IST 4 Information and Logic
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1 IST 4 Information and Logic
2 T = today x= hw#x out x= hw#x due mon tue wed thr fri 30 M1 1 6 oh M1 oh 13 oh 1 oh 2M2M 20 oh oh 2 T Mx= MQx out 27 oh M2 oh oh = office hours oh 3 4 oh oh midterms oh Mx= MQx due 18 oh oh oh 5 oh oh oh
3 MQ2 Memory Due Tuesday 4/28/2015 by 10pm Please PDF lastname-firstname.pdf to
4 Source: Wikipedia The geography of innovation We need the whole world!
5 Last time... The geography of numbers... Algorizmi Fibonacci AD Gottfried Leibniz base 10 base 10 in Europe The binary
6 Binary Adder Gottfried Leibniz The syntax box is a recent concept... digit 1 digit 2 3 bits to 2 bits carry 2 symbol adder carry Represent the number of 1s in the input as two bits in base 2 sum
7 Name this box? d1 d2 Name this box? majority c 2 symbol adder c parity s c d1 d2 c c d1 d2 s Represent the number of 1s in the input as two bits in base 2
8 A language for binary syntax boxes d1 d2 majority c 2 symbol adder c s parity
9 The Magic Box and the Binary Adder A binary syntax box that can compute any binary syntax box Can compute: a b m a b parity M majority d1 d2 m c 2 symbol adder s c
10 Progress starts with the introduction of new languages Languages help in reasoning about the invisible Languages help in reasoning beyond our natural sense Gottfried Leibniz building blocks separation Leibniz was the first person to understand that the key in Information is the LANGUAGE The founder of Information Science
11 Gottfried Leibniz What do you see? Leibniz, Bernoulli and???? separation Jakob Bernoulli Swiss Mathematician Law of large numbers
12 Gottfried Leibniz Leibniz, Bernoulli and Squares Jakob Bernoulli What do you see?
13 Gottfried Leibniz What do you see? Leibniz, Bernoulli and Squares Proof? Statement: Gray bits alternate between 0 and Square of even number is even - Square of odd number is odd In base-2, even number ends with a 0 odd number ends with a Jakob Bernoulli
14 Gottfried Leibniz What do you see? Leibniz, Bernoulli and Squares Statement: The second bit of squares is always Proof? Jakob Bernoulli
15 Gottfried Leibniz Proof Jakob Bernoulli Statement: The second bit of squares is always 0 Even number: Odd number: Key observation: What is the remainder if we divide a square by 4? 0 1
16 Gottfried Leibniz Proof Jakob Bernoulli If we divide a square by 4 the remainder is either 0 or 1 Statement: The second bit of squares is always The two right most bits in the binary representation 16 are the remainder if we divide by
17 Gottfried Leibniz Leibniz The Founder of Information Science Contributed to: Mathematics Physics Logic Probability Computing Philosophy Politics Law History Library science His work was recognized mainly starting 1900
18 Gottfried Leibniz Leibniz Information and Logic Leibniz was born on July 1 st, 1646 in Leipzig When Leibniz was six years old, his father, a Professor of Moral Philosophy at the University of Leipzig, died, leaving a personal library to which h Leibniz i had free access His mother Catharina instilled the love for studying in him, By 12 he had taught himself Latin... He entered the University of Leipzig at age 14, and completed university studies by 20 Source: Wikipedia
19 Gottfried Leibniz Leibniz Information and Logic U. of Leipzig declined to award him a doctorate because of his young age...??... Leibniz submitted the thesis he had intended to submit at Leipzig to the University of Altdorf and obtained his doctorate in law... He never held an academic position, and spent the rest of his life in the service of German noble families
20 Gottfried Leibniz He never held an academic position, and spent the rest of his life in the service of German noble families How does a lawyer/diplomat l become an inventor/major contributor to calculus, algorithms, logic,...???? curiosity passion for learning mentoring
21 Gottfried Leibniz He never held an academic position, and spent the rest of his life in the service of German noble families Went to Paris in 1672 (was 26) on a diplomatic mission, stayed there until 1676 curiosity passion for learning mentoring
22 Paris, Gottfried Leibniz mentoring / inspiration Christiaan Huygens Dutch mathematician French academy of sciences Galileo Galilei Inventor of the pendulum clock... Time keeping for ~300 years... One meter? A 2 seconds degrees Source: Wikipedia
