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This is not the title of our seminar Shikhar Paliwal Varun Suprashanth Asok R Under Guidance of Dr. Pushpak Bhattacharya

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Outline Introducing Gödel, Escher and Bach. Self-Reference. A Brief History Of ‘Logic’. Consistency vs. Completeness. Impact of Gödel's Theorem

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Introducing Bach Johann Sebastian Bach : Bach was a 17 th century German composer who was especially known for his impromptu improvisations. In fact his contemporaries considered his musical prowess with almost a mythical reverence. He was rumored, although falsely, to have rendered an extempore composition containing an 8-part fugue

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Introducing Bach The extent of this exaggeration although bordering stupidity gives us a picture of Bach’s musical capabilities. Douglas R. Hofstadter in his book says, “One could probably liken the task of improvising a six-part fugue to the playing of sixty simultaneous blindfold games of chess, and winning them all. To improvise an eight-part fugue is really beyond human capability.”

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Musical Loops and Riddles Another important aspect of Bach’s music is his use of recursion and musical riddles in his music. Bach sent a composition to the then Prussian King Fredrick II, who was an avid musician himself, along with a letter praising the king. The composition is famous for it’s musical loops

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Musical Loops and Riddles He takes the listener to “increasingly remote provinces of tonality” wherein the listener might think that the notes are now further than the first note. But after exactly 6 such modulations the listener without even noticing is brought back to the first note in a musically agreeable way. The following is a video is a kind of ‘musical palindrome’ by Bach. This is generally called a ‘Crab Cannon’ and was taken from The Musical Offering that Bach sent to the King

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The Letter In the letter that Bach send the King we find a phrase which translates into “As the modulation grows so does the King’s glory”. Bach was basically hinting at the never ending nature of the modulations, cryptically saying that the King’s glory is ever growing

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Introducing Escher Maurits Cornelis Escher M.C. Escher was a 20 th century Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture and tessellations. The special way of thinking and the rich graphic work of M.C. Escher has had a continuous influence in science and art

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Escher’s Paintings This This sketch was inspired by the Penrose stairs that were published by the father-son team of Lionel and Roger Penrose respectively an year before

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Escher’s Paintings

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Escher’s Paintings

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Introducing Gödel Kurt Friedrich Gödel Gödel was a 20 th century Austrian logician, mathematician and philosopher. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age

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Introducing Gödel 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"). This paper effectively stopped further study into ‘perfect’ mathematical systems which don’t fall prey to any paradox and can be automated

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What connects them?

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Self-Reference Self Reference or Strange Loopiness occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly through some intermediate sentence or formula or by means of some encoding. Self Reference is repeatedly seen in Gödel, Escher and Bach’s works

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Self-Reference Self-Reference can be negative or positive. Examples of self-reference: - This is not an example for self reference. - All my countrymen are liars

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Recursive Acronyms Recursive Acronyms is another area where self- reference is liberally used especially by software developers. E.g. : GNU - GNU’s Not Unix BING- BING Is Not Google ACME- Acme Company Makes Everything PNG- PNG’s Not GIF these are only a few examples

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Tupper’s Self-referential Formula Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself

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A Brief History Of ‘Logic’ Ancient Greeks were the first to realize that reasoning was a patterned process and is at least partially governed by statable laws. Aristotle codified syllogisms and Euclid geometry but we had to wait many a century until further progress was made

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A Brief History Of ‘Logic’ The 19 th century saw many noted Mathematicians achieve many wonders. Non-Euclidean geometry was introduced to the shock of the Mathematical community, because it deeply challenged the idea that mathematics studies the real world

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A Brief History Of ‘Logic’ George Boole and De Morgan took Aristotle’s work considerably further by trying to codify deductive reasoning patterns. Gottlob Frege, Giuseppe Peano, David Hilbert, Lewis Carroll were among others the pioneers of this field during the same time

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Cantor’s Theory of Sets But it was Georg Cantor and his Theory of Sets that would revolutionize the Mathematical world. His set theory although powerful was intuition defying. Before long mathematicians came up with a number of paradoxes pertaining to Cantor’s sets. Chief among them was Russell’s Paradox by Bertrand Russell

