Tuesday, May 28, 2013

A personal favorite of mine

"since feeling is first

who pays any attention

to the syntax of things
will never wholly kiss you;
wholly to be a fool
while Spring is in the world

my blood approves,
and kisses are a better fate
than wisdom
lady i swear by all flowers. Don't cry
—the best gesture of my brain is less than
your eyelids' flutter which says

we are for each other: then
laugh, leaning back in my arms
for life's not a paragraph

And death i think is no parenthesis"

--- e. e. cummings

Life is more than the form of its syntax.

Thursday, January 12, 2012

Self-Description in Mathematics

The foundations of mathematics is a strange, strange place - and not only because of philosophical disagreements on constructivism, incompleteness of logical systems, and set-theoretic paradoxes (although all of that is indeed weird). No, one of the strangest things about mathematics is its uncanny ability to describe itself.

Ideally, we like to think of the flow of mathematics almost as a directed acyclic graph: The propositional calculus extends to predicate logic; set theory builds on predicate logic; order theory, algebra, and topology build on set theory; analysis builds on all of these. Where things get weird, however, is when things move backwards through this hierarchy. This can perhaps be seen no more clearly than in order theory, and in particular, lattice theory.

The following relationship exists:
  • A lattice can be described as:
  • Algebraic structures are defined in terms of sets, the axioms of which are defined in terms of predicate and propositional logic.
  • Propositional logic, with its AND and OR operations, form a distributive lattice (specifically a Boolean algebra).
So in other words, algebra and lattices are defined with the help of propositional calculus, lattices can be defined in terms of algebra, and propositional calculus can be defined in terms of algebraic lattices. It is amazing to me how mathematics is continually able to so gracefully "turn around" and, in a sense, define itself. Such self-reference has in some cases resulted in major philosophical difficulties, but in others, such as this one, all appears to work fine. How is it though that mathematics is able to adequately define structures more fundamental than those doing the defining? I personally believe that this exact phenomenon is symptom of mathematics being not a fundamental deductive science which guides the universe, but a careful, consistent, and beautiful language whose precision grants it power far-reaching enough even to predict events which have not yet come to pass (solar eclipses, weather patterns, etc). Interesting stuff, eh?

Wednesday, January 11, 2012

Interesting note on the history of symbolic logic

Very short blurb...While reading through a chapter of Stephen Kleene's "Mathematical Logic", I came across an interesting fact. The symbol in mathematical logical for "inclusive or" (nonexclusive disjunction) is traditionally expressed as "\vee" - that much I knew. What I didn't know is that this symbol is descendant from the Latin word for "inclusive or": vel. Pretty cool stuff.

Saturday, January 7, 2012

The C++ Object Model

I've got a bunch of books lying around that I've never had a chance to thoroughly read through; every once in a while, I pick one up and read a chapter. Sometimes, like with "General Lattice Theory" by George Gr├Ątzer, I try to sprint through a chapter or two just to get a general introductory feel for a subject that I have not yet had much exposure to - after all, one cannot really hope to fully come to terms with an abstract subject matter without dedicating a reasonable amount of time to reading it and tirelessly working through some exercises. I feel that it's good to survey areas of science and engineering with which one has little familiarity, if only to receive a first taste for the material, to have something to contemplate and internalize for a few days, and most importantly, to humble oneself.

Two years ago, I acquired "Inside the C++ Object Model" by Stanley Lippman. The book is a little dated (it was published in 1996), but it remains fairly relevant nonetheless. Back in 2009, the book was more than a little beyond my ability to take in, but I have since then have had much experience with C++ and many of its bizarre features: Multiple inheritance, virtual base classes, pointers-to-member-functions, etc. Thoroughly exhausted with abstract mathematics (at least for a few days), I found myself wanting something a bit more concrete and decided to try my hand reading the book once more.

I have so far read only through the first chapter out of seven, but it has certainly been a pleasure. The book elucidates the truth behind several misconceptions about C++ and is ruthlessly, but necessarily, precise in its description of C++'s memory model for object-oriented programming. Lippman even takes the time to discuss the pros and cons behind alternative memory models which could achieve similar goals, and rationalizes why C++'s model is the way that it is.

C++ is something of an anomaly as a programming language. It's a systems language (yes it is, Steven!). It's an object-oriented language. It's a high-level language. It's a low-level language. It's procedural. It's generic. Hell, thanks to template-metaprogramming, it can even be functional! And yet every one of these adjectives comes with a caveat.

Thursday, January 5, 2012

Classification of Algebraic Structures (Reference)

Below is an incomplete list of common algebraic structures together with a short definition. I've come to realize that I would've loved to have something like this when I was learning it all, so I figured I'd write one for others. I'll update this as I continue to learn. This list is by no means exhaustive, and I'm sure that others have probably done similar things elsewhere with greater success. This is simply partly for my own records and partly for those interested who might not otherwise have the motivation to find such a list on their own.

Wednesday, January 4, 2012

An Interesting View of Differentiation

I've never studied any functional analysis (my background in traditional analysis is not yet nearly strong enough), but I do find the topic quite interesting. It's partly due to its place at the convergence of Linear Algebra and Analysis. It's partly because I love the notion of function spaces. It's partly because it plays a central role in some areas of physics which I find quite interesting. And it's partly (probably mostly) because it just seems so exotic that I can't help but be attracted to it.

Despite having never studied function spaces properly, I occasionally run into them in my readings. Today in particular, I was reading about inner product spaces. I eventually came across the topic of Hilbert Spaces, which are inner product spaces which are also complete metric spaces with respect to the metric induced by the norm implicit in the inner product. One way or another, this got me thinking about derivatives from a viewpoint very different from that usually presented in an undergraduate calculus course.

Monday, January 2, 2012

Implications of Godel's Theorems on a Physical "Theory of Everything"

Disclaimer: I'm no expert on physics. If I've mischaracterized anything in this article, please let me know so I can correct it.

I've been on something of a logic binge lately. For the past several days, I've had Godel's Incompleteness Theorems, Paraconsistent Logic, and more floating through my head. I've always been interested as to what the Incompleteness Theorems mean for the notion of a physical Theory of Everything. After writing my recent post on the subject, I feel that I have a much more solid understanding of the Theorems now than I ever have before, thus I feel somewhat confident in revisiting this idea with a fresh perspective. I'm a bit tired, so I'll keep things pretty simple.

Godel's Theorems consider the inherent limitations of axiomatic systems, and the goal of a Theory of Everything certainly fits under this category. Theoretical physicists have for many years searched for a minimal set of all-guiding universal laws which underlie all physical phenomena. So far, this quest has met little success. According to the Incompleteness Theorems, there are two obvious possibilities: Either the axioms of the universe are incomplete or they are inconsistent - presumably, they are not both.