Philosophy of Mathematics
PHL354H1 S LEC0101 2024
Instructor:
Eamon Darnell
Email:
Course Web Page:
eamon.darnell@mail.utoronto.ca
Quercus.
Class Meetings:
Thursdays: 9:00am–12:00pm.
Room: HS 108
Office Hours:
Fridays: 4:00pm–5:00pm.
Zoom Link: https://utoronto.zoom.us/j/83447491526
Meeting ID: 834 4749 1526
No Passcode Required.
Course Description:
This course gives an introduction to major topics in the philosophy of mathematics. The bulk of the course
will be dedicated to the significant questions, views, and developments in the philosophy of mathematics
during the last century: with an emphasis on the ontology and epistemic foundations of arithmetic. Topics
discussed in the course include Platonism vs. nominalism, the relation between logic and mathematics,
Gödel’s incompleteness theorems, formalism vs. intuitionism, neo-logicism, and structuralism.
Prerequisites:
PHL245/MAT157 and a full course in PHL/CSC/MAT. This course assumes that students have a basic
understanding of first-order logic (e.g., by completing PHL245 or MAT157). Students will be given the
additional technical resources and develop the skills needed to understand the results and theorems that will
be explored in the course.
Mode of Delivery:
This section of PHL354 is an in-person course. All course material will be accessible through Quercus via
weekly modules (and/or the UofT Library). In addition to class, there will be a discussion forum on Piazza
that will be monitored by me.
Communication:
The best way to reach me is by email. I will do my best to respond within 48-hours (not including weekends).
I will not answer emails with questions about particular problems on the assignment. If you have such
questions, you may post them to the Piazza discussion board to get them answered (see below). I will also
not answer emails where the answer can easily be found on Quercus or in other course materials. For issues
such as missed midterm or an extension on the assignment, follow the instructions below.
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You must include PHL354 in the subject of your email
If you come to my office hours, I will be happy to answer any question that you like. This includes requesting
help with assignment questions.
Quercus (required):
Course materials will be posted on Quercus. This includes announcements, lecture slides, recommended
readings, and assignment information. Check Quercus regularly to make sure you get all the content for the
course.
Course Texts (required):
Benacerraf, P., & Putnam, H. (Eds.) (1983). Philosophy of Mathematics: Selected Readings
(Second ed.). Cambridge University Press.
[Available Online via the: UofT Library]
Mancosu, P., Zach, R., & Galvan, S. (2021). An Introduction to Proof Theory: Normalization
Cut-Elimination, and Consistency Proofs (First ed.). Oxford University Press.
[Available Online via the: UofT Library]
Course Readings (required & recommended):
A complete list of readings for the course is posted on Quercus. In each weekly module on Quercus, the
particular readings relevant to that week’s lecture will be listed. All required readings for the course are
available online via the UofT Library (some recommended readings on certain topics are not available online).
Working through the readings each week will complement the class lectures, and will provide you with a
more thorough understanding of the views and topics that we will discuss.
Piazza (required):
Piazza is an online discussion board. If you have questions or wish to opine about course material, post on
the Piazza board. Good questions or discussion topics will be answered or addressed by your fellow students,
or me. This is an excellent resource for getting extra-help and for actively thinking about material covered
in the course.
Your posts should thoughtfully engage with the course material (including assignment problems) or your
fellow students’ posts. Don’t post about basic/administrative course issues that are answered in the syllabus
or in course announcements.
Your posts on Piazza are governed by UofT’s student code of conduct. Click here for more information:
https://governingcouncil.utoronto.ca/secretariat/policies/code-student-conduct-december-13-2019
Evaluation:
There will be one 120-minute midterm test that will occur in-class during our regular class time. The midterm
test will be worth 30% of your final grade.
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There will be one assignment worth worth 25% of your final grade. The assignment details will be provided
on Quercus.
There will be one term-paper due at the end of the semester. The term-paper will be worth 40% of your final
grade. A list of term-paper topics will be given in March. You may write on one of the topics given in the
list, or you may write on a topic of your own choosing if you have your topic approved by me on (or before)
the last day of class (Apr. 4). The total length of the term-paper should not exceed 15 double-spaced pages
(in 11pt–12pt font).
The remaining 5% of your final grade is for monthly participation on Piazza. Making (at least) one thoughtful
post on Piazza each month (Jan., Feb., Mar., and Apr.) is worth 1.25% of your final grade. A “thoughtful”
post will consist of at least two or three sentences that engage with course material and/or a question or
post about course material from someone else in our class.
Monthly Participation on Piazza . . . . . . . . . . . . . . . . . . . . 5%
Midterm Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30%
Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25%
Term-Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40%
Missed Midterm:
No make-up test for the midterm will be given except for in unforeseen and exceptional circumstances. If
you have a legitimate excuse for medical or other reason, then you can request approval to write the make-up
test. To do so, you will need to fill out the MS FORM titled “Missed Midterm Form” on Quercus. You
can upload a copy of documentation there. This must be done within seven days (including weekends and
holidays) of the date of the midterm. If you have an unusual request or you cannot fill out the form for some
reason you may email me or come to my office hours to discuss the matter.
Missing the midterm due to a course conflict is not a legitimate reason.
UofT has an absence declaration policy that you should know. A very important part of the policy is that
“Students should also advise their instructor of their absence. Instructors will not be automatically alerted
when a student declares an absence. It is a student’s responsibility to let instructors know that they have
used the Absence Declaration so that you can discuss any needed consideration, where appropriate.” So, if
you use the UofT absence declaration, which you should, you must still follow the above instructions and
complete the missed midterm form.
