Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics.

Philosophical foundations of mathematics[link]

Platonism[link]

“Platonists, such as Kurt Gödel (1906–1978), hold that numbers are abstract, necessarily existing objects, independent of the human mind”^[1]

The foundational philosophy of Platonist mathematical realism, as exemplified by mathematician Kurt Gödel, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world? ^[2]

Formalism[link]

It has been claimed that “Formalists, such as David Hilbert (1862–1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...”.^[1] Indeed he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms:

"And to what has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear . . . The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated.".^[3]

Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds:".^[3]

"We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise".^[4]

The foundational philosophy of formalism, as exemplified by David Hilbert, is a response to the paradoxes of set theory, and is based on formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic.^[2]

Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl would ask these very questions of Hilbert:

"What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question . . .."^[5]

In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory. As noted by Weyl, Formal logical systems also run the risk of inconsistency; in Peano arithmetic, this arguably has already been settled with several proofs of consistency, but there is debate over whether or not they are sufficiently finitary to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S!

Intuitionism[link]

“Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.”^[1]

The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and more coherently by Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect.^[2]

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.

Logicism[link]

Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Gottlob Frege.

Projective geometry[link]

One of the traps in a deductive system is circular reasoning, a problem that seemed to befall projective geometry until it was resolved by Karl von Staudt. As explained by Laptev & Rosenfeld (1996):

In the mid-nineteenth century there was an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts. Indeed the basic concept that is applied in the synthetic presentation of projective geometry, the cross-ratio of four points of a line, was introduced through consideration of the lengths of intervals.

The purely geometric approach of von Staudt was based on the complete quadrilateral to express the relation of projective harmonic conjugates. Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deducing the properties of a field can be found in either the book by Oswald Veblen and John Young, Projective Geometry (1938), or more recently in John Stillwell's Four Pillars of Geometry (2005). Stillwell writes on page 120

...projective geometry is simpler than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms.

The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon numbers without worry about their basis. However, cross-ratio calculations use metric features of geometry, features not admitted by purists. For instance, in 1961 Coxeter wrote Introduction to Geometry without mention of cross-ratio.

Foundational crisis[link]

The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers – a statement that can be shown to be true, but is not provable by the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove). Meanwhile, the intuitionistic school had not attracted many adherents among working mathematicians, due to difficulties of constructive mathematics.

In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic), they may be avoided. Toward the middle of the 20th century it turned out that set theory (ZFC or otherwise) was inadequate as a foundation for some of the emerging new fields, such as homological algebra, and category theory was proposed as an alternative foundation by Samuel Eilenberg and others.

See also[link]

Notes[link]

1. ^ ^{a b c} Anglin (1994) p. 218
2. ^ ^{a b c} Anglin 1991 p. 218
3. ^ ^{a b} Hilbert 1927 The Foundations of Mathematics in van Heijenoort 1967:475
4. ^ p. 14 in Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkhauser (1992).
5. ^ Weyl 1927 Comments on Hilbert's second lecture on the foundations of mathematics in van Heijenoort 1967:484. Although Weyl the intutionist believed that "Hilbert's view" would ultimately prevail, this would come with a significant loss to philosophy: "I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics" (ibid).

References[link]

• Anglin, W. S. (1996) [1994]. Mathematics. A Concise History and Philosophy (Corrected 2nd printing ed.). New York: Springer Verlag. ISBN 0-387-94280-7.  Chapter 39 Foundations contains concise descriptions, for the 20th century, of Platonism (with respect to Gödel), Formalism (with respect to Hilbert), and Intuitionism (with respect to Brouwer).
• Avigad, Jeremy (2003) Number theory and elementary arithmetic, Philosophia Mathematica Vol. 11, pp. 257–284
• Eves, Howard (1990), Foundations and Fundamental Concepts of Mathematics Third Edition, Dover Publications, INC, Mineola NY, ISBN 0-486-69609-X (pbk.) cf §9.5 Philosophies of Mathematics (pp. 266–271. Eves lists the three with short descriptions prefaced by a brief introduction.
• Goodman, N.D. (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
• Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
• Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
• Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", Hamburger Mathematische Seminarabhandlungen 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
• Katz, Robert (1964), Axiomatic Analysis, D. C. Heath and Company.
• Kleene, Stephen C. (1991) [1952]. Introduction to Meta-Mathematics (Tenth impression 1991 ed.). Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9. 

In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.

