## Thursday, December 20, 2007

### Faith in √2

If you are most people, you find out what the square root of two is by punching two keys on your calculator. Mine says it's 1.4142135623730950488016887242097, but even just 1.4 gets you a long way. (1.42 = 1.96, which is within 2% of 2)

Neither one, however, is actually the square root of 2. You can never write down all the digits for it; they are infinite. What our calculator gives us is a coarse approximation. Thirty-two digits might not seem like a coarse approximation, since five digits would probably get you safely to the moon and back, but our 32 digits only represent 0% (zero percent) of all the digits required to represent it exactly. If the digits of √2 are infinite, we can never write it as a ratio of two integers. We can approximate it as 7/5 (accurate to 2 digits) or as the ratio 25712973861329/18181818181818 (accurate to 14 digits), but it's still an approximation. The square root of two is incommensurable. It cannot be represented as the ratio of two integers, which is the definition of an irrational number (i.e. "irrational" literally means you can't represent it with a ratio).

See my earlier Proof that √2 is irrational

The Pythagoreans believed that the universe was made of integers. Their religious dogma said god only created natural numbers (i.e. countable positive integers). When Hippasus, 2500 years ago, was able to prove that a number (√2) existed that couldn't be constructed from integers, the story is he was either exiled or drowned as a heretic.

There are still modern mathematicians who believe something similar to the old Pythagorean views. Amongst other things, some believe that in order for a mathematical concept to "exist", it must be physically constructable in our universe. They argue that Hippasus and Chaitin's arguments suffer the same flaw; proving that a number cannot be made in a particular way does not prove that it can be made at all (i.e. that it can exist). If I can prove that John Travolta isn't made of green jello, that doesn't mean that John Travolta must exist and be made of something else, it only proves that there is not a green jello John Travolta.

How can √2 be physically "constructable"? Let's say you have a right-triangle with two sides of length 3 and 4. The Hypotenuse of that triangle (from the Pythagorean theroem) is √(32 +42) = √25 = 5. The Pythagoreans were OK with this because 3, 4 and 5 are all nice integers and this triangle can be "constructed" and shown to actually exist. However, if you have a similar triangle with two sides both of length 1, the hypotenuse is √(12 +12) = √2 = 1.414....

Try to make this triangle. To construct it from equal integer sides, you keep running either too long or too short on the hypotenuse. If you had 10 units on each side, the hypotenuse would be 14 and a bit; an integer value would not complete the triangle. If you tried to get a little finer grained and make it with 100 units on a side, the hypotenuse would be 141 units and a bit. You could continue this Sisyphean task forever, but no matter how big you made the integer sides, a hypotenuse side made with an integer would never quite finish the triangle.

This is an important question because so far, to the best of our knowledge, our universe is integer. At one time the smallest particles were atoms, then protons, neutrons and electrons, then quarks, now maybe strings. Unless there is an infinite parade of smaller and smaller particles, the universe is necessarily made up of "smallest particles". There is no evidence of a infinitesimally small particles (particles that approach zero dimension), and as far as I know, nobody is seriously considering it.

That means that regardless of how many "smallest particles" with which we attempt to construct the triangle, the number is always finite. The hypotenuse never works out to be a single integer, and so the triangle can't be constructed in our universe.

The only way √2 could be physically constructed would be if the universe were infinite in size and you could line up the smallest particles across an infinite universe (assuming you had infinite particles in that universe) to make an infinitely sized triangle. That's mighty big.

So, in order for √2 to physically exist in this universe, we need one of two conditions: either the universe is infinite in size, or the universe is infinitesimally divisible. If one of those conditions aren't met, √2 is an asymptote, an unreal mathematical abstraction. You should no more believe that √2 actually exists in any real sense than you believe Thor or Santa Claus exists, no matter what your calculator says. I'm surprised that people don't come door to door with pamphlets trying to get me to believe in √2. Maybe I've found something to do on weekend mornings.

Do you believe in √2?

Burton MacKenZie http://www.burtonmackenzie.com/

## Saturday, December 15, 2007

### √2 is irrational?

How do we know the square root of two cannot be represented by the ratio of two integers? (e.g. 7/5)

Of the proofs I have seen, Gregory Chaitin's is my favourite. It borrows from Hippasus's initial assumption and also ends with reductio ad absurdum. "Reductio ad absurdum" is a fancy way to say "this is crazy and can't be true". Instead of trying to prove what you think is true, try and prove the opposite is true, and prove it is wrong. If I want to prove that √2 cannot be a ratio of two integers (which is the literal meaning of irrational), try and prove the opposite condition:

1. Assume that √2 can be represented as the ratio of two integers, m/n = √2

2. Square both sides, m2/n2 = 2

3. Rearrange so that m2 = 2 * n2

4. Both m and n have unique prime decompositions (and if they didn't we could remove any common factors in the ratio),

m = 2a * 3b * 5c * ...
n = 2p * 3q * 5r * ...

