PHL309 Logic, Language, and Thought
Professor: Craig DeLancey
Office: CC217
Email: delancey@oswego.edu
Current Assignments
4 February
Read from the Galileo selection. You are welcome to start
at the beginning if you can, but I only need you to read
from page 35 to 42. I think there are at least five arguments
that we cannot reason about infinity. Try to identify them,
and see if you can understand them. We'll go over the first
few. I will ask you to explain the last several.
6 February
Homework. On page 40 (editor's number 78), Salviato
explains a paradox having to do with squares and roots
of numbers. Write out this argument as a reductio argument.
What are the necessary steps, what is the conclusion of
the argument? (After Monday, we'll have several examples
of doing this, so you'll have a model to copy.) You should
be able to do this on a page, or at most two. Hand it in
at the beginning of class.
Tentative Assignments (subject to revision)
February
Read the handout with selections from Locke and Berkeley.
Below, I've left the assignments from the last time I taught this
class, so that you can see those assignments.
2010 Assignments
27 January
Keep reading from The Two New Sciences. I sent you
the preferred translation, but as noted it remains
available also here.
Homework: consider the device that is in the online
translation shown as figure 4. In our preferred translation
(the one I emailed you) it is in the margin of page 24 (page
63 by the old page numbers that are in the margin). It looks
like a helmet hanging from a hook connected to a rod sticking
through a weird cylinder. Write a brief (1, at most 2, page)
explanation of what this thing is supposed to be measuring;
and why we would want to measure that; and what we hope to
then learn about our main problem in the dialogue of the first
day (namely, how things hold together). Hand in your explanation
at the beginning of class.
30 January:
Finish reading up to page 44 of the first day of Galileo's
Two New Sciences that I mailed to you. (The original
edition page numbers are on the side margins, and by those we
would like to get to 82.) You're welcome to read farther, but
we won't need to discuss past this point and I won't require
that anyone read past this point. While reading, take some
notes to see if you can understand:
- Galileo has proposed two ways to explain how things
hold together: resistence to void, and some glue-like
substance. What are his objections (regarding fire, for
example) to the glue-like substance?
- Why does Galileo think there is a problem with the
circle-within-circle rotation?
- What does Galileo assume about points, lines, and the
size of lines in making his observations? That is, what is
the relation between points and length, or points and area,
according to Galileo?
- What are Galileo's arguments that it's problematic to
think about infinity? He has several, and they are very
intuitive once you figure them out. Figure them out so that
you can explain them to your roommate.
These would make great pop quiz questions.
Note: if you're the literary type, or just love the theatre as do
I, you might enjoy reading Brecht's famous play "Galileo." A very
cheap podcast of the LA Theatre Works's performance can be had
at Amazon.
Hardcopy of the play is in the library at various places
(PT2603.R397 A27, PT2603.R397 A26, PT2603.R397 L415 1966).
3 February:
I'm going to leave you alone for part of the class. Your
task will be to work in your teams to make the figure suitable
for on Monday February 6 performing the demonstration of the
"rotation" of your figure down a track. You should try to find
a big piece of cardboard before class that you can bring in
to do this. The team list is here.
Team's work together in class to create their demonstrations
of Galileo's proof. We'll perform the demonstration in class
on Monday 6 February. We'll only have a few team assignments
this semester, so don't panic if you don't like teams. Some
of our tasks are just much easier with a team.
This is kind of easy but it will both get you together and
organized, and also it will make it so you remember Galileo's
proof forever!
With your team, create before February 6 so that you
may share in class a recreation of Galileo's geometrical
demonstration of the "gaps" that must exist when we rotate an
n-sided regular figure. This occurs primarily on pages 28-35
of our text. Creativity would be great, but the default way
to do this is to make a big shape out of cardboard (that has
around a common center a smaller version of the same shape)
that you can attach a marker to at each corner, and which you
can rotate against the marker board. If you do that, make a
figure to which markers can be held, that when we rotate
gives us the patterns that Galileo describes; note that there
is always both the whole figure, and then a smaller version
of the same figure with the same center inside it, and that
figure should have a hole or something in at least one corner
or place on its perimeter to stick the marker through also.
We'll measure the "leaps" or bumps in the outline of the
inner figure, and then see if he Galileo got it right.
- Team 1 gets an equilateral triangle;
- Team 2 gets a square;
- Team 3 gets a pentagon;
- Team 4 gets a hexagon;
- Team 5 gets a circle.
