The Pythagorean theorem has at least 370 known proofs^[1]

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.^[2]

Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.

Informal accounts of theorems[link]

Logically, many theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion (A and B can also be denoted the antecedent and consequent). The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that "n is an even natural number" and the conclusion is that "n/2 is also a natural number".

In order to be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader will be able to produce a formal statement from the informal one.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then declare that the theory consists of all theorems provable using those hypotheses as assumptions. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.

Some theorems are "trivial," in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.^[3] A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.

There are other theorems for which a proof is known, but the proof cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search which is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted in recent years. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved.^[4] Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.^{[5][page needed]}

Relation to proof[link]

The notion of a theorem is deeply intertwined with the concept of proof. Indeed, theorems are true precisely in the sense that they possess proofs. Therefore, to establish a mathematical statement as a theorem, the existence of a line of reasoning from axioms in the system (and other, already established theorems) to the given statement must be demonstrated.

Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem. And even though more than one proof may be known for a single theorem, only one proof is required to establish the theorem's validity. The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.

Theorems in logic[link]

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Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. The statements of the language are strings of symbols and may be broadly divided into nonsense and well-formed formulas. A set of deduction rules, also called transformation rules or rules of inference, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises. The set of well-formed formulas may be broadly divided into theorems and non-theorems. However, according to Hofstadter, a formal system will often simply define all of its well-formed formula as theorems.^{[6][page needed]}

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus.

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.

Relation with scientific theories[link]

Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set.

Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatz conjecture has been verified for start values up to about 2.88 × 10¹⁸. The Riemann hypothesis has been verified for the first 10 trillion zeroes of the zeta function. Neither of these statements is considered to be proven.

Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 10¹⁴ have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1.59 × 10⁴⁰, which is approximately 10 to the power 4.3 × 10³⁹. Since the number of particles in the universe is generally considered to be less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search.

Note that the word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory. There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.

Terminology[link]

A number of different terms for mathematical statements exist, these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary and the usage of some terms has evolved over time.

• An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject. Historically these have been regarded as "self evident", but more recently they are considered assumptions that characterize the subject of study. In classical geometry, axioms are general statements while postulates are statements about geometrical objects.^[7] A definition is also accepted without proof since it simply gives the meaning of a word or phrase in terms of known concepts.
• A proposition is a generic term for a theorem of no particular importance. This term sometimes connotes a statement with a simple proof, while the term theorem is usually reserved for the most important results or those with long or difficult proofs. In classical geometry, a proposition may be a construction that satisfies given requirements; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.^[8]
• A lemma is a "helping theorem", a proposition with little applicability except that it forms part of the proof of a larger theorem. In some cases, as the relative importance of different theorems becomes more clear, what was once considered a lemma is now considered a theorem, though the word "lemma" remains in the name. Examples include Gauss's lemma and Zorn's lemma.
• A corollary is a proposition that follows with little or no proof from one other theorem or definition.^[9]
• A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved. For example, the isosceles triangle theorem states that if two sides of a triangle are equal then two angles are equal. In the converse, the given (that two sides are equal) and what is to be proved (that two angles are equal) are swapped, so the converse is the statement that if two angles of a triangle are equal then two sides are equal. In this example, the converse can be proven as another theorem, but this is often not the case. For example, the converse to the theorem that two right angles are equal angles is the statement that two equal angles must be right angles, and this is clearly not always the case.^[10]

There are other terms, less commonly used, which are conventionally attached to proven statements, so that certain theorems are referred to by historical or customary names. For examples:

• Identity, used for theorems which state an equality between two mathematical expressions. Examples include Euler's identity and Vandermonde's identity.
• Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.^[11]
• Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.

A few well-known theorems have even more idiosyncratic names. The division algorithm is a theorem expressing the outcome of division in the natural numbers and more general rings. The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space.

An unproven statement that is believed to be true is called a conjecture (or sometimes a hypothesis, but with a different meaning from the one discussed above). To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. On the other hand, Fermat's last theorem has always been known by that name, even before it was proven; it was never known as "Fermat's conjecture".

Layout[link]

A theorem and its proof are typically laid out as follows:

Theorem (name of person who proved it and year of discovery, proof or publication).

Statement of theorem (sometimes called the proposition).

Proof.

Description of proof.

End mark.

The end of the proof may be signalled by the letters Q.E.D. meaning "quod erat demonstrandum" or by one of the tombstone marks "□" or "∎" meaning "End of Proof", introduced by Paul Halmos following their usage in magazine articles.

The exact style will depend on the author or publication. Many publications provide instructions or macros for typesetting in the house style.

