On Creating Mathematics:
What Arthur and Blaise never knew
Informal questions for
Mathematicians at parties:
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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…
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The Science of Numbers
Problem Solving
Theorem Proving
The Science of Reasoning
The Science of Patterns (Keith Devlin)
Mathematics is like…
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A language
A science
An art
A process
Mathematics is…
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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?
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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)
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Mathematical Discovery/Creation
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Mathematics as a language related to music
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April 17,1999 http://www.npr.org/ramfiles/wesat/19990417.wesat.17.ram
September 9, 2000 http://www.npr.org/ramfiles/wesat/20000909.wesat.15.ram
Applications of Mathematics—Knot Theory & DNA
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February 24, 2001 http://www.npr.org/ramfiles/wesat/20010224.wesat.12.ram
Keith Devlin on the Nature of
Mathematics
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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
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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:
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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…
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What Arthur and Blaise never
knew…
Blaise Pascal (1623-1662)
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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
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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
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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…)
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“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’)
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Timeline:
http://www.norfacad.pvt.k12.va.us/project/pascal/timeline.htm
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Mathematical References:
http://www-groups.dcs.stand.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’)
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References to 53 books and articles:
http://www-groups.dcs.st-and.ac.uk/~history/References/Pascal.html
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General References:
http://www.newadvent.org/cathen/11511a.htm
http://www.ccel.org/p/pascal/pensees/pensees01.htm
Arthur Cayley (1821-1895)
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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)
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Founded the theory of
trees in two papers in the
Philosophical Magazine:
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On the theory of the
analytical forms called
trees.
On the mathematical
theory of isomers.
Applied trees to chemical
structure of saturated
hydrocarbons:
Ck H
Reference: Discrete Mathematics,
Washburn, et.al.
(2k  2) :
C1 H
4,
C2 H
6,
C 3 H 8 ,...
(See overhead of butane structure
and other trees)
James Sylvester (1814-1897)
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Sources:
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)
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…
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Perhaps mathematics is communicated via its
special language…but new mathematical
concepts do not always originate from “a
language.”
Albert Einstein (1879-1955)
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References:
“The words or the
language as they are
written or spoken, do
not seem to play any
role in my mechanism
of thought….”
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
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“…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')
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Reference:
“…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.”
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html
Logic—Patterns of Reasoning
George Boole (1815-1864)
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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 & Axioms
Euclid’s 5 postulates (axioms) for geometry:
1.
2.
3.
4.
5.
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.
Geometry as an Axiomatic
System…Theorems and Models
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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
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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
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Axioms 1’,2’,3, 4 + Elliptic Axiom:
“Two lines always intersect.”
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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)
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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)
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http://www.treasuretroves.com/bios/CantorGeorg.html
“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…
Bertrand Russell sent a letter to
Frege…..
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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
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Consider the following sentences:
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Question:
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“I am now lying to you.”
“This statement is false.”
Are these statements true or false?
Even a biblical example of this conundrum…
…Paul’s comments about Crete
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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…
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“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
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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)
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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 Logicians
Boole
Whitehead
Zermelo
Finsler
de Morgan
Quine
Fraenkel
Scott
Cantor
Bernays
Church
Aczel
Hilbert
von
Neumann
Turing
Russell
Godel
Takeuti
Barwise
Moss
Devlin
Appendix B: Work of Grant
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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)
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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 (selfreferential) presentation.
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On Creating Mathematics: