Transcript Philosophical Principles of Mathematics

Whither Mathematics Education
in the 21st Century?
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by James Nickel,
B.A., B.Th., B.Miss., M.A.
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A Math Exam
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Name and define the Fundamental Rules of
Arithmetic.
District Number 33 has a valuation of $35,000.
What is the necessary levy to carry on a school
seven months at $50 per month and have $104
for incidentals?
Find the interest of $512.60 for 8 months and 18
days at 7 percent.
What is the cost of 40 boards 12 inches wide and
16 feet long at $20 per meter?
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Gulp!
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These questions are part of an 8th grade Final
Exam, Salina, Kansas (1895).
These questions are taken from the original
document on file at the Smokey Valley
Genealogical Society and Library in Salina,
Kansas, and reprinted by the Salina Journal
(www.saljournal.com).
Remember when grandparents and greatgrandparents stated that they only had an 8th
grade education?
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Whither Mathematics Education?
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• Mathematics is a tool to teach:
– Logic.
– Analytical skills.
– The structure and analysis of scientific law.
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• Mathematics is not just a tool … it is a tool
of wonder.
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Questions
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• What perspectives should undergird
mathematics curricula?
• Does God’s Word speak to mathematical
issues?
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The Value of Logic
• “…studied and nearly mastered the six books of
Euclid when he was a member of Congress. He began
a course of rigid mental discipline with the intent to
improve his facilities, especially his powers of logic
and language. Hence his fondness for Euclid, which
he carried with him on the circuit till he could
demonstrate with ease all the theorems in the six
books; often studying far into the night … while his
fellow-lawyers, half a dozen in a room, filled the air
with interminable snoring.”
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Abraham Lincoln (1809-1865), 16th president of the United
States, Biographical sketch, 1860 Presidential campaign.
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The Value of Logic
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“Mathematics as an expression of the human mind
reflects the active will, the contemplative reason,
and the desire for aesthetic perfection. Its basic
elements are logic and intuition, analysis and
construction, generality and individuality … it is
only the interplay [of these elements – JN] ... that
constitute the life, usefulness, and supreme value
of mathematical science.”
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Richard Courant and Harold Robbins, What is
Mathematics? p. xv.
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The Value of Logic
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“[Ratiocination] is the great principle of order in
our thinking; it reduces a chaos into harmony; it
catalogues the accumulation of knowledge; it maps
out for us the relations of its separate departments;
it puts us in the way to correct its own mistakes. It
enables the independent intellects of many, acting
and re-acting on each other, to bring their collective
force to bear upon one and the same subject-matter,
or same question.”
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Logic
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“If language is an inestimable gift to man, the
logical faculty prepares it for our use. Though it
does not go so far as to ascertain truth, still it
teaches us the direction in which truth lies, and how
propositions lie towards each other. Nor is it a slight
benefit to know what is probable, and what is not
so, what is needed for the proof of a point, what is
wanting in theory, how a theory hangs together, and
what will follow, if it be admitted. Though it does
not itself discover the unknown, it is one principal
way by which discoveries are made.”
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John Henry Newman, An Essay in Aid of a Grammar of
Assent, p. 228.
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Logic: Two Aspects
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• Symbolic logic.
• Justification of propositions (as in the
logical flow of mathematical proofs).
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Symbolic Logic
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• Hardware: computer circuitry (i.e.,
integrated circuits).
• Software: Branching statements (If this
condition is true, then do that).
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Problem Solving
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The logical methods of mathematics train
the mind in systematically approaching
and solving, not only mathematical
problems, but …
… other problems as well; especially
those germane to conflict resolution or
ethical infractions.
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Proviso
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This does not mean that mathematical training
guarantees ethical purity or that the ability to
reason guarantees liberty (personally or culturally).
The French Revolution posited the ability to reason
independently is the sole guarantor of liberty.
Truth: Reason is merely a tool of liberty, not the
basis of it.
The deification of reason resulted, not in liberty,
but in heads rolling from the guillotine.
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Failsafe for Logic
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Loving God in Christ (the logos or the
ultimate logic) does not consist of
ratiocination only; it means loving Him
with the totality of your being (Mark
12:29-31).
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Foundational Laws of Logic
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Law of the excluded middle.
Law of contradiction.
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Law of the excluded middle
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For every formal statement B, either B is
true or B is false (thesis/antithesis)
Given B, either B or ~B (B is false).
B: The infinite, personal and Triune God
of Scripture exists and has spoken Truth
in His word …
or ~B.
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Law of contradiction
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A formal statement cannot be both true
and false at the same time.
You cannot say that B and ~B are both
true.
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Thinking antithetically
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The presupposition of antithetical thinking
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Thinking in terms of thesis and antithesis.
