Post #8: The end is nigh

Assignment 3 has just been completed (well, mostly). At this point, we just have to review everything and make sure that we're not leaving out comments or steps. I am putting off this task until later because it requires a lot of to-and-fro scanning, which isn't difficult - it's just tedious. Meanwhile, I have been working really hard this week to try to catch up with everything that has been going on. As the end of the term (and school year) comes to a close, everything is coalescing at the last moment and making the last few days EXTREMELY hectic.

I have started on my problem solving question, and wrote down some preliminary observations and notes. I will continue working on the problem later this week. Excuse any typos I may have made; I rushed the typing of the problem (which is quite long, as you can imagine, since it is part of a storybook). 

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Context - Problem Solving
After several hours of questioning, Alice gathered an enormous amount of data, which she recorded in her memorandum book. She took it all to Humpty Dumpty to see if he could explain it.

“It figures,” said Humpty Dumpty, looking through Alice’s notes, “it figures!”

“What do you mean by that?” asked Alice. “Is this White Knight untruthful?”

“The White Knights never lie, “ replied Humpty Dumpty.

“Then I don’t understand,” replied Alice, “I really don’t understand!”

“Of course not,” responded Humpty Dumpty contemptuously, “you don’t understand Looking-Glass logic!”

“And what is Looking-Glass logic?”

“The kind of logic used by Looking-Glass logicians,” he replied.

“And what is a Looking-Glass logician?” asked Alice.

“Why, one who uses Looking-Glass logic,” he replied. “Surely, you could have guessed that!”

Alice thought this over. Somehow, she didn’t find this explanation very helpful.

“You see,” he continued, “there are certain people here called Looking-Glass logicians. Their statements seem a bit bizarre until you understand the key -- which is really quite simple. Once the key is understood, the whole business makes perfect sense.”

“And what is the key?” asked Alice, more curious than ever.

“Oh, it would never do to tell you the key! However, I will give you some clues. In fact, I will give you the five basic conditions about Looking-Glass logicians from which you can deduce the key.  Here are the conditions:

1. A Looking-Glass logician is completely honest. He will claim those and only those statements which he actually believes.

2. Whenever a Looking-Glass logician claims a statement to be true, then he also claims that he does not believe the statement.

“Just a minute,” interrupted Alice. “Are you not contradicting yourself> According to the first condition, a Looking-Glass logician is always truthful. If, then, he claims a statement to be true, he must really believe that it is true. How then, without lying, can he claim that he doesn’t believe the statement?”
“Good question,” replied Humpty Dumpty. “However, I never said that a Looking-Glass logician is always accurate! Just because he believes something doesn’t mean that he necessarily knows that he believes it, nor even that he necessarily believes that he believes it. Indeed, it could happen that he erroneously believes that he doesn’t believe it.”

“You mean,” replied Alice, utterly astonished, “that a person can actually believe something, and yet believe that he doesn’t believe it?”

“With Looking-Glass lgoicains, yes,” replied Humpty Dumpty, ‘in fact with Looking-Glass logicians this always happens -- this is a direct consequence of the first two conditions.”

“How is that?” asked Alice.

“Well,” replied Humpty Dumpty, “Suppose he believes a statement to be true. Then, by Condition One, he claims the statement is true. Then, by Condition Two, he claims that he doesn’t believe the statement. Hence, again by Condition One, he must believe that he doesn’t believe the statement.

“Anyhow,” continued Humpty Dumpty, “I’m giving you too many hints! Let me finish my list of conditions, and then you should deduce the key to the entire mystery.”

3. Given any true statement, the Looking-Glass logician always claims that he believes the statement.

4. If a Looking-Glass logician believes something, then he cannot also believe its opposite.

5. Given any statement, a Looking-Glass logician either believes the statement or he believes its opposite.

“And that,” concluded Humpty Dumpty proudly, “is the entire list of conditions. From these you should be able to infer just which statements a Looking-Glass logician believes to be true and just which ones he believes to be false. I will now ask you some questions to test your understanding.”

QUESTION

Suppose he believes that all gryphons have wings. Does it follow that there are any gryphons?