• Laptev, B.L. & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 40, Birkhäuser ISBN 3-7643-5048-2 .
• Mancosu, P. (ed., 1998), From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford, UK.
• Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
• Putnam, Hilary (1975), "What is Mathematical Truth?", in Tymoczko (ed., 1986).
• Sudac, Olivier (Apr 2001). "The prime number theorem is PRA-provable". Theoretical Computer Science 257 (1–2): 185–239. DOI:10.1016/S0304-3975(00)00116-X. 
• Troelstra, A. S. (no date but later than 1990), "A History of Constructivism in the 20th Century", http://staff.science.uva.nl/~anne/hhhist.pdf, A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
• Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
• Tymoczko, T. (ed., 1986), New Directions in the Philosophy of Mathematics, 1986. Revised edition, 1998.
• van Dalen D. (2008), "Brouwer, Luitzen Egbertus Jan (1881–1966)", in Biografisch Woordenboek van Nederland. URL:http://www.inghist.nl/Onderzoek/Projecten/BWN/lemmata/bwn2/brouwerle [13-03-2008]
• Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
• Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.

External links[link]

• What is Foundations of Mathematics?
• Logic and Mathematics
• Foundations of Mathematics mailing list

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The Foundations were a British soul band, active from 1967 to 1970. The group, made up of West Indians, White British, and a Sri Lankan, are best known for their two biggest hits, "Baby Now That I've Found You" (a Number One hit in the UK Singles Chart and Canada, and subsequently Top 10 in the US), written by Tony Macaulay and John MacLeod; and "Build Me Up Buttercup" (a number 3 on the Billboard Hot 100 and number 1 in Canada), co-written by Macaulay with Mike d'Abo, at the time the lead vocalist with Manfred Mann. The group was the first multi-racial group to have a number 1 hit in the UK in the 1960s.^[1]

The Foundations are notable for being one of the few label acts to successfully imitate what became known as the Motown Sound. In terms of line-up and musical style, they anticipated the sound of the more successful Hot Chocolate. They were in a similar musical vein as Love Affair, who also topped the UK charts in 1968 with their version of Robert Knight's "Everlasting Love". The Foundations signed to Pye, at the time one of only four big UK record companies (the others being EMI with its HMV, Columbia Records, and Parlophone labels; Decca; and Philips who also owned Fontana).^[2]

Biography[link]

Origins[link]

The Foundations drew much interest and intrigue due to the size and structure of the group. Not only was there a diverse ethnic mix in the group, but there was also diversity in ages and musical backgrounds. The oldest member of the group was Mike Elliott, who was 38 years old. The youngest was Tim Harris, who, at 18, was barely out of school. The West Indian horn section, which consisted of Jamaican-born Mike Elliott and Pat Burke, both saxophonists and Dominican-born Eric Allendale on trombone. They were all highly experienced musicians who came from professional jazz and rock-and-roll backgrounds. Mike Elliott had played in various jazz and rock and roll bands including Tubby Hayes and Ronnie Scott,^[3] the Cabin Boys (led by Tommy Steele's brother, Colin Hicks), and others. Pat Burke, a professional musician, was from the London Music Conservatorium. Eric Allendale had led his own band at one stage as well as having played with Edmundo Ros and being a former member of the Terry Lightfoot^[4] and Alex Walsh bands. Alan Warner, the guitarist, was also an experienced musician for his age, having played with numerous semi-pro groups from the age of 16.^[5] Bassist Peter Macbeth was a former teacher. Tony Gomez, the keyboard player, was a former clerk, while Clem Curtis had been an interior decorator and professional boxer.

The story of the origins of the Foundations can be somewhat surprising and a bit confusing as to who was responsible for choosing the band's name, and various sources give slightly different accounts of their beginnings. One version is that they were originally called The Ramong Sound,^[5][6] or The Ramongs, and there were two lead singers, Clem Curtis and Psychedelic shock rocker Arthur Brown.^[7] Other variations on this are that after they dropped the "Ramong" from their name, they were then called The Foundation Squad or The Foundation Sound.^[5] Apparently, Arthur Brown was only a temporary member for about one month, and by the time the Foundations had signed to Pye, he had left the group.