Try to make your figures with a perimeter or circumference
of about 120 centimeters, and the interior figure with a
perimeter or circumference of about 60 centimeters.
6 February:
Team demonstrations of Galileo's proof!
If we have any time, we'll start talking about Cantor
and about set theory. This might be a good time to
start reading Logicomix.
17 February
Write up an hand in at the beginning of class
the answers to the following questions. Please
work alone.
- Identify three elements of the following set.
{a, b, c, d, e}
- Identify three proper subsets of the following set.
{1, 2, 3}
- Identify an improper subset of the following set.
{11, 13, 17}
- What are the power sets of the following sets?
a. { } (the empty set)
b. { a }
c. { a, b, c }
d. { 1, 2, 3, 4 }
e. { 9, 9 } (trick question!)
f. { {a, b}, {{7}} }
- Which of these sets has the largest cardinality?
Explain what this means and how you know your answer
is correct. You'll probably need to define cardinality,
and describe an on and onto function, and then apply
these notions.
The natural numbers, N.
The odd numbers, O.
The even numbers, E.
- Simplicio argued that a line segment AB that is twice as
long as a line segment CD has twice as many points, but also
AB and CD have infinitely many points, and so AB and CD are
both differenly sized (AB has twice as many points as CD) and
the same sized (AB and CD have the same number of points,
namely infinitely many). How many points are on a line
segment CD? How many are on a line segment AB if it is twice
as long as CD? What does this tell us about Simplicio's
argument. Does Cantor's Claim matter here?
February 24
Now is a good time to read the first 3 sections of Logicomix.
This would take you to page 154. There are many historical inaccuracies
in here, which the authors took in order to simplify and dramatize the
story. The basic thrust is not inaccurate, however; so don't treat it
as a primary source but rather as a dramatic introduction to the portion
of our potted history that we are discussing now. It's a fun book.
27 February
Many of us could use some more practice with the concepts of set
theory. The following homework is recommended. If you don't do it,
I'll simply count your other homeworks more. If you do it, I'll have
a warm feeling towards your effortfulness which is sure to bias my
objective evaluations. To do the homework, please write up the
answers to the following problems, working alone, and hand in the
answers at the beginning of class.
- How does one know if two sets are the same cardinality?
- Are these two sets of the same cardinality?
{1, 2, 3, 4, 5, ....}
{3, 6, 9, 12, 15, ....}
- If the answer to question #2 is yes, what function will
tell us this? Use regular arithmetic notation to describe the
function.
- For each of the following, which is true and which not?
a ∈ {a}
{a} ∈ {a}
{a} ∈ {{a}}
- How many elements will be in the powerset of....
{1}
{1, 2}
{1, 2, 3, 4, 5}
- Suppose the set A = {a, b, c, d, e}. Which of the
following are true?
- a ∈ P(A)
- {a} ∈ P(A)
- {{a}} ∈ P(A)
- {a, b, c, d, e} ∈ P(A)
- {{a, b, c, d, e}} ∈ P(A)
- {{a}, {b}, {c}} ∈ P(A)
- Suppose the set A = {a, b, c, d, e}. Which of the
following are true?
- a ⊆ P(A)
- {a} ⊆ P(A)
- {{a}} ⊆ P(A)
- {a, b, c, d, e} ⊆ P(A)
- {{a, b, c, d, e}} ⊆ P(A)
- {{a}, {b}, {c}} ⊆ P(A)
- What is the powerset of your team?
March 5
Finish Logicomix. We'll review for the test, and
if we have time, we'll review Russell's Logicism.
March 7
Midterm test! Topics can include:
- explain Galileo's "proof" that there can be
infinitely many gaps in a rotated solid;
- explain Galileo's
"proofs" that we cannot reason about infinity;
- explain and apply Cantor's Claim;
- explain Cantor's diagonal argument;
- explain and apply Cantor's Theorem;
- explain how Cantor's Claim answers Galileo's challenges;
- explain and apply some basic
notions of sets (empty set; subset; proper subset;
function; power set);
- explain how unrestricted set formation
leads to the Cantorian antinomy (the contradiction that
we could derive in Cantor's set theory); be very clear about
what the contradiction is;
- explain the goal of logicism
- What is the benefit of an axiomatic system?
- Explain Russell's Contradiction. Why are contradictions
bad?