The make-up, if allowed, will be given at the end of the semester. The mark you earn on it will substitute
for your missed midterm and will count in the weighting system detailed above. Date and time will be
announced later.
Missed/Late Assignment:
An assignment submitted past the due date will be accepted at a penalty of 1/3 of a letter grade, and it will
be penalized an additional 1/3 of a letter grade for every 24 hours past the due date until it is submitted.
If you have a legitimate reason for not submitting the assignment on time, you will need to fill out the
“Assignment Extension Request” form on Quercus. For your extension request you will have to supply
proper documentation. We will be very strict in granting extensions because you will have several weeks to
complete the assignment.
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Missed/Late Term-Paper:
Term-papers submitted past the due date will be accepted at a penalty of 1/3 of a letter grade, and it will
be penalized an additional 1/3 of a letter grade for every 24 hours past the due date until it is submitted.
If you have a legitimate reason for not submitting your term-paper on time, you will need to fill out the
“Assignment Extension Request” form on Quercus. For your extension request you will have to supply proper
documentation.
Policy Regarding the Use of External Aids:
The use of any external aid to write the Assignment or Term-Paper, including generative AI, large language
model applications (e.g., ChatGPT, etc.), is prohibited.
Accessibility and Accommodations:
UofT has an excellent accessibility services: http://www.studentlife.utoronto.ca. If you need any accommodations, please contact accessibility services and then we can work together to make the course a positive
experience for you.
Heath and Wellness:
If you find yourself struggling with the course or any other aspect of university life, please make use of the
resources at the school. Here are some links:
Registrar: https://www.future.utoronto.ca/current-students/registrars
Health and Wellness: https://www.studentlife.utoronto.ca/hwc
Personal Safety: https://safety.utoronto.ca/
Student Support Program: https://www.studentlife.utoronto.ca/cie/myssp
Sexual Violence Prevention and Support Centre: https://www.svpscentre.utoronto.ca/
Campus Police: http://www.campuspolice.utoronto.ca/
Academic Integrity:
The University of Toronto treats cases of academic misconduct very seriously. Academic integrity is a
fundamental value of learning and scholarship at the UofT. Participating honestly, respectfully, responsibly,
and fairly in this academic community ensures that your UofT degree is valued and respected as a true
signifier of your individual academic achievement.
The University of Toronto’s Code of Behaviour on Academic Matters outlines the behaviours that constitute
academic misconduct, the processes for addressing academic offences, and the penalties that may be imposed.
You are expected to be familiar with the contents of this document. Potential offences include, but are not
limited to:
• On tests and exams: Using or possessing any unauthorized aid, including a cell phone. Looking at
someone else’s answers. Letting someone else look at your answers. Misrepresenting your identity.
Submitting an altered test for re-grading.
• Misrepresentation: Falsifying or altering any documentation required by the University, including (but
not limited to) doctor’s notes. Falsifying institutional documents or grades.
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See https://www.academicintegrity.utoronto.ca/ for more information.
Note: You are allowed to work together on the assignment problems. However, if you work with another
person(s) on an assignment problem, you must explicitly include (at the beginning of your solution to that
problem) the name(s) of any individual(s) with whom you collaborated to solve that problem.
Weekly Schedule (Tentative):
Week
Dates
Topics
1
Jan. 11
Introduction; Review of First-Order Logic
Jan. 18
Views about Numbers: Platonism vs. (mathematical) platonism vs. intuitionism;
2
& Frege on Number
3
Jan. 25
Logic and Mathematics I : Frege’s Logicist Project & the Vienna Circle;
Perspectives on Hume’s Principle.
4
Feb. 01
Logic and Mathematics II : Toward Russell and Whitehead’s Principia
Mathematica; G. Peano and Axiomatizing Arithmetic; & Hilbert’s Program.
Feb. 08
5
Logic and Mathematics III : The Language of Arithmetic; Peano Arithmetic;
Arithmetizing Syntax; Representing Arithmetic Relations; & what Arithmetic
Formulas “say” vs. what Arithmetic Formulas say.
6
Feb. 15
†††
Feb. 19–23
Midterm Test
† † † Reading Week † † †
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Weekly Schedule (Tentative):
Week
7
Dates
Topics
Feb. 29
Gödel’s Incompleteness Theorems I : Gödel’s First Incompleteness Theorem;
Gödel’s Second Incompleteness Theorem.
Mar. 07
8
Gödel’s Incompleteness Theorems II : Strengthening the First Incompleteness
Theorem (Rosser’s Theorem); Incompleteness Theorems and Popular Culture;
Provability, Truth and Tarski.
Mar. 14
9
Truth over Mathematics: The role of truth in HIlbert’s Program and Epistemic
Foundations for Arithmetic; Reflection Principles; Identifying and Representing
Epistemic Commitments implicit in (accepting) Arithmetic.
Mar. 21
10
Impacts on Hilbert’s Program I : Goals that survive Gödel’s Theorems; The rise
and influence of L.E.J. Brouwer and his Intuitionism; unique motivations for
Intuitionism; the Logic of Intuitionism.
Mar. 28
11
Impacts on Hilbert’s Program II : Formalisms; Hilbert’s Formalism; Formalism vs.
Structuralism; Hilbert’s Finitist Standpoint: combatting Intuitionism.
Mar. 31
Apr. 04
12
Assignment Due: 11:59pm
Proof Theory: Gerhard Gentzen and the origins of Modern Proof Theory;
Gentzen’s consistency proofs for Pure Arithmetic vs. the Finitist Standpoint;
Gaisi Takeuti’s eliminator method and an upper-bound for Finitism.
Apr. 07
Term-Paper Due: 11:59pm
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