Different Sizes Of Infinity

There’s probably no better way to spend your weekend than pondering infinity with Georg Cantor.  Everybody loves number theory, don’t they?  Might be a physicist thing, I don’t know.  I enjoy thinking about infinity.  Speaking of which, do you know that there are multiple sizes of infinity?  Well, there are!  Being the awesome guy that I am, I’ll take a few moments and quickly demonstrate this incredible result.

When I first heard this, I thought, “That doesn’t even make sense.  How can you have more than infinity? Different sizes of infinity?  How can something that’s boundless even be compared?  Infinity is infinity, isn’t it?”  Well, yes, but there are some subtleties to consider.

Let’s begin with the natural numbers.  What are the natural numbers?  They’re the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and so on, forever.  There are an infinite number of natural numbers.  They never end.  Ok, so far so good.

Next let’s consider the real numbers.  The real numbers are basically any number with an infinite number of digits after the decimal point.  Some examples include:  0.3333333333…, 0.23468828273…, and 12.25000000…, and so on.  The “…” at the end means that the digits continue indefinitely.

Now let’s pair up each natural number to a unique real number.  They don’t have to be in any particular order.  For this example I pair up my natural numbers to real numbers less than 1, but there’s no rule saying this has to be the case.

1 -> 0.33333333333…
2 -> 0.23468828273…
3 -> 0.2500000000…
4 -> 0.22222222222…
5 -> 0.66892735820…

….

And so on.   Here’s the question of the day:  Since we have an infinite amount of natural numbers to work with, is it possible for us to pair up each and every real number to a natural number?   The answer is no, we can’t.  There will be an infinite amount of real numbers we missed, even if we pair each and every natural number to a unique real number.  Let me explain.

I’m going to now construct a real number that is not paired up on your listing, no matter which numbers you choose.  To construct this number, take the first digit from the first number, and change it to something else.  Next take the second digit from the second number, and change it to something else.  Next take the third digit from the third number, and change it to something else. And so on, indefinitely, changing the nth digit of the nth number.  In this example, I’ll go with:  0.44930……  For this particular example, this number will never be paired up with any natural number.  It’ll be left out.  More generally, this process is called a Cantor diagonal and can be viewed as followed:

You should see that there are an infinite number of such diagonals which we’ve missed.  That means that no matter how we pair up our natural numbers to the real numbers, there will be an infinite number of real numbers left over.   There are an infinite number of natural numbers, but in some strange sense, there are infinitely more real numbers.  So, there must be different sizes of infinity!

What does this mean?  What are the implications?  If there is more than one size of infinity, are there possibly others as well?   Cantor proposed the Continuum Hypothesis in 1877, stating:

There is no set whose cardinality is strictly between that of the integers and that of the real numbers.
– Georg Cantor’s Continuum Hypothesis

Cardinality means the number of items in a set.  For example, if you have a set of playing cards, it’s cardinality is 52.  Though I’m only an amateur, I believe this statement is saying that there are no different sizes of infinity between the infinite amount integers (-3, -2, -1, 0, 1, 2, 3, …), and the infinite amount of real numbers.   In this post, I’ve shown that there are an infinite number of natural numbers, and an infinite number of real numbers.  We all know there are also an infinite number of integers, you’re just adding the negative natural numbers as well.  As we’re seeing, in some strange sense, the infinite number of real numbers is greater than the infinite number of natural numbers.  Cantor is proposing that you’re not going to be able to find a set whose size is between those two infinite sizes.   The infinite size of the real numbers is the next “step” from the infinite size of of the integers.  Has your mind exploded yet?

The dilemma plagued the minds of many prominent mathematicians, including Bertrand Russell, Kurt Godel, and David Hilbert, as they struggled to figure out what was going on here.

Just as a matter of protocol, when pondering infinity, you must wear a suit, talk in a distinct British accent, smoke a pipe, and always look profoundly toward the sky in contemplation.  When spoken to, you must reply, “Yes, indeed,” and then take a puff of your pipe.  If you need an example, reference the picture above.

In 1940, Godel was able to demonstrate that the Continuum Hypothesis couldn’t be disproven using mathematical set theory.   A little over twenty years later, a brilliant young mathematician by the name of Paul Cohen proved that mathematical logic is incapable of proving whether the Continuum Hypothesis is true, or false.   I’d like to further reflect on this matter with all of you, but to put it bluntly, mathematicians seem to enjoy making their work as terse and difficult to read as possible, using their own jargon lingo, and complicated symbols to express their ideas.  I’ve been wading through it all for years now, and have slowly been coming to an understanding of a lot of it.  Though I’m not going to take any time discussing this, there are actually an infinite number of sizes of infinity!  Who would have thought?

