Random Words

The leaves. The air always smelled good then. The stained deck. The maple trees. Climb them until you find a nice place to sit. The past floats as your shadow. The past sinks into you. Or you could say it did sink, some time ago. The patchy grass. The smell of leaves. The rubber boots – and the smell of rain.

The end is only the beginning of how you will interpret this life. Everything you’ve seen, everything you’ve done, everywhere you’ve been. They are all colors on your palate. Make them harmonious.

Do not long for the past. If the past seems like a dark time, realize that one moves forward, not backward, and ghosts of the past are but pieces of memory and do not represent current reality. There is only forward. Pick up the pace – or at least walk comfortably.

One year later

Hot damn...

How strange it is that this page has been otherwise abandoned.

Even stranger that the last post here was exactly a year ago (and I just happened to stumble in here tonight) - leading off from the ambitious and failed 'Godel project,' which at some point was intended to be the beginning focus topic of this neglected piece of the interwebs.

And wham! one year later... what's been happening around here? What the hell happened to Godel? I'm sure you're wondering, and I sometimes ask myself the same thing. Well, the soul of Godel lives on, though the degree to which I talk about him here will be most-likely nil. In the end, my desire to expound on the profound importance of Godel's incompleteness theorems came to a close, due to the realization that (a) if I really want to get into some rigorous nitty-gritty in relation to Godel's work, I need to study the nitty-gritty a lot more, and (b), which is ancillary to (a), many individuals have studied the incompleteness theorems for years and still have trouble wrapping their minds around the jungle of mathematical excitement that Godel put together (or took apart, depending on how one looks at it). The original intention was to post content in this area while I was attending a university course on the subject of Godel's work. I ended up dropping the course (see posts below) and can understand on a somewhat rudimentary level, even after reading several books on the topic, the foundations, mechanisms, and significance of Godel's most important and well-known incompleteness theorems.

Nonetheless, while still holding a special place in my day-to-day reflections, Godel has taken a secondary (possibly tertiary) seat in the immediate real estate - of my brain.

I've met many other fine fellows in the last year, though. The least of which was keeper and traveler of the labyrinth, Mr. Friedrich Nietzsche. It is hard for me to type that name without instantly sinking into some combination of profound realization and utter bafflement. Nietzsche is a character that can neither be disregarded or easily understood. The force with which his philosophy speaks to issues of morality, value systems, and religious and/or idealogical dogmatism, to name a select few, is such that one's mind is almost forced to confront all that is the 'reality' of human existence.

So you could say that the aforementioned 'real estate' has acquired a new landlord for the time being.

But really, I think the point of all this is that... well, there are a couple points I want to make here, even if nobody will ever read this and I'm merely affirming to myself.

I typically tend to feed on knowledge in relation to a subject I'm interested in just long enough to understand the central theme(s) of the subject, and by then something else of interest has come my way and I continue feeding on that subject until something else comes along. I assume that many people do this, what with our media-enhanced attention spans - some just continue down one specialized area while others take samples here and there. My intellectual tendencies probably fall somewhere in the middle of these poles - I sample, gain at least a high-level understanding, sample another, gain a high level understanding, and so on; though I always revisit said 'samples' and thus over time gain more focused understanding, just not as quickly as I might if I dedicate my appetite to a more narrow base of subjects. So, point 1: the subjects covered here may not always be of a consistent theme, but over time many subjects will be revisited.

Point 2: I can't say either way whether I'll post anything else here again. Right now I feel like writing - tomorrow I may not. Same goes for the day after, and all subsequent days. No strings attached. I'll write when I do and when I won't write, I won't write. Who knows, maybe I'll wake up tomorrow and by the end of the day there will be a smattering of new editorial hoo-ha to feed the electricity flowing from my brain to my fingertips.

Until then, if then.

Cheers

Electric Thoughts?

Pandora's Box. Opened. Closed. Opened. Closed...

If one could peek inside the young (and old) mind of Kurt Gödel, it might look like this.

"Simple but strange, as one would expect of a proof that draws so close to the edge of self-contradiction, proving that there are true arithmetical propositions that are not provable. . . . The proposition that Gödel’s proof constructs, the one that will simultaneously announce both its own (provable) unprovability and a true (unprovable) arithmetical relationship, has the same sort of double-entendre as Pagliacci's final tragic cry, "La commedia è finita!"-- the comedy is over."-- From Rebecca Goldstein's book, Incompleteness

Ahh, yes. And so the comedy was over. A comedy which exists in the wishful and untenable possibility of creating a completely formalized system of arithmetic, a set of axioms from which one could derive any mathematical truth. Gödel, quiet and reserved as he was, gently crushed the possibility of this ever happening, and hardly anyone noticed at the time. Hogwash, they said.

To be sure, people eventually noticed, and suddenly they weren't laughing anymore. Nope, now the individuals that faithfully stood behind the mission set forth in Hilbert's program gazed in wanting disbelief at the notion that their mission was, at best, a futile one. Hilbert's program involved, among other things, a call to every Math wiz on planet earth to seek ultimate solutions to particularly vexing problems. In the end, Gödel’s incompleteness theorem speaks not only to the limits of pure logic, but to the limits of human inquiry.