23 Cartesian coordinate system Reasoning about motion in 3 dimensions?
24 Rene Descartes Natural boundaries lead to new languages Can you reason about - 4 dimensions? - Motion in 3 dimensions? Analytic geometry: French mathematician who spent most his life in the Dutch Republic was a friend of the father of C. Huygens Use Algebra to reason about Geometry
25 Thales' theorem Thales BC Considered to be the first western philosopher/mathematician Source: Wikipedia
26 Thales' Thm: Proof? Thales BC isosceles triangles Source: Wikipedia
27 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
28 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
29 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
30 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
31 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
32 Thales' Thm: Proof? Rene Descartes Source: Wikipedia
33 Rene Descartes Gottfried Leibniz Isaac Newton Use Algebra to reason about Geometry Led to the invention of Calculus by Leibniz and Newton A new language for describing the physical world!!!
34 Gottfried Leibniz Paris, mentoring / inspiration Christiaan Huygens Blaise Pascal Pierre de Fermat Rene Descartes
35 Gottfried Leibniz Leibniz Information and Computing Blaise Pascal Whil i P i t d th fi t hi bl f While in Paris created the first machine capable of computing multiplication, division, square roots
36 Gottfried Leibniz Leibniz and Language Languages for everything!!!! "The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply py say: Let us calculate [calculemus], without further ado, to see who is right" (The Art of Discovery 1685)
37 Gottfried Leibniz Leibniz and Language Characteristica Universalis: Leibniz s goal was to develop an alphabet of human thought, a universal symbolic language (characteristic) to describe nature Leibniz s dream: 300 years later, it is still a dream... A magic box for nature...
38 Gottfried Leibniz Leibniz was born in 1646 in Leipzig Visited or lived in: Germany France England Austria He died in Hannover in 1716
39 Gottfried Leibniz German food company p y based in Hannover
40 Gottfried Leibniz The Leibniz Cookie by the Bahlsen food company in Hannover Hermann Bahlsen's s original 1891 cookie design
41 A Stamp or a Cookie?
42
43
44 Gottfried Leibniz Leibniz and Language Characteristica Universalis: Leibniz s goal was to develop an alphabet of human thought, a universal Motivated symbolic language by (characteristic) the logic of to describe nature Leibniz s dream: Aittl Aristotle 300 years later, it is still a dream... A magic box for nature...
45 Aristotle Aristotle BC Philosophy?? Love of knowledge Aristotle was a Greek philosopher who wrote on diverse subjects, including physics, metaphysics, poetry, biology and zoology, logic, rhetoric, politics, government, and ethics. He was a student of Plato and teacher of Alexander the Great
46 Aristotle BC Aristotle and Logic Aristotle key motivation for developing logic was to provide a procedure for checking the validity of arguments and to derive new conclusions based on existing premises No homework is fun IST4 assignments are homework IST4 should not have assignments
47 Aristotle BC Let us calculate - syntax the correct answer - semantics A (Greek: "conclusion", "inference"), is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form No homework is fun IST4 assignments are homework IST4 should not have assignments
48 Aristotle BC Aristotle and Logic The Babylonians were wise people Leibniz was a wise person Leibniz i was a Babylonian??? People that are wise are Babylonians Leibniz was wise Leibniz was a Babylonian
49 Aristotle and Logic The Babylonians were wise people Leibniz was a wise person Leibniz was a Babylonian
50 Aristotle and Logic People that are wise are Babylonians Leibniz was wise Leibniz was a Babylonian
51 Aristotle BC Aristotle and Logic A syllogism (Greek: "conclusion", " "inference"), ") Developed an algorithm (syntax based) to solve syllogisms A statement has a rigid form and includes two terms Aristotle considered four types of statements: A: All S is P A: All pirates are thieves E: No S is P I: Some S is P O: Some S is not P E: No pirates are reliable I: Some thieves are pirates O: Some pirates are not reliable Some = at least one
52 - All babies are illogical - Nobody is despised who can manage a crocodile - Illogical persons are despised - All babies are illogical - Nobody is despised who can manage a crocodile - Illogical persons are despised babies cannot manage a crocodile
53 Lewis Carroll
54 ???? 1. All, who neither dance on tight ropes nor eat penny-buns, are old. 2. Pigs that are liable to giddiness are treated with respect. 3. A wise balloonist takes an umbrella with him. 4. No one ought to lunch in public who looks ridiculous and eats penny-buns. 5. Young creatures who go up in balloons are liable to giddiness. 6. Fat creatures who look ridiculous may lunch in public if they do not dance on tight-ropes. 7. No wise creatures dance on tight-ropes if they are liable to giddiness. 8. A pig looks ridiculous carrying an umbrella. 9. All, who do not dance on tight-ropes and who are treated with respect, are fat.
