The Obscurometer was coded by a good friend of mine over the last week. He and I were constantly exchanging tips and tricks about PHP, SQL, math, statistics… a lot of effort went into this little gadget, and I’m stunned at how well it turned out. To be honest I’m a little jealous: here I haven’t released a major project in some time, and now my buddy is going all viral and stuff. Hmm, maybe I should think of something to catch up…

The way the obscurometer works is it queries Last.FM’s API to ask about which bands the user in question has listened to and how often. Then it looks up how many people listen to those bands. The fewer people listen to a band, the more obscurity it gets. The more times you listen to that band, the more obscure you get!

Whether or not you have a Last.FM account you’ll find this page interesting: Statistics. It lists the most obscure listeners that the program has discovered so far, and the least obscure listeners; as well as the most and least popular artists. Hmm, if this gets popular, it could start some kind of Markov chain: just by virtue of being on the “most obscure” list, a group becomes more well-known, paradoxically lowering its obscurity… AHH, my head’s going to explode!

Anyway, go check it out… The Obscurometer

FURTHER READING

Kubla Khan and Dirk Gently’s Holistic Detective Agency
Fighting Music Addiction
The Two Types of Music

That book is the type of book you can read through once and get a lot of enjoyment out of, but the mystery’s plot is a mystery of its own, and it takes many readings and lots of thinking to really understand in all its depth. This is a much subtler and more profound book than Adams’ more well-known Hitchhiker’s Guide series. And part of that is understanding what’s going on with the poem.

When Coleridge published Kublai Khan in the real world, he claimed that it was an unfinished fragment, inspired by an opium trance. As the tale goes, he woke from the trance and was ecstatic to realize he could still remember the full poem. He began writing it and got what we have today, but was interrupted by a “visitor from Porlock”, who distracted him for a full hour. When the nuisance finally left, STC was horrified to discover he’d forgotten the rest of the poem.

Personally, I don’t think this tale is true, at least in the real world (as opposed to Douglas Adams’ world). I think it’s a very clever clue given by the author into the true nature of the poem, which I believe is actually fully complete and coherent as it is. You can read more about that in my analysis.

In the Dirk Gently universe, the myth of the man from Porlock is true… well… eventually. See, there’s a bit of time travel involved.

Early in the novel, Richard (the main protagonist) attends a reading of Kubla Khan at the fictional St. Cedd’s College. The known, published verses are sprinkled throughout this chapter, in between Richard’s reflections on them and other details of the story. The ending line of that chapter is very important but it’s easy to completely miss for the reader unfamiliar with the poem’s history. After the last line– what you and I know as the last line– has already been read, the chapter closes: “And then began the second, entirely more strange, half of the poem.” In Adams’ world, at least pre-time-travel, Coleridge suffers no Porlockian visitor, and publishes a poem roughly twice as long as what we know, and the second half is much stranger than the first (surprising since the “first half”, the poem you and I know, is already so utterly bizarre!)

Long story short, it turns out both Kubla and Rime were actually not authored by Samuel Coleridge at all. Or rather, he authored them, but only as a puppet, influenced by an ancient otherworldly spirit (the opium trance, apparently, made him an easier target). And that spirit is the Ancient Mariner: except instead of a marine ship, he came on a spaceship, he and a whole society of his peers, long before life evolved on Earth. They were forced to make an emergency landing on Earth to do repairs, and the nameless Spirit whom the readers meet, he was the main engineer, and he messed up, blowing up the ship and killing them all in the process. He alone survived, wandering the Earth alone and watching as slimy life began to evolve around him. Hmmm, this sounds a little familiar, doesn’t it… oh right, it’s exactly what happens in the middle of the Rime of the Ancient Mariner!

So where does the Man from Porlock come in to all this? Well, turns out the Kubla Khan poem is really a cryptic set of repair instructions. Samuel Taylor Coleridge was just one cog in an elaborate plot by the ghostly spirit, a plot to go back in time and repair the spaceship correctly and avert an ancient disaster. And it works! With help from Richard and Dirk Gently and co., everything goes according to the “ancient mariner’s” plans– a possessed body, equipped with Coleridge’s hidden instructions, steps through time and is right on course to change history. It’s only then that the remaining protagonists put the pieces together, and realize that if the ancient spaceship is repaired, then life never will evolve on Earth.

They can’t chase after the spirit and stop him: the prehistoric landscape on the other side of the time door is too poisonous for humans to breathe unaided. So instead they use the time machine and visit the year 1797– just in time to interrupt Coleridge and distract him and make him forget the second half of Kubla Khan! Turns out in Adams’ world, the annoying visitor from Porlock was none other than Dirk Gently, Private Eye

Coleridge’s poetry isn’t the only artistic artifact which Adams messes with in his novel. The meddling time travelers are also responsible for the works of Johann Sebastian Bach, among other things– and Douglas manages to tie that in with Kubla Khan as well! “Could I revive within me/ Her symphony and song…” But I’ve already spoiled enough of the story, haven’t I

FURTHER READING

The Kubla Khan Poem
The Rime of the Ancient Mariner
A Modern Version of Genesis 1

Before even cracking the first chapter, I was already surprised by the author’s name. See, most elementary math books are written by people nobody’s ever heard of. But Serge Lang is a household name– at least when “household” means “household of mathematicians”. He’s among the most famous and prolific and talented authors of math textbooks, but the surprising thing is, his books are usually about ridiculous advanced topics you have to go to grad school to even know they exist. That’s why I was shocked to see he wrote about basic mathematics. It’s like if you went to a 6th grade science lab and saw Einstein at the chalkboard.

The most amazing thing about Lang’s venture into high school mathematics: he discusses basic algebra and geometry using the same crisp and precise language that mathematicians use for higher mathematics. It’s a beautiful language and it’s a tragedy that most people will never encounter it, despite years of compulsory education. That would change right away if high schools adopted Lang’s masterpiece.