The Foundations actually did come together in Bayswater, London in January 1967. They practiced and played in a basement club called the Butterfly Club, which they ran.^[8] While managing the club themselves, they played music nightly, and handled the cooking and cleaning. They would get to bed around 6 or 7 a.m., sleep until 4 p.m., get up and begin again to get ready to open at 8 p.m. Sometimes they barely made enough money to pay the rent, let alone feed themselves. At times, they lived off the leftovers and a couple of pounds of rice.^[9] They stayed there until they were eventually forced out by a protection racket gang and then had to move next door to a dingy unused mini-cab office.^[10]

Career from 1967[link]

While the biography on Allmusic states that Barry Class was the first to discover them,^[1] Alan Warner, the original founding member and lead guitarist of The Foundations and former Ramongs member states that it was actually Fairway who saw them in the dingy office and then introduced them to Class.^[10] Either way, both Fairway and Class teamed up to become their managers.^[11] Class took care of the business management, while Fairway went out promoting and trying to secure gigs for the group.^[10]

When they were at the top spot with "Baby, Now That I've Found You", Fairway commented to Melody Maker that most management would pull them out of the "bargain priced dates" that were booked for some time. He expressed gratitude to everyone for their support, and said that they would fulfill every engagement for which they had signed.^[3]

Not long after "Baby, Now That I've Found You" became a hit, rock historian Roger Dopson describes what followed as a "behind the scenes struggle",^[3] where Fairway was "pushed out" and his partner, Barry Class, remained as sole manager of the group. Fairway later attempted to sue the band, alleging that he was wrongfully dismissed, though the band said that he had resigned of his own accord.^[12] Dopson also noted that Fairway also leaked a story to the media saying that the Foundations had broken up which only served to keep the Foundations name in the news headlines.^{[3] [13]} Fairway and Class eventually introduced them to Pye A&R man Tony Macaulay.^[10] The day Macaulay came to hear them play, he was suffering from what he described as the worst hangover of his life. The band was playing so loud he could not judge how good they were, but he decided to give them a chance.^[3] He would later comment in the book, 1000 UK #1 Hits by Jon Kutner and Spencer Leigh, that he woke up that morning with a stinking headache, and when he got to the studio and heard The Foundations, he thought they were pretty terrible. He decided his hangover was to blame, and so he gave them the benefit of the doubt.^[14]

At first they found progress quite slow, and one of their sax players, Pat Burke, had to drop out of the band and take another job while they went through a rough patch. He did rejoin them again later in 1967.^{[15] [16]}

They did a couple of tours backing The Toys and later Motown Records' Edwin Starr.^[10] Curtis doubted if this group called The Toys was the original Toys let alone American. They were noticed by Brian Epstein who added them to the roster of his NEMS Agency but the contract became void when he died.^[17]

When "Baby Now That I've Found You" was first released it went nowhere. Luckily the BBC's newly founded BBC Radio 1 were looking to avoid any records being played by the pirate radio stations and they looked back at some recent releases that the pirate stations had missed. "Baby, Now That I've Found You" was one of them. The single then took off and by November was number one in the UK Singles Chart.^[2] This was the ideal time because of the soul boom that was happening in England since 1965 and with American R&B stars visiting the UK, interest and intrigue in The Foundations was generated. Their second single released in January 1968, "Back On My Feet Again", did not do as well but made it to #18 in the UK,^[2] and #29 in Canada. Also in January 1968 they were invited to put down some tracks for John Peel's radio show. One of the tracks that they laid down was a cover of ? and the Mysterians garage classic 96 Tears.^[18][19] On the same day PP Arnold was in the studio with Dusty Springfield and Madeline Bell as her backing vocalists.^[18]

Around this time after the release of their second single, there were tensions developing between the band and their songwriter/producer, Tony Macaulay. He would not allow them to record any of their own songs.^[1] In an interview, the band's organ player, Tony Gomez, told NME Magazine in an interview that he, Peter MacBeth, and Eric Allendale had some ideas that they wanted to put together. Curtis later recalled that Macaulay was a problem. "Tony Macaulay was very talented, but could be difficult to get on with. When we asked to record some of our own material - just as B sides, we weren't after the A side - he called us 'ungrateful' and stormed out of the studio."^[3] The group felt that Macaulay had reined in their "real" sound, making them seem more pop-oriented than they were.^[1] Tony Macaulay was later to recall, "I was never close to The Foundations. I couldn't stand them, and they hated me! But the body of work we recorded was excellent."^[3]

A third single, also released in 1968 "Any Old Time (You're Lonely and Sad)", reached #48.^[2]

Curtis and Elliott leave group[link]

Original vocalist Curtis left in 1968, because he felt that a couple of the band's members were taking it a bit too easy, thinking that because they had now had a hit, they did not have to put in as much effort as they had previously.^[1] Saxophonist Mike Elliott also left around this time and was never replaced. Curtis hung around and helped them audition a replacement singer. They auditioned 200 singers. It was reported in a New Musical Express article in 1968 that Curtis while being interviewd at a festival had mentioned that they were trying out Warren Davis to replace him. He said he wouldn't leave the band until they found a replacement.^[20] He had become friendly with Sammy Davis, Jr. and was encouraged to try his luck in the United States. He moved to the US for a solo career on the club circuit, encouraged by the likes of Wilson Pickett and Sam & Dave, playing Las Vegas with the Righteous Brothers. His successful replacement was Colin Young.