The Magical World Of Our Neocortex

I’ve been learning a lot about the neocortex lately, and have come across some really fascinating information that I’d like to share. What if I were to tell you I could turn your tongue into a third eye?  Sound bizarre?  You’d be surprised!  Let’s talk a little about a concept neuroscientists call “pattern equivalency.”

The idea that patterns from different senses are equivalent inside your brain is quite surprising, and although well understood, it still isn’t widely appreciated. More examples are in order. The first one you can reproduce at home. All you need is a friend, a freestanding cardboard screen, and a fake hand. For your first time running this experiment, it would be ideal if you had a rubber hand, such as you might buy at a Halloween store, but it will also work if you just trace your hand on a sheet of blank paper. Lay your real hand on a tabletop a few inches away from the fake one and align them the same (fingertips pointed in the same direction, palms either both up or both down). Then place the screen between the two hands so that all you can see is the false one. While you stare at the fake hand, your friend’s job is to simultaneously stroke both hands at corresponding points. For example, your friend could stroke both pinkies from knuckle to nail at the same speed, then issue three quick taps to the second joint of both index fingers with the same timing, then stroke a few light circles on the back of each hand, and so on. After a short time, areas in your brain where visual and somatosensory patterns come together— one of those association areas I mentioned earlier in this chapter— become confused. You will actually feel the sensations being applied to the dummy hand as if it were your own.

Another fascinating example of this “pattern equivalency” is called sensory substitution. It may revolutionize life for people who lose their sight in childhood, and might someday be a boon to people who are born blind. It also might spawn new machine interface technologies for the rest of us.

Realizing that the brain is all about patterns, Paul Bach y Rita, a professor of biomedical engineering at the University of Wisconsin, has developed a method for displaying visual patterns on the human tongue. Wearing this display device, blind persons are learning to “see” via sensations on the tongue.

Here is how it works. The subject wears a small camera on his forehead and a chip on his tongue. Visual images are translated pixel for pixel into points of pressure on the tongue. A visual scene that can be displayed as hundreds of pixels on a crude television screen can be turned into a pattern of hundreds of tiny pressure points on the tongue. The brain quickly learns to interpret the patterns correctly.

One of the first people to wear the tongue-mounted device is Erik Weihenmayer, a world-class athlete who went blind at age thirteen and who lectures widely about not letting blindness stop his ambitions. In 2002, Weihenmayer summited Mount Everest, becoming the first blind person ever to undertake, much less accomplish, such a goal.

In 2003, Weihenmayer tried on the tongue unit and saw images for the first time since his childhood. He was able to discern a ball rolling on the floor toward him, reach for a soft drink on a table, and play the game Rock, Paper, Scissors. Later he walked down a hallway, saw the door openings, examined a door and its frame, and noted that there was a sign on it. Images initially experienced as sensations on the tongue were soon experienced as images in space.

These examples show once again that the cortex is extremely flexible and that the inputs to the brain are just patterns. It doesn’t matter where the patterns come from; as long as they correlate over time in consistent ways, the brain can make sense of them.

– Jeff Hawkins, from his book On Intelligence:  How A New Understanding Of The Brain Will Lead To The Creation Of Truly Intelligent Machines

That blows my mind.  If you vibrate your tongue in the same sorts of patterns and frequencies as visual images, it will become like an eye, and you will soon see images in space.  You can make a blind man see by vibrating his tongue!  I mean, what in the world!  If you don’t understand how the brain works, that will sound almost unbelievable, but it’s true.

I suppose if you wanted a third eye, you could mount a camera on top of your head and start vibrating your tongue appropriately.  Soon you would stop feeling “touch” and “taste” sensations, and would instead see from your tongue.  You could probably do the same thing by wrapping a special device around your leg, vibrating your thighs and legs appropriately, and you would probably begin to see images from another fourth camera, which you could have pointing behind you.  You’d be four eyed monster!  Bwahahahaha!

That’s the very sort of mad scientist stuff I’m into.  That is too cool!

Ok, so why does this work?  How does this work?  I’m glad you asked, my young Padawans.  I will now teach you the dark art of sensory perception!  Get ready!

