w-III

Here is a very nice and good post about week three in the course.

So the third week was about firstly, conjunction, disjunction, and negation, and then about truth tables and then about manipulation of these.

The first section is almost just introduction of conjunction and disjunction as extensions of the ideas of union and intersection. The interesting part comes with negation and the variety of equivalent statements that can be made by moving around the scope of the negation and making appropriate changes to quantifiers, etc. At this point, actually, it becomes simplest to work through these examples through natural language. We all have some sort of idea of how to explain the opposite of some situation, but now we have a better understanding and framework for moving easily between natural language and symbolic language so this becomes the simplest manner of manipulating these operators.

The bit about truth tables is always interesting because sometimes it is easy to want to look at an expression and have an answer but oftentimes a truth table lets us take a step back and slowly work through these expressions in a rigorous way to avoid making mistakes. I always liked the idea that if two truth tables are equivalent then the two expressions are equivalent in a complete and rigorous way. It is always entirely possible that two expressions are equivalent but have entirely different meanings. I would liken it to `the formula for prime numbers' which was mostly just a tricky way of enumerating values using an alternate formulation of Euler's totient function and while looked nice in some sort of closed form, provided no new or interesting information or insight. A truth table for two True/False arguments has only so many possibilities but we can formulate an infinity of absurd expressions that we could demonstrate to be equivalent to some obvious or simple expression. I'm not sure to what ends, but the rigour over a small number of elements is a very `easy' thing to investigate so equivalence proofs become all but trivial.

Finally, the idea that evaluating P=>Q for P False, always evaluating to True. We cannot evaluate this expression with respect to the implication of P when we have `not P', so we must call it true. It reminds me of concepts of open and closed in mathematical sets, maybe in topology. The idea is that sets can be open or closed, but they can also be both or neither. If a set's complement is open then that set is closed and if a set's complement is closed then the set is open. The implication of the converse is what describes the set and complement is a unitary/cyclic operator such that the complement of a set's complement is the set itself. It is just another interesting example about properties that we must be careful to define and manipulate in a very careful way, because while not easily relatable to physical or extant objects, the math itself is complete, consistent and rigorous. If we simply permit weirdness and shirk our human need to relate things to the physical world (an important part of learning about the quantum universe), we can use our math to learn more and extend our knowledge and understanding. We just need to have a little bit of faith. Which is actually pretty scary for a `scientist', I guess.

Okay, cool.

J

w-II

TOP SECRET

THIS IS A POST ABOUT WEEK TWO IN THE COURSE

I'M NOT YELLING, THIS IS JUST HOW COVER PAGES
FOR TOP SECRET DOCUMENTS LOOK

I JUST IMAGINE GEORGE W. BUSH IN HIS TEXAS DRAWL:
"HOW COULD ANYONE IGNORE ALL THESE CAPITAL LETTERS? IT'S IMPOSSIBLE!"

--------------------------

Week two in the course was touching on material that was new to me which is always nice because I like that sort of thing. I am perfectly familiar and comfortable with set notation and the like but predicates are definitely a new thing.

The rest of the lecture was a lot of parsing statements to identify P and Q if they were to be made to match the mathematical statement P=>Q. This started off slow and easy but later progressed to some rather challenging examples that required careful consideration to understand what was being said in plain english and understand the implications in a mathematical (P=>Q) sense. Overall pretty interesting but will likely require me to go review some of the trickier examples. Another horrible failing of the english language. All these equivalent ways of describing a P=>Q situation with some much less clear than the others.

Also a fun example of vacuous truth. Such an interesting mathematical/philosophical phenomenon, and there's another good one about it in the course notes. I guess the idea is that we might run into such a situation when we are programming. Nonetheless they are important to know about and understand.

My final note about the lecture is that there were two characters seated immediately behind me who seemed to think that their unsolicited sarcastic comments were in some way valuable or insightful. Sadly they were mistaken. It was pretty annoying. Almost as annoying as me griping about it on my SLOG, but just not quite.

The final piece of things we did this week that I feel in some way about it the venn diagram with 'x' and 'o' marking. The idea is to draw a venn diagram of two subsets of a third larger set and place 'x' in any area where we know that no element can exist, and 'o' in any area where we know some element must exist. In our tutorial, we were always interested in a single one of the two subsets, and placing indicators 'x' or 'o' in either its intersection with the other subset or its are disjoint with the other subset. This one set in which we were interested, was known to have three elements. Because it is known to be non-empty, when either of the disjoint or intersection (with the other subset) areas has 'x', it is known that the other must have 'o'. Because the union of the disjoint and the intersection section is the whole (sub)set, if one section is empty, the other must necessarily contain any any elements that are a part of that (sub)set, which is at least one (indicated by 'o') by the (sub)set's known non-emptiness. A set's non-emptiness is a required condition to produce a situation to which this 'x' - 'o' complement applies. For our example problems, we can and we must use this to place 'o' (in its correct spot) in any situation where we have placed 'x'. Reviewing the other examples, we typically see that sets are rarely given as non-empty. It is necessary then to take caution when attempting to apply this 'rule'.