New lead singer[link]

With Young the band had two more big hits; "Build Me Up Buttercup" which was their third hit in 1968 and "In the Bad Bad Old Days (Before You Loved Me)" which was a hit in April 1969,^[2] and reached #23 in Canada May 5 that year.

At the height of their popularity, the Foundations management were in negotiations and a UK TV company for a television series that would star members of the band. They had turned down a number of offers to appear in films because of script unsuitability.^[21]

Bassist Peter Macbeth left the band in 1969, to join the group Bubastis with Bernie Living,^[22][23] and was replaced by Steve Bingham.

Beginning of 1970 to the breakup in late 1970[link]

After a successful run of hits, The Foundations broke off with their management and a Bill Graham sponsored tour to support The Temptations at the newly opened Copacabana club. This ended up in disaster and the band came back to the UK in low spirits. It had been previously reported in a publicity sheet around early December 1969 that the band had broken away from their manager Barry Class. Jim Dawson who was formerly their agent and Mike Dolan took over the group's affairs.^[24] The group's final hits were "Born to Live, Born to Die" which was written by Eric Allendale and Tony Gomez.^[25] and "My Little Chickadee", a US only hit which barely made the hot 100. Another member joined the band in 1970. Paul Lockey who had been with Robert Plant in Band Of Joy joined as their bass guitarist.^[26]

"My Little Chickadee" proved to be the band's last hit. In spite of releasing "Take A Girl Like You", the title song to the Oliver Reed and Hayley Mills film, and a heavy blues rock song "I'm Gonna Be A Rich Man", the band split in late 1970.^[27]

1971 to the end of the 1970s[link]

The last record released in the early 1970s as The Foundations was a single "Stoney Ground" b/w "I'll Give You Love" MCA MCA 5075 1971.^[28] By that time the original band had already split up and Alan Warner had been a member of Pluto since joining and helping to form the English progressive rock band in 1970.^[10] There would be two more singles released as The Foundations in the mid to late 1970s.

When Curtis returned to the UK, he formed a new version of the group with little success in spite of releasing several singles, but later had a lucrative spell on the 1960s nostalgia circuit. Curtis' re-formed Foundations have on several occasions and among the many musicians to be part of latter day Foundations were Bill ^[29] and John Springate, the latter becoming a member of The Glitter Band,^[30] Derek "Del" Watson, Paul Wilmot (all members of the band "ELEGY") and Roy Carter who later on joined Heatwave.^[31]

Also in the 1970s there would be a collaborative attempt between two former members of the Foundations. Original Foundations trombonist Eric Allendale attempted to work with original Foundations Foundations drummer Tim Harris.^[32]

In the mid 1970s, while Clem Curtis and The Foundations were on the road, there was also another Foundations line up that was led by Colin Young who were on the road at the same time, who were playing basically the same material. This eventually led to court action which resulted in Curtis being allowed to bill his group as either The Foundations or Clem Curtis & The Foundations. Young was allowed to bill himself as The New Foundations or Colin Young & The New Foundations.^[3]

Also in the mid 1970s, Young and his group, The New Foundations, released a lone single on Pye, "Something For My Baby" / "I Need Your Love".^[3] There were actually two more singles released in the late 1970s as the Foundations. They were "Where Were You When I Needed Your Love" / "Love Me Nice And Easy" and "Closer To Loving You" / "Change My Life" on the Summit and Psycho labels.^[33] These featured Curtis as the lead singer.

Various sources erroneously state that there was an early 1970s English line up that had nothing, or little to do with, the original Foundations. However, Curtis has been leading a new line up of the Foundations since coming back to the UK and reforming the group in the early 1970s.