An intuitive grasp of these results tells us something like this: any formalized system of Mathematics will always be incomplete-- there will always be something missing. No matter how hard we try to strip something of its semantic properties-- thereby exposing purely logical relations-- we will eventually be faced with an incomprehensible vast. It seems to follow from the fact that we are trapped in a world of self-reference, i.e., we have no way of looking at things from outside of our own heads (or in this case, systems of formal logic), that attempting to reduce our experiences to pure logic (*note: logic that contains no semantic properties, and only syntactic properties: order, relation, grammar) will result in a system that speaks of itself, and speaks of itself, and speaks of itself, presumably into infinity. The very idea of an infinite chain of axioms is enough to send the hardened seeker of mathematical Truth (with a big 'T') home crying. It certainly was enough to drive a few of them mad.

Alas, for the love of Plato, there seems to be a reality that exists independently of our ability to percieve it.

The everlasting noumenon.

In the next post I hope to cover a bit of history, providing some background into how we've reached this window of possibly infinite possibilities.

Picking up Momentum. Hit the brakes. Drink some water. Jog a little.

Kurt Gödel’s proof for the existence of god, illustrating the fact that roughly 2% of the population knows how the hell he reached this conclusion.

"If you stop now... if you leave the class knowing what I've told you in these two short periods, you can also know that you are already aware of something of which 98% of the population is completely unaware." -- The professor, speaking during my last class period.

This momentous task has been halted-- originally cruising at 150mph, slowed down to a pace of, oh, about 50mph... on my own terms. I dropped the class, though not because I am uninterested. This 'dropping' was due more to the fact that completing my undergraduate sprint has left me with a desire to focus on certain other things, such as studying for and taking the mindnumbing GRE, and carefully planning my future area of academic study, not to mention getting to some of the multi-volume library of literature that I've acquired in the past couple years.

But fear not.

The Gödel project will continue.

By 'Gödel project', I am referring to the previously mentioned task of exploring, and attempting to explain, the mind-stretching and ingenious incompleteness theorems of which Gödel is suprisingly unnapreciated for (at least in all but 2% of the population). There is some great and accessible literature related to this unique figure of Mathematical Logic, and the implications of his proofs are huge in both areas of Mathematics and Philosophy. The preparation for the aforementioned class has been intriguing, to say the least. Aside from the fact that Gödel was apparently a rather strange individual, there are certain aspects of his life and the work he did in logic that put him at the top of a must-read list for anyone interested in the crumbled foundations of Mathematics (yeah, Gödel did that, though most agree that he didn't set out to put a giant (yet subtle) hole in our methods of reasoning. Quite the opposite, I'm sure.) So, recommended reading (i.e., relevant literature that I've thus far found to be especially lucid and intriguing) includes:

Gödel’s Proof --This book consists of about 120 pages of excellent writing, strictly related to the background and foundations of Mathematical Logic that Gödel nonchalantly proved to be forever incomplete-- easily read in 3-4 hours, even for someone with little or no background in Mathematical logic.

A Profile of Mathematical Logic -- Again, a great introduction, providing a history of formal logic from Aristotle to the present. This book is a little more rigorous, but nonetheless helpful and clear for the novice reader, such as myself.

Incompleteness -- This one focuses more on Gödel’s biographical background, including discussion of the infamous Vienna Circle (of which Gödel was a member), his relationship with Albert Einstein, and a sketch of the methods that Gödel used to engineer his incompleteness theorems. I'm only about 100 pages into this one, but so far has been a great read and, once again, is written for the curious and uninformed observer.

I will most likely be quoting these materials at some length.

And for something unrelated, I wanted to direct readers to CK at Arbitrary Marks, the first individual to link to this blog. Hopefully my current bloggy incarnation will give rise to various threads of interesting conversation.

Cheers!

More on symbols. Absolutely relevant.

The Semantics Differentiation of Minds and Machines

(Via Slashdot)

In the Beginning (there were symbols)

For no good reason


*note this is really for my own personal reference—I figured it would be a good way to cram certain concepts into my noggin, and being the layperson that I am at some of this stuff, having to explain it on this page should help. So… here goes.

Predicate Calculus. Symbols. Symbolization. Commutativity. Gödel.

Okay, so we’ve got some symbols, namely:

‘~’ (tilde) stands for what we would commonly think of as ‘not’ or ‘negation’, i.e., “TheZenFly is ‘not’ a penguin” may be symbolized as “~P”, where “P” stands in place of the predicate.

‘→’ (conditional) usually sits in the middle of a conditional statement (If P then Q), such as: “If TheZenFly is a penguin, then Superman is a monkey”, Thus symbolized as “P→Q”, where ‘P’ stands for ‘penguin’ and ‘Q’ stands for ‘monkey’.