I guess my final point is the role that our use of natural language takes in understanding and thus solving these problems. Suppose we pose the question:

If "some men are doctors", and "some doctors are tall", does it follow that "some men are tall"?

It is maybe not entirely clear, but the answer to the question (in this case) "Are some men tall?" is "no". There is an entire psychological aspect to this problem where if we simply ask "In real life, are some men tall?" everyone would be answering "yes". It is true. We can construct a reasonable definition for what constitutes "tall" based on human height measurements and distributions, and then look at the set of professional basketball players and find at least one single example to demonstrate that there exists a man (an element in the set of men) who is tall. No problem, all done. But at the same time, if we simply replace "men", "doctors" and "tall" and we instead ask:

If "some cats are pets", and "some pets are dogs", does it follow that "some cats are dogs"?  

It might be easier to evoke a correct response of "Of course not! What a ridiculous question! Why would you waste my time even asking that?". But this is because the question that we ask with the foreword "does it follow" is false even with the foreword "in real life". What about further abstraction? What happens when we ask:

If "some A are B", and "some B are C", does it follow that "some A are C"?  

Maybe we can prompt more advanced or specific thought about the question rather than about men and/or medical degrees and/or height and/or cats and/or dogs and/or pets. Just some thoughts. I guess the point is that abstraction to more concrete statements can help us solve questions in our natural language that can often be unclear or confusing for reasons of familiarity and association of these words with extant things or concepts. What about asking:

If "A ∩ B ≠ ∅", and "B ∩ C ≠ ∅", does it follow that "A ∩ C ≠ ∅"?

I'm not sure, but I think it's interesting for sure.

Please someone let me know if I'm on the right track with this whole SLOG thing.

That's all for now.

Thanks for slig-sluggin' it with me,

J

w-I

Spoiler alert: this post is about week one in the course.

The first week lecture was really only 2/3 of a lecture with the first 1/3 being dedicated to reviewing the syllabus. I made sure to accidentally get some coffee on my syllabus to mark it as my own in case some other student tries to make off with it. It was pretty exciting that we get to write a blog about our experiences in the course as opposed to most of my courses where I am forced to sign a waiver stating that I will maintain the secrecy of the goings-on in the course and transgressions of this nature are punishable by death.

Over the course of the lecture we were reminded of the benefits that active participation in the course has. Science has show (or at least the lecture notes have shown) that active participation and engagement in the course is equivalent to punching all of the inactive non-participant students in the face.

I appreciated the idea in the lecture of precision versus ambiguity in human language and that we must strive to be `as precise as as [is] necessary' in order to make communication maximally efficient. This is really just a nice philosophical idea that underlies the ways--importantly the differences in the ways--that we communicate in different settings and with different people. For the past two years I was working on a graduate degree and studying a single subject area in that way almost `warps' your thinking and vernacular to maximal efficiency in that field. It is hard not to think of blood oxygenation level-dependent (BOLD) functional MRI signal when people around me use the word `bold', but this is simply an adaptation made in an attempt to be maximally efficient when discussing functional MRI.

I also enjoyed the `streetcar drama' problem. The problem involves minimal information overheard on a streetcar about the ages of a person's 3 children. We know the product of the ages but the other information is less clear and needs to be interpreted cleverly in order to solve the problem. While we are never given the sum of their ages we can glean from the information given that the for all integer combinations, the sum is degenerate (which is to say, not unique, and not to say that it is in any sense immoral). Lastly, we learn that there is only one eldest child, that is to say one child is strictly older than the rest. Using this information we are able to solve the problem.

Going over the problem in class I remembered that I really wanted to punch someone in the face (metaphorically) so I did put up my hand for this problem and contributed the information that the oldest child must have a unique age. I certainly will remember doing so from the emotional ordeal that doing so has caused (which is what is says in the lecture notes).

A lot of the quantifier stuff I was previously familiar with from other math courses but it is still a good review as I haven't touched that stuff in a long time.

I'm also pretty curious about how SLOG feedback works. Is this a decent post? Maybe even a spectacular one? I would like to know. I don't want to get to the end of the course and then get told that my SLOGs are too irreverent, or I am too good-looking, or some other thing and I am losing marks in the SLOG section. I'm just trying to enjoy it and have fun with it. Maybe that's the motivation behind having us do this at all? I have no idea.

Okay. Cool.

Thanks for sluggin' it in my SLOG,

J