Honors Interdisciplinary Seminar
(175)
Review for the Section Exam
Section 005
Exam will be given in class on Monday, December 7
Your exam will include a number of different formats. As you prepare
for it, keep in mind that there are two distinct but closely connected aspects
of the course: (1) specific mathematical content and (2) what that content
tells us about mathematical truth. The exam will test you on both of these
aspects, in a variety of ways. There will likely be questions of the
true/false, multiple choice, short answer format. I will ask you to
demonstrate that you understand the important mathematical ideas. In
particular, I will ask you to state and prove the Pythagorean Theorem.
However, just as important is what the mathematical ideas tell us about truth.
It is important that you carefully consider how each of the topics in the course
speaks to the issue of truth, specifically mathematical truth. Looking
back over the study questions for each section might prove to be helpful
in this regard. Note that when you handed in your study questions for each
section, we hadn't yet discussed the section, so your ideas likely were not
fully formed. Having finished with the content for the section, I expect
that your thoughts on the issues are more sophisticated and that you have
synthesized the ideas more completely. Please look at those questions
again.
While you are preparing for the exam, remember to talk to me if you run into
difficulties, or if you want to discuss a concept again. (You can find my
weekly office hours online at
http://www.uwosh.edu/faculty_staff/szydliks/schedule.htm). I
will also be available to you before the exam starts on December 7. I
normally have an office hour (in Swart) from 1:00-2:00 pm, and, in addition, I
will be in Clow 209 from a few minutes after 2:00 until classtime, in case you have any last minute
questions.
Exam Topics
Included below are
the major topics we have covered and some of the "Big Ideas" from each section.
The summary does not attempt to capture the entirety of our discussion, so you
should not assume that it is complete. It should, however, help you focus
your studying. Your study questions are very important for helping
you to think about some of the deeper ideas. Refer to them for more
detail. I won't ask any specific questions on Flatland, but you
should feel free to bring that text into the discussion in your essay questions
or explanations, if you find it appropriate.
- Hilbert/"People and Clubs":
What is an axiom? What is a theorem? What is an axiom system?
What are some desirable properties of axiom systems? What is the
axiomatic method? I won't ask you to remember the axioms from Hilbert or to
prove any specific propositions from the axioms. However, you do need to
understand what "People and Clubs" taught us about axiom systems and the
axiomatic method. The example also provides us with a context in which
to speak about other ideas such as independence.
Big Idea: Mathematicians make explicit
assumptions and definitions (or have undefined terms). These assumptions
and definitions form the foundation of the axiomatic system.
- Euclid: Who
was Euclid and when did he live? What specifically made the Elements
such a spectacular accomplishment? What do "People and Clubs" have to do
with Euclid? What are some of the issues with Euclid's definitions (Be
careful here: some of Euclid's language is unfamiliar, so reading the
definitions can be challenging. But this is NOT a flaw in and of
itself)? Referring to the desirable properties of an axiom system, which
properties are called into question by Euclid's axioms? You should be
able to state Euclid's 5 postulates, especially the 5th postulate.
What was the most fundamental issue with Euclid's axioms? How was it
resolved? (Be specific.) What were some of the other issues with
Euclid's axioms?
I will not ask you to prove any of the propositions from the Elements that
were presented in class, with the possible exception of Proposition 1 (see #7
below), the constructions (see #3) and Proposition 47 (see #4).
Big Idea: Euclid gives us a great
example of the power of rigorous proof and of the structure of the
axiomatic method. However, the mathematical truths that one can arrive
at always depend fundamentally on the axioms chosen.
- Constructions: We
spent about an hour going over some Euclidean constructions. These give
us a great sense of how the Greeks did geometry with a straightedge and
compass. You should be able to perform the constructions, carefully
describe the steps in the constructions in words (e.g. "Using my compass, I
made a circle centered at A with radius AB."), and argue how you know that
your constructions are correct. The constructions you are responsible
for are:
- Bisecting a given angle
- Bisecting a given line segment
- Constructing a line perpendicular to a given line through a given point
on the line.
- Copying a given angle onto another given ray.
- Copying a given triangle onto a given side of another line.
Big Idea: Even though our hands and
eyes are not perfect, constructions provide us with a window into the abstract
perfection of a proof. For example, with perfect hands, eyes and tools,
we could bisect any given angle. The constructions are a
way to visualize the processes inherent in some of Euclid's proofs.
- Pythagoras:
The Pythagorean Theorem is the culmination of Book I of the Elements.