In 1978, Vernon Mountcastle, a neuroscientist from John Hopkins University in Baltimore, published a paper entitled An Organizing Principle For Cerebral Function.  He made the observation that no matter which area of the cortex he examined, whether it be Brocas area (responsible for our ability to speak language), auditory cortex (allowing us to hear), visual cortex (allowing us to see and perceive objects in space), or motor cortex (allowing us to control and move our bodies), they all had the same basic organizational structure.  Sure, they varied a little bit, such as in thickness, length of horizontal connections, synapse density, and so forth, but overall, the sections were so similar that they had to be doing the same sort of operation.  Every area of the cortex has the same layering, cell types, and connections.

Interesting.  But what does that mean?  Ever since we were small children, we’ve been taught that we have five senses – sight, sound, touch, hearing, and smell.  But then you look at the brain areas in our cortex which we know produce these senses, and you see that they’re all basically structured the same.  What’s going on?

We can’t experiment with human brains, but we have done some very interesting experiments on animals.  For example, you can take a baby ferret and rewire its brain, sending its visual signals to the auditory sections of its brain, and the auditory signals to the visual area of its brain.  What happens?  The brain area that normally develops into auditory cortex turns into visual cortex, and vice versa.   Scientists have done similar sorts of experiments on rats, running their sense of touch to their visual cortex, and as you’d probably guess, that area specializes for touch, not vision.  The cortex of animals, including ourselves, is very plastic.  It can be transformed into any sensory modality, depending on the inputs.

Our cortex is not specialized for any particular function at birth.  If you’re born deaf,  your brain will use the “auditory” cortex to process visual information.  If you’re born blind, your “visual” cortex will be converted into a very sensitive area for touch, used primarily in reading braille.   If a neocortical area is not receiving the normal inputs it’s “supposed” to receive, it will start to rewire itself, sending out connections, looking for inputs.  So basically, these areas of the brain develop based on the inputs they receive.

So how do we turn your tongue into a third eye?  We vibrate a 2D image onto its surface, having the various vibrational pressure intensities correspond to different colors.  The brain will interpret the changes appropriately, and you will see from your tongue.  Your cortex will begin to rewire itself, and turn your tongue sensory cortex into visual cortex.  This will take a little time though.  At first you’ll feel it as random vibrations on your tongue.  But in time, you’ll become conscious of visual images.  Then when you stop the vibrations and take off the device, it will, in time, revert back to a normal tongue.  No permanent harm done.

I remember watching cartoons when I was a kid and there would be these evil mad scientists who would turn themselves into monsters.  They’d have ten eye-balls, and wings, and a stinging tail, and the good guy would be like, “Dr. X, you monster!  What have you done!”  The villain would then cry out, “We’ll see if you have what it takes to stop me!”  And then an epic battle would take place.  As a kid I wondered how a human being could transform himself or herself into something like that.  Now I kind of know.  I somewhat understand the principles as to how to do that, but there are way too many technicalities.  I’d have to pump blood to my wings, wiring in a vein structure, and linking muscles to motor cortex.   I’d have to alter my heartbeat to accommodate the extra body mass, and so on.  But I can see that if you were smart enough, and had the brain power, you could build yourself any sort of body, with vastly superior intelligence. When our entire cortex is unfolded and stretched out, it’s about the size of a dinner napkin.  Imagine if you grew additional brain cortex, and wired it in appropriately as well.  You could be ten times as smart.

The key point is “wired in appropriately”.  Dolphins and whales have a lot more cortex than we do, though they’re not near as intelligent.  Cortex alone isn’t all there is to it.  It has to be wired into your sensory systems appropriately.  Also, you can see that we already can build eyeballs out of cameras and make the blind see with them.  With time, I wonder when we’ll move beyond biological bodies, and build new bodies out of nanotechnology, using robotic and computer parts.  There’s still a lot of secrets as to why cortex vibrating according to various inputs creates conscious sensations.  There’s very few things in this world which interest me more.  To me, that’s the key to everything.

A while back, I think it was Everett who asked me what I thought about death, and what I believed happened when we died.  This quest to understand consciousness, and what creates it, is the same problem as life and death.  This matter organizes into these patterns, based on self-organizing systems (evolution, etc), and then when coming together into neural networks, we become conscious.  I don’t think “I” die upon death.  “Jason” will die.  All my memories and past experiences, which are stored in my brain, will die.  But the fundamental existence which “I” am is not this physical body.  This physical body creates signals, which vibrates my cortex, which makes me conscious for the moment, but when this body dies and rots in the ground, I don’t see any reason to believe that some other matter, somewhere else in the universe, may not start vibrating, and I wake up as something or someone else.  It sounds like Buddhist or Hindu reincarnation, but it’s what I believe in these days.  I’ll probably wake up as something else, and not remember any of my past forms of existence.   I may be some winged creature on another planet across the cosmos.  Who knows.  Question is, how does my personal subjective consciousness enter and work within my brain’s cortex? That’s what I need to know.

