or false)" axiom, n., definition 2. Oxford English Dictionary Online, accessed 2012-04-28.
5. ^ Posterior Analytics I.2.72a23
6. ^ Ted Sundstorm, "Mathematical reasoning: writing and proof," Prentice Hall, 2003, p 71.
7. ^ Wolff, P. Breakthroughs in Mathematics, 1963, New York: New American Library, pp 47–8
8. ^ Heath, T. 1956. The Thirteen Books of Euclid's Elements. New York: Dover. p200
9. ^ Aristotle, Metaphysics Bk IV, Chapter 3, 1005b “Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it.” W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)|date=June 2011
10. ^ Mendelson, "6. Other Axiomatizations" of Ch. 1
11. ^ Mendelson, "3. First-Order Theories" of Ch. 2
12. ^ Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2
13. ^ Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2

• Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. ISBN 0-534-06624-0

External links[link]

Look up axiom or given in Wiktionary, the free dictionary.

Wikisource has the text of the 1911 Encyclopædia Britannica article Axiom.

• Axiom at PlanetMath
• Metamath axioms page

Little Ray also known as Ray Jimenez, (b. - ) was an East Los Angeles, Chicano rock and brown-eyed soul musician, prominent in the 1960s who sang up tempo Rhythm and Blues. Little Ray was born in Delano, California. His first record was “There’s Something On Your Mind.”

Little Ray sang briefly with Thee Midniters, another East Los Angeles band. He then sang with Little Ray & the Progressions With Paul C Saenz, on Guitar, which became the Little Ray Revue.

External links[link]

• Little Ray - East L.A. Superstar of the 60s
• A Pioneer of the East L.A. Sound Lil' Ray Jimenez, Molina & Firme
• CD Nuestra Navidad

This article relating to Mexican-Americans or Chicanos is a stub. You can help Wikipedia by expanding it. v t e

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2008)

Bill Laswell
Bill Laswell at Moers Festival 2006, Germany
Background information
Born	(1955-02-12) February 12, 1955 (age 57) Salem, Illinois, U.S.
Genres	Avant-garde, art rock, ambient, dub, electronic, experimental, punk jazz, industrial hip-hop
Occupations	Musician, producer, arranger
Instruments	Bass, guitar, keyboard
Associated acts	Method of Defiance, The Golden Palominos, Praxis, Massacre, Material, Buckethead, Painkiller, Ashes, Tabla Beat Science

Bill Laswell (born February 12, 1955 in Salem, Illinois and raised in Albion, Michigan) is an American bassist, producer and record label owner.

Laswell ranks among the most prolific of modern musicians, being involved in hundreds of recordings with many collaborators from all over the world. Laswell's music draws upon many different genres, most notably funk, various world music, jazz, dub and ambient styles. He has also played or produced music from the noisier, more aggressive end of the rock spectrum, such as hardcore punk and metal.

According to music critic Chris Brazier, "Laswell’s pet concept is 'collision music' which involves bringing together musicians from wildly divergent but complementary spheres and seeing what comes out."^[1] The credo of one record label run by Laswell, and which typifies much of his work, is “Nothing Is True, Everything Is Permitted”. Though projects arranged by Laswell may be credited under the same name and often feature the same roster of musicians, the styles and themes explored on different albums can vary dramatically: Material began as a noisy dance music project, but subsequent releases have been centered around hip hop, jazz, or backing spoken word readings by beat generation icon William S. Burroughs. Similarly, most versions of Praxis have featured guitarist Buckethead, but have explored different permutations with each new album.

Though some artists have chafed against Laswell's distinctive recording and production style (most noticeably some of his for-hire production gigs including Motörhead, Swans and White Zombie) many other collaborations, such as with pianist Herbie Hancock and singer Iggy Pop have been lengthier and recurring. He has also worked extensively with his wife, Ethiopian singer Gigi.

Biography[link]

Beginnings[link]

Though starting out as a guitar player, he soon switched to bass. Laswell got his earliest professional experience as a bassist with funk groups in and around Detroit, Michigan as well as Ann Arbor. He often would see shows in Detroit that put together acts such as Iggy and the Stooges (he would work with Pop throughout his career starting in the mid ‘80s), MC5 and Funkadelic (many of whose members are part of his stable of musicians).

Seeing these differing styles of music in his frequent trips to Detroit, as well as being rooted in the African-American music that he grew up immersed in have clearly had an influence on Laswell’s music. His exposure to jazz musicians like John Coltrane, Albert Ayler and particularly Miles Davis’ electric experiments of the mid-1960s to mid-'70s, have also clearly had an impact on his thinking. Laswell’s refusal to pigeon-hole himself, his music, or even the people he works with is arguably his greatest asset as a musician and producer.

