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This is not the title of our seminar Shikhar Paliwal Varun Suprashanth Asok R Under Guidance of Dr. Pushpak Bhattacharya 113-03-2012 Slide 2 Outline Introducing Gdel, Escher and Bach. Self-Reference. A Brief History Of Logic. Consistency vs. Completeness. Impact of Gdel's Theorem. 213-03-2012 Slide 3 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. 313-03-2012 Slide 4 Introducing Bach The extent of this exaggeration although bordering stupidity gives us a picture of Bachs 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. 413-03-2012 Slide 5 Musical Loops and Riddles Another important aspect of Bachs 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 its musical loops. 513-03-2012 Slide 6 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. 613-03-2012 Slide 7 7 Slide 8 The Letter In the letter that Bach send the King we find a phrase which translates into As the modulation grows so does the Kings glory. Bach was basically hinting at the never ending nature of the modulations, cryptically saying that the Kings glory is ever growing. 813-03-2012 Slide 9 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. 913-03-2012 Slide 10 Eschers 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. 1013-03-2012 Slide 11 Eschers Paintings 1113-03-2012 Slide 12 Eschers Paintings 1213-03-2012 Slide 13 Introducing Gdel Kurt Friedrich Gdel Gdel 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. Gdel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age. 1313-03-2012 Slide 14 Introducing Gdel Gdel published his incompleteness theorems in ber formal unentscheidbare Stze 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 dont fall prey to any paradox and can be automated 1413-03-2012 Slide 15 What connects them? 1513-03-2012 Slide 16 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 Gdel, Escher and Bachs works. 1613-03-2012 Slide 17 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. 1713-03-2012 Slide 18 Recursive Acronyms Recursive Acronyms is another area where self- reference is liberally used especially by software developers. E.g. : GNU - GNUs Not Unix BING- BING Is Not Google ACME- Acme Company Makes Everything PNG- PNGs Not GIF these are only a few examples. 1813-03-2012 Slide 19 Tuppers 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. http://en.wikipedia.org/wiki/File:Tupper%27s_self_referential_formula_plot.png 1913-03-2012 Slide 20 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. 2013-03-2012 Slide 21 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. 2113-03-2012 Slide 22 A Brief History Of Logic George Boole and De Morgan took Aristotles 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. 2213-03-2012 Slide 23 Cantors 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 Cantors sets. Chief among them was Russells Paradox by Bertrand Russell. 2313-03-2012 Slide 24 Russells 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]. 2413-03-2012 Slide 25 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. 2513-03-2012 Slide 26 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. 2613-03-2012 Slide 27 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. 2713-03-2012 Slide 28 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. 2813-03-2012 Slide 29 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. 2913-03-2012 Slide 30 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. 3013-03-2012 Slide 31 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. 3113-03-2012 Slide 32 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. 3213-03-2012 Slide 33 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. 3313-03-2012 Slide 34 Theorems of Incompleteness Many mathematicians spent considerable time on this problem. But it was not until Kurt Gdel 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. 3413-03-2012 Slide 35 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! 3513-03-2012 Slide 36 Gdel'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. 3613-03-2012 Slide 37 The Irony The final irony of it all is that the proof of Gdel'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 Gdel came up with. He used this numbering system to bring the Epimenides paradox into Principia Mathematica without violating any of its rules. 3713-03-2012 Slide 38 The Unprovable In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gdel 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. 3813-03-2012 Slide 39 3913-03-2012 Slide 40 Impact of Gdel's Theorem These theorems ended half a century of attempts, beginning with the work of Frge and culminating in Principia Mathematica and Hilberts formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable. 4013-03-2012 Slide 41 Impact of Gdel's Theorem Authors including J. R. Lucas have debated what, if anything, Gdel'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 Gdel's incompleteness theorems would apply to it. 4113-03-2012 Slide 42 References Gdel, Escher, Bach: An Eternal Golden Braid, A Metamorphical Fugue on Minds and Machines in the Spirit of Lewis Carroll by Douglas R. Hofstadter http://en.wikipedia.org/wiki/ http://www.mcescher.com/ http://www.strangepaths.com/it http://www.xkcd.com/ 4213-03-2012