Just reading through Serge’s exposition of multiplication and addition somehow makes me feel smarter– and I’m a guy who’s been studying math for almost fifteen years now. Most authors seem to find it impossible to talk about solving linear equations without assuming a condescending, downright insulting tone. But not this guy. He makes me think I’m sitting with him talking about math, like we’re peers working together to discover the theorems and laws from scratch.

I began searching for good books on elementary math because I’m always getting email from people asking: “What’s a good book to learn math on my own?” It seems to be a common quandary. There’s so much trash out there, what with universities and public school boards creating an unnatural market where material is churned out to maximize profit with little concern for giving the unfortunate students a good deal. You can’t really even blame the big publishing houses in this: they know that their target audience is only buying the book because they have no choice. But people like me and the readers of my blog, we want to learn this stuff, and learn it well, and we deserve a lot better.

Serge Lang doesn’t BS the reader. Here’s an example passage from the book:

It is a tradition in elementary schools to transform a quotient like

into another one in which the square root sign does not appear in the denominator. As far as we are concerned, doing this is not particularly useful in general. It may be useful in special cases, but neither more nor less than other manipulations with quotients, to be determined ad hoc as the need arises. Actually, in many cases it is useful to have the square root in the denominator. We shall give two examples…

This is a polite slam against the ridiculous “rationalize every denominator” policy pervading high-school math courses. I love you Mr. Lang! The whole book is chock full of goodies like this. For anyone who left the high-school math room a little confused, this is like taking the “red pill” and seeing The Matrix for what it is. And I do mean that literally– Chapter 17 is all about Matrix Algebra

What Not To Expect From This Book

Definitely avoid this tome if you’re a fan of modern Textbook Art. You’ll find no picture of a motorcycle on the cover. No human pyramid of grinning old people in suits will introduce the next chapter. No gimmicky “Explore and Discuss” exercises– if you have to differentiate discussion-worthy exercises from non-discussion-worthy ones, it’s time to throw away the latter!

Avoid this textbook if you like books written by ten different coauthors. In defense of standard textbook writers, I’m sure every particular one of them has very good intentions. It’s not any single author’s fault when the book goes to committee. When you take ten different visions of perfection and mix them together, you end up with a brownish muck. Mediocre works are churned out by committees, but it takes a lone man to pen a magnum opus.

Don’t read “Basic Mathematics” if you’re hunting for lots of tedious busywork. The exercises focus more on proving things logically rather than computing and crunching numbers. I love the beauty of true mathematical proof, and I’ve always wanted to share it with the whole world. When I read a particularly beautiful proof I want to grab people on the street and say, “Have you heard this!” If only public schools employed a little less Prentice Hall and a little more Serge Lang Hall, a lot more people would get to share this joy of mine!

Buy Serge Lang’s “Basic Mathematics” Now

FURTHER READING

Five Ways to be Better at Math
“Problems” in Mathematics
James Heisig’s “Remembering The Kanji”
How To Train Your Mathematical Maturity

I noticed that the adhesive ball model of growth actually provides a pretty good analogy for growth in general. When you enter into some new field or area of study or growth, at first everything is difficult and alien, and you have to struggle to understand the most basic of things. That’s kind of like the real life equivalent of rolling up pennies and postage stamps. As you get study and practice, you become more confident and skillful, which is like your katamari is expanding from rolling up new stuff. With a bigger “sphere of knowledge”, it becomes easier and easier to roll up stuff that was completely inaccessible before.

The really interesting thing is that this is a completely fractal process: it’s ultimately the same at every scale. Whether you’re pushing around a 5cm ball rolling up insects, or a 5m ball collecting cars and houses, or a 50m ball demolishing whole countries, the gameplay is basically the same, just with different scenery. Similarly, within some discipline, a grandmaster and a total newbie basically go through the same qualitative process, only the scenery has changed. For example, when a famous mathematician struggles with a difficult proof of a groundbreaking new theorem, the process is the same as when an undergraduate mathematician struggles with the proof of, say, the Heine-Borel Theorem. If the undergraduate tried to tackle the groundbreaking new theorem, it would be like running a 5 foot katamari into a battleship: the katamari would go flying, and the battleship wouldn’t even move But anyone who’s played KD knows, that battleship is doomed as long as the player has the patience to pick up smaller stuff for awhile.

Programming languages are a great example. When I first learned to program (in BASIC), I had to focus on picking up very simple concepts: variables, loops, individual programming commands. These tiny little concepts were like candies or pencils, just the right size for my tiny little katamari. Much later, instead of rolling up individual BASIC commands, I reached a new scale where languages themselves are the right size objects to roll up. Nowadays, I “roll up” entire programming languages with the same ease as when I used to “roll up” single keywords. In KD, you might start the game in an apartment, where the housecat is a looming monster and the couch is like Mt. Everest; but eventually, you can come back and roll up the entire apartment complex, as easily as you once rolled up a tiny little soda can inside the apartment.

Running a successful website is also similar to Katamari Damacy. Initially, you have no hope of making a significant impact on any broad area like “self-development” or “productivity tips” or “travel” or anything. At first, you’re lucky if you can get highly specific traffic (what webmasters refer to as longtail) like “summer youth hostels in takashimadaira tokyo”. These long specific keywords are like tiny little thimbles or gumdrops, perfect for “rolling up” with a tiny little website. Then, as your orb picks up an outer layer of such scruff, you’ll gradually grow reputation and respect which will allow you to penetrate into shorter keyword territory.

What lesson can we take from this analogy? Well, odds are your katamari hasn’t reached the 1000m mark yet where you can roll up the King of All Cosmos himself. In other words, if you’re like everyone else in the world, there’s still lots of room for you to grow. Don’t fret about that. The game is the same on every scale. It doesn’t matter whether you’re a billionaire or just a hundredaire, only the scenery changes.

Get Katamari Damacy from Amazon.com.

FURTHER READING

Skills and Metaskills
How to Train your Mathematical Maturity
The Paradox of the Heap
My Experience with Anime

But what if Brad Pitt wasn’t acting in Fight Club? What if Tyler is really Mr. Pitt’s multiple personality, emerging while the actor thinks himself asleep, just like in the movie? Maybe in his waking life, Brad watches his own films, admiring his own acting without realizing it’s him.