Thinking For Yourself

As I’ve had this blog over the years, I’ve always shared my beliefs, and told what I really thought about things.  When I look back at my old blog posts, I’m embarrassed.  I’ve said some stupid things, and the farther back you go, the dumber I become.  But one thing has always been true of me:  I think for myself.  I keep digging through books, studying how things work, and I come to my own conclusions.

In this world you have to think for yourself.  Nobody can think for you.  If you think your television is going to tell you what to believe, you better think again.  You all may have heard of Cenk Uygur, who hosts an internet news program on YouTube called The Young Turks.  As his popularity increased, MSNBC gave him a prime time news show.  Knowing he could increase his audience and message, he took the position.  Then an interesting thing happened.  After going pretty hard on some big shot politicians, he was pulled into their main office and was told by the head CEO, “Outsiders are cool and all, but we’re part of the establishment.  You’ve made some powerful people in Washington angry, and you need to tone it down.”  That’s paraphrased.  Watch it for yourself.

Even as he was being canned, he was offered a huge sum of hush money, and a minor role as a contributor.  They offered him twice as much money to basically just go away.   If you’re getting your news from the TV or other mainstream sources, I think you should seriously reconsider.

A Look Into The Republican Frontrunners

Tonight I’ve been trying to study neuroscience, in particular our language systems, but I’m unable to concentrate.  My mind is just rushing all over the place.  I want to scream, but there’s nobody to scream to, so I’ll scream here in text.

I’m going to begin by asking you all a few questions.  Do you think that Japan’s current economic problems are due to their emperor having sex with a demonic sun goddess?  Do you think that gays are, by their very nature, violent savages incapable of compassion or virtue?  Do you believe that God possessed Hitler in order to herd the Jews back to Israel?

No?  Really?  *looks at you closely*  Are you sure?

*Pauses*

Quick, come inside.  Hurry!  *Pulls you into a secret room*  You don’t find sane people much these days.  Why do I bring this up?  Take a look at one of the top Republican presidential candidates.

Visit msnbc.com for breaking news, world news, and news about the economy

Is this some sort of crazy dream?  This idiot, Governor Perry, should be laughed off the face of the Earth, but instead he’s a prominent governor and hopeful presidential contender.  Former President George Bush’s team is helping him raise campaign funds.  Seriously!  This man may be our next president?  Am I alone when I think this is insanity?  Literally.  I’m not saying that to be dramatic.  The men featured at Governor Perry’s prayer conference need to be put in a mental asylum and thoroughly examined.  They’re hearing voices in their heads, they’re delusional, and they’re incapable of assessing reality.

Apparently the first amendment doesn’t specify freedom of religion.  Oh no, what it really says is that everybody should be mandated to be a Christian.  And Obama’s healthcare bill?  That’s a plot by the Illuminati to euthanize 93% of the world’s population.  And don’t let statue of liberty fool you.  It’s really a demonic idol, worshiped by our leaders who are working to usher in the anti-christ.  Governor Perry calls for a prayer rally and brings in these guys to be our spiritual moral leaders.  We’re all invited.

What about Republican frontrunner Michelle Bachmann?  She believes the same sort of stuff.  If you ask her what she thinks the biggest issues are for America is today, she’ll start going on about abortion and homosexuality.  Not financial reform, not exploding healthcare costs, not foreign policy or the wars, but abortion and homosexuality, and of course she’s against both.  All those dead babies are crying out and God is punishing our nation.  And if there’s one thing the Lord’s pissed off about, it’s gay sex.  You can kills hundreds of thousands of people in war and God will turn the other way, but when it comes to sex, God starts pouring out His wrath!

Who’s the other frontrunner?  Mitt Romney?  He’s a devout Mormon.  Have you ever heard what Mormons really believe?  Remember Joseph Smith translating the golden tablets by divine inspiration in 1820, or whatever?  I’ll let Christopher Hitchens recount that story.