A statement is true (thesis) or it is not true
(antithesis) …
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subsumes these two laws.
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Transcendental Nature
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These laws are founded upon the
transcendental nature of thesis/antithesis,
not upon the Greek philosophy Aristotle
(384-322 BC). who formulated them.
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You cannot escape …
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To deny the validity of the law of the
excluded middle assumes that it is either
true or false.
You must assume the law of the excluded
middle in order to deny it.
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You cannot escape …
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To deny the law of contradiction means
that you are saying that the law is false.
By denying this law you are proving it.
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You cannot escape …
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Avicenna (980-1037), Islamic physician
and philosopher from Persia, “Anyone
who denies this law should be beaten and
burned until he admits that to be beaten is
not the same as not to be beaten, and to be
burned is not the same as not to be
burned.”
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You cannot escape …
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These two laws are prerequisite for
coherence of the human experience.
If not, then we have no basis for
intelligible communication.
We have the “forks coming forth from the
tongue” scenario.
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More logic
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Informal logical fallacies: ad hominem
(attack of an opponent’s character) and
equivocation.
Formal logical fallacies:
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Asserting the consequent (converse error).
Denying the antecedent (inverse error).
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Conditional Statements
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PQ
Q  P (Converse)
~P  ~Q (Inverse)
~Q  ~P (Contrapositive).
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Logical Equivalence
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P  Q  ~Q  ~P (Contrapositive).
Q  P (Converse)  ~P  ~Q (Inverse)
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More logic
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Disjunctive syllogisms.
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P or Q.
~P.
Therefore Q.
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Reductio ad absurdum argumentation
(reasoning to contradictions or law of
contraposition).
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Resources
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Morris S. Engel, With Good Reason, 6th
ed. (New York: Bedford Books/St.
Martin’s Press, 2000).
Irving M. Copi, Introduction to Logic, 10th
ed. (Englewood Cliffs, NJ: Prentice-Hall,
1998).
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Analytical Skills
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Don’t believe a statement to be true
simply because numbers are attached to it.
Benjamin Disraeli (1804-1881), British
Prime Minister, “There are three kinds of
lies: lies, damned lies, and statistics.”
Rejoinder: “It is easy to lie with statistics,
but it is easier to lie without them.”
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1936 Presidential “Polls”
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Franklin Delano Roosevelt vs. Alf Landon.
10 million voters were sampled by Literary
Digest.
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10 million voters
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1,293,669 said they would vote for Landon
(53%).
972,897 said they would vote for Roosevelt
(47%).
On the basis of these results, the Literary Digest
predicted that Landon would win the election.
Result: Roosevelt won by a landslide 61% to
37%.
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What went wrong?
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Only 20% responded.
The sample of voters relied heavily on
lists of automobile and telephone owners.
In the 1930s, these people would be the
more affluent voters.
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Random Sample Bias
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Under coverage: some groups of the
population are left out of the process of
choosing the sample.
Non response: sample individuals cannot
be contacted or do not cooperate (give
false responses).
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Buyer, Beware!
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Given a dataset, be very cautious when
making extrapolations or interpolations.
If not, sometime in the distant future,
someone will run the mile in 0 seconds.
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Scientific Law: Structure and
Analysis
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Algebra, as a language, teaches structured
thinking; a thinking that resonates with the
patterns revealed in the physical creation
(scientific law).
How many teachers teach Algebra in its
historical context, a context that connects
it directly to the physics of motion?
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Tool of Wonder
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The fact that a honeybee constructs a
honeycomb to maximize packing space
(for honey) given a minimum of packing
material (beeswax) can be confirmed by
the differential calculus.
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Wonder leads to …
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Wonder is the foundation of creativity.
Creativity is the foundation of invention.
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Inventors
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• Intense persistence.
• Optimism.
• Originality of approach combined with an
almost mystical conviction that there are
more effective, more elegant ways of doing
things.
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Keys to Invention/Creativity
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Contemplation (includes analytical and
logical comparison of one thing to
another).
Imagination.
Wonder.
There are more variables in the equation
of invention and creativity than mere logic
alone.
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Sir Isaac Newton (1642-1727)
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Stanley L. Jaki (1924-) notes Newton’s
reflection on the nature of God’s Works,
“Newtonian science was the product of a
truly inventive intellect pondering the
witness of the senses.”
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The Road of Science and the Ways to God, p. 119.
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Big Question
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“Years ago I went into my laboratory and
said, “Dear Mr. Creator, please tell me
what the universe was made for?”
The Great Creator answered, “You want to
know too much for that little mind of
yours. Ask for something more your size,
little man.”