I will ask you to carefully state and prove the Pythagorean Theorem, so make
sure that you have a careful and precise statement and that you understand at
least one of the proofs (any proof will do). Remember the emphasis we
placed on rigor during the presentations. You will need to articulate
where in your proof you use the fact that you have a right triangle, and you
will need to justify all the steps in your argument (e.g. how we know the
squares in Proof #1 are really squares). You can find a copy of
the proof outlines
HERE.
The history of the Pythagoreans provides us with some insight into the nature
of truth. How?
Big Idea: Axiom systems provide tremendous
power. Even a few simple axioms can yield powerful results.
- Fermat’s Last Theorem:
We spent two hours watching a video and discussing Fermat's Last Theorem. You
should have a good understanding of the story. Know meanings of the important
terms: theorem, proof, conjecture, counterexample,
corollary. Note that we went over these terms in class, though we did
not write down the definitions on the board. If you're not sure about
your definitions, make sure to ask. You should be able to state Fermat's
Theorem carefully and to explain in words what it says. Though you don't need
to know the details of the proof of Fermat's Theorem, you should have an
understanding of Wiles' contribution and the nature of his proof. What does
the film say about proof by computer? Why does Wiles cry during the video and
why might I argue that the tears were as much tears of relief as tears of
triumph (be specific)? Could Wiles' proof be the same as the one that Fermat
claimed to have (Why or why not)? Why didn't Wiles study Fermat's theorem in
graduate school? Why then does his choice of elliptic functions to study make
a happy coincidence? Why might we consider Fermat's theorem to be a corollary
(be specific)? If you have forgotten the story or are still confused about the
theorem, a good place to look is on the NOVA website. See the sites
http://www.pbs.org/wgbh/nova/transcripts/2414proof.html or
http://www.pbs.org/wgbh/nova/proof for a transcript of the video or
general information about the theorem. I've also placed a copy of the video on
reserve at Polk Library, so you can watch it again if you think it's
necessary.
Big Idea: Mathematics is a living
subject. The way mathematics is discovered is often very different from
the way it is presented (see Euclid). Sometimes establishing even an
apparently simple truth can be extremely difficult.
- Gödel and
Independence/Non-Euclidean Geometry: What was the major
question regarding Euclid's axiom system? How was it resolved? By
whom? When? What is non-Euclidean geometry? There was some
important terminology in our discussion. What does it mean for two
statements to be equivalent? (Give some statements that are equivalent
to the parallel postulate.) Why do we have "undefined terms" in
geometry? What is a model? What does it mean for a statement to be
independent in an axiom system? How do we show that a statement is
independent? Give some examples of independent statements.
What did Gödel prove? What does his proof tell us about mathematical
truth?
Big Idea: There are limits to the
axiomatic method. In particular, in any arithmetic axiom system, there
will always be statements that can be neither proven nor disproven.
- Lakatos and Uncertainty:
What is Euler's formula? How does Lakatos use Euler's
formula to speak about the nature of proof? What are local and global
counterexamples? How are they used to argue What is "monster barring?"
What does Lakatos tell us about mathematical truth? Be careful here;
when you first read Lakatos, you might have initially arrived at the
conclusion that mathematicians have no certainty at all. However, this
is not the case. Our certainty may not be perfect and absolute, but the
axiomatic method and the nature of mathematical proof is nevertheless
extremely powerful. We must retain a deep and abiding respect for the
axiomatic method.
Big Idea: Although mathematics can
offer us great insights and powerful results, the nature of mathematical proof
is not as certain as we would sometimes like to believe. And as we saw
with Andrew Wiles and the proof of Fermat's Last Theorem, the way mathematics
is discovered can be very different from the way it is presented.
- Cantor and Infinity:
There were lots of big mathematical ideas in this discussion. How
do we show that two infinite sets have the same size? How do we show that they
*don't* have the same size? What does it mean for a set to be
denumerable? How do we show that N, Z, and Q all
have the same size? How did Cantor prove that R is not a
denumerable set? What is the Continuum Hypothesis? What is the
status of the Continuum Hypothesis and why?
Very important, too, is an understanding that although Cantor's ideas are hard
and paradoxical, they are not "made up." Cantor made two very basic,
reasonable definitions (what it means for two sets to have the same size, and
what it means for one set to be "less than" another in size). Given
those definitions, the results (there are at least two different sizes of
infinity, and there are an infinite number of infinities) must follow.
We had some terrific discussion in class comparing the nature of infinity with
the nature of God. But we mustn't confuse that discussion with the rigor
of Cantor.
Big Idea: When we make reasonable
assumptions, we must follow them to their logical conclusions, even if the
results are surprising or controversial.