It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem.

Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes corollaries have proofs of their own which explain why they follow from the theorem.

Lore[link]

It has been estimated that over a quarter of a million theorems are proved every year.^[12]

The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.^[13]

The classification of finite simple groups is regarded by some to be the longest proof of a theorem; it comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and there are several ongoing projects to shorten and simplify this proof.^[14] Another theorem of this type is the Four color theorem whose computer generated proof is too long to be read by a human. It is certainly the longest proof of a theorem whose statement can be easily understood by a layman.

Formalized account of theorems[link]

A theorem may be expressed in a formal language (or "formalized"). A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash

Failed to parse (Missing texvc executable; please see math/README to configure.): S
is often used to indicate that Failed to parse (Missing texvc executable; please see math/README to configure.): S
is a theorem.

Formal theorems consist of formulas of a formal language and the transformation rules of a formal system. Specifically, a formal theorem is always the last formula of a derivation in some formal system each formula of which is a logical consequence of the formulas which came before it in the derivation. The initially accepted formulas in the derivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called a theory.

What makes formal theorems useful and of interest is that they can be interpreted as true propositions and their derivations may be interpreted as a proof of the truth of the resulting expression. A set of formal theorems may be referred to as a formal theory. A theorem whose interpretation is a true statement about a formal system is called a metatheorem.

Syntax and semantics[link]

The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a "true proposition" in which semantics are introduced. Different deductive systems may be constructed so as to yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends on whether or not all of its theorems are also validities. A validity is a formula that is true under any possible interpretation, e.g. in classical propositional logic validities are tautologies. A formal system is considered semantically complete when all of its tautologies are also theorems.

Derivation of a theorem[link]

The notion of a theorem is very closely connected to its formal proof (also called a "derivation"). To illustrate how derivations are done, we will work in a very simplified formal system. Let us call ours Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

Its alphabet consists only of two symbols { A, B } and its formation rule for formulas is:

Any string of symbols of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

which is at least 3 symbols long, and which is not infinitely long, is a formula. Nothing else is a formula.

The single axiom of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

is:

ABBA

The only rule of inference (transformation rule) for Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

is:

Any occurrence of "A" in a theorem may be replaced by an occurrence of the string "AB" and the result is a theorem.

Theorems in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

are defined as those formulae which have a derivation ending with that formula. For example

1. ABBA (Given as axiom)
2. ABBBA (by applying the transformation rule)
3. ABBBAB (by applying the transformation rule)

is a derivation. Therefore "ABBBAB" is a theorem of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS} \,.

The notion of truth (or falsity) cannot be applied to the formula "ABBBAB" until an interpretation is given to its symbols. Thus in this example, the formula does not yet represent a proposition, but is merely an empty abstraction.

Two metatheorems of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

are:

Every theorem begins with "A".

Every theorem has exactly two "A"s.

Interpretation of a formal theorem[link]

Theorems and theories[link]

See also[link]

Logic portal

• Inference
• List of theorems
• Toy theorem
• Metamath - a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems

Notes[link]

1. ^ For full text of 2nd edition of 1940, see Elisha Scott Loomis. "The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs". Education Resources Information Center. Institute of Education Sciences (IES) of the U.S. Department of Education. http://www.eric.ed.gov/PDFS/ED037335.pdf. Retrieved 2010-09-26.  Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics.
2. ^ However, both theorems and theories are investigations. See Heath 1897 Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate"
3. ^ Weisstein, Eric W., "Deep Theorem" from MathWorld.
4. ^ Doron Zeilberger. "Opinion 51". http://www.math.rutgers.edu/~zeilberg/Opinion51.html. 
5. ^ Petkovsek et al. 1996.
6. ^ Hofstadter 1980
7. ^ Wentworth, G.; Smith, D.E. (1913). "Art. 46, 47". Plane Geometry. Ginn & Co.. 
8. ^ Wentworth & Smith Art. 50
9. ^ Wentworth & Smith Art. 51
10. ^ Follows Wentworth & Smith Art. 79
11. ^ The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
12. ^ Hoffman 1998, p. 204.
13. ^ Hoffman 1998, p. 7.
14. ^ An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.