If that’s the case, then what’s this radical activist trying to tell us? What’s the motive behind the roles he plays? Let’s take a look…

The Mysterious Case of Benjamin Button. In this eye-opening film, Pitt– or should I say, Durden– plays a figure who was born an old man, and got younger through the years rather than aging like normal people. In this role, our freedom-fighter friend teaches us quite a few things. Born and raised in one, Mr. Button forces us to confront the retirement home, that societal spectre which we like to pretend doesn’t exist. Shining a personal light into the lives of our elders, Button violently disrupts that pattern which says, “don’t think about old age”. Also, Benjamin engages in a couple elicit affairs with married women, a move to rattle what the status quo declares love and sex should be like.

Ocean’s Eleven. Do we need any more explicit evidence of our shady friend’s plots and schemes? In this Las Vegas thriller, “Rusty Ryan” (as the doppleganger is cast) is one of eleven elite cat burglars. Though he isn’t the lead role here, he does his part to inculcate us into the world of organized crime, a revolutionary world which terrifies our corporate masters. The Eleven are modern-day Robins Hood, robbing the rich to feed the criminals.

Mr. and Mrs. Smith. Here, we see an unlikely couple, each pretending to be “normal”, with “normal jobs”. The truth is, both are super-assassins working for competing agencies. Durden skilfully rips the outer mask off the very meaning of “normal”, and viewers walk away recognizing that none of us is truly “average”, that we’re all harboring dark and awesome secrets. After viewing this film, you can never look at your stereotypical next-door neighbors the same way. Maybe that respectable businessman in the suit is really a spy from North Korea, plotting to blow up Mt. Rushmore. That housewife ahead of you in line at the grocery store? No less than the matriarch of the world’s last true clan of ninjas. Nothing is as it seems!

Se7en. This gruesome roller coaster of sin has one purpose and one purpose only: to desensitize us to the deeds which organized religion most strongly demonizes. While no one would ever emulate the creepy and insane antagonist, this movie is a vaccine: it disgusts us but it leaves us a little more hardened, a little more jaded, conditioned, perhaps, as soldiers. Good citizens don’t watch this sort of movie, responsible citizens shun this kind of entertainment. The stereotypical, smiling black-and-white 50′s spokesman with his suburban, nuclear family would never ever view this flick. Thus, Brad Pitt plunges a dagger into the hearts of these very establishments.

If Brad Pitt really is the genius Tyler Durden, you can rest assured he has his fingers in more pies than just Hollywood. Who knows how many men and women he has under his thumb, warriors in his private crusade for true human liberty? Maybe even I, your humble writer, am a member of the real life fight club, publishing this dangerous article to further the agenda of the underground uprising

FURTHER READING

Invitation to Hedonism
Getting the girl in a movie
How to Contribute to Society
Three Ways to Be More Present

The first hotel and casino we hit was the Stratosphere, taking a taxi to the far north end of the strip. Since it was late, we were disappointed to find the buffet was closed, but we enjoyed some crappy-yet-expensive bar food instead (we hit the buffet the next night, and it was excellent). My girlfriend introduced me to slot machines, something that was previously outside my reality since I was raised with the idea that gambling was a wicked tool of Satan. Actually it’d be more accurate to say it’s a wicked tool of the casino’s shareholders, but in limited doses it’s fun and I’ll write more about that later. In one of the games, I got a near-jackpot (7 “quickhits”, 8 or 9 would be jackpot) but it was with only a five cent bet so I only got about seven dollars or so, a max bet would’ve gotten me over a hundred. We walked the “strip” a little and took advantage of the lack of open-container laws, drinking giant cans of beer while walking the callgirl-card-strewn boulevard.

The next evening was one of the most fun ones of the whole trip: the thrill rides atop the Stratosphere. Since we actually booked in at the hotel, we got a two-for-one deal on an unlimited day pass up the tower and into the rides. This was singularly awesome and if I had to pick just one hotel to stay at, it would definitely be the Stratosphere, exactly because of these adventures.

There are three rides on the tower, roughly a thousand feet above the ocean of Vegas lights. First was the Insanity, where you sit in the grasp of a giant mechanical arm which swings you in wide circles with nothing but emptiness below you. The Insanity experience cannot be described in words: when the radius of rotation reaches its peak, your brain can’t even tell which direction is up and down. We followed that with the X-Scream, a short track of “roller coaster” which abruptly ends after a few dozen feet: if the car were any “normal” roller coaster car, you’d plunge off the end to certain terrifying death. Instead, you jolt to a stop inches before that happens, only to have the track swing upwards and you retrace the path but this time in reverse. All the while, the only thing below you is the narrow track and a whole lot of empty air.

The third ride, oldest of the three but still the most profound, is the Big Shot. This is a floor above the previous two. At the uttermost peak of the actual building, a giant dark metal shaft rises up an additional hundred feet or so. At the bottom of this shaft, the rider takes a seat facing outward, looking down on the city far below. The seats rise slowly at first, only a few feet– and then, within about one second’s time, you’re blasted all the way to the top. Time itself seems to freeze as your brain fumbles to parse this situation, a situation totally unaccounted for in all the millions of years of human evolution. And then, just as fast, the plummet back down turns your heart to dust. I’m not talking freefall, I’m talking significantly faster than freefall, and the mind doesn’t know how to make sense of it, except to be terrified. After the initial ascent and descent, the ride takes you up and down a few more times, but slower and not to the very top; the main thing is that initial rise and fall, which is indescribable.

The next day, we checked out from the ‘sphere. Our next destination would be the Paris Hotel and Casino, further south, in the heart of the strip. We ate breakfast at a very authentic, mom-and-pop Thai restaurant right next to the Stratosphere, and then it was into a taxi and off to our next adventure.

NEXT: The Paris Las Vegas (Not online yet!)

Merlin makes a point of emphasizing being grateful for everything. Your family died in a horrific car wreck? Hallelujah! You got laid off, your spouse ran off, and your house burned down? Thank God! Thank God for cancer, praise heaven for corruption and vice, three cheers for poverty and hunger!