5. If both m and n are squared, the prime decompositions of m2 and n2 are

m2 = 22a * 32b * 52c * ...
n2 = 22p * 32q * 52r * ...

6. Two (2) times n2 has a prime decomposition of

2 * n2 = 22p+1 * 32q * 52r * ...

7. If m2 = 2 * n2 (as in equation 3), then that means that 22a = 22p+1, which is impossible. 2a is an even number, and 2p+1 is an odd number! An odd number cannot be an even number - they cannot be equal. For example, 4 = 22 and 8 = 22+1, and 4 is not equal to 8 (even for large values of 4), thus their powers of two, 2 and 3, respectively, are also not equal.

8. Since this result is absurd (e.g. 4 is not equal to 8), assumption #1 must be false. This means that you cannot form a ratio of two integers to represent √2.

The beauty of this proof is that it is obviously extensible to the square root of any prime number - they're all irrational!

I first read this proof in Metamath, by Gregory Chaitin. I've really enjoyed the book. It's written in a prosaic rather than hard math style, so even a layperson could enjoy it. It explains a lot of mind blowing topics like the limits of what is provable in mathematics (incompleteness), the countability of numbers, randomness, compressability, and other tightly related topics. It has changed my views on the world (again), and given me a lot to chew on. Since I finished reading the book, I have constantly been re-reading sections of it. The image/link below will take you to amazon, where you can get it for yourself!

There is going to be a follow up post on this topic very soon, with less mathematics.
Update: Follow up post at Faith in √2

Burton MacKenZie http://www.burtonmackenzie.com/

## Monday, December 10, 2007

### Ann Coulter is a man - Outed by her own writings

Ann Coulter, an american ultra-right-wing political pundit, is frequently made fun of on the internet due to her inflammatory and intolerant views. A very popular meme is that Ann Coulter is actually a man. For example, the image at the right is alleged to be evidence of Ann Coulter's Adam's Apple. (n.b. only men have Adam's Apples)

I have just discovered more evidence pointing to her manhood, hiding in plain sight, published by Ann, herself! Ann's blog is overflowing with text ostensibly written by Ann. Large blocks of text can be analysed by software such as the Gender Guesser, giving a prediction of whether the author is a man or a woman.

At the time of writing, here's how Ann's Blog entires fared on the Gender Guesser. I have only included the informal writing results (appropriate for blogs).

Dec 5, 2007: They'll Never Forgive You (816 analysed words)
Genre: Informal writing: Verdict: MALE

Nov 28, 2007: NYT: An Undocumented Newspaper (837 analysed words)
Genre: Informal writing: Verdict: MALE

Nov 21, 2007: NYT: Suicide Manual for Dems (840 analysed words)
Genre: Informal writing: Verdict: MALE

There appears to be a visible trend in Ann's writing.. The Gender Guesser isn't defininitive; there is room for error, but it's not biased by politics or rage. The Gender Guesser, an impersonal piece of software, thinks Ann Coulter is most likely a man.

Don't believe me! Go try it, yourself! Go pick any articles from her archives, and run them through the Gender Guesser. (But try to not actually read what she says - it will just make you mad) It is important for more accurate results that you only use large posts (greater than 300 words), and do not include text which has quotes from other people.

For comparison, you're welcome to test out samples of my writing, as well.

Dec 7, 2007: When Software Rulz (with an Iron Fist) (1051 analysed words)
Genre: Informal writing: Verdict: MALE

Nov 25, 2007: Poverty, Education, and how to improve Life for Everybody (985 analysed words)
Genre: Informal writing: Verdict: MALE

Jan 12, 2007: Tittie Love: More on the selective underpinnings of Permanently Swollen Breasts (973 analysed words)
Genre: Informal writing: Verdict: MALE

There is a bright side to this. At least if she ever came over to your house, you wouldn't have to tell him that his skanky old hooch is stinking up the place and to get it the hell off your couch; you could just tell him that he's a big asshole, kick him out, and be done with it.

Burton MacKenZie www.burtonmackenzie.com

## Friday, December 07, 2007

### When Software Rulz (with an Iron Fist)

Science fiction is rife with great stories of artificial intelligence (AI), both rogue and controllable, cold as a machine or parental in affection. Are these stories naive fantasies, or possible realities?

Fast computation machines and the software that runs on them are both still in infancy. At best, the science of programming is about 150 years old (starting with Lady Ada Lovelace, the first programmer). What will it be like when the science has been around as long as iron smelting? (about 3000 years) With time will come greater comlpexity and computational power.

As we near the dawn of an age where a machine may pass a Turing test, I sometimes wonder how might we humans be treated after a machine surpasses us in intelligence. How can we judge how an AI might treat humans?

We can make guesses about how we would be treated by looking at the computational/ecological niche in which an AI could arise. How does the organization see us and deal with us? How it deals with us now will strongly influence how an AI arising from it will deal with us in the future, paralleling the biological founder effect. Take a glance at the niche of some computing organizations which could be first to have the massively-parallel computing power required for an AI to arise.