References[link]

• Heath, Sir Thomas Little (1897). The works of Archimedes. Dover. http://books.google.com/books?id=FIY5AAAAMAAJ&dq=works+of+archimedes&printsec=frontcover&source=bl&ots=PRfqkNAFl-&sig=vRhyLwEr-wNV-bGsOB9eKas-63E&hl=en&ei=Smn_SuL6FZPElAfmj7SSCw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBAQ6AEwAg#v=onepage&q=theorem&f=false. Retrieved 2009-11-15. 
• Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. ISBN 1-85702-829-5. 
• Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. 
• Hunter, Geofrfrey (1996) [1973]. Metalogic: An Introduction to the Metatheory of Standard First Order Logic. University of California Press. ISBN 0-520-02356-0. 
• Mates, Benson (1972). Elementary Logic. Oxford University Press. ISBN 0-19-501491-X. 
• Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). A = B. A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6. http://www.cis.upenn.edu/~wilf/AeqB.html. 

External links[link]

Look up theorem in Wiktionary, the free dictionary.

• Weisstein, Eric W., "Theorem" from MathWorld.
• Theorem of the Day

Richard Cevantis Carrier (born December 1, 1969) is an American historian. He is best known for his writings on Internet Infidels, otherwise known as the Secular Web, where he served as Editor-in-Chief for several years.^[2]

As an advocate of atheism and metaphysical naturalism, he has published articles in books, journals and magazines, and also features on the documentary film The God Who Wasn't There, where he is interviewed about his doubts on the historicity of Jesus.^[3] He currently contributes to The God Contention, a web site comparing and contrasting various worldviews.

Work[link]

Carrier received a PhD in ancient history from Columbia University in 2008. He has published several articles and chapters in books on the subject of history and philosophy. He has also self-published a polemical work on early Christian origins.^[4] He is also a contributor to the Secular Web (one of the main web sites for skeptics of religion on the internet^[5]) writing on a wide range of subjects such as the origins of Christianity and the elements of naturalist and atheist philosophy. He has also engaged in several formal debates, both online and in public, defending subjects as diverse as naturalism as a worldview, natural explanations of early Christian resurrection accounts, and the morality of abortion.^[6]

Defense of metaphysical naturalism[link]

Carrier has defended naturalism as a worldview in his book Sense and Goodness without God,^[7] in a formal online debate with Tom Wanchick,^[8] and in a public debate with Hassanain Rajabali.^[9] Carrier has further defended aspects of naturalism in various articles online.^[10]

Critics and proponents of naturalism have responded to Carrier's philosophy. After briefly assessing Carrier's critique of Mark Steiner's The Applicability of Mathematics as a Philosophical Problem, for example, Russell Howe, professor of mathematics at the Christian academy of Westmont College, argues that Carrier's line of argument plays into the hands of Steiner,^[11] though Carrier disagrees.^[12] On the other hand, Yonatan Fishman of the Department of Neurology at the Albert Einstein College of Medicine, in a paper published for Science & Education, quotes and defends Carrier's demarcation between the natural and the supernatural, which allows supernatural claims (when properly formulated) to be investigated scientifically.^[13]

On the origins of Christianity[link]

Carrier has written a number of polemics on the origins of Christianity and the historical figure of Jesus. In his contribution to The Empty Tomb Carrier argues that the earliest Christians probably believed Jesus had received a new body in the resurrection, and that stories of his old body disappearing from its tomb were developed later. He also argues it is less likely but still possible the original body of Jesus was misplaced or stolen. This work has received critical reviews including those by Philosophy professor Stephen T. Davis in Philosophia Christi^[14] and Christian apologist Norman Geisler.^[15]

Carrier has written and self-published a book arguing against the thesis that Christianity would never have succeeded unless there had been sufficient evidence confirming the supernatural resurrection of Jesus.^[4] Carrier has also questioned the historicity of Jesus in some capacity. Though originally skeptical of the notion, and subsequently more agnostic, since 2005 he has considered it "very probable Jesus never actually existed as a historical person",^[16] yet he also said "though I foresee a rising challenge among qualified experts against the assumption of historicity [of Jesus]...that remains only a hypothesis that has yet to survive proper peer review."^[17]

In news and media[link]

Carrier has played an important role in the national atheist community in the United States. When reports spread of Antony Flew's rejection of atheism in 2004, Carrier engaged in correspondence with Flew to find out what happened and published an extensive analysis of the situation on the Secular Web, finding among other things that Flew changed his belief into there being some sort of "minimal God" (as in Deism). Carrier also came away with the opinion that Flew's changed ideas were not accurately represented in the book Flew co-authored, There is a God.^[18] Carrier's involvement has received national attention in the Associated Press^[19] and New York Times Magazine.^[20] Carrier also appeared on national television in 2004, debating William Lane Craig on Lee Strobel's talk show Faith Under Fire on the PAX network (now ION Television), in a segment on the resurrection of Jesus.^[21] Also in 2006, The Columbus Dispatch reported that Richard Carrier had been selected as the keynote speaker for the Humanist Community of Central Ohio's annual Winter Solstice Banquet in Columbus, Ohio in December of that year, where he spoke on defending naturalism as a worldview.^[22] Carrier is also listed in Who's Who in Hell.^[23]

