Week 12 The Universal Representation: The Computer and Digitalization

1. Week 12 The Universal Representation: The Computer and Digitalization

2. Sources: www.iu.edu/~emusic/361/iuonly/slides/digitalaudio.ppt www.cs.virginia.edu/~evans/cs150/classes/class24/lecture24.ppt www.computinghistorymuseum.org/teaching/.../pptlectures/History.ppt www.educationworld.com/a_lesson/TM/computer%20history1.ppt

3. First Computing Machine:Abacus • 3000 BCE, early form of beads on wires, used in China,… • From semitic abaq, meaning dust. • Still in use today

4. Mechanical Reasoning: Logic Aristotle (~350BC): Organon Codify logical deduction with rules of inference (syllogisms) Every A is a P X is an A X is a P • Every human is mortal. • Gödel is human. • Gödel is mortal.

5. Greek Logic • Euclid (~300BC): Elements • We can reduce Geometry to a few axioms and derive the rest by following rules of • Propositional Logic • Constants: False, True (Binary Logic: Two values) • Symbols 0,1 • Variables: p, q, r, … • Punctuation: ( ) • Connectives: • (not p), • ( p and q), • ( p or q), • ( p implies q, p only if q, if p then q, conditional), • (p if and only if q) • Well-formed formula (wff)

6. Algorithm (825AD) Mathematical “Recipe” for solving a class of problems. Al-Khwārizmī, muslim Persian astronomer and mathematician, wrote a treatise in the arabic language in 825 AD, On Calculation with Hindu–Arabic numeral system.

7. BLAISE PASCAL (1623 - 1662) • In 1642, the French mathematician and philosopher Blaise Pascal invented a calculating device that would come to be called the "Adding Machine".

8. BLAISE PASCAL (1623 - 1662) • Originally called a "numerical wheel calculator" or the "Pascaline", Pascal's invention utilized a train of 8 moveable dials or cogs to add sums of up to 8 figures long. As one dial turned 10 notches - or a complete revolution - it mechanically turned the next dial. • Pascal's mechanical Adding Machine automated the process of calculation. Although slow by modern standards, this machine did provide a fair degree of accuracy and speed.

9. Gottfried Wilhelm von LEIBNIZ(1646-1716) • Computing Machine (1679) • Binary Numbers (1701)

10. Binary Numbers 1. Computers use Binary Numbers.2. What is a Character? 3. What are the Characters in the English Alphabet? A, B, C, …., Z (there are 26 of these)4. We combine these Characters to make Words: CAT, HAT, …5. What are the Characters in the Decimal Number System? 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (there are how many? 10!)6. We combine these to make Decimal Numbers: 12, 34, … (we add columns of 10, 100, … as needed)7. In the Binary Number System, there are only two characters: 0, 1 …(so we add columns of 2, 4, 8, 16, … as needed)8. Now, Let’s learn how to Match a Decimal Number to a Binary Number…

11. Binary Numbers Decimal Binary10’s 1’s 16’s 8’s 4’s 2’s 1’s 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 2 0 0 0 1 0 0 3 0 0 0 1 1 0 4 0 0 1 0 0 0 5 0 0 1 0 1 0 6 0 0 1 1 0 0 7 0 0 1 1 1 0 8 0 1 0 0 0 0 9 0 1 0 0 1

12. Jacquard Loom (1801)Mechanical Computer • first stored program - metal cards • first computer manufacturing • still in use today!

13. Charles Babbage Analytical Engine • Difference Engine c.1822 • huge calculator, never finished • Analytical Engine 1833 • could store numbers • calculating “mill” used punched metal cards for instructions • powered by steam! • accurate to six decimal places

16. Importance of the Difference Engine • 1. First attempt to devise a computing machine that was automatic in action and well adapted, by its printing mechanism, to a mathematical task of considerable importance.

17. born on 10 December 1815. named after Byron's half sister, Augusta, who had been his mistress. Ada Augusta Byron, 1815-1852

18. Ada Augusta Byron, Countess of Lovelace1842 • Translated Menebrea’s paper into English • Taylor’s: “The editorial notes are by the translator, the Countess of Lovelace.” • Footnotes enhance the text and provide examples of how the Analytical Engine could be used, i.e., how it would be programmed to solve problems! • First Algorithm • “world’s first programmer”

19. Logic

20. Mathematics and Mechanical Reasoning • Newton (1687): Philosophiæ Naturalis Principia Mathematica • We can reduce the motion of objects (including planets) to following axioms (laws) mechanically

21. Mechanical Reasoning • Late 1800s – many mathematicians working on codifying “laws of reasoning” • George Boole, Laws of Thought • Augustus De Morgan • Whitehead and Russell, 1911-1913 • Principia Mathematica • Attempted to formalize all mathematical knowledge about numbers and sets

22. All true statements about numbers

23. Perfect Axiomatic System Derives all true statements, and no false statements starting from a finite number of axioms and following mechanical inference rules.

24. Incomplete Axiomatic System incomplete Derives some, but not all true statements, and no false statements starting from a finite number of axioms and following mechanical inference rules.