1980s to present[link]

In or around the late 1980s Curtis and Alan Warner teamed up to recut "Baby, Now That I've Found You" as well as other hits of The Foundations.^[3]

There has also been another line up formed in 1999 that included Young (vocals), Alan Warner (Guitar), Steve Bingham (bass), Gary Moberly (keyboards), Tony Laidlaw (sax) and Sam Kelly then Steve Dixon (drums). This version of the group was reformed due to the popularity of the film There's Something About Mary and the interest created resulting from the 1968 hit "Build Me Up Buttercup" being featured in the film. Some time later Young left this version of the group and was replaced by Hue Montgomery (aka Hugh Montgomery).

Known as Alan Warner's Foundations, this group is on tour in 2012 either in their own right or with The New Honeycombs, Terry Rice-Milton of Cupid's Inspiration, Billie Davis and Keith Powell.

Curtis still appears at venues and tours as Clem Curtis & The Foundations as he has done during the past four decades.

There is a bogus American group using the Foundations name and includes original Foundations hits as part of their repertoire. This group has no connection with the group that recorded the hits on Pye.^{[citation needed]}

Former personnel[link]

The Foundations[link]

• Clem Curtis: lead vocals - born 28 November 1940,^[34] Trinidad, West Indies.
• Colin Young: lead vocals - b. 12 September 1944, Barbados, West Indies - replaced Clem Curtis in 1968.
• Arthur Brown: vocals - b. 24 June 1942 Whitby, Yorkshire, Member for approximately one month in 1967
• Alan Warner: lead guitar - b. 21 April 1947,^[34] Paddington, west London.
• Peter Macbeth: bass guitar - b. Peter McGrath, 2 February 1941,^[34] Marylebone, North London.
• Steve Bingham: bass guitar - b. 4 April 1949, Solihull, Warwickshire.
• Tim Harris: drums - b. 14 January 1948,^[34] St John's Wood, North London - Died 2007
• Tony Gomez: keyboards - b. 13 December 1938,^[34] Colombo, Ceylon - (now Sri Lanka).
• Pat Burke: tenor saxophone/flute - b. 9 October 1937,^[34] Kingston, Jamaica, West Indies.
• Mike Elliott: tenor saxophone - b. 6 August 1929,^[34] Jamaica, West Indies. - Left in 1968
• Eric Allandale: trombone - b. Eric Allandale Dubuisson, 4 March 1936,^[34] Dominica, West Indies — died 23 August 2001.
• Paul Lockey: bass guitar^[35] - joined in 1970 for nine months

Guests[link]

• Mike D'Abo :piano - b. Michael David D'Abo, 1 March 1944, Betchworth, Surrey. Co-wrote and guested on "Build Me Up Buttercup" contributing piano.
• John Mcleod: piano

Clem Curtis & The Foundations[link]

• Clem Curtis: vocals
• James Colah: keyboards
• Michael J. Parlett: saxophone
• Alan Warner: guitar ^[3]
• Roy Carter: bass guitar ^[36]
• George Chandler: backing vocals
• Valentine Pascal: electric guitar

1970s line-up[link]

• Clem Curtis: Vocals
• Bill Springate: guitar
• John Springate: Vocals
• Del Watson: Bass
• Paul Wilmot : Drums
• John Paul: ? ^[37]

1977 line-up[link]

• Clem Curtis: Vocals
• Leroy Carter
• John Savile
• Valentine Pascal
• Georges Delanbanque ^[38]

Discography[link]

Summary of single releases

From the bands beginning to their breakup near the end of 1970, the Foundations released ten singles in the UK including two versions of the same song. A good deal of the songs on the singles were composed by Tony Macaulay and John Macleod. They had four significant hits from these plus a minor hit with one of their own compositions, "Born To Live, Born To Die" and another minor hit with "My Little Chickadee" in the United States. There were other titles announced that were either never recorded or were never released. They were "Our Love Went Thataway",^[39] "Tear Jerker, Music-worker, You" which was to be released around the same time as "Better By Far" by Lulu and "No Place On Earth Could Find You" ^[40][41] In 1971 the single "Stoney Ground" was released. It is believed that this single was actually by Colin Young and his new backing band Development. It seems quite likely as the Colin Young and Development debut single "Any Time At All" pre-dates "Stoney Ground". In the mid and late seventies there were two more singles released under the Foundations name. They were "Where Were You When I Needed Your Love" and "Closer To Loving You" which featured the Northern Soul classic "Change My Life" as the B side. These last two singles to bear the Foundations name featured Clem Curtis once more as the lead vocalist.