Move to New York[link]

In the late 1970s Laswell made the move to New York city, immersing himself in the thriving New York scene. He moved into famed producer Giorgio Gomelsky’s loft and became part of a group of musicians that would eventually become the first (and only even remotely consistent) incarnation of Material.

Aside from Laswell’s first known recording on one side of a Michael Blaise and the Cheaters 7” called Scoring Power in 1978, Laswell and Material became the backing band for Daevid Allen and New York Gong, appearing on some recordings and embarking on a small tour. Material, primarily consisting of Laswell, keyboardist Michael Beinhorn and drummer Fred Maher, also cut a number of 12” releases for Red Records and others. They were usually supplemented by guitarists, notably either Cliff Cultreri and occasionally Robert Quine. Living in the East Village also put Laswell at the center of a group of musicians both up and coming such as John Zorn and established, such as Fred Frith and Brian Eno. His persistence in asking Eno to work with him paid off in the form of contributions to Eno and David Byrne’s album My Life in the Bush of Ghosts as well as Eno’s own On Land. Brian Eno also contributed a song to the Material album, One Down.

Celluloid Records[link]

Within a few years of moving to New York, Laswell soon founded a recording studio with producer/engineer Martin Bisi (of later indie rock renown) and hooked up with Jean Karakos and his fledgling label Celluloid Records. Under the Material moniker (now also a production unit consisting of Laswell and Beinhorn – Maher being long gone - and by 1984 consisting solely of Laswell) Laswell became the de facto house producer for Celluloid until the sale of the label in the later ‘80s. During this fruitful time in the early to mid 80s, Laswell was able to record some of his Material excursions (which ran the gamut from experimental jazz/funk to pop and R&B, featuring everyone from avant-jazz figures Henry Threadgill and Sonny Sharrock to Archie Shepp and pop star Whitney Houston) as well as projects such as Massacre, with Fred Frith and Fred Maher.

His association with Celluloid allowed some of his first forays into this so-called ‘collision music' - the term was coined for Laswell by the British writer Chris May, then editor of Black Music & Jazz Review and later a Celluloid staff member - and forays into world music. Recordings with The Golden Palominos and production on albums by Shango, Toure Kunda and Fela Kuti all appeared on the label. Celluloid also released a slew of 12” devoted to Hip-Hop, becoming a pre-cursor to the popularity the form enjoyed starting in the mid 80s. Fab 5 Freddy, Phase II and Afrika Bambaataa all appeared on the label. Criminally forgotten, Laswell also put together the very successful 12” World Destruction which paired PiL’s John Lydon with Afrika Bambaataa – years before the Run DMC/Aerosmith collaboration broke down the rock/hip-hop barrier. 1982 also saw Laswell’s solo debut, Baselines.

Also recording a Laswell-helmed solo album for Celluloid was Ginger Baker whom Laswell coaxed out of semi-retirement, giving the drummer's career a new boost. He likewise brought Sonny Sharrock out of semi-retirement and produced some of the guitarist's most acclaimed recordings starting with the solo LP Guitar.

Breakthrough[link]

Laswell's artistic and commercial breakthrough came via jazz icon Herbie Hancock's Future Shock album (1983); Laswell produced the album, played bass on all the songs, and co-wrote most of the material. Its track "Rockit" has frequently been regarded as a pivotal moment in the influence of hip hop and turntablism (via Grand Mixer D.ST). The track was the first hit song to feature turntable scratching. The collaboration has led to three other albums by Herbie Hancock, as well as numerous Hancock appearances on Laswell productions through the early 2000s.

Post-Celluloid 1980s[link]

Concurrent to and post-Celluloid, Laswell became a hot producer in demand, due to the success of Hancock's "Rockit". The often lucrative pay-to-produce nature of some of these projects helped fund much of Laswell's work.

The remainder of the 80’s saw Laswell produce albums for people like Sly & Robbie (whom Laswell continues to work with) Mick Jagger, PiL, Motörhead, The Ramones, Iggy Pop and Yoko Ono. Many of these projects afforded Laswell the opportunity to bring in some of his normal working crew to record on more mainstream records. PiL's 1986 release Album (later CD) has no notes on who performed, but over time, various people have confirmed that no PiL personnel other than singer John Lydon were involved, some of the musicians included drummers Tony Williams and Ginger Baker, bassists Jonas Hellborg, Laswell himself, guitarist Steve Vai and others. Lydon claims that Miles Davis actually recorded parts for the album which were never used.