When something happens in our life, we have the choice of viewing it as a blessing or as a curse. This is entirely subjective. There is no universal arbiter to define these terms. When something is a blessing, the status of “blessingness” is not a tangible thing existing somewhere in the physical world. You can’t put it on a scale or under a telescope. It’s a status in your head. Likewise when something is a curse. Certain things tend to be classed one way or the other by concensus– winning the lotto is a blessing, losing a friend is a curse– but this “boon or bane” decision is ultimately arbitrary, and you can choose your labels freely.

Choose to view everything as a boon, and you are presented a very different world. At first, when you begin training your inner Merlin, it takes deliberate effort, and you feel silly. “Hmm, late fee for missing the electric bill.. have to remember to be thankful for this.. thanks for this late fee!” However, as time passes, it becomes automatic, and you really see things more positively. Not because you’re forcing yourself to– your very personality transforms into a more positive and optimistic one.

When I read the main Merlin Carothers book, Prison To Praise, it transformed me in more ways than one. Not only did I become innately more positive, I also bit into the proselytization with which the book is laced. This speaks for the power of the book: all my youth, I was constantly being proseletized to, by my parents, my parents’ church, my high school, by whatever entertainment slipped through my parents’ filter. None of it touched me at all. I was a Christian by upbringing, in the same sense that I was American by upbringing. It was a sports team to cheer for as it competed against the Arab Muslims. There was certainly never any spirit or power there.

But when I read Carothers– I was junior high age at the time– it ushered in a brief period of institutional Christianity in me. I was really serious about it. I even tried to convert my two best friends, in a tear-filled confrontation which I’m very glad they haven’t brought up since. Heck, one of the scariest things I did as a kid, I actually made a feeble attempt to talk Jesus to my friend’s psychic dad. Just before that, he was making noise about forbidding his son from hanging out with me because of my sudden radicalism, but after I confronted him, I think he recognized I was temporarily confused. Heh, he was a psychic, maybe he foresaw me writing this very article

Thankfully, I didn’t end up a bible thumper. Merlin Carothers led me through a very different and new phase in life, and it’s one I disagree with now, but I’m glad I took it. I’ve come to recognize that the more “phases” I go through, the more I grow as a person. It’s reality expansion. I wouldn’t care if the Big MC had converted me to Scientology, Flying Spaghetti Monster, or even the ultra-crazy Fujitaisekiji. It was a learning experience and gave me that much extra reference experience, thus making me a more intelligent person in the end. Besides converting me temporarily into a fan of his religion football team, Merlin also taught me about the power of praise, and that’s a lesson that’s benefited me all my life.

Order Merlin Carothers’ Prison To Praise online.

FURTHER READING

I gave a Toastmasters Speech titled Unconditional Thanksgiving, which talks more about giving thanks for absolutely everything. I talk more there about the theory behind the philosophy.

Raised by Christians, I have a lot of knowledge about Christian trivia. People who know me today can actually be startled by the depth of my knowledge of my parents’ theology. Since I’m not a fan of the football team myself, though, I tend to have very different opinions than most Christians. Read, for example, my article on Sex Before Marriage.

For another book review, read my review of Steve Pavlina’s Personal Development For Smart People.

A few quarters back, standing outside an auditorium before giving a calculus final exam, some of my students asked what I was reading. It was “The Higher Infinite”, a very advanced tome of mathematics by Akihiro Kanamori. My students were understandably pretty interested to hear about the idea of more than one kinds of infinity. In fact, there are infinitely many levels of infinity. And that’s even pretty well-known, at least among people who read about math. What Kanamori writes about, in rather difficult mathematical language (the book is meant for advanced graduate students or math PhDs), are some lesser-known infinities, which are so big that they literally bend the basic facts of mathematics. It’s some pretty trippy stuff.

WHAT IS INFINITY?

In formal set theory, there is no single entity called “infinity”. In mathematics, when we say that the answer to some question is “infinity”, we really mean that any finite answer would be too small. The “entity” of infinity is just a kind of shorthand for expressing this idea. But there are infinite sets– that is, collections which are infinite. When we talk about multiple “levels of infinity”, we’re really talking about collections of different, infinite, sizes.

Here’s an example. Take the collection of all natural numbers (the natural numbers are the numbers 0, 1, 2, 3, 4, and so on; the numbers you use to count). How big is this collection? Any finite answer, is too small. For example, if we guess that the collection of natural numbers has size one million, then that’s too small because there are more than a million counting numbers. Since any finite answer is too small, we say the collection is infinite.

What about the set of all real numbers, with any number of decimal places, including infinitely many non-repeating decimal places, like in the number pi=3.1415…? If we look at this “continuum” of real numbers, again it’s infinite, because any finite size would be too small. It turns out, that the set of all real numbers, is actually larger than the set of all counting numbers. And that’s the prototypical example to show that there are more than one sizes of infinity. But, what does it mean to say that one infinite collection is larger than another?

To understand how the sizes of infinite collections can be compared, it’s first necessary to understand how the sizes of finite collections are compared. One way to compare the sizes of finite collections, is to just count them and compare the numbers. But that doesn’t generalize to infinite collections, because we can’t count an infinite collection– if we could, it would not be infinite.

A better way– or at least, a more generalizable way– to compare the sizes of finite collections, is to check whether we can marry their elements up together in a nice one-to-one way. If two finite collections have the same size, then we can think of one collection as being the collection of “males”, think of the other as being the collection of “females”, and marry them up so everyone has exactly one partner. If the collections had different sizes, we couldn’t do this, someone would be left out.

This “marriage” idea is referred to in mathematics as a “bijection”. A bijection is just a fancy, smart-sounding way of saying, you assign each member of the first collection to a unique member of the other, so that nothing is left out and nothing is matched up twice. If two collections have a bijection between them, then they have the same size, and if not, then they have different sizes.

