Sorry for the recent delay (due to inclement weather that one of the instructors had been experiencing) in posting lecture materials. Normal scheduling shall resume soon
This course will cover elementary algebra, geometry, and some aspects of precalculus and calculus, but it shall be approaching these subjects from the perspective of higher mathematics and philosophy of mathematics. The mathematics covered will be at the high school to first year undergraduate level, so people with very little background in mathematics, but an interest in what philosophers and mathematicians do and in learning mathematics should get a lot out of the course. We will introduce some concepts in higher mathematics, including set theory, combinatorics, measure theory, number theory, analysis, proof theory and abstract algebra, and we shall show how the elementary concepts relate to these topics. We will also introduce concepts in the philosophy of mathematics, such as what mathematical objects and relations and properties are (mathematical metaphysics), whether we discover or construct them, and how we know about them (mathematical epistemology), and how to go about mathematics (normative mathematics).
- What is a number?
Make a Khan Academy Account and add me and ange1obear as a coach by clicking on your user name in the top right and adding our email to your "coaches" list (firstname.lastname@example.org). After doing this, watch Kahn Academy's entire Addition and Subtraction series doing all the practice module problems in between (they are the starred items in the side bar). We will not be able to track your progress if you don't do this!
Try to think about how we might define a number. Also register on the Canvas page so that you can get the first reading.
We are literally starting you at 1+1=2 (per Khan Academy's practice module), so very little mathematical background is required here. The Higher Mathematics and Philosophy is mostly conceptual, but a background in either is helpful (e.g. if you have already taken a mathematics course and are taking this one to brush up, you are in an excellent position to get a lot from the course, ditto if you've taken a philosophy of mathematics course). You will also require adobe acrobat reader or a browser which can read pdfs, and an email so you can use canvas.
There are three aspects to this course. The first is the elementary mathematics (algebra, geometry, precalculus, etc) aspect, which shall consist of mostly third party material (Kahn Academy, other Ureddit courses in introductory mathematics, etc). Each week (and sometimes more than once a week), a lecture shall be presented on a given topic in elementary mathematics, and a problem set/quiz with our selected problems shall appear on the Canvas page for students to do (more on Canvas below). The second aspect of the course is the higher mathematics aspect. We will introduce a higher mathematical concept in a followup lecture which relates to the elementary concept in that week's lecture. We will then assign a problem or two that would be relevant to understanding that concept and how mathematicians use it. The third aspect is the philosophy of mathematics, which shall consist of a reading authored by a philosopher of mathematics on a subject relevant to the higher mathematics concept that week or the lower mathematics one. One or two problems shall be assigned which are relevant to that topic as well. Live conferences shall occasionally be held over Canvas depending on student demand for either review or learning new material in a more discussion based format.
The main course texts (which all include available on the internet for free) in each aspect are (abbreviations included after titles):
Elementary Mathematics: Elementary Mathematics by WWL Chen and XT Duong [EM] (this text is mostly supplementary, Khan Academy's practice module and other courses shall be providing most of the material in video, not text, format. Doing as many problems as you can is essential for building your mathematical intuition, so it's good to read these resources for practice problems).
Higher Mathematics: Elements of Abstract and Linear Algebra by Edwin H. Connell [EAA] and William F. Trench's Introduction to Real Analysis [IRA] are available online for students to consult if they wish. The lectures introducing these concepts shall probably be sufficient for you to do the problems assigned on them, and we will not be getting in as deep as these books do, but students who are interested in exploring higher math on a deeper level during this course are encouraged to make use of the "How to Become a Pure Mathematician or Statistician" [HBPMS] blog and the ureddit course on group theory.
Philosophy of Mathematics: Philosophy of Mathematics: Selected Readings by Benacerraf and Putnam [PM] and Russell's An Introduction to Mathematical Philosophy [IMP]. Readings from PM will be posted in the subreddit.
Along with reading material, this course will also include some video and teaching module resources, including:
Canvas: Canvas is a teaching module, similar to blackboard and sakai, which allows instructors to give online quizzes to students, do webconferences, and give and receive assignments as well as update a class. The assignments (e.g. problem sets and quizzes) shall all be posted on canvas (due to its latex functionality) and web conferences shall be done on Canvas. To use the canvas site, you will have to email an instructor your email.
Khan Academy [KA]: We really owe everything to this guy. He's very committed to free information, and his practice module for mathematics (click on "practice" after making an account) is very intuitive and simple. We'll be tracking your progress on this as you go through the course, and using Khan's videos to introduce many elementary mathematics concepts.
"How to Become a Pure Mathematician or Statistician" [HBPMS]: This is an excellent blog for someone wanting to self-study any level of mathematics. Many free resources are included. We will be using material from this site mostly for practice problems.
The Stanford Encyclopedia of Philosophy [SEP]: is the foremost reference in academic philosophy today for a good reason. Each article is written by a notable philosopher on the article's subject, and the non-philosophy stuff isn't bad either (linked here is the introductory article on set theory, which is a very good resource for you to study right now to learn topics on the first part of the course).
The Course Outline
Lower Math portions are denoted by [LM] at the front, higher math portions are denoted by [HM] and Philosophy portions are denoted by [P]
What is a Number?
- [LM]: Addition, Subtraction, Multiplication, Division, Order of Operations, Fractions, and Exponents.
- [HM]: Addition is defined, and the arguments it takes (numbers) are questioned as to their appropriate definitions. Set Theory is introduced.
- [P]: What do we mean when we give a mathematical definition of something? The notion of an axiom and a theorem are fleshed out. Grounding relations are explored and logicism is introduced.
Where are the numbers?
The e-mail for reaching either of us is email@example.com
Ange1obear is a graduate student in philosophy at a top 20 department (Leiter) in the US. He has bachelor's degrees in physics and philosophy from a top 5 university (US News). Gnomic is an upperclassman biochemistry major at a university whose Mathematics department is among the top 20 in the US (US News) and whose Philosophy department is among the top 10 (Leiter).