55 NO WISE PIGS GO UP IN BALLONS NO!!
56 Gottfried Leibniz is a small number need a number system for logic... Leibniz tried very hard to solve this problem!!
57 ~2000 years later , Algebra of Logic George Boole a number system for logic...
58 Syllogism to Algebra George Boole, 1847 Separation: The validity of algebra depends d only on syntax Syntax Semantics
59 Syllogism to Algebra George Boole, 1847 Criticizing the fact that Mathematics is perceived as only the Science of Magnitude
60 Syllogism to Algebra George Boole, 1847 Calculus of Logic is part of Mathematics!!
61 Syllogism to Algebra George Boole, 1847 Connection to the past - Aristotle A: All S is P E: No S is P I: Some S is P O: Some S is not P
62 Syllogism to Algebra George Boole, 1847 In the class = 1 Not in the class = 0 X not-x If x is 1, (1-x) is 0 If (1-x) is 1, x is 0
63 Syllogism to Algebra George Boole, 1847 In the class = 1 Not in the class = 0 X not-x If x is 1, (1-x) is 0 If (1-x) is 1, x is 0
64 Syllogism to Algebra George Boole, 1847 A: All X is Y If x is 1, y has to be 1 If x is 0, y is either 0 or 1 X Y
65 Syllogism to Algebra G B l 1847 George Boole, 1847 A: All X is Y x y A is true If x is 1, y has to be 1 If x is 0, y is either 0 or 1 X Y
66 Syllogism to Algebra G B l 1847 George Boole, 1847 A: All X is Y x y A is true
67 Theodore Hailperin February 5, 2014 Calculus of Logic can express Aristotle s syllogism, and MORE However, there were many issues with Boole s formalization... Need a new Algebra! Boolean Algebra
68 Historical notes: Boolean Algebra Pre-Boole (16xx): Leibniz; universal language for reasoning Inception (18xx): Boole, Jevons, Peirce, Venn, Schroder, De Morgan Next step: Huntington 1904; concise set of axioms (four) Sheffer 1913: Five axioms and one binary operation Huntington/Robbins 1933: three axioms, conjecture Recent progress : McCune 1996; Robbins conjecture proved!
69 The Algebra (Boolean Calculus) Boole, Jevons, Peirce, Schroder (18xx) Axiomatic System: Huntington t (1904) Algebraic system: set of elements B, two binary operations + and B has at least two elements (0 and 1) If the following axioms are true then it is a Boolean Algebra: A1. identity A2. complement A3. commutative A4. distributive
70 The Algebra (Boolean Calculus) Boole, Jevons, Peirce, Schroder (18xx) Axiomatic System: Huntington t (1904) Algebraic system: set of elements B, two binary operations + and Questions??? Distributive axiom looks strange? B has at least two elements (0 and 1) If The the complement following axioms is not well are defined, true unique? then it is a Boolean Algebra: A1. identity A2. complement A3. commutative Associativity? A4. distributive ( ) ( ) ( ) a=1, b=1, c=1
71 Two Values or not Two Values? Algebraic system: set of elements B, two binary operations + and More questions??? B has at least two elements (0 and 1) Yes Can a Boolean algebra have more than two elements? No Can a Boolean algebra have three elements? Yes Can a Boolean algebra have four elements?
72 Axioms to Theorems Algebraic system: set of elements B, two binary operations + and B has at least two elements (0 and 1) Axioms: Building blocks for developing the theory of Boolean Algebra
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