Feynman point

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999 and so on.

Gödel's universe

Kurt Gödel found a solution to Einstein's equations for General Relativity where time is meaningless, because every point in spacetime can access every other point in spacetime through Closed Timelike Curves.

In other words, he found a solution to Einstein's equation where you can trivially time travel to any point in history from any point in space, a clear violation of Einstein (and most every other physicist's) views on the nature of time.

It can be defined as follows:




where ω is a nonzero real constant, which turns out to be the angular velocity, as measured by a nonspinning observer riding any one of the arbitrary points.

Gödel never explained how he found his solution, but there are many possible derivations. Let's see one here:


Start with a simple frame in a cylindrical type chart, featuring two undetermined functions of the radial coordinate:

 

Think of the timelike unit vector field e0 as a tangent to the lines of arbitrary points.

Following Gödel, we can interpret the arbitrary points as galaxies, so that the Gödel interpretation becomes a cosmological model of a rotating universe. Because this model exhibits no Hubble expansion, it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe which would in principle be allowed by general relativity (if one admits the legitimacy of a nonzero cosmological constant).

Russel's Paradox over morning tea

The following is an unfinished post I found gathering dust in my drafts folder. I'm too lazy to finish it, it's too interesting to delete, and leaving it to fester just seems wrong so I'm posting it in its unpolished, incomplete form. Maybe it'll be interesting to someone.

This morning, as I settled with some tea to check my RSS feed collection with Netvibes, I noticed that a recent Dictionary.com Word of the Day was circumlocution. I found this to be rather delightful, as I've always been a fan of the word.

I first happened upon the word in the midst of some forgotten book which I read in my teens. Upon reading its definition - The use of many words to express an idea that might be expressed by few - I found it endlessly charming that the word circumlocution itself is, in fact, an example of the opposite of what it describes; that is to say, it is an example of using one word to express an idea that would otherwise require the use of many. Lexically speaking, circumlocution is an example of its own antonym. For the sake of clarity, that antonym could be taken to be conciseness or terseness.

I became an instant fan of the word, not least of all because of its scholarly Latin character and elegantly flowing phonetic quality. Much to the chagrin of anyone unfortunate enough to be in my company, I'd attempt to use it in casual conversation as often (and sometimes, it would seem, as inappropriately) as possible.

In other news, I'm possibly the most humdrum person you'll ever meet.

I'm fascinated by the concept of self-reference

These words are not autantonyms (also known as Janus words). Autantonyms are homographs with inherently incompatible binary definitions or, more strictly, polysemes. All this befuddling lingo-babble just means that autantonyms are words that have more than one meaning, and two of those meanings happen to be opposites. The word "fast", for example, can mean "quick", but also "fixed in place" - which is the reverse of the former definition. For our purposes, our words are, from a completely self-referential perspective, themselves examples of their own antonyms. Autological words are words which describe themselves. Heterological words are words which do not describe themselves. There is, as far as I am aware, no term for words which strictly describe the opposite of itself. Or, as stated above more clearly, examples of their own antonym. For the sake of convenience, I shall coin these antautological words. The distinction between heterological words and antautological words is subtle, but important.

The word "abbreviation", for example, means "reduction" or "shortening". With the average length of a word in the English language being approximately five letters, it's safe to say that "abbreviation, weighing in at a comparatively marathon twelve letters, is clearly a long word.

You have two sets, "words which are examples of their own antonyms" and "words which are not examples of their own antonyms" (Set A and Set B respectively).

Set A contains words like "abbreviation" (for the sake of convenience we'll say that at twelve letters it falls outside our objective criteria for a "short" word).

The Grelling-Nelson paradox holds for contrautological words. If "contrautological" is an autological word, it applies to itself. The paradox is introduced. If "contrautological" is a contrautological word, then - by its definition - it must be an autological word. The paradox is again introduced. Is "contrautological" both a contrautological and an autological word at the same time? Think it over after getting some sleep.