Laswell has stated in numerous interviews that he met with Davis a number of times and discussed working together, but busy schedules kept them from arranging such a recording before Davis’ death, though Laswell's chief engineer^[2] reports an unreleased Davis recording session from 1986.

1986 saw the formation of Last Exit. Laswell and Sonny Sharrock co-founded the metal and hardcore punk-flavored free jazz supergroup along with drummer Ronald Shannon Jackson and saxophone player Peter Brötzmann. Aside from one album that Laswell cobbled together in-studio, the band was primarily a live one. The group showed up at gigs and played wild sets with no rehearsal. The first time the four members played together was on stage at their first show.

The later part of the ‘80s also saw Laswell completely sever ties with the Celluloid label, which has since been sold several times: the catalog’s various releases seem to be in constant reissue on one label or another. Many of the labels are known for poor practice in securing rights to recordings and are often rumored to not be paying royalties to anyone other than whomever is licensing the material to them.

Greenpoint Studios, Axiom Records and the 1990s[link]

1990 marked a watershed year in Laswell’s control and ability to produce high-quality recordings controlled by himself. In addition to purchasing his own studio (Greenpoint Studio in Brooklyn), Chris Blackwell, founder of Island Records and longtime Laswell booster, gave Laswell the opportunity to begin a new label with the backing of Island Records. Thus, Axiom Records was born.

Axiom played the ‘Nothing Is True, Everything Is Permitted’ credo to its fullest. With a sizable budget and minimal interference from Island executives, Laswell had the means to make arguably some of the most important music of his career. In addition to albums by Material that featured players ranging from Sly & Robbie, William S. Burroughs, Wayne Shorter, Bootsy Collins and Bernie Worrell, he produced and released albums by drummer and Ornette Coleman acolyte Ronald Shannon Jackson, Sonny Sharrock (featuring Pharoah Sanders and Elvin Jones), Laswell main-stay Nicky Skopelitis, Last Poets' Umar Bin Hassan and Ginger Baker.

Axiom also released a slew of well-produced recordings from musicians around the world. Among the studio based albums, Palestinian oud and violin prodigy Simon Shaheen recorded an album of music by Egyptian composer Mohammed Abdel Wahab. Gambian virtuoso Foday Musa Suso recorded an album of futuristic dance music featuring his electric Kora and Turkish saz master Talip Özkan recorded an album. The real coup was in the series of pristine field recordings that Axiom allowed Laswell the ability to produce. A major-league budget and new, more portable recording technology gave rise to recordings by the Master Musicians of Jajouka (done in their village in the Rif Mountains), Mandinka and Fulani music (recorded at Suso’s family compound in the Gambia) and Gnawa music from Morocco.

The most successful project and one of the few still in print on Axiom – where the first release was produced, was Praxis. Originally the moniker that an experimental Celluloid 12” by Laswell was released under in 1984, Praxis now became a full-fledged band, featuring enigmatic guitarist Buckethead. The release, Transmutation (Mutatis Mutandis) featured Buckethead, drummer Brain (whom Laswell worked with previously with the Limbomaniacs), Bernie Worrell, Bootsy Collins and Af Next Man Flip (Afrika Baby Bam from the Jungle Brothers). The album was a racous blend of funk grooves and metal riffs, overseen with many tracks co-written by Laswell. The project has spawned other releases, never with the same line-up twice, generally consisting of the core trio of Buckethead, Brain and Laswell supplemented by others.

1994/1995 saw a bit of a slow-down in Axiom’s output, but a number of genre-shattering 2CD compilation sets were released. Axiom Funk’s Funkcronomicon saw previously released tracks by Praxis and Nicky Skopelitis paired with a host of tracks mainly featuring various members of the Parliament/Funkadelic crew. George Clinton, Bootsy Collins, Bernie Worrell and the last recordings of Eddie Hazel are featured prominently. The album also features contributions from Last Poets' Umar Bin Hassan and Abiodun Oyewole as well as Torture (now Sensational) and DXT (formerly D.ST). Axiom Dub was another compilation featuring tracks in a new-skool dub style from Laswell along with The Orb, Jah Wobble, Sly & Robbie, Mad Professor, Techno Animal, the WordSound crew, WE and others.^[3] Laswell also remixed the whole of the Axiom catalog into a 2 disc ambient mix called Axiom Ambient, subtley blending seemingly disparate tracks from the catalog into a seamless in the mix translation. Laswell released some of the music recorded in those sessions as a sample library for other musicians to use as raw material when making recordings, on a CD he titled Sample Material - International Free Zone.^[4][5]