March 9
A discussion of platonism vs. conventionalism.
March 19
Review of test. Finishing our aside about conventionalism.
For the test, the grades ranged from 2 to 40 (out of 40).
The average was 27, the standard deviation 10 (so there was
a lot of variation). I consider any grade in the 30s very
good (Aish) and in the 20s decent (Bish). If you did not
do well and you felt you understood the material, we just
need to work to make sure you are able to explain and
demonstrate your understanding.
I will post here the
rough rubric I used to grade the questions. The main
difficulty I saw was not really knowing the underlying idea
to some proof or problem or challenge; or (often totally)
failing to explain a proof or claim. The point of the
questions of course was to ask you to explain the problems,
to show you could explain the material.
I'm a bit concerned about the grade average, so I will try
to give us more time to practice. I'll be sure that we have
at least five more homeworks before the end of the semester.
One of these will be a review of things we already learned.
There will also be another test, and a paper, so if you
didn't do well you have a chance to make it up if you work
hard.
March 21, 23
Introduction to Godel.
If you got the book by Casti, Godel, read chapters
1-4. Additional resources include the Stanford
Encyclopedia entry on Godel; this is often technical,
but by skipping the hard parts you could still learn much.
26 March
Homework due. I've attached it as a pdf here, and will also bring some
copies to class.
March 30
If you chose to buy Godel, now is a good time to read
(up to and including) chapters 6 and 7.
April 2
Working alone, you will write descriptions for 2 turing
machines. You will need to provide for each: the rule
table, a description of how the tape will be interpreted
and configured (including how to interpret the output),
and where the head starts on the tape.
The machines should not be universal!
It can be helpful to use a Turing machine simulation. Here's
one that may work for you:
http://math.hws.edu/TMCM/java/labs/xTuringMachineLab.html.
You could test your machine on it.
Try to do the machines with only 1s, 0s (or 1s and blanks),
and as few other characters as you need as your alphabet
(sometimes it makes things a lot easier to add a character,
for example to mark the end of a number, so do that if it
helps you!).
The machines should be:
- A machine that tells us whether a number is odd or
even. (You can write a number n as n "1"s on the tape;
that's the easiest way to do it.)
- An addition machine for numbers n, m where always m >
0 but n could be 0. (You can write a number n as n "1"s
on the tape and m as m "1"s on the tape; that's the
easiest way to do it. Consider always starting with n.)
For each problem, clearly describe the proper input and its
interpretation, and the expected output and its
interpretation. That is, where does the tape head start and
finish in relation to the data, etc.
April 2
Our in class topic will be the halting problem and
Turing's diagonal argument.
April 4
We will review your Turing machines, which were mostly
very very far from being effective procedures.
If we have time left, our in class topic will be the halting
problem and the impossible machine argument.
April 6
No class.
April 9 & 11
I'll be giving a talk in Tucson, but during class time and
also at other times, you have an independent task and also
a group task. Both will be due April 13. These are:
April 13
1. Individual assignment due at the beginning of
class. Play with one of the Elizas, like this one
or this
one or just google "Eliza simulator." Write up in a
paragraph how you think it works. Work of your impression of
playing with it (that is, don't do any research).
2. Team project using Turing machines due at the
beginning of class. You are going to make two Turing machines
(none universal!) and write up their specification. You'll
hand in that specification.
It will be very helpful to use a Turing machine simulation.
Here's one that may work for you:
http://math.hws.edu/TMCM/java/labs/xTuringMachineLab.html.
Try to do the machines with only 1s and 0s (or 1s and
blanks), and as few other characters as you need as your
alphabet (sometimes it makes things a lot easier to add a
character, for example to mark the end of a number, so do
that if it helps you!).
Here's one way to get started. Let a number be interpreted
to be the quantity of 1s in a string, so that 4 is
"...011110..." (or, you might find it easier to let 4 be
"...01111x..." or "...x11110..." or whatever). The machine
should start with the head on the leftmost 1 of the left
number (unless that number is zero, in which case it starts
on a zero with another zero between it and the next number to
the right); the two numbers should be represented as 1s
separated by a single 0 and surrounded by 0s; and the machine
should end on the leftmost square of your answer. The
machines should be:
- an addition machine; it should be able to
handle 0+0 or 0+ any number or any number + 0. (If you
are using 1s and blanks, then a zero is a blank;
and 0+5 would be written on the tape as (starting
leftmost under the tape head) "blank blank 11111".
