Overview

What is truth? When we say that a particular statement is true, what do we really mean? A physicist might tell you that the statement "If you drop a ball somewhere on earth, it will fall towards the centre of the Earth" is a true statement. They'll explain the theory of gravity and it will sound convincing and will match up with any situation in your life that you've observed a falling object on Earth. But is it really a true statement? How we do know this will always happen? The theory of gravity seems convincing and certainly explains a lot of what we see, but how do know that's true? How do we know it's not possible to observe anything other than what we've already observed? 

In contrast, consider the statement "Suppose a right-angled triangle has side lenghts a,b,c, with c being the length of the hypotenuse. Then: a^2+b^2=c^2." This is a true statement known as Pythagoras' Theorem. How do we know that it's true? Well, consider the figure below. We see two big squares, each of which has a side length of a+b, and therefore both large squares have the same area, namely ab. Inside of each large square are four right-angled triangles with side-lengths a,b,c, but positioned differently. Since each of these triangles have the same area, the total area of white space in each large square must be equal. In the left picture, we see the white space has area c^2; in the right, the white space has area a^2+b^2. It follows that a^2+b^2=c^2. 

The picture and explanation above consitute a proof of Pythagoras' theorem. It is an argument grounded in logic that explains why Pythagoras' theorem is true. It shows that, as long as we're working with plane geometry, this statement is always true--full stop. There is no ambiguity about whether or not this statement it is true, because we have proved it. 

Is the proof given here a good proof? Well, that depends on you, the audience. The proof is correct/logically sound, so in this sense, yes, it's a vaild proof. Did you understand the proof? If you didn't, then I'd imagine you don't consider this a very good proof at this time. A better proof, for you, might include more details about some of the intermediate assertions I made. On the other hand, if you did understand the proof, you may find more details to be tedious and distracting to you. 

Mathematics is the pursuit of absolute truth (which, in itself, needs to be defined carefully). The way that we establish truth in mathematics is to produce a proof that is logically sound. This proof serves a secondary purpose: to communicate with others why a particular assertion is true. If your audience has difficulty understanding your proof, then you have failed to convince your audience that you are actually speaking truth and therefore: is it really true for them?

Math 1000 is a course designed to introduce you to proof writing. It is a writing intensive course (and is a designated WRIT course) in which you will write many or your own proofs, and learn to give, received, and incorporate feedback from your peers--a wonderful test audience for your proofs. In addition to exploring a variety of proof techniques and what goes into producing a valid proof, we'll be talking about what goes into writing a good proof (how should we define this by the way?).

By the end of Math 1000, you will be able to:

1. identify and use a variety of proof structures and techniques such as: proof by contradiction, proof by induction, and proof by contrapositive;

2. self-asses your quality of writing and validity of proofs produced throughout the course;

3. provide construct feedback to your peers on their proofs;

4. incorperate peer and instructor feedback to refine your arguments;

5. demonstrate a working understanding of the core course topics (mostly set-theory);

6. curate a portfolio that documents the development of your proof-writing skills.

In addition to these concrete learning goals that ground Math 1000, I hope you leave Math 1000 with a sense of ownership of your mathematical narrative and the confidence to pursue further students at Brown, whether it be future course work or independent studies. The problem-solving, critical thinking, and logical writing skills that you develop here will no doubt be of use to you in your discipline--even if it's not mathematics!