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Smaller Question
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Then I asked. “Please, Mr. Creator, tell me
what man was made for?”
Again the Great Creator replied, “You are
still asking too much. Cut down on the
extent and improve the intent.”
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Tiny Question
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So then I asked, “Please, Mr. Creator, will
you tell me why the peanut was made?”
“That’s better, but even then it’s infinite.
What do you want to know about the
peanut?”
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The Tiny Peanut
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“Mr. Creator, can I make milk out of the
peanut?”
“What kind of milk do you want? Good
Jersey milk or just plain boarding house
milk?”
“Good Jersey milk.”
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The Peanut
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And then the Great Creator taught me to take
the peanut apart and put it together again.
And out of the process have come for all
these products!
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Cited in Ethel Edwards, Carver of Tuskegee
(Cincinnati: Ethel Edwards & James T.
Hardwick, 1971), pp. 114-117.
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The Peanut
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George Washington Carver (ca. 18641943) discovered nearly 300 derivative
products from his investigation of the
peanut.
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Mathematics Curriculum
What to look for?
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The how and the why.
Unity in diversity.
Induction and deduction.
History.
Operational Science.
Philosophy.
Remove isolation.
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The How and the Why
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A balance between the how (pure
mechanics) and the why (reasoned
justifications).
For example, what is the logical
justification for (-1)(-1) = +1?
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Unity in Diversity
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Show connections.
 (pi), the ratio of the circumference to
the diameter of a circle, and Probability.
Breadsticks, tile floors, and the
calculation of .
Conic Sections, Quadratic equations, and
the Physics of Motion.
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Induction and Deduction
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Balance between induction and
deduction.
Induction: search for patterns (engenders
discovery and wonder).
Deduction: justification of patterns
induced (engenders logical analysis).
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Induction and Deduction
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Johannes Kepler (1571-1630) discovered
the three laws of planetary motion using
induction.
Isaac Newton, starting from his
Universal Law of Gravitation, deduced
Kepler’s three laws!
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Who flunked?
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Martin Gardner (1914-), in his introduction to
the textbook Mathematics: A Human Endeavor
(by Harold R. Jacobs), “An old anecdote tells
of a meeting of the entire faculty of a major
university. The president said his listeners
would be amused to know the names of two
new students who had flunked freshman
classes. A lad named Cicero had failed Latin.
Everybody laughed.”
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Who is Gauss?
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“Another student named Gauss had
failed mathematics. Only the
mathematicians and physicists laughed.
This would not have been the case if the
members of the liberal arts faculty had
explored the world of mathematics with
Harold Jacobs.”
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History
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We must connect mathematical
propositions to the people who
discovered them.
We must know the critical place of
mathematics in the development of
Western Culture.
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Operational Science
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Mathematics is the handmaiden of
science.
Dr. Morris Kline (1908-1992), in
Mathematics and the Search for
Knowledge, p. 144, “Radio waves and
light waves operate in a physical
darkness illuminated only for those who
would carry the torch of mathematics.”
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Philosophy (the love of wisdom)
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Two primary questions.
Cosmological question: What is the
origin, nature, and destiny of the cosmos
(all things)?
Anthropological question: What is the
origin, nature, and destiny of man?
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Four Subsidiary Questions
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The ontological question: What is the
nature of existence?
The epistemological question: What is
the nature and limits of knowledge?
The axiological question: What is
ultimate value?
The teleological question: Where are we
going?
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Answers
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The answers to these four philosophical
questions determine how one answers the two
primary questions.
The answers to the two primary questions
determine how one structures the totality of
life.
The propositions of Scripture generate a
worldview that provides enduring answers to
the larger questions which are applicable to
the totality of life.
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Truth and Consequences: The
totality of life
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Civil/social.
Law.
Economics.
International politics.
Aesthetics.
Ecclesiastical.
Educational …
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Mathematics included!
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Resources
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• Dr. Glenn R. Martin (1935-2004), Prevailing
Worldviews of Western Society Since 1500
(Marion: Triangle Publishing), 2006
(www.trianglepublishing.com).
• The works of Francis Schaeffer (1912-1984):
The God Who is There, Escape from Reason,
He is There and He is Not Silent.
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Remove isolation!
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Never teach mathematics in isolation;
knowing mathematics should open the
door to:
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Remove isolation!
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The wonders of Creation.
The heritage (indeed, providential) of
history.
The appreciation of philosophical
presuppositions governing the JudeoChristian foundations of Western Culture.
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Review
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Mathematics and logic.
Mathematics and wonder.
Mathematics curricula.
Mathematics and truth.
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Conclusion in Truth
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Proverbs 1:7, “The fear of the LORD is
the beginning of knowledge; fools despise
wisdom and instruction.”
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