That is, the first blank is zero, the second
is interpreted as separating your two numbers.
0+0 would be "blank blank blank".)
- A subtraction machine, where the rightmost number is
always smaller than or equal to the left number.
For both problems, clearly describe the proper input and its
interpretation, and the expected output and its
interpretation. That is, where does the tape head start and
finish in relation to the data, etc.
In class: we will discuss the Halting problem, and if we
have time the Turing test.
April 16
Review of your turing machines and write-ups, and then
discussion of the Turing test, and also proposals that there
are exceptions to the Church-Turing thesis (especially: the
Lucas-Penrose argument).
April 18
Quest.
April 20
Play with one of the life worlds. A good one can be found
here.
Homework: describe (draw) a stable unmoving structure, and a
stable moving structure. In class: a discussion of
predictability versus determinism.
April 23
We'll (1) review the halting problem proofs; (2) review
the difference between deterministic and predictable;
and (3) introduce the formal definition of Kolmogorov
Complexity and Chaitin Randomness.
Your homework: read
Chaitin's Introduction to his book, The Limits of
Mathematics
. This is very good because it also reviews much of the material
we have discussed in the last half of the semester.
April 25
Team project. For each of the following strings, create the
smallest turing machine that can print the string. To
really demonstrate Kolmogorov complexity, we should be using a
UTM, and writing programs, but that would be a bit tricky for us
since it would require learning to program some UTM, so we'll just
get the flavor of Kolmogorov complexity by using raw Turing
machines. We will define size by the number of rules you use,
plus the alphabet, plus the number of characters on your tape at
the start. The third machine must halt with a tape that shows
exactly this string to get full credit; to make things easier for
you I'm not actually requiring that the first two machines halt --
just that if we stop them at some time ourselves, it will have the
pattern we want in progress of being constructed. You'll need to
hand in a sheet with your rule table, and with your tape's
starting contents, and with a tape interpretation (that is: how do
we start it up, where is the head?).
- 10101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010....(and so on, forever)
- 10110111011110111110111111011111110111111110111111111011111111110111111111110111111111111011111111111110111111111111110111111111111111011111111111111110....(and so on, forever)
- 10010101010010111110010000001110110110010010010110100101001000011001001001011101001001000110100100010110101010001001001110110000010101010101011001001000001110100111100000001101111101100100101101001111
Remember: you task is foremost to get the string on the tape!
If, for number 3, the machine is in halt with the answer on
the tape, you're done. When in doubt, just remember that.
Since I've been told that #2 is hard, I will consider it optional/
extra-credit. But it's not so hard if you allow yourself an extra
symbol, like x, to mark the 1s you've counted alread.
April 27
Quiz. Short answer questions. Questions could include:
- Describe Cantor's diagonal argument.
- What is Godel's First Incompleteness Theorem?
- What is Godel's Second Incompleteness Theorem?
- What is a common sense interpretation of the
Godel sentence?
- What is a Turing machine? What is a Universal
Turing Machine?
- Write/describe a Turing machine to accomplish
a very simple task.
- What is the Turing test?
- What is the Church-Turing Thesis?
- What is the Halting Problem? What is the answer
to the problem? Describe the diagonal argument showing
this. Describe the impossible machine argument showing
this.
- What is the Kolmogorov Complexity of a description?
April 30
Review of quiz and our K-complexity approximating Turing
machines. Proof of the undecidability of randomness.
The average was a 33, the standard deviation 10.5. I would
typically consider then at 35 a B, 40 and above an A, and C
would hover around 23.
May 2
Continuation of the undecidability of randomness.
Will Trevor ever coordinate with his friends to
get to the Manilow concert? Tune in to find out.
May 4
Last day of classes.
Discussion of reason as akin to empirical work.
This is the last day that I will be willing to accept
drafts of your final exam paper.
May 7
Final papers officially due before 2:00 p.m. in my office.
However, I will accept them without penalty up to but no
later than 2:00 p.m. on Wednesday May 9. These cannot be
emailed. Here is the recommended topic:
- We have seen how great progress was driven by the
discovery of seeming paradoxes (such as Galileo's
"proof" that the squares are fewer and the same number
as the natural numbers) or real ones (such as Russell's
paradox in Frege's system). Each of you has a field of
study outside of this class. Describe a paradox or
seeming paradox in your field that spurned progress.
