What Integration is and isn’t
Russell W. Howell
Westmont College
Some Approaches
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William Hasker
–
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“Faith-learning integration may be briefly described as a scholarly
project whose goal is to ascertain and to develop integral
relationships which exist between the Christian faith and human
knowledge, particularly as expressed in the various academic
disciplines.”
Karl Barth
–
“Where confession is serious and clear, it must be fundamentally
translatable.”
Some Approaches

Arthur Holmes
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The Idea of a Christian College: Four approaches
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Attitudinal (Augustine, Trueblood)
Ethical (“Middle level” concepts and fact-value relationships)
Foundational (Philosophical perspectives)
Worldview (Pluralistic; open-ended)
In retrospect, Holmes thinks “contribution” may have
been a better choice of words than “integration” when
used in the phrase integration of faith and learning.
Some Implications
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Integration is not
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Integration is more than
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Indoctrination
A defensive apologetic
A trivialized mixing of discipline with faith
Prayer or devotionals before class
An articulated position on a particular issue
Integration is
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A living dialogue between faith and discipline
An infusion of faith into all areas of life
David Hilbert, Early Career
Ph.D. (Königsberg) on February 7, 1885
Topics for defense against “opponents”
• The method of determining absolute
electromagnet resistance by
experiment
• The a priori nature of arithmetic
Immanuel Kant (1724 – 1804):
Proposed the “synthetic a priori”
nature of space and number
David Hilbert, 1886
Kant and the Synthetic A Priori
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Analytic truths
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Synthetic truths
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Those whose predicate is not contained in the subject.
E.g., “Sacramento is the capital of California.”
A priori truths
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Those whose predicate is contained in the subject.
E.g., “A bachelor is an unmarried male.”
Those that are known independently of experience
E.g., “If it’s either raining or snowing and it’s not raining, then it’s snowing.”
A posteriori truths
–
–
Those that are known on the basis of experience.
E.g., “All men are mortal.”
Putting the Terms Together
Analytic
A
p
r
i
o
r
i
A
p
o
s
t
e
r
i
o
r
i
A bachelor is an
unmarried male
The morning star is the
evening star (Kripke)
(A necessary truth—Venus
is necessarily identical to
itself—rather than an
analytic truth)
Synthetic
2+3 = 5
(or, 2+3 have one sum)
Sacramento is the
capital of California
Early Mathematical Triumphs
The Solution to “Gordan’s Problem”
• Every subset of the polynomial ring
k[z1, z2, …, zk] has a finite ideal basis
• Hilbert’s proof was one of existence,
not one of construction
Gordan’s Reaction:
“Das ist nicht Mathematik.
Das ist Theologie!”
David Hilbert, 1890
Georg Cantor
Cantor’s 1874 Result:
The irrationals are uncountable
Hilbert on Cantor’s work:
“...the finest product of mathematical
genius and one of the supreme
achievements of purely intellectual
human activity.”
Leopold Kronecker
“God created the integers, all
else is the work of man.”
Hilbert:
“No one will drive us out of
this paradise that Cantor
has created for us!”
Lutzen Brouwer
Published in topology
Founder of intuitionism
• Troubled by set theory paradoxes
• Insisted on strict constructions
• Denied the validity of the excluded
middle principle
Interaction during a Göttingen lecture:
Student: “You say that we can’t know
whether in the decimal representation of
 ten 9’s occur in succession. Maybe we
can’t know—but God knows!”
Brouwer: “I do not have a pipeline to God.”
Expanding Disciplinary Boundaries
Possibilities for discussion
 Does
God know whether the continuum
hypothesis is true or false?
 Yes:
Mathematical realism
 No: Mathematics as a human construction
P ≠ NP, could God create a polynomial-time
algorithm for the traveling salesman?
 Are the truths of mathematics eternal and
necessary?
 If
Faith-Learning and Collateral Reading
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Elementary Calculus
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Berkeley’s The Analyst
Multivariable Calculus, Linear Algebra
–Edwin
Abbott’sOR,
Flatland
THEAbbott
ANALYST;
A DISCOURSE

Addressed to an Infidel
MATHEMATICIAN.
–Michio
Kaku’s HyperspaceWHEREIN It is examined whether the
Object, Principles, and Inferences of the modern Analysis are
–James Gleick’s Chaos
more distinctly conceived, or more evidently deduced, than
Probability
and
Statisticsand Points of Faith.
Religious
Mysteries
–Plantinga’s
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An Evolutionary Argument Against Naturalism
Real Analysis
–Hardy’s
A Mathematician’s Apology
–Kanigel’s
The Man Who Knew Infinity
Faith-Learning and Collateral Reading
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Artificial Intelligence
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Automata and Formal Languages
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Rudy Rucker’s Infinity and the Mind
Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid
Introductory Programming
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Kurtzweil’s The Age Of Spiritual Machines
Dreyfus’ What Computers Can’t Do
Really, too many to mention
Weizenbaum’s Computer Power and Human Reason
Gene Chase’s article, What does a Computer Program Mean?
See also
http://www.messiah.edu/acdept/depthome/mathsci/acms/bibliog.htm
Analogical Opportunities for Faith
Discussions in Mathematics/CS
Level’s of infinity
 Density of rationals, irrationals vs. inability to
put them in a 1 – 1 correspondence
 Halting Problem
 Das ist nicht Mathematik. Das ist Theologie!
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