The 1990s also saw a number of other labels owned by or thoroughly associated with Laswell, come and go. The most prolific of these was Subharmonic, conceived by Laswell and ex-Celluloid A&R Robert Soares. Though not owned by Laswell, the label was essentially a release house for his projects, most of which fell into the ambient or ambient-dub categories. The label also licensed a few releases from European labels for American re-release, notably Psychonavigation (with Pete Namlook) and Cymatic Scan (with Tetsu Inoue) from Pete Namlook's FAX label, and Somnific Flux (with Mick Harris – there as MJ Harris) and Cold Summer (by Lull – a Mick Harris project) from the Sentrax label. Other collaborators included Jonah Sharp and Terre Thaemlitz. The label also released albums from Painkiller, Praxis and Laswell’s new project, Divination, an ambient dub project (first appearing almost as a project title, and then an umbrella moniker for releases of ambient compilations). Additionally, a sub-label called Strata was created containing five releases mostly in what could be deemed a more experimental dub/noise/ambient vein. Each of these releases (Death Cube K, Cypher 7, Azonic and two under his alias Automaton) came housed in a solid black jewel case with the name of the project and album title printed on the front.

Three other short-lived labels were also created around the time of the demise of the Subharmonic deal. Meta, which was intended to be a spoken word label, and SubMeta. Submeta managed four releases before folding. Meta (co-created by Janet Rienstra) released only one album, Baptism of Solitude, - of Paul Bowles reading excerpts from his work over soundscapes by Laswell. Meta would appear periodically again, distributed by other labels, over the next few years until it came back in full as its own entity as a spiritual/yogic label run by Janet Rienstra, though Laswell still heavily figures in Meta’s output. Black Arc was also created as an associated label of Rykodisc, focusing on ‘Black Rock, Cyber Funk and Future Blues’, according to a released sampler. The label featured a number of P-Funk alumni on most of the albums, as well as releasing albums by Parliament/Funkadelic members Bootsy Collins (under his Zillatron moniker), Bernie Worrell (Japan-only), Mutiny (Jerome Brailey) and Billy Bass.

Late 1990s[link]

Always one to be courting controversy due to his alleged radical treatment of music, Laswell released two albums of remixes from dead artists – Bob Marley's Dreams of Freedom on Axiom and Miles Davis’ Panthalassa. The first contained airy, ambient dub translations of some of Marley’s Island catalog, largely sans Marley’s voice. Chris Blackwell, largely the man responsible for bringing Marley to the masses in the ‘70s requested the album as part of a planned series of remix albums by various producers who were rooted in the reggae/dub tradition. Blackwell’s departure from Island killed any further albums.

For Panthalassa, Laswell took the tapes from Miles’ ‘electric period’ and re-imagined them. The impetus for the project being that the original releases were just mixes made by Teo Macero from long in studio sessions. Nothing originally released was necessarily exactly what was done in the studio, but rather a cut-up and remix to begin with. Needless to say, critic and fan responses varied wildly with Laswell and Macero conducting a public feud in the media.

The late ‘90s saw two other major changes. As noted before, Chris Blackwell left Island Records. Although he took the Axiom imprint with him to his new Palm Pictures label, the back catalog stayed with Island. Many of the albums are now out of print, efforts to obtain master recordings and new distribution has been unsuccessful. The other change came in the form of studio space. Laswell, seeing that Greenpoint had turned into a sort living space for hanger-ons moved his studio to West Orange, New Jersey, calling it Orange Music Sound Studios.

21st century[link]

With Palm Pictures slowly moving into film and away from music with the changing landscape of the industry, Laswell lost a major supporter of his more high-concept albums as well as the Axiom imprint. Under Palm’s umbrella, though, four highly regarded albums and a DVD set were released. Of those releases there was a DVD set, a studio release and a live 2-disc set from Tabla Beat Science. Tabla Beat Science is a project that revolves somewhat around the tabla virtuoso Zakir Hussain, son of the late Alla Rakha. The studio release also featured Karsh Kale, Trilok Gurtu, Ustad Sultan Khan and Talvin Singh. This very popular and well received grouping has become a primarily live project playing everywhere from the US to Beirut to Japan over the years. The core of Laswell, Kale, Kahn and Hussain are usually supplemented by other musicians, which have included at various times Gigi, DJ Disk, Serj Tankian from System of a Down, Sussan Deyhim, visual artists Petulia Mattioli and others. 2001 saw the release of the album Life Space Death with Japanese trumpeter Toshinori Kondo, Laswell on bass, guitar and keyboards and words by The 14th Dalai Lama,^[6] interviewed by Kondo.^[7]

At the request of Chris Blackwell, 2001 also had Laswell overseeing Ethiopian singer Gigi’s debut release for Palm Pictures. Supplementing Gigi’s multilingual, Ethiopian rooted vocals with a vast array of well respected musicians such as Wayne Shorter, Herbie Hancock and Laswell himself, they created a strong release that was very well received. Laswell and Gigi also became romantically involved and were later married. She has figured in a number of his releases and concerts over the years, and he has produced further outings by Gigi such as her Abyssinia Infinite grouping and her second solo release for Palm, Gold & Wax.

1999 also saw the first release on Laswell’s new label, called Innerythmic (Eraldo Bernocchi and Toshinori Kondo’s Charged project). After a brief inactive period, the label re-started again in earnest in 2001, releasing over the next few years a slew of innovative albums from the likes Nicky Skopelitis/Raoul Björkenheim, James Blood Ulmer, Shin Terai and Gonervill among others. Innerhythmic also released a live Praxis recording and re-issued some of the Black Arc releases from the ‘90s including Zillatron, The Last Poets Holy Terror and Buddy Miles’ Hell & Back. The label tends to go through extended periods of inactivity but still releases albums here and there, the next planned release being a studio recording featuring Raoul Björkenheim, Bill Laswell and Morgan Ågren anticipated in the summer of 2011 according to Björkenheim's website.^[8]

Though touching on the realm of drum and bass in the 1990s with his Oscillations releases and the compilation Tetragramaton: Submerged - conceived along with Soares - the last few years have seen Laswell step up his work in this area with Soares as a collaborator in concept/A&R for drum & bass artist development. Starting with Brutal Calling, a hard drum 'n' bass release with Ohm Resistance label owner Submerged (Kurt Gluck), a series of releases and live dates have cropped up. Laswell’s new project in this vein is Method of Defiance (the actual Method of Defiance name and the original idea of a futurist, cyborg drum 'n' bass driven group consisting of laptops breakbeat artists, electric instrumentalists and trumpeter Graham Haynes was conceived by Soares). The first release focused on the core of Laswell and Submerged once again, entitled "The only way to go is down", featuring photos of Soares, with contributions from Toshinori Kondo and drummer Guy Licata. The second release under the moniker, though, was more of a compilation style project, though still focusing on drum 'n' bass. Inamorata stretched the concept out, pairing Laswell’s bass with a different combination of respected jazz and world musicians and drum 'n' bass producers linked to Soares on each track. Artists including Herbie Hancock, John Zorn, Pharoah Sanders, Nils Petter Molvaer, Toshinori Kondo and Buckethead were paired with drum 'n' bass producers including AMIT, Paradox, Submerged, Future Prophecies, Karsh Kale, Evol Intent, SPL, Outrage, Fanu, and Corrupt Souls. To that end, Laswell’s last collaboration with Soares was a full-on recording with Finnish drum 'n' bass maestro Fanu on Ohm Resistance and Karl Records, entitled Lodge, which includes contributions from Molvaer and Bernie Worrell amongst others. After collaborating with Laswell for 15 years, Soares left the crew in 2008. The concept of the group has once again morphed into a full band concept. In 2009, Rare Noise Records released Live in Nihon, which showcased this new direction/grouping. The group now consisted of Laswell, Guy Licata, Dr. Israel, Toshinori Kondo and Bernie Worrell.

Along with frequent live dates around the world with Method of Defiance, Material, Painkiller and the reformed in the late ‘90s Massacre (with This Heat's Charles Hayward now in the drum chair) Laswell still makes numerous trips to Japan each year for various recordings and live dates, including his ongoing Tokyo Rotation^[9] mini-festivals at the Shinjuku Pit-Inn, which is now a yearly occurrence. In addition to Tokyo Rotation being an actual mini-festival, the moniker has been used as a sort of umbrella to include general operations revolving around activities in Japan, as witnessed by the usage of Tokyo Rotation Presents in relation to the website's announcement of other gigs outside the Pit Inn shows as well being noted on Method of Defiance's release Nihon.