25. Inconsistent Axiomatic System Derives all true statements, and some false statements starting from a finite number of axioms and following mechanical inference rules. some false statements

26. Principia Mathematica • Whitehead and Russell (1910– 1913) • Three Volumes, 2000 pages • Attempted to axiomatize mathematical reasoning • Define mathematical entities (like numbers) using logic • Derive mathematical “truths” by following mechanical rules of inference • Claimed to be complete and consistent • All true theorems could be derived • No falsehoods could be derived

27. Russell’s Paradox • Some sets are not members of themselves • set of all Even Numbers • Some sets are members of themselves • set of all things that are non-Even Numbers • S = the set of all sets that are not members of themselves • Is S a member of itself?

28. Russell’s Paradox • S = set of all sets that are not members of themselves • Is S a member of itself? • If Sis an element of S, then Sis a member of itself and should not be in S. • If S is not an element of S, then Sis not a member of itself, and should be in S.

29. Epimenides Paradox Epidenides (a Cretan): “All Cretans are liars.” Equivalently: “This statement is false.” Russell’s types can help with the set paradox, but not with these.

30. Kurt Gödel • Born 1906 in Brno (now Czech Republic, then Austria-Hungary) • 1931: publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme(On Formally Undecidable Propositions of Principia Mathematica and Related Systems)

31. Gödel’s Solution All consistent axiomatic formulations of number theory include undecidable propositions. undecidable – cannot be proven either true or false inside the system.

32. Gödel’s Theorem In the Principia Mathematica system, there are statements that cannot be proven either true or false.

33. Gödel’s Theorem In any interesting rigid system, there are statements that cannot be proven either true or false.

34. Proof – General Idea • Theorem: In the Principia Mathematica system, there are statements that cannot be proven either true or false. • Proof: Find such a statement

35. Gödel’s Statement G: This statement does not have any proof in the system of Principia Mathematica. G is unprovable, but true!

36. Gödel’s Proof Idea G: This statement does not have any proof in the system of PM. If G is provable, PM would be inconsistent. If G is unprovable, PM would be incomplete. Thus, PM cannot be complete and consistent!

37. On Computable Numbers with an application to the Entscheidungs-problem (1936) Code breaking: Enigma Alan Turing (1912-1954)

38. Turing Machines, 1936 • Universal Computing machine. • Precise vocabulary: 0, 1 • Class of primitive • operations: • Read • Write • Shift Left • Shift Right • Well Formed Sequences • Correctness • Completeness • Equivalence • Complexity

39. http://aturingmachine.com/

40. Herman Hollerith (1860-1929)

41. Herman Hollerith • Born: February 29, 1860 • Civil War: 1861-1865 • Columbia School of Mines (New York) • 1879 hired at Census Office • 1882 MIT faculty (T is for technology!) • 1883 St. Louis (inventor) • 1884 Patent Office (Wash, DC) • 1885 “Expert and Solicitor of Patents”

42. Census • Article I, Section 2: Representatives and direct Taxes shall be apportioned among the several states...according to their respective numbers...(and) every ...term of ten years • 1790: 1st US census • Population: 3,929,214 • Census Office

43. Population Growth: • 1790 4 million • 1840 17 million • 1870 40 million • 1880 50 million fear of not being able to enumerate the census in the 10 intervening years • 1890 63 million

45. Computing Tabulating Recording Company,(C-T-R) • 1911: Charles Flint • Computing Scale Company (Dayton, OH) • Tabulating Machine Company, and • International TimeRecordingCompany (Binghamton, NY)

46. IBM (1924) • Thomas J. Watson (1874-1956) hired as first president • In1924, Watson renames CTR as International Business Machines

47. Vacuum Tubes - 1941 - 1956 • First Generation Electronic Computers used Vacuum Tubes • Vacuum tubes are glass tubes with circuits inside. • Vacuum tubes have no air inside of them, which protects the circuitry.

48. HOWARD AIKEN (1900 - 1973) • Aiken thought he could create a modern and functioning model of Babbage's Analytical Engine. • He succeeded in securing a grant of 1 million dollars for his proposed Automatic Sequence Calculator; the Mark I for short. From IBM. • In 1944, the Mark I was "switched" on. Aiken's colossal machine spanned 51 feet in length and 8 feet in height. 500 meters of wiring were required to connect each component.

49. HOWARD AIKEN • The Mark I did transform Babbage's dream into reality and did succeed in putting IBM's name on the forefront of the burgeoning computer industry. From 1944 on, modern computers would forever be associated with digital intelligence.

50. ENIAC 1946 • Electronic Numerical Integrator And Computer • Under the leadership of J. Presper Eckert (1919 - 1995) and John W. Mauchly (1907 - 1980) the team produced a machine that computed at speeds 1,000 times faster than the Mark I was capable of only 2 years earlier. • Using 18,00-19,000 vacuum tubes, 70,000 resistors and 5 million soldered joints this massive instrument required the output of a small power station to operate it.