We have the conditions: 

1. A Looking-Glass logician is completely honest. He will claim those and only those statements which he actually believes.

2. Whenever a Looking-Glass logician claims a statement to be true, then he also claims that he doesn’t believe the statement.

3. Given any true statement, the Looking-Glass logician always claims that he believes the statement.

4. If a Looking-Glass logician believes something, then he cannot also believe its opposite.

5. Given any statement, a Looking-Glass logician either believes the statement or he believes its opposite.

To do: tweed out the differences between claim and believe (because they are definitely different -- one is more declarative while the other isn't)

To do: translate things into symbols.

BELIEVE [TRUE FALSE] and CLAIM [TRUE FALSE] into predicates, in order to evaluate the statements above. 

I will follow up on this soon!

Post #7: Post-test stress

I'm talking about syllabic stress (aw yeah, let's get the poetry started) : 

white pages and empty spaces/

the vast expanse of a test undone/

the clock uncogs and so begins the hum/

the familiar tap tap of pencils like drums/

without time to stall the students scrawl/

the answer to the first question/

obvious to those who paid attention/

to solve it meant algebraic expansion/

moving on to the second question/

two out of three in tidy succession/

the logic was clear and without/

fear  I deftly attacked the problem/

pencil in hand and feeling in command/

the third question now lay before me/

who was to say that the previous day/

I had forgotten to look at floors and ceilings?/

nonetheless I can say I tried my best/

to deduce the answer through reasoning/

but tick tock announced the clock/

as classmates finished triumphant/

pound pound my heart was a-flutter/

amid the backdrop of constant mutters/

the end was nigh and with a final sigh/

the test was returned unfinished/

had I not been so careless and aloof/

I would have finished the very last proof

It turns out that last question was extremely straightforward (it was by far the shortest proof out of the three). I figured out how to prove it 5 minutes after handing in my test. Sigh, mais c'est la vie.

Post #6: In which I talk about the prettiness of proofs

Today another milestone was reached -- assignment 2 is now finished!

For better or for worse, chapter 3 (proofs) is now officially a thing of the past. Well, actually, this is not entirely true. Proofs will reoccur in the next chapter (algorithm analysis) and feature heavily in the rest of my math and computer science career. However, it does feel good knowing that the completion of assignment 2 marks the end of the proofs chapter. There is always comfort in closure.

Assignment 2 was very well-designed in that it required us to tackle a variety of proofs using a variety of different techniques. Going through the process of deciding which technique to use for what proof has helped me recognize the subtle differences in the expressions which make some directions easier to work with than others. Since the scope of the questions covered diverse topics from modulus, inequalities, to the greatest common divisor of a pair of numbers, I found that doing the proofs reinforced my understanding of these often (overlooked and forgotten, but also extremely important) mathematical properties and definitions.

In retrospect, the questions were not too hard -- although my group did spend a considerable amount of time on question #1. Oddly enough, we found this to be the most challenging question. We mainly ran into the trouble of finding a starting point in manipulating one side to look like the other. The proof statement looked easy enough, but the right "step" that sets the momentum of the proof was hard for us to discern immediately. Contrariwise, I personally thought that the algebraic proofs were the easiest, because the intermediate steps were straightforward as long as you saw the end product and tweaked things here and there to achieve what you wanted from the very beginning.

Having been exposed to a formal study of proofs, I now appreciate the creativity and beauty inherent in the process of proving even the simplest of statements. By writing down all the links and connections made in the body of the proof, you are acknowledging the sometimes disparate concepts and ideas from every corner of mathematics, and unifying them in a single context. As the proof-writer, you also play the dual role of a teacher and a student. You are, at each step in the proof, trying to convince both yourself and readers that your line of reasoning is correct. Indeed, until you  actually reach the end of the proof, each intermediate step could either take you to the next vital step, or deter you from reaching the conclusion (in which case you'll have to sit down and  brood and ponder and stare blankly at white walls until the answer comes to you). This is what makes proofs so interesting as well as rewarding; all the insightful "aha!" moments interspersed with the frustration of trying to see where things are leading make you at once vulnerable, and in-control. With this new-found zeal for proofs, I'll move forward in my studies with an open mind.