Proofs and Puzzles

These past few days, this course has only been concerned with proofs. Although the proofs so far are not difficult, I predict they will eventually become fairly difficult. Proofs have always been one of the hardest aspects of mathematics. As opposed to calculations, which eventually become fairly mechanic, proofs require a high level of intelligence and understanding. It is not simply a matter of following the same algorithm over and over again. Usually, the middle part is the most difficult. It's the part that requires some intuition, some intense reasoning, and (excuse the cliche) the ability to think outside the box.

It's not that proofs are particularly difficult for me. Not to say that I am particularly good at proofs, but I tend to get along on math test. The way I've always studied for math is to do lots of problems. Even when it comes to proofs, to study for those, I would do a lot proof problems. So far, it's working pretty well, but maybe this course can give me more effective methods at tackling proofs.

Anyways, I might as well use this opportunity to record a problem solving scenario. The problem I had to solve was the tutorial 3 office hour puzzle. Since this puzzle was decently easy, in the future, I will probably post another problem solving scenario.

Anyways, the link to the tutorial can be found here: http://www.cdf.toronto.edu/~heap/165/W10/Tutorial/t3/t3.pdf

Understand the problem:

This problem is fairly straightforward to understand. Basically, you are given a bunch of information, and from that information, you have to figure out the order the students came to the office and what each students wants to do.

The second step is to plan a solution, and for this fairly simple problem, there isn't going to be a complex plan of any sort. Basically, I am going to list all the given information one by one and try to deduce further information from the given information. Basically, the plan is to just connect the dots.

So, we know that the last two are not interested in computer games and one of them is interested in designing programs to create music. We also know that Alex came to office hours first, so he could be interested in computer games, but he is not interested in computer generated music. We know that Kim did not arrive second and did not arrive first either, and is not interested in video games, which makes sense given what we know. Since someone came after Lee, we know that Lee could be second or third. And finally, we know that the last person wants to study bioinformatics.

Now, from this, we know that Lee could not have been third, since the person who came after him wants to design databases, and the last person wants to study bioinformatics. Hence, Lee could only have been second. We also know that the last person is female and that Kim is male, and hence, Kim could not have been fourth. Hence, Kim is third and that leaves Robin to be fourth.

Since Robin is fourth, she is interested in bioinformatics. Since the person who came after Lee is interested in databases, that means that Kim is interested in databases. And, since Alex is not interested in computer generated music, that means that Lee is, and that Alex is interested in video games. Thus, we have:

Alex: first, computer games

Lee: second, music

Kim: third, databases

Robin: last, bioinformatics

However, I guess it can be argued that we don't know what Alex is interested in. All the problem says is that the last two are not interested in computer games and that neither is Alex. That does not necessarily mean that anybody is interested in computer games. But, I guess it is implied.

Looking back, I realize that this problem is just a matter of connecting the dots. Next time I'm faced with a similar problem, I'll also begin by listing all the given information and try to deduce further information from that. Now, I get my opiods.

Law of Excluded Middle

Hmm, so we're doing proofs now. I sort of liked logic more, but maybe we'll start doing logic proofs. So far the proofs are pretty easy, so I'll have to see. I sort of find the indenting and commenting and necessity of introducing very obvious statements somewhat tedious. In Calculus, the proofs are less formal without all that indenting and I don't have to explain so many steps.

I was recently getting distracted by Wikipedia and read the article on the law of excluded middle: either P or not P. Basically, a statement is either true or false, and obviously can't be both. This reminded me of an essay I read about Bertrand Russell. The essay started with the statement: "The king of France is bald." Clearly, this statement is not true, which according to the law of excluded middle, means that the opposite must be true: "The king of France is not bald." However, this does not seem to be any more true, since France does not have a king. Yet, we can't just discard these statements and call them meaningless. They do have a clear logical form. Logic seems so much more clear when you are only using symbols. When applied to natural language, logic seems to screw up a lot of times. In any case, Russell's solution to treat sentences of the form "The F is a G" as three separate claims: "There is an F", "no more than one thing is the F", "and if anything is an F, then it is G." Basically, he tries to treat it as universally quantified. In the case of the king of France, this resolves the problem, because there is no king of France. But, what if I made the statement: "Santa Claus is jolly." Once again, there is no Santa Claus, but intuitively, even though there is no Santa Claus, this seems to be a true statement to me, especially in contrast to its negation. In any case, Russell's solution appears in an essay On Denoting, which I have never read. Maybe he addresses the issue of fictional characters. I would have to read his essay to find out.

Another thing that is also bugging me is this pair of sentences: "The next sentence is false. The previous sentence is true." If every statement is either true or false, then which one of these statements is true and which one is false? Or, is it meaningless to assign truth and false values to these statements; that these statements are neither true nor false? Even if it is meaningless to do assign truth and false values, I don't think these statements themselves are meaningless.

First Post

Before taking this course, I really did not know what it was going to be about. "Mathematical Expressions and Reasoning for CS"; what the hell does that mean? So far, this course seems to be an introduction to symbolic logic, which I'm quite excited about. I honestly enjoy this course very much, and I'm not trying to be fawning. Since I'm doing a double major in computer science and philosophy, it's really not surprising that I'm interested in logic. I think logic is the only discipline that ties philosophy and computer science together. It's actually one of the fields of philosophy that I admire the most. Unlike some other fields, especially metaphysics, logic isn't filled with BS. As interesting as people like Sartre and Nietzsche are to read, I can't help but feel that their ideas are entirely unfounded. As a consequence, people might think that logic is dry, but this is really a matter of personal taste. To me, not only is logic so clear and precise, it is a field that all other fields of philosophy depend on. To make good philosophical arguments, one needs to have a good control of logic. To even get by in life, one needs to have at least some understanding of logic.

I was first introduced to logic when I read about Wittgenstein and Russell: Russell's paradox, Russell's essay "On Denoting", Wittgenstein's "the beetle in the box" thought experiment, etc. Actually, a lot of their work on logic is also related to language. In any case, before taking this course, I learned about logic in philosophy courses. In those courses, I mostly learned about how logic relates to argument: modus ponens/tollens, logical fallacies (both formal and informal), translating natural language into logical arguments, soundness and validity, etc. Basically, I learned about applying the rules of logic to everyday language in order to construct powerful arguments. This course takes a different approach in that it tries to express logical sentences using symbols and applying rules to manipulate those symbols. It's approach is certainly more mathematical and more complex. I've always wanted to take a course like this, so I'm pretty happy. Well, that's it for now. Let's see how this course goes.