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Russell’s Paradox Russell's paradox is a famous example of an impredicative construction, namely the set of all sets which do not contain themselves. Clearly, every set is either a member of itself or not a member of itself and no set can be both. E.g. : – Set of all sets [self-swallowing]. – Set of all humans [ordinary]

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The Paradox Now nothing prevents us from inventing R, the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox

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‘Principia Mathematica’ Again we see self-reference creeping up. In fact many of the Mathematicians back then thought that if one could come up with a system that would not allow self reference it could be the perfect system. Russell and Whitehead did subscribe to this view. ‘Principia Mathematica’ was their attempt to rid all mathematical entities of self reference

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‘Principia Mathematica’ The idea of their system was basically this. A set of the lowest "type" could contain only "objects" as members not sets. A set of the next type up could only contain objects, or sets of the lowest type. In general, a set of a given type could only contain sets of lower type, or objects

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‘Principia Mathematica’ Clearly, no set could contain itself because it would have to belong to a type higher than its own type. The theory of types handled Russell's paradox, but it did nothing about other cases like the Epimenides paradox or Grelling's paradox. Eg: Take the two step Epimenides loop The following sentence is false. The preceding sentence is true

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‘Principia Mathematica’ The first sentence, since it speaks of the second, must be on a higher level than the second. But by the same token, the second sentence must be on a higher level than the first. More precisely, such sentences simply cannot be formulated at all in a system based on a strict hierarchy of languages

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Concluding History At this juncture Mathematics and Logic were not treated that different. People considered one to be the subset of the other or the other way around depending on who they were. This period was also when people were uncertain whether the rules they were operating under would fall apart as soon as another paradox sprouts up

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Consistency vs. Completeness This was the goal of ‘Principia Mathematica’, which purported to derive all of mathematics from logic, and, to be sure, without contradictions! David Hilbert came up with the following questions to test ‘Principia Mathematica’ 1.all of mathematics really was contained in the methods delineated by Russell and Whitehead, or 2.the methods given were even self-consistent

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Consistency vs. Completeness This is a rather perplexing question to ask since it basically wants the system to prove from within. ‘It is like lifting yourself by your own shoelaces’. Hilbert was fully aware of this dilemma and therefore expressed the hope that a demonstration of consistency or completeness could be found which depended only on "finitistic" modes of reasoning. These were a small set of reasoning methods usually accepted by mathematicians

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Consistency vs. Completeness In this way, Hilbert hoped that mathematicians could partially lift themselves by their own bootstraps. The sum total of mathematical methods might be proved sound, by invoking only a smaller set of methods

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Theorems of Incompleteness Many mathematicians spent considerable time on this problem. But it was not until Kurt Gödel aged 25 and hardly an year past his Doctorate from the University of Vienna bursts onto the center stage of mathematics with his paper on the Theorems of Incompleteness

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Theorems of Incompleteness This paper revealed there were irreparable "holes" in the axiomatic system proposed by Russell and Whitehead. Also more generally, that no axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsistent system!

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Gödel's Incompleteness Theorem The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers, that 1. If the system is consistent, it cannot be complete. 2. The consistency of the axioms cannot be proven within the system

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The Irony The final irony of it all is that the proof of Gödel's Incompleteness Theorem involved importing the Epimenides paradox right into the heart of ‘Principia Mathematica’, a bastion supposedly invulnerable to the attacks of Strange Loops! This was done using a new numbering system that Gödel came up with. He used this numbering system to bring the Epimenides’ paradox into ‘Principia Mathematica’ without violating any of it’s rules

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The Unprovable 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

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Impact of Gödel's Theorem These theorems ended half a century of attempts, beginning with the work of Frège 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

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Impact of Gödel's Theorem Authors including J. R. Lucas have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centres on whether the human mind is equivalent to a Turing Machine, or by the Church-Turing Thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it

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References Gödel, Escher, Bach: An Eternal Golden Braid, A Metamorphical Fugue on Minds and Machines in the Spirit of Lewis Carroll by Douglas R. Hofstadter

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