Summary of album releases

During the 1960s the Foundations recorded and released four LPs in the United Kingdom. Before the release of their debut album, it was originally announced, in the October 1967 of Beat Instrumental Monthly that the debut album's title was to be Sound Basis.^[9] However, when it was released on Pye, it had the title of From The Foundations. The American version of this album, which was released on the UNI label, was given the title of Baby, Now That I've Found You. This album featured Curtis on lead vocals. The next release was in 1968. It was a live LP called Rocking The Foundations, and also featured Curtis on lead vocals plus two instrumentals "The Look of Love" and "Coming Home Baby". Also in 1968, another LP was released on the Marble Arch label. This featured re-recordings of their previous hits and songs, but with Young on vocals instead of Curtis. It also featured a version of "Build Me Up Buttercup". There was also an American album called Build Me Up Buttercup. This was a compilation of Foundations tracks. Side one consisted of tracks from their Rocking The Foundations album; and side two consisted of "Build Me Up Buttercup", the B side of that single, plus some earlier Foundations tracks. The group's last LP release was in 1969, which featured their hit "In The Bad Bad Old Days", and the minor US hit "My Little Chickadee". Since then there have been various compilations of the Foundations songs, released on both the Golden Hour and PRT labels.

UK singles[link]

• "Baby, Now That I've Found You" / "Come On Back to Me" - Pye 7N 17366 - 1967 - UK #1, US #11
• "Back On My Feet Again" / "I Can Take Or Leave Your Loving" - Pye 7N 17417 - 1968 - UK #18
• "Any Old Time (You're Lonely And Sad)" / "We Are Happy People" - Pye 7N 17503 -1968 - UK #48
• "Build Me Up Buttercup" / "New Direction" Pye 7N 17636 - 1968 - UK #2, US #3
• "In the Bad Bad Old Days (Before You Loved Me)" / "Give Me Love" - Pye 7N 17702 - 1969 - UK #8
• "Born to Live, Born to Die" / "Why Did You Cry" - Pye 7N 17809 1969 - UK #46
• "Baby, I Couldn't See" / "Penny Sir" - Pye 7N 17849 - 1969
• "Take A Girl Like You" / "I'm Gonna Be A Rich Man" - Pye - 7N 17904 - 1970
• "I'm Gonna be a Rich Man" / "In The Beginning" Pye 7N 17956 - 1970
• "Stoney Ground" / "I'll Give You Love" - MCA MKS 5075–1971
• "Where Were You When I Needed Your Love" / "Love Me Nice And Easy" - Summit SU 100
• "Closer To Loving You" / "Change My Life" - Psycho P 2603–1978
• "Baby Now That I've Found You" / "Build Me Up Buttercup" - Old Gold OG9407 - (1979 Re-release)
• "Baby Now That I've Found You" / "Build Me Up Buttercup" - Flashback FBS 6 - (1979 Re-release)
• "Baby Now That I've Found You" (1989 Re Mix) / "Build Me Up Buttercup" - PRT PYS 24 - 1989
• "Build Me Up Buttercup" - CD single - (1998 Re-release) - UK #71

^{[2] [33]}

UK albums[link]

• From the Foundations - Pye NSPL 18206 - 1967
• Rocking the Foundations - Pye NSPL 18227 - 1968
• The Foundations - Marble Arch MALS 1157–1968
• Digging the Foundations - Pye NSPL 18290 - 1969
• Golden Hour of the Foundations (Greatest Hits) - GH 574 - 1973
• Back to the Beat - PRT DOW7 - 1983
• Best Of - PRT PYL 4003–1987

UK EPs 7"[link]

• "It's All Right" - Pye NEP24297 - 1968
• "Mini Monster" - Pye PMM.103

UK EPs 12"[link]

• "Baby, Now That I've Found You" - Pye Big Deal BD 107 - (4 tracks)
• "Baby, Now That I've Found You" - PRT Pyt 24 - 1989 - (3 tracks incl remix)

UK CDs[link]

• Golden Hour Of The Foundations - Knight Records KGH CD 104 - 1990
• Strong Foundations - The Singles and More - Music Club - MCCD 327 - 1997
• Build Me Up Buttercup - Castle Select SELCD 527 - 1998
• Baby, Now That I've Found You - Sequel Records - NEECD 300 - 1999
• Build Me Up Buttercup (The Complete Pye Collection) [Remastered] - Castle - 2004

US singles[link]

