On Creating Mathematics:

1. On Creating Mathematics: What Arthur and Blaise never knew

2. Informal questions for Mathematicians at parties: • What was the title of your dissertation? (snicker-snicker) • What was your research?… (i.e., What is left to study in mathematics? A new way to add or multiply? • What exactly do mathematicians do?

3. Mathematics is… • The Science of Numbers • Problem Solving • Theorem Proving • The Science of Reasoning • The Science of Patterns (Keith Devlin)

4. Mathematics is like… • A language • A science • An art • A process

5. Mathematics is… • Vast (Mac Lane’s Connections) • Performed in a wide variety of ways • By a wide variety of people (See overhead of the connections within Calculus; also the overhead on the historical development of Probability)

6. What do mathematicians do? • Add, Multiply, Subtract, Divide, etc. • Do Algebra, Make Geometry T-Proofs • Solve Problems, Model Nature • Experiment, Conjecture, Prove • Precisely identify assumptions (axioms) • Precisely define terms • Categorize, Classify, Generalize, Reason

7. Is New Mathematics … Discovered, Invented, or Created?

8. Keith Devlin on Mathematics:(The “Math Guy” with Scott Simon on NPR’s Weekend Edition) • Mathematical Discovery/Creation • April 17,1999 http://www.npr.org/ramfiles/wesat/19990417.wesat.17.ram • Mathematics as a language related to music • September 9, 2000 http://www.npr.org/ramfiles/wesat/20000909.wesat.15.ram • Applications of Mathematics—Knot Theory & DNA • February 24, 2001 http://www.npr.org/ramfiles/wesat/20010224.wesat.12.ram

9. Keith Devlin on the Nature of Mathematics • Mathematics is the Science of Patterns • Not only the patterns of numbers (arithmetic)… • But also the patterns of shapes (geometry), reasoning (logic), motion (calculus), surfaces and knots (topology), etc. Reference: Mathematics--The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library)

10. Mathematics is like…Music • Both appreciated by many professional scientists and mathematicians • Similar tasks in learning: practice, drill, learn a language, learn to sight-read, learn aesthetics • In Tasks and Roles: • Teach/Study • Compose (Experiment-Conjecture-Prove, Invent new mathematical ideas) • Conduct (Seminar Presentation at a Conference) • Perform (trained student of mathematics) • Improvise (problem solve: do all of the above…)

11. On Creating Mathematics… • What Arthur and Blaise never knew…

12. Blaise Pascal (1623-1662) • French mathematician, philosopher, and religious figure • Projective geometry • Mechanical adding machine • Religious perspective Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html

13. Pascal’s Calculating Machine • 1642-1645 Designed a mechanical calculator to assist his father’s role of examining all tax records of the Province of Normandy. • Provided a monopoly (“patent”) in 1649 by the king of France. Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html

14. Pascal--The Mathematical Prodigy • At age sixteen Blaise published an Essay pour les coniques. • “This consisted of only a single printed page—but one of the most fruitful pages in history.” • “It contained the proposition, described by the author as mysteriumhexagrammicum…” Source: Carl Boyer, A History of Mathematics

15. Pascal’s Mystic Hexagram Reference: The MacTutor History of Mathematics archive— http://www-groups.dcs.st-and.ac.uk/~history/index.html

16. Pascal’s Spiritual side…Memorial de Pascal FIRE “In the year of Grace, 1654, On Monday, 23rd of November, Feast of St. Clement, Pope and Martyr, and of others in the Martyrology, Virgil of Saint Chrysogonus, Martry, and others, From about half past ten in the evening until about half past twelve… Source: Emile Cailliet, Pascal: The emergence of genius

17. Pascal’s Spiritual side…Memorial de Pascal, (cont’) FIRE “God of Abraham, God of Isaac, God of Jacob, not of the philosophers and scholars. Certitude. Certitude. Feeling. Joy. Peace. God of Jesus Christ…. ‘Thy God shall be my God.’… Joy, joy, joy, tears of joy…Total submission to Jesus Christ… Eternally in joy for a day’s exercise on earth.” Source: Emile Cailliet, Pascal: The emergence of genius

