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.

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:
• A partial order in which each pair of elements has a unique infimum and supremum.
• An algebraic structure which consists of a set together with two operations, "\wedge" and "\vee" such that the operations obey laws of associativity, commutativity, idempotence, and absorption.
• 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?

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.

Philosophical Digression on Paraconsistent Logic

While reading about Godel's Incompleteness Theorems before writing my post on it, I encountered a strange idea: The notion of "paraconsistent logic". As I explain in my Godel post, the Principle of Explosion states that "from a contradiction, all else follows"; if a formal theory is capable of expressing a contradiction, then both a statement and its negation can be used for proofs. This often leads to all statements being true in a theory. Such a theory is termed inconsistent.

But certainly not all contradictions lead to the principle of explosion! If I simultaneously believe, for instance, that an apple is both green and red, this will not lead me to the contradiction that a number can be both odd and even. Therefore, the Principle of Explosion applies only within the scope of what objects the contradiction speaks of (directly or indirectly). The study of paraconsistent logic considers the implications of having rigorous logic theories in which some contradictions are possible.