*sigh*

I haven’t been keeping up with politics lately, but I’ve been hearing about Sarah Palin possibly running for president.  Remember the MILF whose foreign policy credentials consist solely in the fact that she was governor of Alaska, which is next to Russia?  She consistently gets her facts wrong and finds, “What periodicals do you read?”  to be a “gotcha” question, just another liberal elitist “lamestream” media attack on her character because we don’t like her Christian virtues.  She feels our involvement in the middle-east is part of a divine plan, ordered from God.  That’s just what we need, a war-mongering religious bimbo running the country.

And even though she bombs every interview, never has anything substantial to say beyond cliche talking points, and shows herself to be a complete moron, incapable of handling even minor pressure, she may well be our next president.  How dumb is the American public?  I mean seriously.  How dumb are they?  We all need to make sure to follow her every tweet, cover every little thing she does, and even idolize her daughter who has never accomplished or done anything.

We’re living in the movie Idiocracy.  Somehow, the average IQ must has dropped about 40 points over the past 30~40 years.  As Bill Maher recently stated in an interview, this Republican party is not your father’s Republican party.  These people are religious extremists.  I can’t even make stuff up that’s as ridiculous as what they believe and are saying.

Did you hear what Rick Perry’s group was saying about atheists?  They may do all sorts of humanitarian work, and be kind, and otherwise good people, but don’t buy into it!  They have an evil agenda!  *sits dumbfounded*  What!?  What evil agenda?  Then the camera cuts to this young man listening intently, heeding the warning call.  And Oprah, oh my goodness, she’s apparently the most evil villain of them all.  She rejects the authenticity of Christ!  The true power of the blood!  Yes, we can’t have forgiveness and morality without true, authentic human sacrifice.  Old man Newt Gingrich is saying the same sorts of things.  Atheists have an agenda to take over the nation!

Is it any wonder why everything’s falling apart around us?  Is it any wonder why our infrastructure is crumbling, our children are failing in science and mathematics, and why Wall Street gangsters are running all over us?  These people have NO IDEA what’s going on.  I don’t think the American public has any idea what’s going on, or what our real problems are, either.  They have to be complete morons to even consider voting for candidates like these.  I can’t even get into the minds of people so dumb.

While I’m on my ranting tirade, I might as well let it all out.  The mainstream public is a bunch of cowards, fearful of events which are so unlikely to happen that it’s ridiculous.  Why do we have the Patriot Act?  Why are we digitally strip-searched in our airports, with TSA agents ogling our girlfriends’ nude bodies?  Because people are idiots and they’re cowards.  Land of the free and home of the brave?  Past generations maybe, but not us.  Not this America.

Do you ever just lose hope?

Christians Need Us All To Eventually Die

The other day I was speaking with my father, who, as many of you already know, is a pastor of a protestant church.  Somehow we got to speaking about medicine and recent advances in technology, and I began telling him about synthetic organs.  Naturally I’m very excited about them.  I recently came across an article where a patient received a synthetically engineered windpipe, grown from the patient’s own stem cells.

Dad acknowledged the technology as being a great thing, but he was worried at the idea that mankind will eventually escape death, or at least attempt it.  The thought that we could stop aging, and replace any damaged internal organs by growing new ones didn’t seem to fly well with him.  He believed that God will have to intervene before that happens.  I wondered how we’re supposed to value human life, and do all we can to preserve it, and then not, both at the same time.

For example, he doesn’t believe in abortion.  That’s wrong.  No questions asked, it’s murder.  You ask him about assisted suicide, such as terminally ill patients, and that too is murder.  You do that and both the doctor and the patient are sent to hell for sure (absent the physician later repenting).  We must keep people alive for as long we can; that is our duty, and anything less is murder in the eyes of God.  Ok.  So then us scientists keep working on better and better technology, extending the human lifespan further and further.  Considering the human body is pretty much just a machine, and we’re learning how all its parts works, it’s inevitable that we’ll eventually end aging and death.  Then he gets uncomfortable.

Ultimately it comes down to this:  Christians want to save and preserve life, but only to a certain extent.  Eventually people need to die so that they can be judged by God for their actions.  Dad argued that he wouldn’t want evil people living forever.

Is there any room in Christian thinking for technology keeping us alive indefinitely?  I don’t think so.  They want us both to live and to die, and as technology improves, this going to become more and more of a problem.  True eternal life can only be granted by God, after you’ve been properly judged.  The Christian concept of heaven is a world where all the bad people have been jailed in hell, and only the saints are allowed into heaven.  Technology keeping us alive indefinitely ruins this entire paradigm, and even the need for a savior.  Maybe not all Christians think this way, but I’d be interested to hear how they intend to solve this problem.