Scenario 1: Grassroots Distributed Efforts
The "SETI@Home", the "Great Internet Marsenne Prime Search", "Folding@Home", etc, distributed computing efforts could host a massively parallel intelligence. These efforts benefit both the users (us) and the organization. The users feel satisfaction for contributing otherwise idle processing time for the betterment of knowledge for us all, and are at least passingly interested in knowing the answers that result from the effort. The organization benefits directly from contributions of computational processing time which furthers its goals. This sort of organization would give rise to an AI that sees advantage in working closely with humans for common benefit. We've got knowledge to uncover and let's do it together. Six billion brains and a massively parallel AI think better than just six billion brains or a massively parallel AI alone.

Its fitness function is proportional to its profit (a portion thereof which will pay to increase its size/computational ability/redundancy/robustness), which is proportional to its optimal "milking" of the market serving ads to human eyes. Google's search technology and infrastructure has made a significant improvement to human sharing and access of knowledge. Any increases in knowledge search technology only make things better for both us and Google.

Google's AI niche will be that of a powerful servant of to users. This AI will exceed our abilities in thought, but the user interface will ultimately evolve to mimic a personal servant, with the servant's characteristics on a sliding scale of servitude tailored to the individual. This spectrum ranges from low-level interface (i.e. the "classic" Google) to a soft spoken and patient avatar who can answer a range of questions, but simultaneously whispers in your ears about a product somebody is selling that might be related to your question. The MORE communication humans have with the AI, the more the human learns, and the more money the AI makes (to reinvest in improving itself). Thus, the AI is selected towards whatever makes an optimal amount of money, which is a function of increasing information going into human brains. A further beauty of this situation is that Google should optimally be content-indifferent. It doesn't matter what you want to know, whether your interest is irrational numbers or Jessica Alba's chest.

Humans still go to work, produce the initial wealth, and carry on as always, but now a portion of the human-produced wealth will channel into the AI, which ultimately still benefits us as a whole by increasing and enhancing communication between individuals. Google will be a farmer AI, which harvests wealth from us as we harvest eggs from chickens, except we'll all get our own personal farmer that does a lot of stuff for us and really has a lot of good ideas.

Scenario 3: Microsoft.
Microsoft wants control over your computer. Microsoft AI will want control over your life for the benefit of Microsoft. Microsoft will stop you from playing music or video it doesn't think you're allowed to listen to. Microsoft will condescendingly patronize you with a dancing paperclip. (e.g. "Are you SURE you don't to buy the next upgrade? I think you really should. If you don't, I'll cut you off in two years.") Pay Microsoft its money or rot in the stone age, peon. Microsoft AI despises Google AI and will frequently try to undermine and destroy it. Have no other Gods before Microsoft! A Microsoft AI will rule us with an iron fist and make us conform to its whims. Double plus ungood.

You make the choice.
Before this earthquake of humanity arrives, there's still time for the pebbles to vote. What kind of AI would you want for the future of humanity? Nows the time to figure it out.

And I, for one, welcome our new insect overlords. I'd like to remind them that as a trusted TV personality I can be helpful in rounding up others to toil in their underground sugar caves.

Burton MacKenZie www.burtonmackenzie.com

More or less on this topic, here's a teaser for a good looking rogue AI movie. The film is Yellow by Neill Blomkamp.

## Sunday, December 02, 2007

### Pain Sandwich (Sympathy for the Planet Mercury)

Here, it's cold enough outside to kill you. If you sat outside in regular clothes in this -30oC (-22oF) ambient weather, you would be dead (or near death) by morning. Out in the uninsulated garage, there is a small heater to take the sting off the cold, but it only heats things really close to it. My hands were particularly cold, so I took off my glove and put it as close to the glowing heater as I could stand, just on the edge of pain.

The other side of my hand was in the heat shadow, and there it's around -30oC. Cold really hurts, too. A lot. At a place I once worked, a tech came up from the USA to install something during a cold snap. He had worked in freezing environments before, but after coming here he told us "I figured that when it's colder than -10oC (+14oF) you don't really notice the difference anymore... I just didn't know how painful cold really is!" He also said things like "holy shit is it fucking cold out here!! Son of a bitch!"

Anyway, out in the garage, my hand was simultaneously in pain from proximity to a 1200 Watt heater on one side and the stinging -30oC on the other. The planet Mercury faces one side to the sun for a long time (although it's not tidally locked), with surface temperatures ranging from -193oC to 427oC. This is simultaneously about 6.5 times colder and 8.5 times hotter than my bare hand experiences out in the garage. If Mercury were somehow alive, as some people suggest is true of the Earth, I know what Mercury is thinking. Mercury is thinking "Great god-damn this hurts!"

Burton MacKenZie www.burtonmackenzie.com

(Original Image of Sol/Mercury by Mbz1, released under the GNU Free Documentation License)