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 Lanes 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 NPRs Weekend Edition)

• Mathematical Discovery/Creation
• April 17,1999 http//www.npr.org/ramfiles/wesat/19
  990417.wesat.17.ram
• Mathematics as a language related to music
• September 9, 2000 http//www.npr.org/ramfiles/wesa
  t/20000909.wesat.15.ram
• Applications of MathematicsKnot Theory DNA
• February 24, 2001 http//www.npr.org/ramfiles/wesa
  t/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 likeMusic

• 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/histor
y/Mathematicians/Pascal.html
13
Pascals Calculating Machine

• 1642-1645 Designed a mechanical calculator to
  assist his fathers 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/histor
y/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 pagebut
  one of the most fruitful pages in history.
• It contained the proposition, described by the
  author as mysterium hexagrammicum
• Source Carl Boyer, A History of Mathematics

15
Pascals Mystic Hexagram
Reference The MacTutor History of Mathematics
archive http//www-groups.dcs.st-and.ac.uk/histo
ry/index.html
16
Pascals Spiritual sideMemorial 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
Pascals Spiritual sideMemorial 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 joyTotal submission to
  Jesus Christ
• Eternally in joy for a days exercise on earth.
• Source Emile Cailliet, Pascal The emergence of
  genius

18
Pascals 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/Mathe
  maticians/Pascal.html
• http//www.treasure-troves.com/bios/Pascal.html
• http//www.maths.tcd.ie/pub/HistMath/People/Pasca
  l/RouseBall/RB_Pascal.html

21
Pascal (cont)

• References to 53 books and articles
• http//www-groups.dcs.st-and.ac.uk/history/Refer
  ences/Pascal.html
• General References
• http//www.newadvent.org/cathen/11511a.htm
• http//www.ccel.org/p/pascal/pensees/pensees01.ht
  m

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/Cayle
  y.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/Sy
lvester.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 languagebut 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 Hadamards 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 Hadamards The Psychology of Invention in
  the Mathematical Field.

28
Albert Einsteinanother 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
LogicPatterns 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 UniverseLife by the Numbers)

• Source http//www-groups.dcs.st-and.ac.uk/histor
  y/Mathematicians/Boole.html

32
Geometry as an Axiomatic SystemUndefined terms
Axioms

• Euclids 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.
• (Playfairs 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 SystemTheorems and
Models

• Question Is Euclids 5th Axiom independent of
  the first fouror 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 paralleland they are destined
  to meet at two pointsjust 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.ht
  ml

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 finebut then

38
Bertrand Russell sent a letter to Frege..

• Russells 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!)
• Freges work to prove the consistency of his
  system of logic fell apart
• This problem in foundations became known as
  Russells 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
Pauls comments about Crete

• Titus 112
• Even one of their own prophets has said,
  Cretans are always liars, evil brutes, and lazy
  gluttons.
• This testimony is true.
• The logicians 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
  setsnot 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
  1980s
• 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 Devlins Contemporary Set
  Theory,
• and my overheads of graphs and trees that model
  sets.)

45
Appendix A Other Logicians
Boole Whitehead Zermelo Finsler
de Morgan Quine Fraenkel Scott
Cantor Bernays Church Aczel
Hilbert von Neumann Turing Barwise Moss
Russell Godel Takeuti Devlin
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.

47
Appendix C Other Contacts
Takeuti Van Lambalgan
Henson Yury Serdyuk
Barwise Scott
Moss Devlin
Hajek