You can think of counting small (smaller than size eleven) sets as establishing a bijection between the set and between a certain subset of your fingers. You count “one, two, three, four,” holding up a finger with each utterance, and you’re implicitly “marrying” a certain set of your fingers to elements of the set you’re counting.

The bijection idea naturally generalizes to infinite sets with no extra work. Two sets, whether they be finite or infinite, are said to have the same size if there’s some way to link their elements together bijectively, so that each element of the first gets associated with exactly one element of the second.

Right away, some trippy examples come up. For example, the set of all even counting numbers (0,2,4,6,8,…) has the same size as the set of all counting numbers. That’s because if you take any even counting number and divide it in half, you get a counting number, and the result is unique and no other even number gives the same result. The act of dividing in half, is a bijection from the set of even counting numbers, to the set of all counting numbers.

THERE ARE MORE THAN ONE SIZES OF INFINITY

The famous example which first shows that there are more than one sizes of infinity, is the fact that there are more real numbers than there are counting numbers. A real number being, any number, positive or negative, with any number of decimal places. Including numbers like pi or the square root of two, where there are infinitely many non-repeating decimal places.

The fact that there are more real numbers than counting numbers was a big surprise to mathematicians; after all, both sets have infinitely many elements, so a mathematician back in the old times might have said both sets have size infinity.

Saying that there are more reals than naturals comes down to saying that there aren’t enough naturals to associate with the reals; if we try to marry each real number to a unique natural number, with no naturals getting married twice, then there won’t be enough natural numbers to go around and some (in fact, most) of the reals will be left out. In other words, there is no bijection from the naturals to the reals. Since the naturals are a subset of the reals, that is, every natural is a real, there are at least as many real numbers as there are natural numbers; so if there’s no bijection between them, then the set of real numbers must be strictly bigger.

Georg Cantor, the mathematician who discovered a lot of this stuff about different infinities, published a famous proof that there’s no bijection between the naturals and the real numbers. You can read his proof, the famous “diagonal argument”, at Wikipedia. I’ll give a different proof, one which is in my opinion more acid-trippy and fun. Anyway the diagonal argument has been beaten into the ground by a million other people so it’s better to give a less well-known proof anyway.

My proof uses a certain infinite sum which is related to one of the Zeno’s paradoxes, if you’ve ever read about those. Basically, in order to walk across a distance of size 1 meter, first you have to walk one half meter. Then, you have to walk one fourth meter. Then, you have to walk one eighth meter, then one sixteenth meter, and so on. Adding up all those partial walks, each one half as long as the previous, you get the total distance of one meter. This gives the infinite sum, 1/2+1/4+1/8+…=1. What does this have to do with proving there are more reals than naturals? Hold onto your hat…

Suppose that there was a bijection from the naturals to the reals. Then we could “count” the reals, saying, this real number is the first real; this real number is the second real; this real number is the third real; and so on. Using the association from the counting numbers, to the reals. And it would hit every real number, since, according to the assumption, each real number has a natural number associated with it.

Now I’m gonna show a way that you can cover the entire real number line, with a covering of length 1. That’s ridiculous, because the real number line is infinitely long. There’s no way to cover it with just a covering of length one. For instance the real line contains the interval from -1 to 1, which has length 2 just by itself.

Well, take the “first” real number (i.e., the number which is associated to the counting number 0). Cover it with a cover of length 1/2. Next, take the “second” real number, and cover it with a cover of length 1/4. Take the “third” real number, and cover it with a cover of length 1/8, then cover the “fourth” real number with a cover of length 1/16, and so on. Even if none of these covers overlapped, the total area of the cover would be at most 1/2+1/4+1/8+…=1. In fact, the covers overlap a lot, so the total area of these covers ends up being less than 1. But every real number is the nth real number for some n; that’s the assumption we made, that we had a bijection between counting numbers and reals. So, every real number gets covered. I’ve covered the whole real line, with a covering scheme where the covers have a total length no longer than 1. Impossible, that’s not even enough to cover just the part of the real line from -1 to 1.

I started out assuming there was a bijection from the naturals to the reals. Then, I showed that the assumption allows me to do something ridiculous. So the assumption must be wrong, and there is no bijection between the naturals and the reals. They have different sizes, and the reals contain the naturals, so there are more reals than naturals.

(Actually the proof is missing a little detail, since I’m assuming some common sense notions about how lengths work. The details are filled in in an advanced branch of math called “measure theory”, but I figured it’d be worth the reduced details to post this alternate proof that there’s no bijection.)

THERE ARE INFINITELY MANY LEVELS OF INFINITY

I just established there are more than one levels of infinity, by showing that the set of real numbers is bigger than the set of natural numbers. But the infinitudes get much, much bigger, and there are far more sizes of infinity than just the size of the set of naturals and the size of the set of reals.

If you take any collection of objects, it makes sense to talk about subsets of that collection. Like, the set of even numbers is a subset of the set of all numbers. Well, once you have the notion of subsets, you can ask about the collection of all subsets of a set. For example, you can ask about the set of all subsets of the natural numbers. This set-of-all-subsets is called the power set. The power set of the natural numbers is the set of all sets of natural numbers.

Here’s the big breakthrough which leads to infinitely many levels of infinity. It’s a fundamental truth discovered by Georg Cantor, and it totally turned mathematics on its head. Cantor showed: if you have any set whatsoever– empty, finite, or any level of infinite– then the power set is even bigger.

For example, the power set of the natural numbers– the set of all sets of natural numbers– is bigger than the set of natural numbers. So that gives another proof that there are more than one levels of infinity. (Incidentally, which set is bigger, the power set of the natural numbers, or the set of real numbers? Or are they the same size? It turns out this question is unanswerable. Read more at Continuum Hypothesis)

To get infinitely many levels of infinity, you can just repeat the process. You can take the power set of the power set of the set of naturals, and get something even bigger. And then you can take the power set of that. The process never ends, and it provides infinitely many levels of infinity.

SO HOW MANY LEVELS OF INFINITY ARE THERE?