• "Baby Now That I've Found You" / "Come On Back To Me" - Uni 55038 - 1967
• "Back On My Feet Again" / "I Can Take Or Leave Your Loving" - Uni 55058 - 1968
• "Any Old Time (You're Lonely And Sad)" / "We Are Happy People" - Uni 55073 -1968
• "Build Me Up Buttercup" / "New Direction" - Uni 55101 - 1968
• "In The Bad Bad Old Days (Before You Loved Me)" / "Give Me Love" - Uni 55117 - 1969
• "Born To Live, Born To Die" / "Why Did You Cry" - Uni 55162 - 1969 - UK #46
• "My Little Chickadee" / "Solomon Grundy" - Uni 55137 - 1969
• "Take A Girl Like You" / "I'm Gonna Be A Rich Man" - Uni 55210 - 1970
• "Stoney Ground" / "I'll Give You Love" - Uni 55315 - 1971
• "Build Me Up Buttercup" / "Baby, Now That I've Found You" - Eric 192 (Re-release)

US albums[link]

• Baby Now That I've Found You - Uni 3016 (Mono)/73016 (Stereo) - 1967
• Build Me Up Buttercup - Uni 73043—1968
• Digging The Foundations - Uni 73058—1969

Canadian singles[link]

• "Baby Now That I've Found You" - Pye 827—1968 - #1
• "Back On My Feet Again" - Pye 833 - #29
• "Build Me Up Buttercup" - Pye 17636 - #1
• "In the Bad Bad Old Days (Before You Loved Me)" - Pye 17702 - #23

References[link]

Reference links and other information[link]

• The Crazy World Of Arthur Brown http://www.richieunterberger.com/brown.html
• Alex Gitlin Pluto page http://www.alexgitlin.com/npp/pluto.htm
• Gary Moberley info http://web.archive.org/web/20091027104142/http://geocities.com/SunsetStrip/Diner/2674/moberley_gary_a.htm
• Pluto Website http://www.pluto-rockband.com/70141/info.php?p=2&pno=0
• Alan Warner site page http://www.alan-warner.com/10102/info.php?p=4&pno=0
• Ride Your Pony http://rideyourpony-twighlightzone.blogspot.com/2007/09/group-made-up-of-west-indians.html
• Mike D'Abo Foundations http://www.mikedabo.com/foundations.htm
• Buddah Records @ Disco-Disco.com http://www.disco-disco.com/labels/buddah.shtml
• www.javirunner-brightsideofthemoon.blogspot.com Info on Barry Class
• www.tamworthbands.com Info on Barry Class

External links[link]

• Clem Curtis & The Foundations web site
• Alan Warner's Foundations official website
• The Foundations at Allmusic
• The Foundations discography at Discogs
• Clem Curtis web site
• Clem Curtis on Myspace
• Alan Warner site page
• Interview with Clem Curtis

Kurt Gödel
Born	(1906-04-28)April 28, 1906 Brno, Moravia, Austria–Hungary
Died	January 14, 1978(1978-01-14) (aged 71) Princeton, New Jersey, United States
Fields	Mathematics, Mathematical logic
Institutions	Institute for Advanced Study
Alma mater	University of Vienna
Doctoral advisor	Hans Hahn
Known for	Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel metric, Gödel's ontological proof
Notable awards	Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1974) Fellow of the Royal Society[1]
Signature

Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German pronunciation: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian/American logician, mathematician, and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.^[2]

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. There are many important results whose only known proofs use the axiom of choice, which is equivalent to Zorn's lemma. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Life[link]

Childhood[link]

Gödel was born April 28, 1906, in Brno, Austria-Hungary into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh).^[3] At the time of his birth the town had a slight German-speaking majority,^[4] and this was the language of his parents.^[5] The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".^[6]

Although he spoke very little Czech himself, Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I. According to his classmate Klepetař, "Gödel considered himself always Austrian and an exile in Czechoslovakia" ("ein Österreicher im Exil in der Tschechoslowakei") during this time.^[7] He chose to become an Austrian citizen at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Gödel attended the Evangelische Volksschule, a Lutheran school in Brno from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna[link]

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.^[8] Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."^[9]

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?

This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

The Incompleteness Theorem[link]

In 1931 and while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

1. If the system is consistent, it cannot be complete.
2. The consistency of the axioms cannot be proven within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

In his two-page paper Zum intuitionistischen Aussagenkalkül (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

The mid 1930s: further work and visits to the US[link]

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel.^[11] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.^[12]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.^[13] He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures which have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him, and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.

Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

Relocation to Princeton, Einstein and US citizenship[link]

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the Institute for Advanced Study (IAS).