‘&’ or ‘^’ (“and”) is used to connect two statements (yeah, English grammar), such as “P ^ Q”… or “TheZenFly is a penguin ‘and’ Superman is a monkey”.

‘V’ stands for the common usage of the word ‘or’, again connecting two separate statements, i.e., “Either TheZenFly is a penguin ‘or’ Superman is a monkey” (P V Q).

One should note that ‘or’ is used in the sense of ‘one or the other, or both’—in this sense, it is an ‘inclusive or’, meaning that statements on either side of the connector (V) could both occur to make the whole true. The symbol ‘V’ with a line under it would indicate a function of ‘one or the other, but not both’—in this case the connector (V) would be considered ‘exclusive’.

I should note that this symbol isn’t really a V, but nonetheless resembles one, and this is all I currently have at my disposal for proper symbolization…

‘↔’ stands for ‘if and only if’, and again, sits between two propositions, i.e., “TheZenFly is a penguin ‘if and only if’ Superman is a monkey” (P ↔ Q)

digging deeper…

Variables, typically symbolized by lowercase letters (x, y, z, a, b, c,…), are meant specify individuals, or ‘subjects’, in the grammatical sense. In the above examples, ‘TheZenFly’ and ‘Superman’ could be symbolized as such: ‘x (TheZenFly) is a P’, and ‘y (Superman) is a Q’ respectively.

Okay, a couple more—noting that I can’t symbolize existential quantifiers because my word processor insists on typing in Greek after entering one (I hope to resolve this minor annoyance at some point). So we’ll just say that:

“For All” is typically symbolized with an upside-down ‘A’, and is meant to specify, as the name implies, all members, or subjects (propositional variables) of a particular set of objects, hence: “For All x, Px” (for all ZenFlys, ZenFly is a penguin, or ‘Ax (Px)’)

“There Exists” is typically symbolized with a backwards ‘E’, and is meant to specify a particular member of a particular set of objects, e.g., “There Exists an x, such that Px” (There exists a ZenFly such that TheZenFly is a penguin, or ‘Ex (Px)’

I’ll have to come up with some better examples…

Also, it occurs to me that I’ve read entire books to prepare myself for this, and it is hard to explain without writing a book myself. In other words, there is a lot of jargon and other hoo-ha that I won’t be able to spend time explaining, though I’ll do my best to stay on planet earth for whoever might be reading. The above mentioned symbols/concepts are essential, so please, bear with me.

Okay, so… certain Mathematical operations can be symbolized using the above mentioned terms. For example we know that addition and multiplication is commutative (1 + 2 = 2 +1). When speaking in terms of propositions instead of numbers, we can say the same of propositional variables, e.g., AxAy (For All x and for All y) (x^y = y^x).

Similarly, we could say that multiplication distributes over addition with the symbolization: AxAyAz, x(y^z) = (x .y) ^(x .z)

So what’s the point? Well, such a system of symbolization can be used to exemplify the logical relations between propositions, kind of like packing propositions into little boxes for the purpose of ridding a set of a propositions of all the gobbly-gook, so that once one is aware of the concepts involved with the symbols, they can easily check the logical validity of such relations. Gödel, the main focus of this project, came up with a couple proofs (involving his incompleteness theorems), showing that any kind of axiomatic system with which one would claim to base any kind of Mathematical truth is necessarily incomplete, or lacking, as it were. In other words, there is no such thing as a logically ‘airtight’ system—there will always be a hole in one’s theories. This is apparently pretty huge. Why? Because it calls everything into question. Ultimately, all of the Mathematical ‘truths’ that we claim to be ‘absolute’ or ‘irrefutable’ are necessarily refutable in some way. I’m guessing this has something to do with the issue of self-reference and the assumptions that one must start with to get their proverbial feet off the ground (this, of course, is just an intuitive guess).

Alas, that’s about as far as I’ve gotten, and Gödel’s proofs are extremely complex (really, I’ve hardly begun to understand), so I shan’t continue to oversimplify. More to come soon…

Scratch Pad. Memorial Tablet.

In the near future (next week), I will be using this space to profile the grueling-- yet profoundly interesting!-- course that will consume my thought processes for the next few months. Topics will include:

Incompleteness Theorems (Gödel)

Set Theory (this is the set of all sets containing blog posts about set theory)

Predicate Calculus (e.g., For all x, there exists an A, such that x is an A. . . . stuff like that)

Paradox (e.g., 'This sentence is false', or, 'I am lying')

Good times-- stay tuned if you feel like heading down the road that drove many others to complete madness! Ultimately, the question to be answered here is, "why will I never be able to prove that I am who I am?" Or rather, why must any such axiomatic version of proof for this matter be necessarily incomplete?

(some kind of explanation forthcoming)

Vessel

The Keyboard is my Vessel.
Plasthmatic Fingers slap plastic.
Send signals.
Into the system-- the 1010110101001011.
Words invisible. Words amplified-- from 0100111100101110101001
Send signals back.
Is this what I'm seeing? Is this what I'm doing?

The 1 is the blood that makes what you see.
The 0 is the empty space in everything.