18. Pascal’s scientific/mathematical interests after Memorial Renunciation: • Pascal refrains from publishing mathematical treatises already printed. • During his lifetime nothing more will appear under his name. • Mathematical treatises were published in 1658 and in 1659 anonymously under the name of “Amos Dettonville.” Source: Emile Cailliet, Pascal: The emergence of genius

19. Pascal: Mathematics & Religion (and the Sociology of Mathematics…) • “Desargues was the prophet of projective geometry, but he went without honor in his day largely because his most promising disciple, Blaise Pascal, abandoned mathematics for theology.” --Carl Boyer in A History of Mathematics

20. Pascal (cont’) • Timeline: http://www.norfacad.pvt.k12.va.us/project/pascal/timeline.htm • Mathematical References: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html http://www.treasure-troves.com/bios/Pascal.html http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html

21. Pascal (cont’) • References to 53 books and articles: http://www-groups.dcs.st-and.ac.uk/~history/References/Pascal.html • General References: http://www.newadvent.org/cathen/11511a.htm http://www.ccel.org/p/pascal/pensees/pensees01.htm

22. Arthur Cayley (1821-1895) • A brilliant English mathematician • With an “uncanny memory” • An avid mountain climber and novel reader • Did extensive work in algebra and pioneered the study of matrices • Unified metric and projective geometries Source: http://www.treasure-troves.com/bios/Cayley.html

23. Arthur Cayley (1821-1895) • Founded the theory of trees in two papers in the Philosophical Magazine: • On the theory of the analytical forms called trees. • On the mathematical theory of isomers. • Applied trees to chemical structure of saturated hydrocarbons: (See overhead of butane structure and other trees) Reference: Discrete Mathematics, Washburn, et.al.

24. James Sylvester  (1814-1897) • An eccentric and gifted English mathematician • A close friend of and collaborator with Cayley • “Absent-minded” • Accomplished as a poet and a musician • Created the notion of differential invariants (at age of 71) Sources:http://www.treasure-troves.com/bios/Sylvester.html and http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html

25. Einstein quote tempers the language metaphor… • Perhaps mathematics is communicated via its special language…but new mathematical concepts do not always originate from “a language.”

26. Albert Einstein  (1879-1955) • “The words or the language as they are written or spoken, do not seem to play any role in my mechanism of thought….” References:http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html and Jacques Hadamard’s The Psychology of Invention in the Mathematical Field.

27. Albert Einstein   • “…The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily produced and combined. These elements are, in my case, of visual and muscular type. Conventional words have to be sought for laboriously.” References:http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html and Jacques Hadamard’s The Psychology of Invention in the Mathematical Field.

28. Albert Einstein—another quote “If I were to have the good fortune to pass my examinations, I would go to Zurich. I would stay there for four years in order to study mathematics and physics. I imagine myself becoming a teacher in those branches of the natural sciences, choosing the theoretical part of them….” Reference:http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html

29. Albert Einstein—(cont')  • “…Here are the reasons which lead me to this plan. Above all, it is my disposition for abstract and mathematical thought, and my lack of imagination and practical ability.” Reference:http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html

30. Logic—Patterns of Reasoning

31. George Boole (1815-1864) • Enjoyed Latin, languages, and constructing optical instruments. • Laid the foundation for modern computing… (See video of Devlin on Boole and our Mathematical Universe—Life by the Numbers) Source:http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Boole.html

32. Geometry as an Axiomatic System…Undefined terms & Axioms Euclid’s 5 postulates (axioms) for geometry: • We can draw a (unique) line segment between any two points. • Any line segment can be continued indefinitely. • A circle of any radius and any center can be drawn. • Any two right angles are congruent. • (Playfair’s Version) Through a given point not on a given line can be drawn exactly one line not intersecting the given line.