Gödel very quickly resumed his mathematical work. In 1940, he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory which is a classic of modern mathematics. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.

Albert Einstein was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely...to have the privilege of walking home with Gödel".^[14]

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.^[15][16]

Later years and death[link]

Gödel became a permanent member of the Institute of Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.

In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday.^[17] These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed.^[18] To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, Adele was hospitalized for six months and could no longer prepare Gödel's food. In her absence, he refused to eat, eventually starving to death.^[19] He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.^[20]

Gravestone of Kurt Gödel in the Princeton, NJ, cemetery

Religious views[link]

Gödel was a convinced theist. He rejected the notion of others like his friend Albert Einstein that God was impersonal.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world [Welt] is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."^[21]

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."^[22] He said about Islam: "I like Islam. It is a consistent [or consequential] idea of religion and open-minded".^{[23] [24]}

Legacy[link]

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association of Symbolic Logic has invited an annual Kurt Gödel lecture each year since 1990.

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's Nachlass, and the final two include correspondence.

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".^[25]

Douglas Hofstadter wrote a popular book in 1979 called Gödel, Escher, Bach: An Eternal Golden Braid to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

Important publications[link]

In German:

• 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Mathematik und Physik 37: 349–60.
• 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173–98.
• 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.

In English:

• 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
• 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470–85.
• 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 1: 175–81

In English translation:

• Kurt Godel, 1992. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
• Kurt Godel, 2000.^[26] On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
• Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
  • 1930. "The completeness of the axioms of the functional calculus of logic," 582–91.
  • 1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).
  • 1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596–616.
  • 1931a. "On completeness and consistency," 616–17.
• "My philosophical viewpoint", c. 1960, unpublished.
• "The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.

• Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
  • Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback:ISBN 978-0-19-514720-9,
  • Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback:ISBN 978-0-19-514721-6,
  • Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback:ISBN 978-0-19-514722-3,
  • Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5,
  • Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

See also[link]

	Biography portal
	Logic portal

Notes[link]

References[link]

• Dawson, John W., 1997. Logical dilemmas: The life and work of Kurt Gödel. Wellesley MA: A K Peters.
• 1911 Encyclopædia Britannica/Brünn. (2007, September 19). In Wikisource, The Free Library. Retrieved 10 pm EST March 13, 2008.
• Rebecca Goldstein, 2005. Incompleteness: The Proof and Paradox of Kurt Gödel. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.

Further reading[link]

• John L. Casti and Werner DePauli, 2000. Gödel: A Life of Logic, Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.
• John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. AK Peters, Ltd., 1996.
• John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", Scientific American, vol. 280 num. 6, pp. 76–81
• Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
• Jaakko Hintikka, 2000. On Gödel. Wadsworth.
• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
• Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint ca. 2001.
• Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. ISBN 978-0-923891-57-2)
• J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
• Ernest Nagel and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
• Procházka, Jiří, 2006, 2006, 2008, 2008, 2O1O. Kurt Gödel: 1906–1978: Genealogie. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl.Volume V,Brno,Princeton 2O1O, ISBN 8O-9O3476-9-X.In Germ.,Engl.
• Ed Regis, 1987. Who Got Einstein's Office? Addison-Wesley Publishing Company, Inc.
• Raymond Smullyan, 1992. Godel's Incompleteness Theorems. Oxford University Press.
• Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.
• Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press.
• Hao Wang, 1996. A Logical Journey: From Godel to Philosophy. MIT Press.
• Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
• Yourgrau, Palle, 2004. A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), p 861–868):

External links[link]

Wikimedia Commons has media related to: Kurt Gödel

• O'Connor, John J.; Robertson, Edmund F., "Kurt Gödel", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Godel.html .
• Kurt Gödel at the Mathematics Genealogy Project
• Weisstein, Eric W., Gödel, Kurt (1906–1978) from ScienceWorld.
• Kennedy, Juliette. "Kurt Gödel." In Stanford Encyclopedia of Philosophy.
• Time Bandits: an article about the relationship between Gödel and Einstein by Jim Holt
• "Gödel and the limits of logic" by John W Dawson Jr. (June 2006)
• Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue
• Paul Davies and Freeman Dyson discuss Kurt Godel
• "Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
• It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.
• Dangerous Knowledge Google Video of a BBC documentary made by David Malone about the life and work of Kurt Gödel and other revolutionary mathematical thinkers.
• Gödel photo g.
• Kurt Gödel at Find a Grave