For example, a geology major might describe how the
discover of marine fossils on mountain tops seemed
paradoxical until theories of mountain formation,
continental drift, and so on, were developed. A
wellness major might write about the fact that people
often retain or even gain weight when they go on a diet.
And so on.
Here are some of the kinds of alternate possible paper
topics, which students have written on in the past.
- Is there some issue in your field of study or field
of interest (perhaps where you hope to make a career)
where limits of reason might arise in a way that you can
describe with a bit of rigor? For example, if you are
studying software engineering, how might specific limits
of reason affect that field? Or, as another example,
how does chaos affect meteorology? Such limits are
likely to concern chaos or perhaps complexity. We could
kick around ideas together if you think possibly. In
such a case, we would need to explain the problem, show
why there may be a limit issue, and discuss the
consequences.
- Defenders of "Intelligent Design" claim some
structures are too complex to have evolved. Use the
tools of Kolmogorov complexity to describe this claim.
Evaluate the claim. You must take into consideration
the role that randomness -- if there is any randomness
-- can play.
- A theory of a thing or kind of thing can be
thought of as in part a method of forming compressed
descriptions of the relevant kind. For example, we see
certain patterns in motion, and have now a science of
dynamics. That physical theory of dynamics, if it is
complete, is a way to produce compressed descriptions of
the relevant kind of phenomena. Note that our brains
have a certain size limit; we can only "hold" some
number of bits of information. What happens if we aim
to understand a phenomenon that is complex than the
standard human brain capacity? Are we likely ever to
confront such a challenge? Where? Would we ever know
if we were hitting such a limit (this last question
seems easy to answer with a "no," but think hard about
how we might)?
- The Fermi paradox is that we do not hear any
extraterrestrial messages, but it seems we should. It
seems we should because there are just so many stars out
there. Come up with even very pessimistic estimates of
how frequent the right kinds of planets are, of how
likely life is to emerge on those planets, of how likely
intelligence is to evolve, how likely intelligence is to
use electromagnetic communication methods (radio, TV),
and so on. Multiply these together and get some
terribly small frequency of radio-using intelligent
life. Still, there are so many damn stars in our galaxy
alone that it seems there should be many such
civilizations even just in the Milky Way. So the
paradox is, where are they? Why is everything so quiet?
Many answers have been proposed. One might be (I don't
know who first proposed this!) that very advanced
civilizations surround us, and they are transmitting
radio messages, but that they use such complex languages
that their transmissions appear like noise to us; thus,
we hear their radio messages whenever we point a
radiotelescope into space, but cannot distinguish their
communications from background radiation/noise. Is this
possible (does this describe a situation consistent with
complexity theory)? And, if so, how could distant
civilizations come to use such a common "language"?
That is, we must presume both that faster than light
travel is impossible and that these civilizations did
not all come from the same place (do not presume
hyperdrives or subspace radio or anything presently
thought impossible); thus they have only ever
communicated by radio with some years of delay in their
signals. Could they get a conversation even going if
they are using very complex codes? Or, if not, must we
presume that such conversations could only arise slowly
over many centuries (they start talking in very simple
language, and work up to a very complex?). Think
through these scenarios. And, in any case, is this a
plausible solution to the Fermi paradox?
- Create and fully document a universal Turing
machine. Check with me about the Turing machine framework you
will use.
For your final paper, I would like you to aim for 5-6
pages, 12-point courier font, 1 inch margin, double
spaced throughout. The goal for you is to write on how
the limits of reason that we have discovered in this
class might affect some topic of interest to you.
Please clear your topic beforehand if it is not the
recommended topic, by emailing a paragraph or so to
me on what you hope to write on as soon as you can. In
that paragraph, identify the central hypothesis that you
will defend. Your answer should try to explore as
clearly and rigorously as possible an answer. Write
just as much as is necessary to do that brilliantly.
You
must follow the rules and guidelines laid out in
my
philosophy paper format.
You can pass the course without doing well on this final
paper, but to get an A you must show some insight on
this final.
I will look at hard copies of drafts up to an including
the last day of class. Do me a favor please and label
your draft "DRAFT." Please do not write "FINAL DRAFT"
or "DRAFT FINAL" on any papers, since I can't tell then
if you are giving me (1) a draft of the final, or (2)
the ultimate paper.
Wikipedia is not an acceptable source for academic
research and should not be cited as such.