33. Geometry as an Axiomatic System…Theorems and Models • Question: Is Euclid’s 5th Axiom independent of the first four…or can we prove it from the first four? • Answer: Independent because there is a valid mathematical model that will satisfy the first four but not the fifth…

34. Hyperbolic Geometry • Axioms 1-4 + Hyperbolic Axiom: “Through a given point, not on a given line, at least two lines can be drawn that do not intersect the given line.”

35. Elliptic (or Spherical) Geometry • Axioms 1’,2’,3, 4 + Elliptic Axiom: “Two lines always intersect.” • The Model: “Draw straight lines on a spherical globe.” • To be straight they must follow great circles. • Start them off “parallel”…and they are destined to meet at two points…just as the lines of longitude meet at the two poles. (See overhead of great circles on a sphere)

36. Georg Cantor (1845-1918) • Developed a systematic study of the “infinite” and transfinite numbers. • Developed new concepts: ordinals, cardinals, and topological connectivity. • His highly original views were vigorously attacked by contemporaries. (See overhead of Cantor in the balance) • http://www.treasure-troves.com/bios/CantorGeorg.html

37. “Naïve” Axiom of Set Theory Comprehension: “From any clearly defined property P, We may specify the set of all sets that have that property.” Examples: E = Empty set = { x | x is not equal to x} (Read: The set of all x such that x is not equal to x.) U = Universal set = {x | x = x} Note: E is not an element of E. U is an element of U. This looked fine…but then…

38. Bertrand Russell sent a letter to Frege….. • Russell’s set = R = {x | x is not an element of x} • Question: Is R in R? Is R not in R? • Neither can be true…(Check it!) • Frege’s work to prove the consistency of his system of logic fell apart… • This problem in foundations became known as “Russell’s Paradox.”

39. Related Semantic Paradoxes • Consider the following sentences: • “I am now lying to you.” • “This statement is false.” • Question: • Are these statements true or false? • Even a biblical example of this conundrum…

40. …Paul’s comments about Crete • Titus 1:12 “Even one of their own prophets has said, ‘Cretans are always liars, evil brutes, and lazy gluttons.’ This testimony is true.” The logician’s half serious question for the Apostle Paul: “Was the prophet lying?”

41. Paradox and Mystery… • “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.” --Albert Einstein

42. Zermelo Fraenkel Set theory • ZF and ZFC are generally assumed to be consistent. • They only allow Separation from already existent sets…not complete comprehension. • Much of the mathematical work in set theory of the past century has involved extending the axiom base, and proving issues of independence and relative consistency (See overheads of list of axioms)

43. Paul Finsler (1894-1970) • Student of Hilbert and Caratheodory • Cartan named a book and a geometric space in his honor • Differential Geometer interested in Logic and Set Theory • Work in Set Theory most widely recognized in 1980’s • His work was later extended by Dana Scott, Peter Aczel, Jon Barwise, and Larry Moss. References: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Finsler.html

44. Two of my research projects:(Extending the ideas of Finsler, Scott, et. al.) • GST: Graph-isomorphism-based Set Theory • (where graph isomorphisms of “element-hood digraphs” • determine set equality) • Bi-AFA: Blending the ideas of Church with those of • Finsler/Scott yields a new set theory with a universal set. • (See overhead of Devlin’s Contemporary Set Theory, • and my overheads of graphs and trees that model sets.)

45. Appendix A: Other Logicians

46. Appendix B: Work of Grant • Type-set articles using TeX • Read, wrote, networked, considered new topics… • Developed Mathematica animations for some concepts of geometry related to logic. (and then convert them to QuickTime format) • Presented parts of this work at a national conference in Symbolic Logic in New York City • Presented other parts at a Bluffton College mathematics seminar as well as during this (self-referential) presentation.