M.O.D. Technologies[link]

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In 2010, Laswell created his newest label, M.O.D. Technologies, in partnership with Giacomo Bruzzo of Rare Noise and John Brown, his longtime Manager. The label is centered around the activities of the principles of the now-solidified Method of Defiance lineup, and is intended to be one of Laswell's main creative vehicles moving forward, being seen as the successor to his Axiom label. The label kicked off with the release of three albums toward the end of 2010: two from Method of Defiance - Jahbulon (a reggae/dancehall/vocal affair featuring Hawk and Dr. Israel) and the instrumental album Incunabula, which features Herbie Hancock on one track and DJ Krush throughout. The third release was a new live album from Gigi w/Material entitled Mesgana Ethiopia. In 2011, the label released an album by Praxis (a US/European version of the Profanation album, with bonus tracks), as well as a new collaboration with Lee 'Scratch' Perry featuring Gigi, Dr. Israel, Hawk, Sly Dunbar, Bernie Worrell, TV on the Radio's Tunde Adebimpe and more, in May in the US. A remix album focusing on Method of Defiance tracks from Jahbulon and Incunabula, with new mixes by Mad Professor, Laswell, Scientist and more, came out in October. 2012 will see a further movement of many of Laswell's projects to the M.O.D. Technologies umbrella. In addition to Garrison Hawk's album "Survive" released on April 17, 2012 will see the first in an intended batch of releases that focus on Ethiopian music and artists. The first project is Jano, a young rock band that was the first band recorded in Addis Ababa at the new studio that Laswell built.

Frequent collaborators[link]

Even though many Laswell-produced albums have featured dozens of musicians, he tends to work with a small group of collaborators who appear on most of his recordings. Such musicians include bassists Jah Wobble, Jonas Hellborg and Bootsy Collins; guitarists Buckethead and Nicky Skopelitis; keyboardists Jeff Bova and Bernie Worrell; and percussionists Aïyb Dieng and Karsh Kale. Laswell has also frequently worked with musicians from the sprawling P-funk camp.

In addition, Laswell has relied on the expertise of a small number of engineers over the years. Robert Musso (a producer, musician and label-owner in his own right) has been Laswell's chief engineer for close to 25 years. Oz Fritz has occasionally filled the role as well over almost the same time period, though (particularly in the last few years after a move to the West Coast) Fritz is usually Laswell's live engineer of choice, known for his stellar live mixing technique. In addition, a small core of assistants have come through over time, the most recent mainstay being James Dellatacoma.

Over the years, Laswell has also been an in demand remixer and purveyor of what is usually noted as ‘mix translation’. Remixes (released and unreleased) have been done for artists including Sting, Nine Inch Nails, Almamegretta, Scorn, Ozzy Osbourne, Tori Amos. In addition he is often hired for his skills at the board doing straight mixes of albums. In recent years he has done much work in this area for various projects on John Zorn's Tzadik record label.

In 2005, Laswell was invited to appear on the PBS series Soundstage. The show featured a host of the musicians he has played with over the years including incarnations of his Praxis and Tabla Beat Science projects. In addition to some of the core performers from these projects, Pharoah Sanders, Foday Musa Suso, Bootsy Collins and Catfish Collins and many others participated. Though Laswell mixed the show in 5.1 surround sound, to date no DVD or official recording has been released. The hour-long aired version (part of a much longer show) has popped up on file-sharing sites.

Laswell has also participated on a collaboration with Sony Creative Software on a box set Loop Library called The Bill Laswell Collection.^[10]

See also[link]

• Axiom (record label)

References[link]

External links[link]

• Bill Laswell on Myspace
• M.O.D. Technologies on Facebook
• M.O.D. Technologies on Myspace
• Official M.O.D. Technologies site
• Official Method of Defiance site
• Official Tokyo Rotation site
• Innerhythmic.com (record label)
• discography at silent-watcher.net
• discography at turning-groove.de
• Sacred Dub fan site/community site, includes podcasts
• Bill Laswell discography at Discogs
• Last.FM: Bill Laswell pages
• 2011 Innerviews interview
• 1999 Innerviews interview
• Method of Defiance "Inamorata" review by Mark Jenkins, The Washington Post, December 21, 2007
• Bill Laswell Interview by The Wire Magazine. December, 1994
• Part 1 of Interview with Directors of Destroy All Rational Thought DVD - which celebrates the work of Brion Gysin and William Burroughs. Contains music/text by Bill Laswell
• Part 2 of Interview with Directors of Destroy All Rational Thought DVD - which celebrates the work of Brion Gysin and William Burroughs. Contains music/text by Bill Laswell