I’ve shown you how there are infinitely many different levels of infinity, and a natural question which you might ask is, what is the size of the set of all levels of infinity? So far, I’ve showed how to get infinitely many different levels of infinity, but the infinities we can create using just power set, can be placed into a natural association with the natural numbers. I can say that the 0th infinity corresponds to the number of natural numbers. And then I can say the 1st infinity is the power set of the 0th. And the 2nd is the power set of the 1st, and so on. This hits all the infinities we get with just repeated power setting of the naturals. If we just look at repeated power sets, we get as many levels of infinity as there are natural numbers.

But is that all of them?

No. There are infinities so big that no matter how many times I apply the power set operation, I’ll never reach them. Here’s an example. What if we take the set of naturals, and then the power set of that, and combine them into one big set. And then, we throw in everything in the power set of the power set of the naturals. And then, throw in everything in the power set of that. And keep going, forever. So, in other words, we get the set of all things which show up anywhere in any of the repeated power sets starting with the naturals. Since this set contains all those power sets as subsets, it must be bigger than all of them. It’s one mind-bendingly, insanity-destroyingly huge set! (But, it’ll turn out it’s still “tiny” in the world of mathematical logic)

Ok, but I’m beating around the bush. The question is, how many infinities are there? The answer is unsettling. It turns out, there are so many levels of infinity, that no level of infinity is enough to answer the question. No matter how hard anyone tries, to come up with some incomprehendably large level of infinity, there are more levels of infinity than that.

LARGE CARDINALS

Congratulations, if you’ve read this far, you know just about as much about building really-big-freaking-infinities as the average mathematician. Now strap yourself in, I’m gonna talk about how to go to a whole new level, making the infinities we’ve talked about so far look like tiny insects. This is actually cutting-edge stuff I’m sharing with you, and is mostly only known by people who actually specialize in mathematical logic.

Logicians use the term “LARGE CARDINAL” to refer to some levels of infinity so big that they literally transcend math. These levels of infinity are so awesomely, divinely, insanely huge, the term LARGE CARDINAL must always be written in all capitals, because a LARGE CARDINAL is that freakin’ ginormous.

In order to get an idea what a LARGE CARDINAL is, you have to understand a little of the very foundations of modern mathematics. You already know there is no absolute truth and no absolute provable theorem, because all proofs must ultimately call upon some unprovable assumptions. Usually those assumptions are so “obvious” that no-one would ever question them, but nevertheless they can’t be proven (and if they could somehow be proven, it would only be by relying on even more unprovable assumptions). It’s kind of how a dictionary can’t really define every word, without eventually falling into circular definitions (try defining the word “the”, using only words which can be defined without using “the”).

In modern mathematics, the “unprovable assumptions” have been very neatly organized into a system called ZFC. Basically, mathematicians know deep down that their proofs have holes in them, that math would fail if ZFC fails. But that’s okay, because if ZFC fails, then unicorns are real and dreams come true and everyone has a million dollars.

One of the things about ZFC, proved by Godel in his famous “Incompleteness Theorem”, is that you can’t prove ZFC using ZFC. It is possible to prove ZFC, if you make other assumptions above and beyond ZFC. But then, your proof is in some extended system, and you can’t prove that system, unless you make even more assumptions, and so on.

ZFC is really a “statement about the existence of a universe”. It doesn’t say outright that “math is true”, so much as it says, “there exists a universe in which math is true”. Then we prove theorems about such a universe. Presumably, math is true in the real universe, so these theorems become true in the real universe. But if ZFC fails, it really means there’s no universe where math is true, and all the theorems proved from ZFC are vacuous.

One thing about universes, is that you can have universes within universes. For example, going philosophical for a moment, suppose the Real Life universe is made up of two parts, a “close” part which humans can navigate, see, or in some way detect. And a “far” part which humans can never reach, see, or even detect. Then if we looked at the “close” part of the universe, even though it would not be all of the “Real Universe”, it would be a universe by itself. There’d be no way for us to tell that the “close universe” wasn’t the entire universe.

The “close universe” in my example is closed and consistent because the non-close part of the Real universe is undetectable, and so if humans assumed the “close universe” was the entire universe, they’d never reach any contradiction. (If they could reach a contradiction, they’d have detected the far universe, but by definition the far universe is undetectable)

Now, suppose we assume that there’s a universe which contains certain sets of objects, and that these sets satisfy ZFC. Ok, so far we just have vanilla math. But let’s also assume that among those sets, there’s a set which, by itself, satisfies ZFC. In otherwords, the elements of this set are themselves sets, and ZFC holds among them if we pretend they’re all the sets in the universe. Then we’ve gone beyond ZFC itself into an extended system, and this extended system proves ZFC, because the set in which ZFC holds can be taken as a “close universe” where ZFC is true.

In this extended assumption system, the set where ZFC is true is a LARGE CARDINAL. It’s a set so big that people “living inside it” and following normal rules of math can’t detect that there’s anything else in the universe.

What’s more, normal math (ie, math in ZFC) cannot itself prove that the LARGE CARDINAL exists. Because, if ZFC proved the LARGE CARDINAL exists, then ZFC would prove ZFC, since ZFC is true inside the LARGE CARDINAL. And ZFC can’t prove itself.

All the levels of infinity which we can possibly conceive using normal math, must exist within the LARGE CARDINAL, because ZFC– and thus, all of “normal math”– exists within the LARGE CARDINAL. Consequently, the LARGE CARDINAL is bigger than any level of infinity that we can construct using modern mathematics.

(And yet, I just constructed it! Isn’t that a contradiction? No, because the construction steps outside conventional mathematics by assuming things outside ZFC)

Let’s give a name to this new system, the system where we assumed ZFC together with the existence of a set where ZFC is true. Let’s call it ZFC+. Then we can repeat the process: assume there exists some universe, with certain sets of objects, and that ZFC+ holds in this universe, and assume further that among those sets, there’s a set which, by itself, satisfies ZFC+. Then that set is another LARGE CARDINAL, and this time, it’s so big that it can’t even be constructed with all the supernatural powers of ZFC+. It can only be constructed in the even more ridiculous system, ZFC++ if you will.

