Set theory axioms
The following are the axioms of ZFC (ZermeloFraenkel plus choice) developed between 1908 and
the 1920s.
1. Axiom of extensionality; 2. Axiom of regularity
(also called the Axiom of foundation); 3. Axiom
schema of specification (also call
Consistency and
existence
SQ12. What is the relationship between consistency
and existence?
Hankel: the mathematician counts as impossible
only what is self-contradictory.
Frege: A concept is still admissible even though its
defining characteristics do co
Zero
SQ7. What is 0?
That a concept contains a contradiction is not
always obvious without investigation, but to
investigate it we must first possess it, and, in logic,
treat it like any other. All that can be demanded of a
concept from the point of view
Notes
Auditing: people not formally enrolled in the class
may sit in or audit, but if you are not enrolled, please
ensure that enrolled students all have seats before
you take one yourself.
Please bring the text were discussing to class.
Drinks are fine b
Philosophy of Mathematics
Patricia Marino
Fall 2012
Paper topics, second paper assignment. Write 1500-1800 words on ONE of the topics
below. The paper is due Dec 11; email it to me at pmarino@uwaterloo.ca before 5:30
that day. Be sure to give your paper a
Philosophy of Mathematics
Patricia Marino
Fall 2012
Paper topics, first paper assignment. Write 900-1200 words on ONE of the topics below.
The paper is due Oct 4; email it to me in pdf format before class starts at 5:30. Be sure to
give your paper a title
The mathematical
practice objection
In contrast with CH, this question [of Lebesgue
measurability of certain sets of reals] concerns only
a limited class of definable sets of reals, sets whose
definitions have concrete geometric interpretations,
and it in
Two talks about phil
of science
Who Speaks for the Global Climate? Institutional
Pluralism and Democratic Representation. Mark
B. Brown, California State University, Sacramento,
Thursday, October 25, 2012 Great Hall, Conrad
Grebel College University 7:00
Quine, On What
There Is
Quine is a proponent of naturalism.
Naturalism: "the abandonment of the goal of first
philosophy," "the recognition that it is within science
itself that reality is to be identified and described."
-Quine, quoted in Shapiro, Thinki
Dates etc.
Optional rewrites for papers are due Thur Nov. 1 by
email before 5:30. If you have questions about your
grade, comments, etc., do not hesitate to ask me.
First test is Thur Oct 18. Format: five short answer
questions (youll choose five from a s
Quine, Two Dogmas
of Empiricism
Two dogmas: analytic/synthetic distinction;
reductionism
Quine: these are ill-founded.
Results: metaphysics and science run together, and
support for pragmatism.
Quine against the
analytic/synthetic
dist.
1. How does Quine
Objection from an
interlocutor
But surely theres a fact of the matter of whether l=1,
even if we do not know it yet?
Heyting: to say this is to rely on metaphysics, which
should be banished from mathematics (p. 68).
Intuitionistic math as
infererior to m
Brouwer, Intuitionism
and Formalism
Intuitionists and formalists agree that there is an
exactness to mathematics that does not hold in the
rest of science.
The question where mathematical exactness does
exist, is answered differently by the two sides; the
Legitimacy of ideal
elements
SQ9. On what conditions is the method of ideal
elements legitimate?
That condition is a proof of consistency, for the
extension of a domain by the addition of ideal
elements is legitimate only if the extension does not
cause c
Naturalism and
applicability
Sorin Bangu, Wigners Puzzle for Mathematical
Naturalism
Two challenges for
Maddys views
1. What two challenges does Bangu think the Mnaturalist will have trouble meeting simultaneously?
I begin by noting that M-naturalism itse
Maddy on second
philosophy
These days as more and more philosophers count
themselves as naturalists, the term has come to
mark little more than a vague science-friendliness .
My goal in this book is to delineate and practice a
particularly austere form of
Interlude: the nature
of realism
There are two general aspects to realism, illustrated
by looking at realism about the everyday world of
macroscopic objects and their properties. First, there
is a claim about existence. Tables, rocks, the moon,
and so on,
Burgess
Burgess, Mathematics and Bleak House.
A consideration of mathematical fictionalism.
Bleak House is a novel by Charles Dickens that
involves an endless court case, Jarndyce vs.
Jarndyce.
Destructive and
Reconstructive
We expressed some surprise tha
Bedrock
SQ6. How is the Bedrock community different from
ours and what's the point of this example?
Imagine that for us, Platonism is mandatory: it
would be irrational not to believe in abstract objects.
Imagine that in Bedrock, things are similar to our
Why math and not
other practices?
SQ2. How does Maddy respond to the challenge of
why we view mathematical practice as different
from, say, astrological practice?
The question has to do with why we trust
mathematical methods to tell us about mathematical
Maddy, Three Forms
of Naturalism
Idea: to contrast the naturalisms of Quine, Maddy,
and Burgess.
Can a naturalist do
more than describe?
Neurath has likened science to a boat which, if we
are to rebuild it, we must rebuild plank by plank
while staying afl
Physics is written in
mathematics
It is not true, however, as is so often stated, that
this had to happen because mathematics uses the
simplest possible concepts and these were bound to
occur in any formalism. As we saw before, the
concepts of mathematics
Philosophy of Mathematics, Tue and Thur 5:30-6:50, HH 150
Professor: Patricia Marino
Office: HH 332
Office Hours: will be posted on LEARN page
Email: pmarino@uwaterloo.ca
Course Description:
Philosophy of mathematics asks (among other things): Do mathemat
Patricia Marino
Paper writing guidelines
- Have a clear thesis that you argue for. Do not merely summarize, or write a report.
- State your thesis early on, argue for it, then give a brief conclusion.
- Be as methodical, organized, clear and concise as yo
Patricia Marino, paper grading information
Typically, for this paper, the above will correlate with the paper features below,
though doing an exceptionally good or bad job in one area can move your grade up
or down a category.
90-100
The paper does all th
Philosophy of Mathematics Fall 2012
Benacerraf, "Mathematical Truth"
1. What are the two kinds of conditions an account of mathematical truth ought to be
faithful to?
2. What is the "standard" semantical account?
3. What are "combinatorial" accounts?
4. W
Philosophy of Mathematics
Study Questions week 12
Patricia Marino
Sorin Bangu, Wigners Puzzle for Mathematical Naturalism
1. What two challenges does Bangu think the M-naturalist will have trouble meeting
simultaneously?
2. What are the two interpretation
Philosophy of Mathematics
Study Questions week 11
Patricia Marino
Colyvan, "The Miracle of Applied Mathematics"
1. How does Colyvan suggest the problem of applied mathematics has been usually seen
as a problem for anti-realist theories of mathematics?
2.
Philosophy of Mathematics
Study Questions week 10
Patricia Marino
Yablo, The Myth of the Seven
1. What is the purity problem? What is the alternative problem about applicability that
Yablo wants to focus on?
2. What does it mean to take numbers as represe
Philosophy of Mathematics
Study Questions week 9
Patricia Marino
Penelope Maddy, "Three Forms of Naturalism"
1. How do Maddy's and Burgess's versions of naturalism differ from that of Quine?
How are they different from each other?
2. How does Maddy respon