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What Integration is and isn’t. Russell W. Howell Westmont College. Some Approaches. William Hasker

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What Integration is and isn’t

Russell W. Howell

Westmont College

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Some Approaches

  • William Hasker

    • “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.”

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Some Approaches

  • Arthur Holmes

    • The Idea of a Christian College: Four approaches

      • 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.

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Some Implications

  • Integration is not

    • Indoctrination

    • A defensive apologetic

    • A trivialized mixing of discipline with faith

  • Integration is more than

    • Prayer or devotionals before class

    • An articulated position on a particular issue

  • Integration is

    • A living dialogue between faith and discipline

    • An infusion of faith into all areas of life

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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

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Kant and the Synthetic A Priori

  • Analytic truths

    • Those whose predicate is contained in the subject.

    • E.g., “A bachelor is an unmarried male.”

  • Synthetic truths

    • Those whose predicate is not contained in the subject.

    • E.g., “Sacramento is the capital of California.”

  • A priori truths

    • 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.”

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Putting the Terms Together





2+3 = 5

(or, 2+3 have one sum)

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)

Sacramento is the capital of California

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Gordan’s Reaction:

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

“Das ist nicht Mathematik.

Das ist Theologie!”

David Hilbert, 1890

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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.”

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Leopold Kronecker

“God created the integers, all else is the work of man.”


“No one will drive us out of this paradise that Cantor has created for us!”

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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.”

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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

  • If P≠NP, could God create a polynomial-time algorithm for the traveling salesman?

  • Are the truths of mathematics eternal and necessary?

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Faith-Learning and Collateral Reading

  • Elementary Calculus

    • Berkeley’s The Analyst

  • Multivariable Calculus, Linear Algebra

    • Edwin Abbott Abbott’s Flatland

    • Michio Kaku’s Hyperspace

    • James Gleick’s Chaos

  • Probability and Statistics

    • Plantinga’s An Evolutionary Argument Against Naturalism

  • Real Analysis

    • Hardy’s A Mathematician’s Apology

    • Kanigel’s The Man Who Knew Infinity

THE ANALYST; OR, A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.

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Faith-Learning and Collateral Reading

  • Artificial Intelligence

    • Kurtzweil’s The Age Of Spiritual Machines

    • Dreyfus’ What Computers Can’t Do

    • Really, too many to mention

  • Automata and Formal Languages

    • Rudy Rucker’s Infinity and the Mind

    • Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid

  • Introductory Programming

    • Weizenbaum’s Computer Power and Human Reason

    • Gene Chase’s article, What does a Computer Program Mean?

  • See alsohttp://www.messiah.edu/acdept/depthome/mathsci/acms/bibliog.htm

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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!