This process can be continued on and on. By going to new systems of math, we can build levels of infinity which transcend the old systems of math, and then we can repeat the process, as often as you like.

These LARGE CARDINALS I’ve talked about are just one type of large cardinal. The general process is: take ZFC (normal modern math) and extend it with some new assumption, strong enough to prove ZFC. Since ZFC is really a statement of the form “a universe exists in which math is true”, and since the enhanced system proves ZFC, the enhanced system must prove that a universe exists where ZFC-math is true. But ZFC-math isn’t provable in ZFC-math, so that universe where ZFC-math is true, must be larger than anything that can be built in the un-enhanced system of ZFC.

IS THERE ANYTHING BIGGER THAN LARGE CARDINALS?

Yes, it’s possible to go even bigger than LARGE CARDINALS! All throughout the article so far, I keep talking about “sets”. So far, I’ve just been doing “naive set theory”, treating sets intuitively as “collections of objects”, but this naive treatment actually leads to certain paradoxes.

The most prototypical example of a paradox of naive set theory, goes like this. Let X be the set of all sets which don’t contain themselves. That is, if you look at any set, it lies in X if and only if it does not lie in itself. The question is, is X an element of itself? If X is an element of itself, then by definition, it’s NOT an element of itself. Ridiculous. But if X is not an element of itself, then by definition, it IS an element of itself. If we assume X is in itself, we get that X is not in itself, and if we assume X is not in itself, we get that it IS in itself. ARRGH, brain explode!

That paradox is known as “Russel’s Paradox“. It basically says we have to be more careful with what “sets” are allowed to exist. The system of ZFC I discussed above is basically a code of laws specifying which sets actually exist. Under ZFC, it’s impossible to construct “the set of all sets which don’t contain themselves”, so the paradox is avoided.

In real world applications, no one really cares about “the set of all sets that don’t contain themselves”, so we’re not missing out on much by not having it. However, there are some useful “collections” which it’s nice to talk about, which are also un-constructable using ZFC. The most obvious example is, the set of all sets. Since mathematicians and logicians spend so much time talking about sets, it would certainly be nice to talk about the set of all sets, just to organize our work. And, unfortunately, you can’t build the set of all sets using ZFC. (In fact, you can show that if you could build the set of all sets, then you could build the set of all sets which don’t contain themselves, and then you’d get Russel’s paradox)

Mathematicians get around this in a kind of cheating way, by introducing “classes”. What’s a class? It’s a collection of objects. How is that different than a set? Well, the only difference is that a set is restricted by ZFC. A class is also restricted by different assumptions, but the restrictions are less stringent, so that it’s possible to talk about the class of all sets.

Classes are just a linguistic trick to let us talk about “the set of all sets” (in essence), without invoking paradoxes. If we had a set of all sets, we could use ZFC assumptions to reach a paradoxical contradiction. But ZFC assumptions don’t say anything about classes– they can’t “touch” classes, and so even though we have the class of all sets, we can’t pull a paradox from it.

So what’s the size of the class of all sets? It must be infinite, since there are infinitely many sets. If the class of all sets had the same size as any particular set, then that would give us a foothold to “touch” the class with ZFC assumptions, and we could get a paradox. The class of all sets is bigger than any particular set, it’s even bigger than a LARGE CARDINAL.

Whereas the LARGE CARDINALS are awesome and stupefying in their size, the “proper classes”, such as the class of all sets, are bigger but somehow they feel like cheating, like we got them through some linguistic tricks. To a realist mathematician who believes that math “exists” somewhere out there, the LARGE CARDINALS are actual “existent” entities, whereas the proper classes are just linguistic formality. In some sense, they only “exist” because of Russel’s Paradox, and some mathematicians don’t like to think about Russel’s Paradox, since it’s the closest we’ve ever been to the Death of Math. Nevertheless, proper classes are an example of something even bigger than LARGE CARDINALS.

CONCLUSION

Next time you’re in a contest to name the biggest infinity, after your enemy says “infinity times infinity plus a million”, you can say “smallest LARGE CARDINAL”, or “size of the class of all sets”. Of course, they can just counter by saying, “power set of what you said!”…

Here are some other things I wrote. Soon, glowingfaceman.com will contain infinity+1 articles!

Ergative Verbs
Introduction To Toastmasters
My Shorthand Writing
Researching English On Books.google.com

CRITERIA FOR PRODUCTIVE PERSONAL DEVELOPMENT

In his book “Personal Development For Smart People”, Steve starts out by putting forth criteria that a self-development strategy should have, if it’s legitimate and not just a cheap way to sell books. These criteria aren’t rocket science, in fact they seem so obvious it makes you go, “I should’ve thought of that!” But what Steve’s done is he’s pushed the field of self-development to a new level by going meta. Most self-development books will focus on some specific details, like finances, self esteem, willpower, social skills, or some combination of these. What Steve does is he shifts the focus away from specific details and looks for the deeper underlying patterns underlying it all.

Here are some of the criteria Steve identified, criteria which a personal development philosophy should enjoy in order to be really worth following:

• Universality – The high-level guiding principles of a philosophy should apply equally well to all areas of life. I learned how important this was when I started working to improve my social skills. Some guys think they can change their whole personalities when they go to a club or coffee shop, like their deep inner selves are just an accessory in their outfit. And then when they return to their job, it’s back to their old chode selves. Obviously that’s completely incongruent. It’s not universal.
• Internal consistence – A philosophy for self-development absolutely must be internally consistent. When the bullets are flying and real life trials are rearing their ugly faces, the last thing anyone needs is two chapters giving conflicting info on what to do.
• Completeness – If a follower of a certain philosophy takes a correct action, that action should be explainable in a natural way from the fundamental tenets of the philosophy. For example, suppose a philosophy teaches that the entire path to happiness comes from reciting a certain mantra every morning. Then, based on that law, I should be able to go rob banks and wage crimes on humanity, and still be happy, as long as I recite the mantra every morning. “Oh no!” cry the gurus of the philosophy, “You can’t do that!” Then the philosophy isn’t complete, is it!

