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On Creating Mathematics:
What Arthur and Blaise never knew
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?
Mathematics is… The Science of Numbers Problem Solving Theorem Proving The Science of Reasoning The Science of Patterns (Keith Devlin)
Mathematics is like… A language A science An art A process
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)
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
Is New Mathematics …
Discovered, Invented, or Created?
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
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)
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…)
On Creating Mathematics…
What Arthur and Blaise never knew…
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
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
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 mysterium hexagrammicum…”
Source: Carl Boyer, A History of Mathematics
Pascal’s Mystic Hexagram
Reference: The MacTutor History of Mathematics archive— http://www-groups.dcs.st-and.ac.uk/~history/index.html
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
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
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
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
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
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
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
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)
,...8 3 ,6 2 ,4 1 :)22( HCHCHCHC kk
Reference: Discrete Mathematics, Washburn, et.al.
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
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.”
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.
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.
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
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
Logic—Patterns of Reasoning
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
Geometry as an Axiomatic System…Undefined terms & AxiomsEuclid’s 5 postulates (axioms) for geometry:1. We can draw a (unique) line segment between any two
points.
2. Any line segment can be continued indefinitely.
3. A circle of any radius and any center can be drawn.
4. Any two right angles are congruent.
5. (Playfair’s Version) Through a given point not on a given line can be drawn exactly one line not intersecting the given line.
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…
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.”
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)
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
“Naïve” Axiom of Set TheoryComprehension:
“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…
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.”
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…
…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?”
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
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)
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
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.)
Appendix A: Other LogiciansBoole Whitehead Zermelo Finsler
de Morgan Quine Fraenkel Scott
Cantor Bernays Church Aczel
Hilbert von Neumann
Turing Barwise
Moss
Russell Godel Takeuti Devlin
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.