Publications[link]

Selected articles[link]

• "The Guarded Tomb of Jesus and Daniel in the Lion’s Den: An Argument for the Plausibility of Theft." Journal of Higher Criticism 8.2 (Fall 2001).
• "Pseudohistory in Jerry Vardaman’s Magic Coins: The Nonsense of Micrographic Letters." Skeptical Inquirer 26.2 (March-April 2002) and 26.4 (July-August 2002).
• "The Function of the Historian in Society." The History Teacher 35.4 (August 2002).
• "Hitler’s Table Talk: Troubling Finds." German Studies Review 26.3 (October 2003).
• "The Argument from Biogenesis: Probabilities Against a Natural Origin of Life." Biology & Philosophy 19.5 (November 2004).
• "Whence Christianity? A Meta-Theory for the Origins of Christianity." Journal of Higher Criticism 11.1 (Spring 2005).
• "Fatal Flaws in Michael Almeida’s Alleged ‘Defeat’ of Rowe’s New Evidential Argument from Evil." Philo 10.1 (Spring-Summer 2007).

Books and chapters[link]

• Entries on “Epicurus,” “Lucretius,” “Philodemus,” “Second Sophistic,” “Soranus of Ephesus.” In Encyclopedia of the Ancient World (edited by Thomas J. Sienkewicz). Salem Press (2002). ISBN 0-89356-038-3.
• “The Spiritual Body of Christ and the Legend of the Empty Tomb,” “The Plausibility of Theft,” “The Burial of Jesus in Light of Jewish Law.” In The Empty Tomb: Jesus Beyond The Grave (edited by Robert M. Price and Jeffery Jay Lowder) Prometheus Books (2005) ISBN 1-59102-286-X
• Sense and Goodness without God: A Defense of Metaphysical Naturalism. AuthorHouse (2005) ISBN 1-4208-0293-3.
• “Abortion Cannot be Regarded as Immoral.” In The Abortion Controversy (edited by Lucinda Almond) Greenhaven Press (2007) ISBN 0-7377-3274-1.
• "Bayes’s Theorem for Beginners: Formal Logic and Its Relevance to Historical Method," in Sources of the Jesus Tradition: Separating History from Myth ed. R. Joseph Hoffmann (Prometheus Press, 2010).
• Not the impossible faith, Why Christianity didn't need a miracle to succeed Lulu.com (2009) ISBN 978-0-557-04464-1
• Chapters: "Christianity's success was not incredible", "Neither life nor the universe appear intelligently designed", "Moral facts naturally exist (and science could find them)" in The End of Christianity edited by John W. Loftus (Amherst, NY Prometheus Books 2011) ISBN 978-1-61614-413-5.
• Chapters: "Why the resurrection is unbelievable", "Christianity was not responsible for modern science" in The Christian Delusion edited by John W. Loftus (Amherst, NY Prometheus Books 2010) ISBN 978-1-61614-168-4.

See also[link]

• Anti-supernaturalism
• Christ myth theory
• Criticism of Christian origins
• Internet Infidels
• Historicity of Jesus
• List of American historians
• List of atheist activists and educators

References[link]

External links[link]

• Richard Carrier's homepage (including a full list of publications)
• Richard Carrier's blog
• Richard Carrier's writings at Internet Infidels
• Richard Carrier discusses Metaphysical Naturalism on The Polyschizmatic Reprobates Hour
• Richard Carrier's Atheist column at The God Contention

Terry Gilliam
Gilliam at the 36th Deauville American Films Festival.
Born	Terrence Vance Gilliam (1940-11-22) 22 November 1940 (age 71) Minneapolis, Minnesota, United States
Occupation	Actor, animator, director, producer, screenwriter
Years active	1967–present
Spouse	Maggie Weston (1973–present)
Website
terrygilliamweb.com (Official Site)

Terrence Vance "Terry" Gilliam ( /ˈɡɪliəm/; born 22 November 1940) is an American-born British screenwriter, film director, animator, actor and member of the Monty Python comedy troupe. Gilliam is also known for directing several films, including Brazil (1985), The Adventures of Baron Munchausen (1988), The Fisher King (1991), 12 Monkeys (1995), and Fear and Loathing in Las Vegas (1998). The only "Python" not born in Britain, he took British citizenship in 1968.