By outlining these kinds of criteria in his introduction, Pavlina essentially sets up hurdles for himself to clear. In this way, he’s really forcing himself to perform in the rest of the book, because once the reader reads his criteria, the reader is going to be scrutinizing the following text to see whether Pavlina practices what he preaches. This is very characteristic of Steve Pavlina- he is not a man to take the easy route of declaring himself a guru and asking readers to follow blindly. In fact that would violate his very philosophy, which is very much based on consciousness and alertness and questioning things.

THE THREE AND THE SEVEN

Having established criteria, and thus implicitly setting a certain minimum expectation, Steve follows through by laying down three core virtues, and deriving four more by combining them in different ways, to get a total of seven virtues which he claims form the cornerstone of self-development.

Steve claims to have derived his system of seven virtues by looking for general patterns in the vast, disconnected literature of self-development- including major world religions and philosophical movements and belief systems of all shapes and colors.

Here are the seven virtues which Steve describes in Personal Development For Smart People:

1. Truth Truth means acknowledging what IS. It means having an accurate, clear map of reality. Truth is light amidst the darkness of the world. If we are not honest with ourselves, all our other actions cannot help but be confused and unbalanced, coming as they are from a misguided view of the world, like a blind man driving a car.
2. Love Love is what separates our actions from the actions of robots. Not just love for other humans, but love for what we do, love for the beauty of the world around us, love for our own selves, love for the blessings in our lives.
3. Power Power refers to our ability to interact with the world around us and actually influence it. We are not mere observers, we actually dwell in this world and take part in its dramas and adventures, and all the truth and love in the world wouldn’t do anything to change our lives if we didn’t take action on them. The most powerful action is action which is lucid, conscious and deliberate.
4. Oneness = Truth + Love Combining Truth and Love, we get the virtue of oneness. The best way to describe this is through the analogy of cells in a body. In a body, good cells act in the best interest of other cells and of the body as a whole. A “selfish” cell is a parasite at best, a disease at worst. Oneness goes a long way to bring meaning and passion to life in a world of selfish, self-interested pursuits and values.
5. Authority = Truth + Power Combining Truth and Power, we get the virtue of authority. To understand this, think of the tragic cartoon character Dilbert, a brainy engineer with a stupid boss. Dilbert has all the truth and none of the power, and so he accomplishes nothing. Dilbert’s boss has all the power and none of the truth, and so he just wastes everyone’s time. Neither has real authority. If either gained the positive virtue of the other, the result would be authority.
6. Courage = Love + Power Combining Love and Power, we get perhaps my favorite virtue, the virtue of courage. If power was deliberate, conscious, lucid action, courage is power in the face of danger (real or imagined). Courage is what lets a man approach a beautiful woman and start a conversation. Courage is what lets a salaryman quit his day job and start his own business. To be honest, I can’t really understand the logic behind this particular combination (love plus power), although it certainly makes sense in the “man approaching pretty woman” example.
7. Intelligence = Truth + Love + Power The ultimate virtue, according to Steve, is “intelligence”, the combination of Truth, Love, and Power. I think really this is just a convenient name for a virtue that doesn’t really have a name of its own; certainly Steve’s “intelligence” is not the same as Webster’s Dictionary’s “intelligence”. But then again, intelligence is kind of difficult to define, and Steve’s is, if less orthodox, perhaps more practical and useful than the standard definition. If you look at human beings as artifacts of evolution, then intelligence really is our ultimate feature, what puts us at the top of the food chain, so this definition works well in an evolutionary sense.

Probably the biggest benefit of this structured system is that it adds order to self-development, a field which is usually far-flung and chaotic, and often incomplete. Whether or not Steve Pavlina’s structure is “objectively true” in some verifiable scientific sense, isn’t really important. It’s just a convenient and elegant thread for uniting the disconnected pieces of self-development.

Moreover, because of how most the combinations make a lot of logical sense, it creates a very practical hierarchy: if you want to improve your Authority, for example, Steve gives specific exercises in his Authority chapter, but beyond that, you can also work on your Truth or your Power, and building either of those will naturally hit your Authority as well.

APPLICATIONS OF THE SEVEN VIRTUES

In the second half of Personal Development For Smart People, Steve Pavlina applies the seven virtues to discuss six aspects of life. Here are the six areas which Steve applies the seven virtues to:

1. Habits
2. Career
3. Money
4. Health
5. Relationships
6. Spirituality

If you just want a “vanilla self-help book” where a guru tells you what you should do with little explanation, you could take P.D.F.S.P. and skip the first part entirely and just read the “Applications”. I don’t really recommend that, though.

The most interesting application chapter, appropriately, is the climactic final chapter, “Spirituality”. “Spirituality” (which, btw, can be read perfectly well through an NLP viewpoint with no actual voodoo or magic involved at all) could be more accurately named “Belief Systems”. In this chapter, Steve talks about belief systems– religions, atheism, political ideologies, philosophies, etc.– and brilliantly compares them to “lenses” which let us perceive reality.

The idea Steve conveys in “Spirituality” is that different “lenses” (belief systems) provide different viewpoints of reality. Through one lens, we may see certain things clearly, while others are obscured. To live life peering exclusively through one lens would be akin to purposefully depriving ourselves of full sensory input. Think, for example, when someone takes a political ideology too seriously and seems to become “blind” to reality.

The most interesting thing about the “Spirituality” chapter– and this is something you won’t even read in the book itself– is that you can apply it “recursively” to Steve’s own writings. So, in Steve Pavlina’s own language, Steve Pavlina’s own writings are just another “lens” to view reality. Though, in the opinion of Glowing Face Man, Steve provides a very powerful and enlightening lens.

Buy Personal Development for Smart People: The Conscious Pursuit of Personal Growth

Here are some other articles I’ve written. I poured a lot of Truth, Love, and Power into writing these articles.
How to Consciously Choose Health
Metabolism As An Example Of Accepting What Is
30 Day Article-A-Day Challenge Completed!
How To Be A Better Teacher